diff --git a/dev/.documenter-siteinfo.json b/dev/.documenter-siteinfo.json new file mode 100644 index 0000000..b45c5e5 --- /dev/null +++ b/dev/.documenter-siteinfo.json @@ -0,0 +1 @@ +{"documenter":{"julia_version":"1.10.4","generation_timestamp":"2024-06-21T05:08:41","documenter_version":"1.4.1"}} \ No newline at end of file diff --git a/dev/advanced/index.html b/dev/advanced/index.html new file mode 100644 index 0000000..79221bf --- /dev/null +++ b/dev/advanced/index.html @@ -0,0 +1,27 @@ + +Advanced details · MRVCModels
GitHub

Advanced details

Estimation

For the MM algorithm, the updates in each iteration are

\[\begin{aligned} +\text{vec}\ \boldsymbol{B}^{(t)} &= [(\boldsymbol{I}_d \otimes \boldsymbol{X}^T) \boldsymbol{\Omega}^{-(t)} (\boldsymbol{I}_d \otimes \boldsymbol{X})]^{-1} (\boldsymbol{I}_d \otimes \boldsymbol{X}^T) \boldsymbol{\Omega}^{-(t)} \text{vec}\ \boldsymbol{Y} \\ +\boldsymbol{\Gamma}_i^{(t + 1)} &= \boldsymbol{L}_i^{-(t)T}[\boldsymbol{L}_i^{(t)T}\boldsymbol{\Gamma}_i^{(t)}(\boldsymbol{R}^{(t)T}\boldsymbol{V}_i\boldsymbol{R}^{(t)})\boldsymbol{\Gamma}_i^{(t)}\boldsymbol{L}_i^{(t)}]^{1/2} \boldsymbol{L}_i^{-(t)}, +\end{aligned}\]

where $\boldsymbol{\Omega}^{(t)} = \sum_{i=1}^m \boldsymbol{\Gamma}_i^{(t)} \otimes \boldsymbol{V}_i$ and $\boldsymbol{L}_i^{(t)}$ is the Cholesky factor of $\boldsymbol{M}_i^{(t)} = (\boldsymbol{I}_d \otimes \boldsymbol{1}_n)^T [(\boldsymbol{1}_d \boldsymbol{1}_d^T \otimes \boldsymbol{V}_i) \odot \boldsymbol{\Omega}^{-(t)}] (\boldsymbol{I}_d \otimes \boldsymbol{1}_n)$, while $\boldsymbol{R}^{(t)}$ is the $n \times d$ matrix such that $\text{vec}\ \boldsymbol{R}^{(t)} = \boldsymbol{\Omega}^{-(t)} \text{vec}(\boldsymbol{Y} - \boldsymbol{X} \boldsymbol{B}^{(t)})$. $\odot$ denotes the Hadamard product.

For the EM algorithm, the updates in each iteration are

\[\begin{aligned} +\text{vec}\ \boldsymbol{B}^{(t)} &= [(\boldsymbol{I}_d \otimes \boldsymbol{X}^T) \boldsymbol{\Omega}^{-(t)} (\boldsymbol{I}_d \otimes \boldsymbol{X})]^{-1} (\boldsymbol{I}_d \otimes \boldsymbol{X}^T) \boldsymbol{\Omega}^{-(t)} \text{vec}\ \boldsymbol{Y} \\ +\boldsymbol{\Gamma}_i^{(t + 1)} &= \frac{1}{r_i} \boldsymbol{\Gamma}_i^{(t)} ( \boldsymbol{R}^{(t)T} \boldsymbol{V}_i \boldsymbol{R}^{(t)} - \boldsymbol{M}_i^{(t)} ) \boldsymbol{\Gamma}_i^{(t)} + \boldsymbol{\Gamma}_i^{(t)}, +\end{aligned}\]

where $r_i = \text{rank}(\boldsymbol{V}_i)$. As seen, the updates for mean effects $\boldsymbol{B}$ are the same for these two algorithms.

Inference

Standard errors for our estimates are calculated using the Fisher information matrix, where

\[\begin{aligned} +\text{E} \left[- \frac{\partial^2}{\partial(\text{vec}\ \boldsymbol{B})^T \partial(\text{vec}\ \boldsymbol{B})} \mathcal{L} \right] &= (\boldsymbol{I}_d \otimes \boldsymbol{X}^T) \boldsymbol{\Omega}^{-1} (\boldsymbol{I}_d \otimes \boldsymbol{X}) \\ +\text{E} \left[ - \frac{\partial^2}{\partial (\text{vech}\ \boldsymbol{\Gamma}_i)^T \partial (\text{vec}\ \boldsymbol{B})} \mathcal{L} \right] &= \boldsymbol{0} \\ +\text{E} \left[ - \frac{\partial^2}{\partial (\text{vech}\ \boldsymbol{\Gamma}_j)^T \partial (\text{vech}\ \boldsymbol{\Gamma}_i)} \mathcal{L} \right] &= \frac{1}{2} \boldsymbol{U}_i^T (\boldsymbol{\Omega}^{-1} \otimes \boldsymbol{\Omega}^{-1}) \boldsymbol{U}_j +\end{aligned}\]

and $\boldsymbol{U}_i = (\boldsymbol{I}_d \otimes \boldsymbol{K}_{nd} \otimes \boldsymbol{I}_n) (\boldsymbol{I}_{d^2} \otimes \text{vec}\ \boldsymbol{V}_i) \boldsymbol{D}_{d}$. Here, $\boldsymbol{K}_{nd}$ is the $nd \times nd$ commutation matrix and $\boldsymbol{D}_{d}$ the $d^2 \times \frac{d(d+1)}{2}$ duplication matrix. $\text{vech}\ \boldsymbol{\Gamma}_i$ creates an $\frac{d(d+1)}{2} \times 1$ vector from $\boldsymbol{\Gamma}_i$ by stacking its lower triangular part.

Special case: missing response

In the setting of missing response, the adjusted MM updates in each interation are

\[\begin{aligned} +\text{vec}\ \boldsymbol{B}^{(t)} &= [(\boldsymbol{I}_d \otimes \boldsymbol{X}^T) \boldsymbol{\Omega}^{-(t)} (\boldsymbol{I}_d \otimes \boldsymbol{X})]^{-1} (\boldsymbol{I}_d \otimes \boldsymbol{X}^T) \boldsymbol{\Omega}^{-(t)} \text{vec}\ \boldsymbol{Z}^{(t)} \\ +\boldsymbol{\Gamma}_i^{(t + 1)} &= \boldsymbol{L}_i^{-(t)T}[\boldsymbol{L}_i^{(t)T}\boldsymbol{\Gamma}_i^{(t)}(\boldsymbol{R}^{*(t)T}\boldsymbol{V}_i\boldsymbol{R}^{*(t)} + \boldsymbol{M}_i^{*(t)})\boldsymbol{\Gamma}_i^{(t)}\boldsymbol{L}_i^{(t)}]^{1/2} \boldsymbol{L}_i^{-(t)}, +\end{aligned}\]

where $\boldsymbol{Z}^{(t)}$ is the completed response matrix from conditional mean and $\boldsymbol{M}_i^{*(t)} = (\boldsymbol{I}_d \otimes \boldsymbol{1}_n)^T [(\boldsymbol{1}_d \boldsymbol{1}_d^T \otimes \boldsymbol{V}_i) \odot (\boldsymbol{\Omega}^{-(t)} \boldsymbol{P}^T \boldsymbol{C}^{(t)}\boldsymbol{P}\boldsymbol{\Omega}^{-(t)})] (\boldsymbol{I}_d \otimes \boldsymbol{1}_n)$, while $\boldsymbol{R}^{*(t)}$ is the $n \times d$ matrix such that $\text{vec}\ \boldsymbol{R}^{*(t)} = \boldsymbol{\Omega}^{-(t)} \text{vec}(\boldsymbol{Z}^{(t)} - \boldsymbol{X} \boldsymbol{B}^{(t)})$. Additionally, $\boldsymbol{P}$ is the $nd \times nd$ permutation matrix such that $\boldsymbol{P} \cdot \text{vec}\ \boldsymbol{Y} = \begin{bmatrix} \boldsymbol{y}_{\text{obs}} \\ \boldsymbol{y}_{\text{mis}} \end{bmatrix}$, where $\boldsymbol{y}_{\text{obs}}$ and $\boldsymbol{y}_{\text{mis}}$ are vectors of observed and missing response values, respectively, in column-major order, and the block matrix $\boldsymbol{C}^{(t)}$ is $\boldsymbol{0}$ except for a lower-right block consisting of conditional variance. As seen, the two MM updates are of similar form.

Special case: $m = 2$

When there are $m = 2$ variance components such that $\boldsymbol{\Omega} = \boldsymbol{\Gamma}_1 \otimes \boldsymbol{V}_1 + \boldsymbol{\Gamma}_2 \otimes \boldsymbol{V}_2$, repeated inversion of the $nd \times nd$ matrix $\boldsymbol{\Omega}$ can be avoided and reduced to one $d \times d$ generalized eigen-decomposition per iteration. Without loss of generality, if we assume $\boldsymbol{V}_2$ to be positive definite, the generalized eigen-decomposition of the matrix pair $(\boldsymbol{V}_1, \boldsymbol{V}_2)$ yields generalized eigenvalues $\boldsymbol{d} = (d_1, \dots, d_n)^T$ and generalized eigenvectors $\boldsymbol{U}$ such that $\boldsymbol{U}^T \boldsymbol{V}_1 \boldsymbol{U} = \boldsymbol{D} = \text{diag}(\boldsymbol{d})$ and $\boldsymbol{U}^T \boldsymbol{V}_2 \boldsymbol{U} = \boldsymbol{I}_n$. Similarly, if we let the generalized eigen-decomposition of $(\boldsymbol{\Gamma}_1^{(t)}, \boldsymbol{\Gamma}_2^{(t)})$ be $(\boldsymbol{\Lambda}^{(t)}, \boldsymbol{\Phi}^{(t)})$ such that $\boldsymbol{\Phi}^{(t)T} \boldsymbol{\Gamma}_1^{(t)} \boldsymbol{\Phi}^{(t)} = \boldsymbol{\Lambda}^{(t)} = \text{diag}(\boldsymbol{\lambda^{(t)}})$ and $\boldsymbol{\Phi}^{(t)T} \boldsymbol{\Gamma}_2^{(t)} \boldsymbol{\Phi}^{(t)} = \boldsymbol{I}_d$, then the MM updates in each iteration become

\[\begin{aligned} +\text{vec}\ \boldsymbol{B}^{(t)} &= [(\boldsymbol{\Phi}^{(t)T}\otimes \tilde{\boldsymbol{X}})^T (\boldsymbol{\Lambda}^{(t)} \otimes \boldsymbol{D} + \boldsymbol{I}_d \otimes \boldsymbol{I}_n)^{-1} (\boldsymbol{\Phi}^{(t)T}\otimes \tilde{\boldsymbol{X}})]^{-1} \\ +&\quad \cdot (\boldsymbol{\Phi}^{(t)T}\otimes \tilde{\boldsymbol{X}})^T (\boldsymbol{\Lambda}^{(t)} \otimes \boldsymbol{D} + \boldsymbol{I}_d \otimes \boldsymbol{I}_n)^{-1} \text{vec}(\tilde{\boldsymbol{Y}} \boldsymbol{\Phi}^{(t)}) \\ +\boldsymbol{\Gamma}_i^{(t + 1)} &= \boldsymbol{L}_i^{-(t)T}[\boldsymbol{L}_i^{(t)T}\boldsymbol{N}_i^{(t)T}\boldsymbol{N}_i^{(t)}\boldsymbol{L}_i^{(t)}]^{1/2} \boldsymbol{L}_i^{-(t)}, +\end{aligned}\]

where $\tilde{\boldsymbol{X}} = \boldsymbol{U}^T \boldsymbol{X}$, $\tilde{\boldsymbol{Y}} = \boldsymbol{U}^T \boldsymbol{Y}$, $\boldsymbol{L}_1^{(t)}$ is the Cholesky factor of $\boldsymbol{\Phi}^{(t)}\text{diag}(\text{tr}(\boldsymbol{D}(\lambda_k^{(t)}\boldsymbol{D} + \boldsymbol{I}_n)^{-1}), k = 1,\dots, d)\boldsymbol{\Phi}^{(t)T}$, $\boldsymbol{L}_2^{(t)}$ is the Cholesky factor of $\boldsymbol{\Phi}^{(t)}\text{diag}(\text{tr}((\lambda_k^{(t)}\boldsymbol{D} + \boldsymbol{I}_n)^{-1}), k = 1,\dots, d)\boldsymbol{\Phi}^{(t)T}$, $\boldsymbol{N}_1^{(t)} = \boldsymbol{D}^{1/2}\{[(\tilde{\boldsymbol{Y}} - \tilde{\boldsymbol{X}}\boldsymbol{B})\boldsymbol{\Phi}^{(t)}]\oslash(\boldsymbol{d}\boldsymbol{\lambda}^{(t)T} + \boldsymbol{1}_n\boldsymbol{1}_d^T) \} \boldsymbol{\Lambda}^{(t)}\boldsymbol{\Phi}^{-(t)}$, and $\boldsymbol{N}_2^{(t)} = \{[(\tilde{\boldsymbol{Y}} - \tilde{\boldsymbol{X}}\boldsymbol{B})\boldsymbol{\Phi}^{(t)}]\oslash(\boldsymbol{d}\boldsymbol{\lambda}^{(t)T} + \boldsymbol{1}_n\boldsymbol{1}_d^T) \} \boldsymbol{\Phi}^{-(t)}$. $\oslash$ denotes the Hadamard quotient.

In this setting, the Fisher information matrix is equivalent to

\[\begin{aligned} +\text{E} \left[- \frac{\partial^2}{\partial(\text{vec}\ \boldsymbol{B})^T \partial(\text{vec}\ \boldsymbol{B})} \mathcal{L} \right] &= (\boldsymbol{\Phi}^{T}\otimes \tilde{\boldsymbol{X}})^T (\boldsymbol{\Lambda} \otimes \boldsymbol{D} + \boldsymbol{I}_d \otimes \boldsymbol{I}_n)^{-1} (\boldsymbol{\Phi}^{T}\otimes \tilde{\boldsymbol{X}}) \\ +\text{E} \left[ - \frac{\partial^2}{\partial (\text{vech} \ \boldsymbol{\Gamma}_i)^T \partial (\text{vec}\ \boldsymbol{B})} \mathcal{L} \right] &= \boldsymbol{0} \\ +\text{E} \left[ - \frac{\partial^2}{\partial (\text{vech}\ \boldsymbol{\Gamma}_j)^T \partial (\text{vech}\ \boldsymbol{\Gamma}_i)} \mathcal{L} \right] &= \frac{1}{2} \boldsymbol{D}_d^T(\boldsymbol{\Phi}\otimes \boldsymbol{\Phi}) \text{diag}(\text{vec}\ \boldsymbol{W}_{ij}) (\boldsymbol{\Phi} \otimes \boldsymbol{\Phi})^T\boldsymbol{D}_d, +\end{aligned}\]

where $\boldsymbol{W}_{ij}$ is the $d \times d$ matrix that has entries

\[\begin{aligned} +(\boldsymbol{W}_{11})_{kl} &= \text{tr}(\boldsymbol{D}^2(\boldsymbol{\lambda}_k \boldsymbol{D} + \boldsymbol{I}_n)^{-1}(\boldsymbol{\lambda}_l \boldsymbol{D} + \boldsymbol{I}_n)^{-1}) \\ +(\boldsymbol{W}_{12})_{kl} &= \text{tr}(\boldsymbol{D}(\boldsymbol{\lambda}_k \boldsymbol{D} + \boldsymbol{I}_n)^{-1}(\boldsymbol{\lambda}_l \boldsymbol{D} + \boldsymbol{I}_n)^{-1}) \\ +(\boldsymbol{W}_{22})_{kl} &= \text{tr}((\boldsymbol{\lambda}_k \boldsymbol{D} + \boldsymbol{I}_n)^{-1}(\boldsymbol{\lambda}_l \boldsymbol{D} + \boldsymbol{I}_n)^{-1}). +\end{aligned}\]

diff --git a/dev/api/index.html b/dev/api/index.html new file mode 100644 index 0000000..3fc1671 --- /dev/null +++ b/dev/api/index.html @@ -0,0 +1,19 @@ + +API · MRVCModels
GitHub

API

MultiResponseVarianceComponentModels.MRVCModel — Method
MRVCModel(
+    Y::AbstractVecOrMat,
+    X::Union{Nothing, AbstractVecOrMat},
+    V::Union{AbstractMatrix, Vector{<:AbstractMatrix}}
+    )
+MRTVCModel(
+    Y::AbstractVecOrMat,
+    X::Union{Nothing, AbstractVecOrMat},
+    V::Vector{<:AbstractMatrix}
+    )

Create a new MRVCModel or MRTVCModel instance from response matrix Y, predictor matrix X, and kernel matrices V. For MRTVCModel instance, the number of variance components must be two.

Keyword arguments

se::Bool        calculate standard errors; default true
+reml::Bool      pursue REML estimation instead of ML estimation; default false
source
MultiResponseVarianceComponentModels.fisher_B! — Method
fisher_B!(model::MRVCModel)

Compute the sampling variance-covariance model.Bcov of regression coefficients model.B, assuming inverse of covariance matrix model.Ω is available at model.storage_nd_nd.

source
MultiResponseVarianceComponentModels.fisher_Σ! — Method
fisher_Σ!(model::MRVCModel)

Compute the sampling variance-covariance model.Σcov of variance component estimates model.Σ, assuming inverse of covariance matrix model.Ω is available at model.storage_nd_nd.

source
MultiResponseVarianceComponentModels.fit! — Method
fit!(model::MRVCModel)
+fit!(model::MRTVCModel)

Fit a multivariate response variance components model by MM or EM algorithm.

Keyword arguments

maxiter::Int        maximum number of iterations; default 1000
+reltol::Real        relative tolerance for convergence; default 1e-6
+verbose::Bool       display algorithmic information; default true
+init::Symbol        initialization strategy; :default initializes by least squares, while
+    :user uses user-supplied values at model.B and model.Σ
+algo::Symbol        optimization algorithm; :MM (default) or EM (for MRVCModel)
+log::Bool           record iterate history or not; default false
source
MultiResponseVarianceComponentModels.h2 — Method
h2(model::MRVCModel)

Calculate heritability estimates and their standard errors, assuming that all variance components capture genetic effects except the last term. Also return total heritability from sum of individual contributions and its standard error.

source
MultiResponseVarianceComponentModels.loglikelihood! — Method
loglikelihood!(model::MRVCModel)

Overwrite model.storage_nd_nd by inverse of the covariance matrix model.Ω, overwrite model.storage_nd by U' \ vec(model.R), and return the log-likelihood. Assume model.Ω and model.R are already updated according to model.Σ and model.B.

source
MultiResponseVarianceComponentModels.loglikelihood_miss! — Method
loglikelihood_miss!(model::MRVCModel)

Overwrite model.storage_nd_nd by inverse of the covariance matrix model.Ω, overwrite model.storage_nd by U' \ vec(model.R), overwrite model.storage_n_miss_n_miss_1 by conditional variance, precompute model.storage_n_miss_n_obs_1 for conditional mean, and return the value of surrogate Q-function of log-likelihood. Assume model.Ω and model.R are already updated according to model.Σ and model.B.

source
MultiResponseVarianceComponentModels.lrt — Method
lrt(model1::MRVCModel, model0::MRVCModel)

Perform a variation of the likelihood ratio test for univariate variance components models as in Molenberghs and Verbeke 2007 with model1 and model0 being the full and nested models, respectively.

source
MultiResponseVarianceComponentModels.permute — Method
permute(Y::AbstractVecOrMat)

Return permutation P such that vec(Y)[P] rearranges vec(Y) with missing values spliced after non-missing values. Also return inverse permutation invP such that vec(Y)[P][invP] = vec(Y).

source
MultiResponseVarianceComponentModels.update_B_miss! — Method
update_B_miss!(model::MRVCModel)

Update the regression coefficients model.B, assuming inverse of covariance matrix model.Ω is available at model.storage_nd_nd and model.storage_n_miss_n_obs_1 for conditional mean is precomputed.

source
MultiResponseVarianceComponentModels.update_Γk! — Method
update_Γk!(model, k)

Update the parameters model.Γ[k] and model.Ψ[:,k] assuming a diagonal plus low rank structure for model.Σ[k]. Assumes covariance matrix model.Ω is available at model.storage_nd_nd and model.Ω⁻¹R precomputed.

source
MultiResponseVarianceComponentModels.update_Σ! — Method
update_Σ!(model::MRVCModel)

Update the variance component parameters model.Σ, assuming inverse of covariance matrix model.Ω is available at model.storage_nd_nd. If missing response, assume conditional variance model.storage_n_miss_n_miss_1 is precomputed.

source
MultiResponseVarianceComponentModels.update_Σk! — Method
update_Σk!(model::MRVCModel, k, Val(:EM))

EM update the model.Σ[k] assuming it has full rank d, inverse of covariance matrix model.Ω is available at model.storage_nd_nd, and model.Ω⁻¹R is precomputed.

source
MultiResponseVarianceComponentModels.update_Σk! — Method
update_Σk!(model::MRVCModel, k, Val(:MM))

MM update the model.Σ[k] assuming it has full rank d, inverse of covariance matrix model.Ω is available at model.storage_nd_nd, and model.Ω⁻¹R is precomputed.

source
MultiResponseVarianceComponentModels.update_Σk_miss! — Method
update_Σk_miss!(model::MRVCModel, k, Val(:MM))

MM update the model.Σ[k] assuming it has full rank d, inverse of covariance matrix model.Ω is available at model.storage_nd_nd, model.Ω⁻¹R is precomputed, and Ω⁻¹P'CPΩ⁻¹ is available at model.storage_nd_nd_miss.

source
diff --git a/dev/assets/MRVC.png b/dev/assets/MRVC.png new file mode 100644 index 0000000..85e4add Binary files /dev/null and b/dev/assets/MRVC.png differ diff --git a/dev/assets/documenter.js b/dev/assets/documenter.js new file mode 100644 index 0000000..c6562b5 --- /dev/null +++ b/dev/assets/documenter.js @@ -0,0 +1,1050 @@ +// Generated by Documenter.jl +requirejs.config({ + paths: { + 'highlight-julia': 'https://cdnjs.cloudflare.com/ajax/libs/highlight.js/11.8.0/languages/julia.min', + 'headroom': 'https://cdnjs.cloudflare.com/ajax/libs/headroom/0.12.0/headroom.min', + 'jqueryui': 'https://cdnjs.cloudflare.com/ajax/libs/jqueryui/1.13.2/jquery-ui.min', + 'katex-auto-render': 'https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.8/contrib/auto-render.min', + 'jquery': 'https://cdnjs.cloudflare.com/ajax/libs/jquery/3.7.0/jquery.min', + 'headroom-jquery': 'https://cdnjs.cloudflare.com/ajax/libs/headroom/0.12.0/jQuery.headroom.min', + 'katex': 'https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.8/katex.min', + 'highlight': 'https://cdnjs.cloudflare.com/ajax/libs/highlight.js/11.8.0/highlight.min', + 'highlight-julia-repl': 'https://cdnjs.cloudflare.com/ajax/libs/highlight.js/11.8.0/languages/julia-repl.min', + }, + shim: { + "highlight-julia": { + "deps": [ + "highlight" + ] + }, + "katex-auto-render": { + "deps": [ + "katex" + ] + }, + "headroom-jquery": { + "deps": [ + "jquery", + "headroom" + ] + }, + "highlight-julia-repl": { + "deps": [ + "highlight" + ] + } +} +}); +//////////////////////////////////////////////////////////////////////////////// +require(['jquery', 'katex', 'katex-auto-render'], function($, katex, renderMathInElement) { +$(document).ready(function() { + renderMathInElement( + document.body, + { + "delimiters": [ + { + "left": "$", + "right": "$", + "display": false + }, + { + "left": "$$", + "right": "$$", + "display": true + }, + { + "left": "\\[", + "right": "\\]", + "display": true + } + ] +} + + ); +}) + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery', 'highlight', 'highlight-julia', 'highlight-julia-repl'], function($) { +$(document).ready(function() { + hljs.highlightAll(); +}) + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +let timer = 0; +var isExpanded = true; + +$(document).on("click", ".docstring header", function () { + let articleToggleTitle = "Expand docstring"; + + debounce(() => { + if ($(this).siblings("section").is(":visible")) { + $(this) + .find(".docstring-article-toggle-button") + .removeClass("fa-chevron-down") + .addClass("fa-chevron-right"); + } else { + $(this) + .find(".docstring-article-toggle-button") + .removeClass("fa-chevron-right") + .addClass("fa-chevron-down"); + + articleToggleTitle = "Collapse docstring"; + } + + $(this) + .find(".docstring-article-toggle-button") + .prop("title", articleToggleTitle); + $(this).siblings("section").slideToggle(); + }); +}); + +$(document).on("click", ".docs-article-toggle-button", function (event) { + let articleToggleTitle = "Expand docstring"; + let navArticleToggleTitle = "Expand all docstrings"; + let animationSpeed = event.noToggleAnimation ? 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+$(document).ready(function () { + $("#documenter .docs-navbar").headroom({ + tolerance: { up: 10, down: 10 }, + }); +}); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +$(document).ready(function () { + let meta = $("div[data-docstringscollapsed]").data(); + + if (meta?.docstringscollapsed) { + $("#documenter-article-toggle-button").trigger({ + type: "click", + noToggleAnimation: true, + }); + } +}); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +/* +To get an in-depth about the thought process you can refer: https://hetarth02.hashnode.dev/series/gsoc + +PSEUDOCODE: + +Searching happens automatically as the user types or adjusts the selected filters. +To preserve responsiveness, as much as possible of the slow parts of the search are done +in a web worker. Searching and result generation are done in the worker, and filtering and +DOM updates are done in the main thread. The filters are in the main thread as they should +be very quick to apply. This lets filters be changed without re-searching with minisearch +(which is possible even if filtering is on the worker thread) and also lets filters be +changed _while_ the worker is searching and without message passing (neither of which are +possible if filtering is on the worker thread) + +SEARCH WORKER: + +Import minisearch + +Build index + +On message from main thread + run search + find the first 200 unique results from each category, and compute their divs for display + note that this is necessary and sufficient information for the main thread to find the + first 200 unique results from any given filter set + post results to main thread + +MAIN: + +Launch worker + +Declare nonconstant globals (worker_is_running, last_search_text, unfiltered_results) + +On text update + if worker is not running, launch_search() + +launch_search + set worker_is_running to true, set last_search_text to the search text + post the search query to worker + +on message from worker + if last_search_text is not the same as the text in the search field, + the latest search result is not reflective of the latest search query, so update again + launch_search() + otherwise + set worker_is_running to false + + regardless, display the new search results to the user + save the unfiltered_results as a global + update_search() + +on filter click + adjust the filter selection + update_search() + +update_search + apply search filters by looping through the unfiltered_results and finding the first 200 + unique results that match the filters + + Update the DOM +*/ + +/////// SEARCH WORKER /////// + +function worker_function(documenterSearchIndex, documenterBaseURL, filters) { + importScripts( + "https://cdn.jsdelivr.net/npm/minisearch@6.1.0/dist/umd/index.min.js" + ); + + let data = documenterSearchIndex.map((x, key) => { + x["id"] = key; // minisearch requires a unique for each object + return x; + }); + + // list below is the lunr 2.1.3 list minus the intersect with names(Base) + // (all, any, get, in, is, only, which) and (do, else, for, let, where, while, with) + // ideally we'd just filter the original list but it's not available as a variable + const stopWords = new Set([ + "a", + "able", + "about", + "across", + "after", + "almost", + "also", + "am", + "among", + "an", + "and", + "are", + "as", + "at", + "be", + "because", + "been", + "but", + "by", + "can", + "cannot", + "could", + "dear", + "did", + "does", + "either", + "ever", + "every", + "from", + "got", + "had", + "has", + "have", + "he", + "her", + "hers", + "him", + "his", + "how", + "however", + "i", + "if", + "into", + "it", + "its", + "just", + "least", + "like", + "likely", + "may", + "me", + "might", + "most", + "must", + "my", + "neither", + "no", + "nor", + "not", + "of", + "off", + "often", + "on", + "or", + "other", + "our", + "own", + "rather", + "said", + "say", + "says", + "she", + "should", + "since", + "so", + "some", + "than", + "that", + "the", + "their", + "them", + "then", + "there", + "these", + "they", + "this", + "tis", + "to", + "too", + "twas", + "us", + "wants", + "was", + "we", + "were", + "what", + "when", + "who", + "whom", + "why", + "will", + "would", + "yet", + "you", + "your", + ]); + + let index = new MiniSearch({ + fields: ["title", "text"], // fields to index for full-text search + storeFields: ["location", "title", "text", "category", "page"], // fields to return with results + processTerm: (term) => { + let word = stopWords.has(term) ? null : term; + if (word) { + // custom trimmer that doesn't strip @ and !, which are used in julia macro and function names + word = word + .replace(/^[^a-zA-Z0-9@!]+/, "") + .replace(/[^a-zA-Z0-9@!]+$/, ""); + + word = word.toLowerCase(); + } + + return word ?? null; + }, + // add . as a separator, because otherwise "title": "Documenter.Anchors.add!", would not + // find anything if searching for "add!", only for the entire qualification + tokenize: (string) => string.split(/[\s\-\.]+/), + // options which will be applied during the search + searchOptions: { + prefix: true, + boost: { title: 100 }, + fuzzy: 2, + }, + }); + + index.addAll(data); + + /** + * Used to map characters to HTML entities. + * Refer: https://github.com/lodash/lodash/blob/main/src/escape.ts + */ + const htmlEscapes = { + "&": "&", + "<": "<", + ">": ">", + '"': """, + "'": "'", + }; + + /** + * Used to match HTML entities and HTML characters. + * Refer: https://github.com/lodash/lodash/blob/main/src/escape.ts + */ + const reUnescapedHtml = /[&<>"']/g; + const reHasUnescapedHtml = RegExp(reUnescapedHtml.source); + + /** + * Escape function from lodash + * Refer: https://github.com/lodash/lodash/blob/main/src/escape.ts + */ + function escape(string) { + return string && reHasUnescapedHtml.test(string) + ? string.replace(reUnescapedHtml, (chr) => htmlEscapes[chr]) + : string || ""; + } + + /** + * Make the result component given a minisearch result data object and the value + * of the search input as queryString. To view the result object structure, refer: + * https://lucaong.github.io/minisearch/modules/_minisearch_.html#searchresult + * + * @param {object} result + * @param {string} querystring + * @returns string + */ + function make_search_result(result, querystring) { + let search_divider = `
`; + let display_link = + result.location.slice(Math.max(0), Math.min(50, result.location.length)) + + (result.location.length > 30 ? "..." : ""); // To cut-off the link because it messes with the overflow of the whole div + + if (result.page !== "") { + display_link += ` (${result.page})`; + } + + let textindex = new RegExp(`${querystring}`, "i").exec(result.text); + let text = + textindex !== null + ? result.text.slice( + Math.max(textindex.index - 100, 0), + Math.min( + textindex.index + querystring.length + 100, + result.text.length + ) + ) + : ""; // cut-off text before and after from the match + + text = text.length ? escape(text) : ""; + + let display_result = text.length + ? "..." + + text.replace( + new RegExp(`${escape(querystring)}`, "i"), // For first occurrence + '$&' + ) + + "..." + : ""; // highlights the match + + let in_code = false; + if (!["page", "section"].includes(result.category.toLowerCase())) { + in_code = true; + } + + // We encode the full url to escape some special characters which can lead to broken links + let result_div = ` + +
+
${escape(result.title)}
+
${result.category}
+
+

+ ${display_result} +

+
+ ${display_link} +
+ + ${search_divider} + `; + + return result_div; + } + + self.onmessage = function (e) { + let query = e.data; + let results = index.search(query, { + filter: (result) => { + // Only return relevant results + return result.score >= 1; + }, + }); + + // Pre-filter to deduplicate and limit to 200 per category to the extent + // possible without knowing what the filters are. + let filtered_results = []; + let counts = {}; + for (let filter of filters) { + counts[filter] = 0; + } + let present = {}; + + for (let result of results) { + cat = result.category; + cnt = counts[cat]; + if (cnt < 200) { + id = cat + "---" + result.location; + if (present[id]) { + continue; + } + present[id] = true; + filtered_results.push({ + location: result.location, + category: cat, + div: make_search_result(result, query), + }); + } + } + + postMessage(filtered_results); + }; +} + +// `worker = Threads.@spawn worker_function(documenterSearchIndex)`, but in JavaScript! +const filters = [ + ...new Set(documenterSearchIndex["docs"].map((x) => x.category)), +]; +const worker_str = + "(" + + worker_function.toString() + + ")(" + + JSON.stringify(documenterSearchIndex["docs"]) + + "," + + JSON.stringify(documenterBaseURL) + + "," + + JSON.stringify(filters) + + ")"; +const worker_blob = new Blob([worker_str], { type: "text/javascript" }); +const worker = new Worker(URL.createObjectURL(worker_blob)); + +/////// SEARCH MAIN /////// + +// Whether the worker is currently handling a search. This is a boolean +// as the worker only ever handles 1 or 0 searches at a time. +var worker_is_running = false; + +// The last search text that was sent to the worker. This is used to determine +// if the worker should be launched again when it reports back results. +var last_search_text = ""; + +// The results of the last search. This, in combination with the state of the filters +// in the DOM, is used compute the results to display on calls to update_search. +var unfiltered_results = []; + +// Which filter is currently selected +var selected_filter = ""; + +$(document).on("input", ".documenter-search-input", function (event) { + if (!worker_is_running) { + launch_search(); + } +}); + +function launch_search() { + worker_is_running = true; + last_search_text = $(".documenter-search-input").val(); + worker.postMessage(last_search_text); +} + +worker.onmessage = function (e) { + if (last_search_text !== $(".documenter-search-input").val()) { + launch_search(); + } else { + worker_is_running = false; + } + + unfiltered_results = e.data; + update_search(); +}; + +$(document).on("click", ".search-filter", function () { + if ($(this).hasClass("search-filter-selected")) { + selected_filter = ""; + } else { + selected_filter = $(this).text().toLowerCase(); + } + + // This updates search results and toggles classes for UI: + update_search(); +}); + +/** + * Make/Update the search component + */ +function update_search() { + let querystring = $(".documenter-search-input").val(); + + if (querystring.trim()) { + if (selected_filter == "") { + results = unfiltered_results; + } else { + results = unfiltered_results.filter((result) => { + return selected_filter == result.category.toLowerCase(); + }); + } + + let search_result_container = ``; + let modal_filters = make_modal_body_filters(); + let search_divider = `
`; + + if (results.length) { + let links = []; + let count = 0; + let search_results = ""; + + for (var i = 0, n = results.length; i < n && count < 200; ++i) { + let result = results[i]; + if (result.location && !links.includes(result.location)) { + search_results += result.div; + count++; + links.push(result.location); + } + } + + if (count == 1) { + count_str = "1 result"; + } else if (count == 200) { + count_str = "200+ results"; + } else { + count_str = count + " results"; + } + let result_count = `
${count_str}
`; + + search_result_container = ` +
+ ${modal_filters} + ${search_divider} + ${result_count} +
+ ${search_results} +
+
+ `; + } else { + search_result_container = ` +
+ ${modal_filters} + ${search_divider} +
0 result(s)
+
+
No result found!
+ `; + } + + if ($(".search-modal-card-body").hasClass("is-justify-content-center")) { + $(".search-modal-card-body").removeClass("is-justify-content-center"); + } + + $(".search-modal-card-body").html(search_result_container); + } else { + if (!$(".search-modal-card-body").hasClass("is-justify-content-center")) { + $(".search-modal-card-body").addClass("is-justify-content-center"); + } + + $(".search-modal-card-body").html(` +
Type something to get started!
+ `); + } +} + +/** + * Make the modal filter html + * + * @returns string + */ +function make_modal_body_filters() { + let str = filters + .map((val) => { + if (selected_filter == val.toLowerCase()) { + return `${val}`; + } else { + return `${val}`; + } + }) + .join(""); + + return ` +
+ Filters: + ${str} +
`; +} + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +// Modal settings dialog +$(document).ready(function () { + var settings = $("#documenter-settings"); + $("#documenter-settings-button").click(function () { + settings.toggleClass("is-active"); + }); + // Close the dialog if X is clicked + $("#documenter-settings button.delete").click(function () { + settings.removeClass("is-active"); + }); + // Close dialog if ESC is pressed + $(document).keyup(function (e) { + if (e.keyCode == 27) settings.removeClass("is-active"); + }); +}); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +$(document).ready(function () { + let search_modal_header = ` +
+
+

+ + + + +

+
+
+ +
+
+ `; + + let initial_search_body = ` +
Type something to get started!
+ `; + + let search_modal_footer = ` + + `; + + $(document.body).append( + ` + + ` + ); + + document.querySelector(".docs-search-query").addEventListener("click", () => { + openModal(); + }); + + document + .querySelector(".close-search-modal") + .addEventListener("click", () => { + closeModal(); + }); + + $(document).on("click", ".search-result-link", function () { + closeModal(); + }); + + document.addEventListener("keydown", (event) => { + if ((event.ctrlKey || event.metaKey) && event.key === "/") { + openModal(); + } else if (event.key === "Escape") { + closeModal(); + } + + return false; + }); + + // Functions to open and close a modal + function openModal() { + let searchModal = document.querySelector("#search-modal"); + + searchModal.classList.add("is-active"); + document.querySelector(".documenter-search-input").focus(); + } + + function closeModal() { + let searchModal = document.querySelector("#search-modal"); + let initial_search_body = ` +
Type something to get started!
+ `; + + searchModal.classList.remove("is-active"); + document.querySelector(".documenter-search-input").blur(); + + if (!$(".search-modal-card-body").hasClass("is-justify-content-center")) { + $(".search-modal-card-body").addClass("is-justify-content-center"); + } + + $(".documenter-search-input").val(""); + $(".search-modal-card-body").html(initial_search_body); + } + + document + .querySelector("#search-modal .modal-background") + .addEventListener("click", () => { + closeModal(); + }); +}); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +// Manages the showing and hiding of the sidebar. +$(document).ready(function () { + var sidebar = $("#documenter > .docs-sidebar"); + var sidebar_button = $("#documenter-sidebar-button"); + sidebar_button.click(function (ev) { + ev.preventDefault(); + sidebar.toggleClass("visible"); + if (sidebar.hasClass("visible")) { + // Makes sure that the current menu item is visible in the sidebar. + $("#documenter .docs-menu a.is-active").focus(); + } + }); + $("#documenter > .docs-main").bind("click", function (ev) { + if ($(ev.target).is(sidebar_button)) { + return; + } + if (sidebar.hasClass("visible")) { + sidebar.removeClass("visible"); + } + }); +}); + +// Resizes the package name / sitename in the sidebar if it is too wide. +// Inspired by: https://github.com/davatron5000/FitText.js +$(document).ready(function () { + e = $("#documenter .docs-autofit"); + function resize() { + var L = parseInt(e.css("max-width"), 10); + var L0 = e.width(); + if (L0 > L) { + var h0 = parseInt(e.css("font-size"), 10); + e.css("font-size", (L * h0) / L0); + // TODO: make sure it survives resizes? + } + } + // call once and then register events + resize(); + $(window).resize(resize); + $(window).on("orientationchange", resize); +}); + +// Scroll the navigation bar to the currently selected menu item +$(document).ready(function () { + var sidebar = $("#documenter .docs-menu").get(0); + var active = $("#documenter .docs-menu .is-active").get(0); + if (typeof active !== "undefined") { + sidebar.scrollTop = active.offsetTop - sidebar.offsetTop - 15; + } +}); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +// Theme picker setup +$(document).ready(function () { + // onchange callback + $("#documenter-themepicker").change(function themepick_callback(ev) { + var themename = $("#documenter-themepicker option:selected").attr("value"); + if (themename === "auto") { + // set_theme(window.matchMedia('(prefers-color-scheme: dark)').matches ? 'dark' : 'light'); + window.localStorage.removeItem("documenter-theme"); + } else { + // set_theme(themename); + window.localStorage.setItem("documenter-theme", themename); + } + // We re-use the global function from themeswap.js to actually do the swapping. + set_theme_from_local_storage(); + }); + + // Make sure that the themepicker displays the correct theme when the theme is retrieved + // from localStorage + if (typeof window.localStorage !== "undefined") { + var theme = window.localStorage.getItem("documenter-theme"); + if (theme !== null) { + $("#documenter-themepicker option").each(function (i, e) { + e.selected = e.value === theme; + }); + } + } +}); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +// update the version selector with info from the siteinfo.js and ../versions.js files +$(document).ready(function () { + // If the version selector is disabled with DOCUMENTER_VERSION_SELECTOR_DISABLED in the + // siteinfo.js file, we just return immediately and not display the version selector. + if ( + typeof DOCUMENTER_VERSION_SELECTOR_DISABLED === "boolean" && + DOCUMENTER_VERSION_SELECTOR_DISABLED + ) { + return; + } + + var version_selector = $("#documenter .docs-version-selector"); + var version_selector_select = $("#documenter .docs-version-selector select"); + + version_selector_select.change(function (x) { + target_href = version_selector_select + .children("option:selected") + .get(0).value; + window.location.href = target_href; + }); + + // add the current version to the selector based on siteinfo.js, but only if the selector is empty + if ( + typeof DOCUMENTER_CURRENT_VERSION !== "undefined" && + $("#version-selector > option").length == 0 + ) { + var option = $( + "" + ); + version_selector_select.append(option); + } + + if (typeof DOC_VERSIONS !== "undefined") { + var existing_versions = version_selector_select.children("option"); + var existing_versions_texts = existing_versions.map(function (i, x) { + return x.text; + }); + DOC_VERSIONS.forEach(function (each) { + var version_url = documenterBaseURL + "/../" + each + "/"; + var existing_id = $.inArray(each, existing_versions_texts); + // if not already in the version selector, add it as a new option, + // otherwise update the old option with the URL and enable it + if (existing_id == -1) { + var option = $( + "" + ); + version_selector_select.append(option); + } else { + var option = existing_versions[existing_id]; + option.value = version_url; + option.disabled = false; + } + }); + } + + // only show the version selector if the selector has been populated + if (version_selector_select.children("option").length > 0) { + version_selector.toggleClass("visible"); + } +}); + +}) diff --git a/dev/assets/logo.svg b/dev/assets/logo.svg new file mode 100644 index 0000000..df89daf --- /dev/null +++ b/dev/assets/logo.svg @@ -0,0 +1,169 @@ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +≈ ++ ++…+ + diff --git a/dev/assets/themes/documenter-dark.css b/dev/assets/themes/documenter-dark.css new file mode 100644 index 0000000..1d71701 --- /dev/null +++ b/dev/assets/themes/documenter-dark.css @@ -0,0 +1,7 @@ +html.theme--documenter-dark .pagination-previous,html.theme--documenter-dark .pagination-next,html.theme--documenter-dark .pagination-link,html.theme--documenter-dark .pagination-ellipsis,html.theme--documenter-dark .file-cta,html.theme--documenter-dark .file-name,html.theme--documenter-dark .select select,html.theme--documenter-dark .textarea,html.theme--documenter-dark .input,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input,html.theme--documenter-dark .button{-moz-appearance:none;-webkit-appearance:none;align-items:center;border:1px solid transparent;border-radius:.4em;box-shadow:none;display:inline-flex;font-size:1rem;height:2.5em;justify-content:flex-start;line-height:1.5;padding-bottom:calc(0.5em - 1px);padding-left:calc(0.75em - 1px);padding-right:calc(0.75em - 1px);padding-top:calc(0.5em - 1px);position:relative;vertical-align:top}html.theme--documenter-dark .pagination-previous:focus,html.theme--documenter-dark .pagination-next:focus,html.theme--documenter-dark .pagination-link:focus,html.theme--documenter-dark .pagination-ellipsis:focus,html.theme--documenter-dark .file-cta:focus,html.theme--documenter-dark .file-name:focus,html.theme--documenter-dark .select select:focus,html.theme--documenter-dark .textarea:focus,html.theme--documenter-dark .input:focus,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input:focus,html.theme--documenter-dark .button:focus,html.theme--documenter-dark .is-focused.pagination-previous,html.theme--documenter-dark .is-focused.pagination-next,html.theme--documenter-dark .is-focused.pagination-link,html.theme--documenter-dark .is-focused.pagination-ellipsis,html.theme--documenter-dark .is-focused.file-cta,html.theme--documenter-dark .is-focused.file-name,html.theme--documenter-dark .select select.is-focused,html.theme--documenter-dark .is-focused.textarea,html.theme--documenter-dark .is-focused.input,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.is-focused,html.theme--documenter-dark .is-focused.button,html.theme--documenter-dark .pagination-previous:active,html.theme--documenter-dark .pagination-next:active,html.theme--documenter-dark .pagination-link:active,html.theme--documenter-dark .pagination-ellipsis:active,html.theme--documenter-dark .file-cta:active,html.theme--documenter-dark .file-name:active,html.theme--documenter-dark .select select:active,html.theme--documenter-dark .textarea:active,html.theme--documenter-dark .input:active,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input:active,html.theme--documenter-dark .button:active,html.theme--documenter-dark .is-active.pagination-previous,html.theme--documenter-dark .is-active.pagination-next,html.theme--documenter-dark .is-active.pagination-link,html.theme--documenter-dark .is-active.pagination-ellipsis,html.theme--documenter-dark .is-active.file-cta,html.theme--documenter-dark .is-active.file-name,html.theme--documenter-dark .select select.is-active,html.theme--documenter-dark .is-active.textarea,html.theme--documenter-dark .is-active.input,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.is-active,html.theme--documenter-dark .is-active.button{outline:none}html.theme--documenter-dark .pagination-previous[disabled],html.theme--documenter-dark .pagination-next[disabled],html.theme--documenter-dark .pagination-link[disabled],html.theme--documenter-dark .pagination-ellipsis[disabled],html.theme--documenter-dark .file-cta[disabled],html.theme--documenter-dark .file-name[disabled],html.theme--documenter-dark .select select[disabled],html.theme--documenter-dark .textarea[disabled],html.theme--documenter-dark .input[disabled],html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input[disabled],html.theme--documenter-dark .button[disabled],fieldset[disabled] html.theme--documenter-dark .pagination-previous,html.theme--documenter-dark fieldset[disabled] .pagination-previous,fieldset[disabled] html.theme--documenter-dark .pagination-next,html.theme--documenter-dark fieldset[disabled] .pagination-next,fieldset[disabled] html.theme--documenter-dark .pagination-link,html.theme--documenter-dark fieldset[disabled] .pagination-link,fieldset[disabled] html.theme--documenter-dark .pagination-ellipsis,html.theme--documenter-dark fieldset[disabled] .pagination-ellipsis,fieldset[disabled] html.theme--documenter-dark .file-cta,html.theme--documenter-dark fieldset[disabled] 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.button.is-white{background-color:#fff;border-color:#fff;box-shadow:none}html.theme--documenter-dark .button.is-white.is-inverted{background-color:#0a0a0a;color:#fff}html.theme--documenter-dark .button.is-white.is-inverted:hover,html.theme--documenter-dark .button.is-white.is-inverted.is-hovered{background-color:#000}html.theme--documenter-dark .button.is-white.is-inverted[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-white.is-inverted{background-color:#0a0a0a;border-color:transparent;box-shadow:none;color:#fff}html.theme--documenter-dark .button.is-white.is-loading::after{border-color:transparent transparent #0a0a0a #0a0a0a !important}html.theme--documenter-dark .button.is-white.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--documenter-dark .button.is-white.is-outlined:hover,html.theme--documenter-dark .button.is-white.is-outlined.is-hovered,html.theme--documenter-dark .button.is-white.is-outlined:focus,html.theme--documenter-dark .button.is-white.is-outlined.is-focused{background-color:#fff;border-color:#fff;color:#0a0a0a}html.theme--documenter-dark .button.is-white.is-outlined.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--documenter-dark .button.is-white.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-white.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-white.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-white.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #0a0a0a #0a0a0a !important}html.theme--documenter-dark .button.is-white.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-white.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--documenter-dark 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html.theme--documenter-dark .button.is-white.is-inverted.is-outlined{background-color:transparent;border-color:#0a0a0a;box-shadow:none;color:#0a0a0a}html.theme--documenter-dark .button.is-black{background-color:#0a0a0a;border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-black:hover,html.theme--documenter-dark .button.is-black.is-hovered{background-color:#040404;border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-black:focus,html.theme--documenter-dark .button.is-black.is-focused{border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-black:focus:not(:active),html.theme--documenter-dark .button.is-black.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(10,10,10,0.25)}html.theme--documenter-dark .button.is-black:active,html.theme--documenter-dark .button.is-black.is-active{background-color:#000;border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-black[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-black{background-color:#0a0a0a;border-color:#0a0a0a;box-shadow:none}html.theme--documenter-dark .button.is-black.is-inverted{background-color:#fff;color:#0a0a0a}html.theme--documenter-dark .button.is-black.is-inverted:hover,html.theme--documenter-dark .button.is-black.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--documenter-dark .button.is-black.is-inverted[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-black.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#0a0a0a}html.theme--documenter-dark .button.is-black.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--documenter-dark .button.is-black.is-outlined{background-color:transparent;border-color:#0a0a0a;color:#0a0a0a}html.theme--documenter-dark .button.is-black.is-outlined:hover,html.theme--documenter-dark .button.is-black.is-outlined.is-hovered,html.theme--documenter-dark .button.is-black.is-outlined:focus,html.theme--documenter-dark .button.is-black.is-outlined.is-focused{background-color:#0a0a0a;border-color:#0a0a0a;color:#fff}html.theme--documenter-dark .button.is-black.is-outlined.is-loading::after{border-color:transparent transparent #0a0a0a #0a0a0a !important}html.theme--documenter-dark .button.is-black.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-black.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-black.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-black.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--documenter-dark .button.is-black.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-black.is-outlined{background-color:transparent;border-color:#0a0a0a;box-shadow:none;color:#0a0a0a}html.theme--documenter-dark .button.is-black.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--documenter-dark .button.is-black.is-inverted.is-outlined:hover,html.theme--documenter-dark .button.is-black.is-inverted.is-outlined.is-hovered,html.theme--documenter-dark .button.is-black.is-inverted.is-outlined:focus,html.theme--documenter-dark .button.is-black.is-inverted.is-outlined.is-focused{background-color:#fff;color:#0a0a0a}html.theme--documenter-dark .button.is-black.is-inverted.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-black.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-black.is-inverted.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-black.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #0a0a0a #0a0a0a !important}html.theme--documenter-dark .button.is-black.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-black.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--documenter-dark .button.is-light{background-color:#ecf0f1;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--documenter-dark .button.is-light:hover,html.theme--documenter-dark .button.is-light.is-hovered{background-color:#e5eaec;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--documenter-dark .button.is-light:focus,html.theme--documenter-dark .button.is-light.is-focused{border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--documenter-dark .button.is-light:focus:not(:active),html.theme--documenter-dark .button.is-light.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(236,240,241,0.25)}html.theme--documenter-dark .button.is-light:active,html.theme--documenter-dark .button.is-light.is-active{background-color:#dde4e6;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--documenter-dark .button.is-light[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-light{background-color:#ecf0f1;border-color:#ecf0f1;box-shadow:none}html.theme--documenter-dark .button.is-light.is-inverted{background-color:rgba(0,0,0,0.7);color:#ecf0f1}html.theme--documenter-dark .button.is-light.is-inverted:hover,html.theme--documenter-dark .button.is-light.is-inverted.is-hovered{background-color:rgba(0,0,0,0.7)}html.theme--documenter-dark .button.is-light.is-inverted[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-light.is-inverted{background-color:rgba(0,0,0,0.7);border-color:transparent;box-shadow:none;color:#ecf0f1}html.theme--documenter-dark .button.is-light.is-loading::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--documenter-dark .button.is-light.is-outlined{background-color:transparent;border-color:#ecf0f1;color:#ecf0f1}html.theme--documenter-dark .button.is-light.is-outlined:hover,html.theme--documenter-dark .button.is-light.is-outlined.is-hovered,html.theme--documenter-dark .button.is-light.is-outlined:focus,html.theme--documenter-dark .button.is-light.is-outlined.is-focused{background-color:#ecf0f1;border-color:#ecf0f1;color:rgba(0,0,0,0.7)}html.theme--documenter-dark .button.is-light.is-outlined.is-loading::after{border-color:transparent transparent #ecf0f1 #ecf0f1 !important}html.theme--documenter-dark .button.is-light.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-light.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-light.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-light.is-outlined.is-loading.is-focused::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--documenter-dark .button.is-light.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-light.is-outlined{background-color:transparent;border-color:#ecf0f1;box-shadow:none;color:#ecf0f1}html.theme--documenter-dark .button.is-light.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);color:rgba(0,0,0,0.7)}html.theme--documenter-dark .button.is-light.is-inverted.is-outlined:hover,html.theme--documenter-dark .button.is-light.is-inverted.is-outlined.is-hovered,html.theme--documenter-dark .button.is-light.is-inverted.is-outlined:focus,html.theme--documenter-dark .button.is-light.is-inverted.is-outlined.is-focused{background-color:rgba(0,0,0,0.7);color:#ecf0f1}html.theme--documenter-dark .button.is-light.is-inverted.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-light.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-light.is-inverted.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-light.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #ecf0f1 #ecf0f1 !important}html.theme--documenter-dark .button.is-light.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-light.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--documenter-dark .button.is-dark,html.theme--documenter-dark .content kbd.button{background-color:#282f2f;border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-dark:hover,html.theme--documenter-dark .content kbd.button:hover,html.theme--documenter-dark .button.is-dark.is-hovered,html.theme--documenter-dark .content kbd.button.is-hovered{background-color:#232829;border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-dark:focus,html.theme--documenter-dark .content kbd.button:focus,html.theme--documenter-dark .button.is-dark.is-focused,html.theme--documenter-dark .content kbd.button.is-focused{border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-dark:focus:not(:active),html.theme--documenter-dark .content kbd.button:focus:not(:active),html.theme--documenter-dark .button.is-dark.is-focused:not(:active),html.theme--documenter-dark .content kbd.button.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(40,47,47,0.25)}html.theme--documenter-dark .button.is-dark:active,html.theme--documenter-dark .content kbd.button:active,html.theme--documenter-dark .button.is-dark.is-active,html.theme--documenter-dark .content kbd.button.is-active{background-color:#1d2122;border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-dark[disabled],html.theme--documenter-dark .content kbd.button[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-dark,fieldset[disabled] html.theme--documenter-dark .content kbd.button{background-color:#282f2f;border-color:#282f2f;box-shadow:none}html.theme--documenter-dark .button.is-dark.is-inverted,html.theme--documenter-dark .content kbd.button.is-inverted{background-color:#fff;color:#282f2f}html.theme--documenter-dark .button.is-dark.is-inverted:hover,html.theme--documenter-dark .content kbd.button.is-inverted:hover,html.theme--documenter-dark .button.is-dark.is-inverted.is-hovered,html.theme--documenter-dark .content kbd.button.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--documenter-dark .button.is-dark.is-inverted[disabled],html.theme--documenter-dark .content kbd.button.is-inverted[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-dark.is-inverted,fieldset[disabled] html.theme--documenter-dark .content kbd.button.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#282f2f}html.theme--documenter-dark .button.is-dark.is-loading::after,html.theme--documenter-dark .content kbd.button.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--documenter-dark .button.is-dark.is-outlined,html.theme--documenter-dark .content kbd.button.is-outlined{background-color:transparent;border-color:#282f2f;color:#282f2f}html.theme--documenter-dark .button.is-dark.is-outlined:hover,html.theme--documenter-dark .content kbd.button.is-outlined:hover,html.theme--documenter-dark .button.is-dark.is-outlined.is-hovered,html.theme--documenter-dark .content kbd.button.is-outlined.is-hovered,html.theme--documenter-dark .button.is-dark.is-outlined:focus,html.theme--documenter-dark .content kbd.button.is-outlined:focus,html.theme--documenter-dark .button.is-dark.is-outlined.is-focused,html.theme--documenter-dark .content kbd.button.is-outlined.is-focused{background-color:#282f2f;border-color:#282f2f;color:#fff}html.theme--documenter-dark .button.is-dark.is-outlined.is-loading::after,html.theme--documenter-dark .content kbd.button.is-outlined.is-loading::after{border-color:transparent transparent #282f2f #282f2f !important}html.theme--documenter-dark .button.is-dark.is-outlined.is-loading:hover::after,html.theme--documenter-dark .content kbd.button.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-dark.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .content kbd.button.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-dark.is-outlined.is-loading:focus::after,html.theme--documenter-dark .content kbd.button.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-dark.is-outlined.is-loading.is-focused::after,html.theme--documenter-dark .content kbd.button.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--documenter-dark .button.is-dark.is-outlined[disabled],html.theme--documenter-dark .content kbd.button.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-dark.is-outlined,fieldset[disabled] html.theme--documenter-dark .content kbd.button.is-outlined{background-color:transparent;border-color:#282f2f;box-shadow:none;color:#282f2f}html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined:hover,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined:hover,html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-hovered,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-hovered,html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined:focus,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined:focus,html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-focused,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-focused{background-color:#fff;color:#282f2f}html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-loading:hover::after,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-loading:focus::after,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-loading.is-focused::after,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #282f2f #282f2f !important}html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined[disabled],html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined,fieldset[disabled] html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--documenter-dark .button.is-primary,html.theme--documenter-dark .docstring>section>a.button.docs-sourcelink{background-color:#375a7f;border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-primary:hover,html.theme--documenter-dark .docstring>section>a.button.docs-sourcelink:hover,html.theme--documenter-dark .button.is-primary.is-hovered,html.theme--documenter-dark .docstring>section>a.button.is-hovered.docs-sourcelink{background-color:#335476;border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-primary:focus,html.theme--documenter-dark 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.button.is-warning.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-warning.is-outlined{background-color:transparent;border-color:#ad8100;box-shadow:none;color:#ad8100}html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined:hover,html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined.is-hovered,html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined:focus,html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined.is-focused{background-color:#fff;color:#ad8100}html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #ad8100 #ad8100 !important}html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--documenter-dark .button.is-warning.is-light{background-color:#fffaeb;color:#d19c00}html.theme--documenter-dark .button.is-warning.is-light:hover,html.theme--documenter-dark .button.is-warning.is-light.is-hovered{background-color:#fff7de;border-color:transparent;color:#d19c00}html.theme--documenter-dark .button.is-warning.is-light:active,html.theme--documenter-dark .button.is-warning.is-light.is-active{background-color:#fff3d1;border-color:transparent;color:#d19c00}html.theme--documenter-dark .button.is-danger{background-color:#9e1b0d;border-color:transparent;color:#fff}html.theme--documenter-dark 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.button.is-danger.is-inverted:hover,html.theme--documenter-dark .button.is-danger.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--documenter-dark .button.is-danger.is-inverted[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-danger.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#9e1b0d}html.theme--documenter-dark .button.is-danger.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--documenter-dark .button.is-danger.is-outlined{background-color:transparent;border-color:#9e1b0d;color:#9e1b0d}html.theme--documenter-dark .button.is-danger.is-outlined:hover,html.theme--documenter-dark .button.is-danger.is-outlined.is-hovered,html.theme--documenter-dark .button.is-danger.is-outlined:focus,html.theme--documenter-dark .button.is-danger.is-outlined.is-focused{background-color:#9e1b0d;border-color:#9e1b0d;color:#fff}html.theme--documenter-dark .button.is-danger.is-outlined.is-loading::after{border-color:transparent transparent #9e1b0d #9e1b0d !important}html.theme--documenter-dark .button.is-danger.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-danger.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-danger.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-danger.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--documenter-dark .button.is-danger.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-danger.is-outlined{background-color:transparent;border-color:#9e1b0d;box-shadow:none;color:#9e1b0d}html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined:hover,html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined.is-hovered,html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined:focus,html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined.is-focused{background-color:#fff;color:#9e1b0d}html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #9e1b0d #9e1b0d !important}html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--documenter-dark .button.is-danger.is-light{background-color:#fdeeec;color:#ec311d}html.theme--documenter-dark .button.is-danger.is-light:hover,html.theme--documenter-dark .button.is-danger.is-light.is-hovered{background-color:#fce3e0;border-color:transparent;color:#ec311d}html.theme--documenter-dark .button.is-danger.is-light:active,html.theme--documenter-dark .button.is-danger.is-light.is-active{background-color:#fcd8d5;border-color:transparent;color:#ec311d}html.theme--documenter-dark .button.is-small,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.button{font-size:.75rem}html.theme--documenter-dark .button.is-small:not(.is-rounded),html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.button:not(.is-rounded){border-radius:3px}html.theme--documenter-dark .button.is-normal{font-size:1rem}html.theme--documenter-dark .button.is-medium{font-size:1.25rem}html.theme--documenter-dark .button.is-large{font-size:1.5rem}html.theme--documenter-dark .button[disabled],fieldset[disabled] html.theme--documenter-dark .button{background-color:#8c9b9d;border-color:#5e6d6f;box-shadow:none;opacity:.5}html.theme--documenter-dark .button.is-fullwidth{display:flex;width:100%}html.theme--documenter-dark .button.is-loading{color:transparent !important;pointer-events:none}html.theme--documenter-dark .button.is-loading::after{position:absolute;left:calc(50% - (1em * 0.5));top:calc(50% - (1em * 0.5));position:absolute !important}html.theme--documenter-dark .button.is-static{background-color:#282f2f;border-color:#5e6d6f;color:#dbdee0;box-shadow:none;pointer-events:none}html.theme--documenter-dark .button.is-rounded,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.button{border-radius:9999px;padding-left:calc(1em + 0.25em);padding-right:calc(1em + 0.25em)}html.theme--documenter-dark .buttons{align-items:center;display:flex;flex-wrap:wrap;justify-content:flex-start}html.theme--documenter-dark .buttons .button{margin-bottom:0.5rem}html.theme--documenter-dark .buttons .button:not(:last-child):not(.is-fullwidth){margin-right:.5rem}html.theme--documenter-dark .buttons:last-child{margin-bottom:-0.5rem}html.theme--documenter-dark .buttons:not(:last-child){margin-bottom:1rem}html.theme--documenter-dark .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large){font-size:.75rem}html.theme--documenter-dark .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large):not(.is-rounded){border-radius:3px}html.theme--documenter-dark .buttons.are-medium .button:not(.is-small):not(.is-normal):not(.is-large){font-size:1.25rem}html.theme--documenter-dark .buttons.are-large .button:not(.is-small):not(.is-normal):not(.is-medium){font-size:1.5rem}html.theme--documenter-dark .buttons.has-addons .button:not(:first-child){border-bottom-left-radius:0;border-top-left-radius:0}html.theme--documenter-dark .buttons.has-addons .button:not(:last-child){border-bottom-right-radius:0;border-top-right-radius:0;margin-right:-1px}html.theme--documenter-dark .buttons.has-addons .button:last-child{margin-right:0}html.theme--documenter-dark .buttons.has-addons .button:hover,html.theme--documenter-dark .buttons.has-addons .button.is-hovered{z-index:2}html.theme--documenter-dark .buttons.has-addons .button:focus,html.theme--documenter-dark .buttons.has-addons .button.is-focused,html.theme--documenter-dark .buttons.has-addons .button:active,html.theme--documenter-dark .buttons.has-addons .button.is-active,html.theme--documenter-dark .buttons.has-addons .button.is-selected{z-index:3}html.theme--documenter-dark .buttons.has-addons .button:focus:hover,html.theme--documenter-dark .buttons.has-addons .button.is-focused:hover,html.theme--documenter-dark .buttons.has-addons .button:active:hover,html.theme--documenter-dark .buttons.has-addons .button.is-active:hover,html.theme--documenter-dark .buttons.has-addons .button.is-selected:hover{z-index:4}html.theme--documenter-dark .buttons.has-addons .button.is-expanded{flex-grow:1;flex-shrink:1}html.theme--documenter-dark .buttons.is-centered{justify-content:center}html.theme--documenter-dark .buttons.is-centered:not(.has-addons) .button:not(.is-fullwidth){margin-left:0.25rem;margin-right:0.25rem}html.theme--documenter-dark .buttons.is-right{justify-content:flex-end}html.theme--documenter-dark .buttons.is-right:not(.has-addons) .button:not(.is-fullwidth){margin-left:0.25rem;margin-right:0.25rem}@media screen and (max-width: 768px){html.theme--documenter-dark .button.is-responsive.is-small,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.is-responsive{font-size:.5625rem}html.theme--documenter-dark .button.is-responsive,html.theme--documenter-dark .button.is-responsive.is-normal{font-size:.65625rem}html.theme--documenter-dark .button.is-responsive.is-medium{font-size:.75rem}html.theme--documenter-dark .button.is-responsive.is-large{font-size:1rem}}@media screen and (min-width: 769px) and (max-width: 1055px){html.theme--documenter-dark .button.is-responsive.is-small,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.is-responsive{font-size:.65625rem}html.theme--documenter-dark .button.is-responsive,html.theme--documenter-dark .button.is-responsive.is-normal{font-size:.75rem}html.theme--documenter-dark .button.is-responsive.is-medium{font-size:1rem}html.theme--documenter-dark .button.is-responsive.is-large{font-size:1.25rem}}html.theme--documenter-dark .container{flex-grow:1;margin:0 auto;position:relative;width:auto}html.theme--documenter-dark .container.is-fluid{max-width:none !important;padding-left:32px;padding-right:32px;width:100%}@media screen and (min-width: 1056px){html.theme--documenter-dark .container{max-width:992px}}@media screen and (max-width: 1215px){html.theme--documenter-dark .container.is-widescreen:not(.is-max-desktop){max-width:1152px}}@media screen and (max-width: 1407px){html.theme--documenter-dark .container.is-fullhd:not(.is-max-desktop):not(.is-max-widescreen){max-width:1344px}}@media screen and (min-width: 1216px){html.theme--documenter-dark .container:not(.is-max-desktop){max-width:1152px}}@media screen and (min-width: 1408px){html.theme--documenter-dark .container:not(.is-max-desktop):not(.is-max-widescreen){max-width:1344px}}html.theme--documenter-dark .content li+li{margin-top:0.25em}html.theme--documenter-dark .content p:not(:last-child),html.theme--documenter-dark .content dl:not(:last-child),html.theme--documenter-dark .content ol:not(:last-child),html.theme--documenter-dark .content ul:not(:last-child),html.theme--documenter-dark .content blockquote:not(:last-child),html.theme--documenter-dark .content pre:not(:last-child),html.theme--documenter-dark .content table:not(:last-child){margin-bottom:1em}html.theme--documenter-dark .content h1,html.theme--documenter-dark .content h2,html.theme--documenter-dark .content h3,html.theme--documenter-dark .content h4,html.theme--documenter-dark .content h5,html.theme--documenter-dark .content h6{color:#f2f2f2;font-weight:600;line-height:1.125}html.theme--documenter-dark .content h1{font-size:2em;margin-bottom:0.5em}html.theme--documenter-dark .content h1:not(:first-child){margin-top:1em}html.theme--documenter-dark .content h2{font-size:1.75em;margin-bottom:0.5714em}html.theme--documenter-dark .content h2:not(:first-child){margin-top:1.1428em}html.theme--documenter-dark .content h3{font-size:1.5em;margin-bottom:0.6666em}html.theme--documenter-dark .content h3:not(:first-child){margin-top:1.3333em}html.theme--documenter-dark .content h4{font-size:1.25em;margin-bottom:0.8em}html.theme--documenter-dark .content h5{font-size:1.125em;margin-bottom:0.8888em}html.theme--documenter-dark .content h6{font-size:1em;margin-bottom:1em}html.theme--documenter-dark .content blockquote{background-color:#282f2f;border-left:5px solid #5e6d6f;padding:1.25em 1.5em}html.theme--documenter-dark .content ol{list-style-position:outside;margin-left:2em;margin-top:1em}html.theme--documenter-dark .content ol:not([type]){list-style-type:decimal}html.theme--documenter-dark .content ol.is-lower-alpha:not([type]){list-style-type:lower-alpha}html.theme--documenter-dark .content ol.is-lower-roman:not([type]){list-style-type:lower-roman}html.theme--documenter-dark .content ol.is-upper-alpha:not([type]){list-style-type:upper-alpha}html.theme--documenter-dark .content ol.is-upper-roman:not([type]){list-style-type:upper-roman}html.theme--documenter-dark .content ul{list-style:disc outside;margin-left:2em;margin-top:1em}html.theme--documenter-dark .content ul ul{list-style-type:circle;margin-top:0.5em}html.theme--documenter-dark .content ul ul ul{list-style-type:square}html.theme--documenter-dark .content dd{margin-left:2em}html.theme--documenter-dark .content figure{margin-left:2em;margin-right:2em;text-align:center}html.theme--documenter-dark .content figure:not(:first-child){margin-top:2em}html.theme--documenter-dark .content figure:not(:last-child){margin-bottom:2em}html.theme--documenter-dark .content figure img{display:inline-block}html.theme--documenter-dark .content figure figcaption{font-style:italic}html.theme--documenter-dark .content pre{-webkit-overflow-scrolling:touch;overflow-x:auto;padding:0;white-space:pre;word-wrap:normal}html.theme--documenter-dark .content sup,html.theme--documenter-dark .content sub{font-size:75%}html.theme--documenter-dark .content table{width:100%}html.theme--documenter-dark .content table td,html.theme--documenter-dark .content table th{border:1px solid #5e6d6f;border-width:0 0 1px;padding:0.5em 0.75em;vertical-align:top}html.theme--documenter-dark .content table th{color:#f2f2f2}html.theme--documenter-dark .content table th:not([align]){text-align:inherit}html.theme--documenter-dark .content table thead td,html.theme--documenter-dark .content table thead th{border-width:0 0 2px;color:#f2f2f2}html.theme--documenter-dark .content table tfoot td,html.theme--documenter-dark .content table tfoot th{border-width:2px 0 0;color:#f2f2f2}html.theme--documenter-dark .content table tbody tr:last-child td,html.theme--documenter-dark .content table tbody tr:last-child th{border-bottom-width:0}html.theme--documenter-dark .content .tabs li+li{margin-top:0}html.theme--documenter-dark .content.is-small,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.content{font-size:.75rem}html.theme--documenter-dark .content.is-normal{font-size:1rem}html.theme--documenter-dark .content.is-medium{font-size:1.25rem}html.theme--documenter-dark .content.is-large{font-size:1.5rem}html.theme--documenter-dark .icon{align-items:center;display:inline-flex;justify-content:center;height:1.5rem;width:1.5rem}html.theme--documenter-dark .icon.is-small,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.icon{height:1rem;width:1rem}html.theme--documenter-dark .icon.is-medium{height:2rem;width:2rem}html.theme--documenter-dark .icon.is-large{height:3rem;width:3rem}html.theme--documenter-dark .icon-text{align-items:flex-start;color:inherit;display:inline-flex;flex-wrap:wrap;line-height:1.5rem;vertical-align:top}html.theme--documenter-dark .icon-text .icon{flex-grow:0;flex-shrink:0}html.theme--documenter-dark .icon-text .icon:not(:last-child){margin-right:.25em}html.theme--documenter-dark .icon-text .icon:not(:first-child){margin-left:.25em}html.theme--documenter-dark div.icon-text{display:flex}html.theme--documenter-dark .image,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img{display:block;position:relative}html.theme--documenter-dark .image img,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img img{display:block;height:auto;width:100%}html.theme--documenter-dark .image img.is-rounded,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img img.is-rounded{border-radius:9999px}html.theme--documenter-dark .image.is-fullwidth,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-fullwidth{width:100%}html.theme--documenter-dark .image.is-square img,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-square img,html.theme--documenter-dark .image.is-square .has-ratio,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-square .has-ratio,html.theme--documenter-dark .image.is-1by1 img,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-1by1 img,html.theme--documenter-dark .image.is-1by1 .has-ratio,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-1by1 .has-ratio,html.theme--documenter-dark .image.is-5by4 img,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-5by4 img,html.theme--documenter-dark .image.is-5by4 .has-ratio,html.theme--documenter-dark #documenter 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+ Maintainer: @highlightjs/core-team + Website: https://highlightjs.org/ + License: see project LICENSE + Touched: 2021 +*/pre code.hljs{display:block;overflow-x:auto;padding:1em}code.hljs{padding:3px 5px}.hljs{background:#F3F3F3;color:#444}.hljs-comment{color:#697070}.hljs-tag,.hljs-punctuation{color:#444a}.hljs-tag .hljs-name,.hljs-tag .hljs-attr{color:#444}.hljs-keyword,.hljs-attribute,.hljs-selector-tag,.hljs-meta .hljs-keyword,.hljs-doctag,.hljs-name{font-weight:bold}.hljs-type,.hljs-string,.hljs-number,.hljs-selector-id,.hljs-selector-class,.hljs-quote,.hljs-template-tag,.hljs-deletion{color:#880000}.hljs-title,.hljs-section{color:#880000;font-weight:bold}.hljs-regexp,.hljs-symbol,.hljs-variable,.hljs-template-variable,.hljs-link,.hljs-selector-attr,.hljs-operator,.hljs-selector-pseudo{color:#ab5656}.hljs-literal{color:#695}.hljs-built_in,.hljs-bullet,.hljs-code,.hljs-addition{color:#397300}.hljs-meta{color:#1f7199}.hljs-meta .hljs-string{color:#38a}.hljs-emphasis{font-style:italic}.hljs-strong{font-weight:bold}.gap-4{gap:1rem} diff --git a/dev/assets/themeswap.js b/dev/assets/themeswap.js new file mode 100644 index 0000000..9f5eebe --- /dev/null +++ b/dev/assets/themeswap.js @@ -0,0 +1,84 @@ +// Small function to quickly swap out themes. Gets put into the tag.. +function set_theme_from_local_storage() { + // Initialize the theme to null, which means default + var theme = null; + // If the browser supports the localstorage and is not disabled then try to get the + // documenter theme + if (window.localStorage != null) { + // Get the user-picked theme from localStorage. May be `null`, which means the default + // theme. + theme = window.localStorage.getItem("documenter-theme"); + } + // Check if the users preference is for dark color scheme + var darkPreference = + window.matchMedia("(prefers-color-scheme: dark)").matches === true; + // Initialize a few variables for the loop: + // + // - active: will contain the index of the theme that should be active. Note that there + // is no guarantee that localStorage contains sane values. If `active` stays `null` + // we either could not find the theme or it is the default (primary) theme anyway. + // Either way, we then need to stick to the primary theme. + // + // - disabled: style sheets that should be disabled (i.e. all the theme style sheets + // that are not the currently active theme) + var active = null; + var disabled = []; + var primaryLightTheme = null; + var primaryDarkTheme = null; + for (var i = 0; i < document.styleSheets.length; i++) { + var ss = document.styleSheets[i]; + // The tag of each style sheet is expected to have a data-theme-name attribute + // which must contain the name of the theme. The names in localStorage much match this. + var themename = ss.ownerNode.getAttribute("data-theme-name"); + // attribute not set => non-theme stylesheet => ignore + if (themename === null) continue; + // To distinguish the default (primary) theme, it needs to have the data-theme-primary + // attribute set. + if (ss.ownerNode.getAttribute("data-theme-primary") !== null) { + primaryLightTheme = themename; + } + // Check if the theme is primary dark theme so that we could store its name in darkTheme + if (ss.ownerNode.getAttribute("data-theme-primary-dark") !== null) { + primaryDarkTheme = themename; + } + // If we find a matching theme (and it's not the default), we'll set active to non-null + if (themename === theme) active = i; + // Store the style sheets of inactive themes so that we could disable them + if (themename !== theme) disabled.push(ss); + } + var activeTheme = null; + if (active !== null) { + // If we did find an active theme, we'll (1) add the theme--$(theme) class to + document.getElementsByTagName("html")[0].className = "theme--" + theme; + activeTheme = theme; + } else { + // If we did _not_ find an active theme, then we need to fall back to the primary theme + // which can either be dark or light, depending on the user's OS preference. + var activeTheme = darkPreference ? primaryDarkTheme : primaryLightTheme; + // In case it somehow happens that the relevant primary theme was not found in the + // preceding loop, we abort without doing anything. + if (activeTheme === null) { + console.error("Unable to determine primary theme."); + return; + } + // When switching to the primary light theme, then we must not have a class name + // for the tag. That's only for non-primary or the primary dark theme. + if (darkPreference) { + document.getElementsByTagName("html")[0].className = + "theme--" + activeTheme; + } else { + document.getElementsByTagName("html")[0].className = ""; + } + } + for (var i = 0; i < document.styleSheets.length; i++) { + var ss = document.styleSheets[i]; + // The tag of each style sheet is expected to have a data-theme-name attribute + // which must contain the name of the theme. The names in localStorage much match this. + var themename = ss.ownerNode.getAttribute("data-theme-name"); + // attribute not set => non-theme stylesheet => ignore + if (themename === null) continue; + // we'll disable all the stylesheets, except for the active one + ss.disabled = !(themename == activeTheme); + } +} +set_theme_from_local_storage(); diff --git a/dev/assets/warner.js b/dev/assets/warner.js new file mode 100644 index 0000000..3f6f5d0 --- /dev/null +++ b/dev/assets/warner.js @@ -0,0 +1,52 @@ +function maybeAddWarning() { + // DOCUMENTER_NEWEST is defined in versions.js, DOCUMENTER_CURRENT_VERSION and DOCUMENTER_STABLE + // in siteinfo.js. + // If either of these are undefined something went horribly wrong, so we abort. + if ( + window.DOCUMENTER_NEWEST === undefined || + window.DOCUMENTER_CURRENT_VERSION === undefined || + window.DOCUMENTER_STABLE === undefined + ) { + return; + } + + // Current version is not a version number, so we can't tell if it's the newest version. 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GitHub

Examples

Simulate data

julia> using MultiResponseVarianceComponentModels, LinearAlgebra, Random
julia> Random.seed!(6789)Random.TaskLocalRNG()
julia> n = 1_000;  # n of observations
julia> d = 4;      # n of responses
julia> p = 10;     # n of covariates
julia> m = 5;      # n of variance components
julia> X = rand(n, p);
julia> B = rand(p, d)10×4 Matrix{Float64}:
+ 0.866192  0.739958   0.951693   0.716391
+ 0.157873  0.633772   0.916356   0.068644
+ 0.818581  0.333021   0.0713489  0.451072
+ 0.707496  0.924568   0.0636189  0.618679
+ 0.357756  0.173016   0.982384   0.54638
+ 0.722163  0.0683381  0.343516   0.616161
+ 0.441486  0.64316    0.862809   0.400984
+ 0.812486  0.275565   0.477624   0.482087
+ 0.889787  0.669932   0.103051   0.582507
+ 0.331484  0.989292   0.785077   0.456195
julia> V = [zeros(n, n) for _ in 1:m]; # kernel matrices
julia> Σ = [zeros(d, d) for _ in 1:m]; # variance components
julia> for i in 1:m
+           Vi = randn(n, n)
+           copy!(V[i], Vi' * Vi)
+           Σi = randn(d, d)
+           copy!(Σ[i], Σi' * Σi)
+       end
julia> Ω = zeros(n * d, n * d); # overall nd-by-nd covariance matrix Ω
julia> for i = 1:m
+           Ω += kron(Σ[i], V[i])
+       end
julia> Ωchol = cholesky(Ω);
julia> Y = X * B + reshape(Ωchol.L * randn(n * d), n, d)1000×4 Matrix{Float64}:
+ -270.382    -66.0256   233.653     158.94
+  168.935    354.423     14.8585    -54.193
+   85.8007   117.902   -217.707      65.1394
+ -136.574    -42.6014  -234.087      75.6404
+ -190.312   -210.051    105.117     -10.0791
+  153.414    187.179     20.691     -62.3589
+  164.97     -22.4103   -13.6301    161.682
+   21.9438  -184.643    231.526    -136.723
+ -284.549     79.7798   -37.1967   -136.943
+  -59.2492  -124.37    -160.587     112.235
+    ⋮
+  -82.4606   200.557     -4.16006   -47.2201
+  170.0       95.8103   222.034    -179.011
+   98.1247   -33.0932   -67.4875    -86.9752
+ -167.871    110.395   -184.76      -71.029
+  111.644    -23.6517   113.328    -219.276
+  -77.2094   168.859     65.7808     22.939
+  129.34    -155.123   -251.557     120.656
+ -359.121    -82.2576   -82.0063     10.8301
+  -77.7126   -90.7158   -79.5722     92.9695
Note

In the case of heritability and genetic correlation analyses, one can use classic genetic relationship matrices (GRMs) for $\boldsymbol{V}_i$'s, which in turn can be constructed using SnpArrays.jl.

Maximum likelihood estimation

julia> model = MRVCModel(Y, X, V)A multivariate response variance component model
+   * number of responses: 4
+   * number of observations: 1000
+   * number of fixed effects: 10
+   * number of variance components: 5
julia> @timev fit!(model)[ Info: Running MM algorithm for ML estimation
+iter = 0, logl = -26845.09612499673
+iter = 1, logl = -25695.041826875506
+iter = 2, logl = -25293.37779878975
+iter = 3, logl = -25159.24906847003
+iter = 4, logl = -25107.52993220223
+iter = 5, logl = -25081.38428962063
+iter = 6, logl = -25064.352419795112
+iter = 7, logl = -25051.625744592016
+iter = 8, logl = -25041.61006910993
+iter = 9, logl = -25033.604422863456
+iter = 10, logl = -25027.18042662134
+iter = 11, logl = -25022.017854787307
+iter = 12, logl = -25017.85980986688
+iter = 13, logl = -25014.498019681763
+iter = 14, logl = -25011.76503965001
+iter = 15, logl = -25009.527835987446
+iter = 16, logl = -25007.68180780364
+iter = 17, logl = -25006.145330198324
+iter = 18, logl = -25004.854995838356
+iter = 19, logl = -25003.761633006412
+iter = 20, logl = -25002.827079011357
+iter = 21, logl = -25002.02162797271
+iter = 22, logl = -25001.322047241047
+iter = 23, logl = -25000.710054799467
+iter = 24, logl = -25000.171160151207
+iter = 25, logl = -24999.693786243588
+iter = 26, logl = -24999.26860570048
+iter = 27, logl = -24998.888038955105
+iter = 28, logl = -24998.545873977288
+iter = 29, logl = -24998.236977041215
+iter = 30, logl = -24997.95707160835
+iter = 31, logl = -24997.702568235134
+iter = 32, logl = -24997.470432811737
+iter = 33, logl = -24997.258083718323
+iter = 34, logl = -24997.063310912818
+iter = 35, logl = -24996.884211761542
+iter = 36, logl = -24996.719139735058
+iter = 37, logl = -24996.566663066376
+iter = 38, logl = -24996.425531184308
+iter = 39, logl = -24996.294647261133
+iter = 40, logl = -24996.173045607276
+iter = 41, logl = -24996.05987293569
+iter = 42, logl = -24995.95437274284
+iter = 43, logl = -24995.855872212444
+iter = 44, logl = -24995.7637711789
+iter = 45, logl = -24995.67753277965
+iter = 46, logl = -24995.596675504443
+iter = 47, logl = -24995.520766401234
+iter = 48, logl = -24995.449415250514
+iter = 49, logl = -24995.38226954833
+iter = 50, logl = -24995.31901017304
+iter = 51, logl = -24995.259347628595
+iter = 52, logl = -24995.203018777862
+iter = 53, logl = -24995.14978399378
+iter = 54, logl = -24995.09942466663
+iter = 55, logl = -24995.051741018295
+iter = 56, logl = -24995.006550180144
+iter = 57, logl = -24994.963684497296
+iter = 58, logl = -24994.922990031344
+iter = 59, logl = -24994.88432523246
+iter = 60, logl = -24994.84755976209
+iter = 61, logl = -24994.812573442905
+iter = 62, logl = -24994.779255325782
+iter = 63, logl = -24994.74750285241
+iter = 64, logl = -24994.717221108924
+iter = 65, logl = -24994.688322154096
+iter = 66, logl = -24994.660724417583
+iter = 67, logl = -24994.63435215651
+iter = 68, logl = -24994.609134967097
+iter = 69, logl = -24994.585007342
+[ Info: Updates converged!
+[ Info: Calculating standard errors
+ 78.850210 seconds (15.63 M allocations: 992.266 MiB, 0.18% gc time, 7.40% compilation time)
+elapsed time (ns):  78850209625
+gc time (ns):       145010500
+bytes allocated:    1040466060
+pool allocs:        15597659
+non-pool GC allocs: 26821
+malloc() calls:     931
+realloc() calls:    107
+free() calls:       335
+minor collections:  1
+full collections:   0
+Converged after 70 iterations.

Then variance components and mean effects estimates can be accessed through

julia> model.Σ5-element Vector{Matrix{Float64}}:
+ [2.8964235833864374 -0.9547701186106777 1.6856925389781934 0.8185656173741515; -0.9547701186106778 7.474693619948036 -4.744935846247633 -0.601253052759175; 1.6856925389781932 -4.744935846247633 8.639624062398605 0.4049276157435974; 0.8185656173741512 -0.601253052759175 0.4049276157435973 1.0145168589537352]
+ [3.776260289140289 2.3919496497788035 -0.3666324933910243 -1.365148782876448; 2.391949649778804 1.9914356452751707 -0.2630505419731652 -1.5892702567044454; -0.36663249339102433 -0.26305054197316524 4.0912210911875855 -2.287177667967102; -1.365148782876448 -1.5892702567044454 -2.287177667967102 4.526163931034507]
+ [3.1157472905367367 -0.08262940589305556 -0.27318718635822953 0.4179078660737166; -0.08262940589305548 3.4556143634179275 -0.20341922639890309 0.6648870938984194; -0.27318718635822953 -0.20341922639890314 1.9155083377403541 0.49426216391703903; 0.41790786607371666 0.6648870938984194 0.49426216391703903 0.5435108874298807]
+ [3.6027030953820782 1.2366936673894326 0.3637181414535267 0.06851794976775202; 1.2366936673894326 0.9715515927327851 1.6405295499121633 0.5971896668720472; 0.3637181414535266 1.6405295499121637 5.828012183031251 0.9752007240509573; 0.06851794976775206 0.5971896668720472 0.9752007240509573 0.9502953338899155]
+ [5.669808082536949 -2.01083534935516 1.1222606709505223 1.4405860817195557; -2.0108353493551605 6.032248061565413 9.439926228569771 -2.150730923690082; 1.1222606709505227 9.439926228569773 19.179102992380216 -3.054827357292586; 1.4405860817195555 -2.150730923690082 -3.054827357292586 3.445429926626015]
julia> model.B10×4 Matrix{Float64}:
+   7.76641     0.0581691   30.0176     10.8424
+  -1.70871    17.0653      26.3436      5.24079
+  -4.99575    28.325      -20.9966    -13.1088
+ -10.2939     -6.00867     12.3106    -16.3959
+  -3.07093     0.733837    15.8845      1.17996
+  14.1045     -3.92946     11.5408     11.2497
+  -1.47949    -8.30862      5.49722    -0.213233
+   6.03975     5.27422      0.189634    3.90586
+  -0.632607  -14.288      -29.194       4.9749
+   6.17054   -13.2489     -23.6247     -3.52297
julia> hcat(vec(B), vec(model.B))40×2 Matrix{Float64}:
+ 0.866192    7.76641
+ 0.157873   -1.70871
+ 0.818581   -4.99575
+ 0.707496  -10.2939
+ 0.357756   -3.07093
+ 0.722163   14.1045
+ 0.441486   -1.47949
+ 0.812486    6.03975
+ 0.889787   -0.632607
+ 0.331484    6.17054
+ ⋮
+ 0.068644    5.24079
+ 0.451072  -13.1088
+ 0.618679  -16.3959
+ 0.54638     1.17996
+ 0.616161   11.2497
+ 0.400984   -0.213233
+ 0.482087    3.90586
+ 0.582507    4.9749
+ 0.456195   -3.52297
julia> reduce(hcat, [hcat(vec(Σ[i]), vec(model.Σ[i])) for i in 1:m])16×10 Matrix{Float64}:
+  2.59353    2.89642    3.32588    …  3.6027      5.37835    5.66981
+ -0.908817  -0.95477    2.21659       1.23669    -1.43756   -2.01084
+  1.58682    1.68569   -0.0345923     0.363718    1.65263    1.12226
+  0.365451   0.818566  -1.36959       0.0685179   0.983537   1.44059
+ -0.908817  -0.95477    2.21659       1.23669    -1.43756   -2.01084
+  7.12022    7.47469    1.75764    …  0.971552    6.84955    6.03225
+ -4.72969   -4.74494   -0.0950186     1.64053     9.98583    9.43993
+ -0.430129  -0.601253  -1.55052       0.59719    -1.98954   -2.15073
+  1.58682    1.68569   -0.0345923     0.363718    1.65263    1.12226
+ -4.72969   -4.74494   -0.0950186     1.64053     9.98583    9.43993
+  6.22778    8.63962    4.06617    …  5.82801    19.2083    19.1791
+  0.969228   0.404928  -2.51336       0.975201   -3.5885    -3.05483
+  0.365451   0.818566  -1.36959       0.0685179   0.983537   1.44059
+ -0.430129  -0.601253  -1.55052       0.59719    -1.98954   -2.15073
+  0.969228   0.404928  -2.51336       0.975201   -3.5885    -3.05483
+  1.2446     1.01452    5.02933    …  0.950295    2.85342    3.44543

Standard errors

Sampling variance and covariance of these estimates are

julia> model.Σcov50×50 Matrix{Float64}:
+  0.47412      -0.00272341    0.108306     …  -0.00114518   -0.000725468
+ -0.00272341    0.280366     -0.0353382       -0.000510353   0.000627446
+  0.108306     -0.0353382     0.483991        -0.00663509    0.000814462
+  0.0597015    -0.0241462    -0.00476179      -0.00287831   -0.00410306
+ -0.000531375  -0.000740352  -0.00227336       0.00175773   -0.000550985
+ -5.46266e-6    0.0615992    -0.0084444    …   0.00754037   -0.000713865
+ -7.80744e-5    0.0347709    -0.00356761      -0.00609519    0.00360991
+  0.0240561    -0.0132044     0.213683         0.0166641    -0.000929684
+  0.0133385    -0.00919492    0.0562126       -0.0423021     0.00471451
+  0.00739022   -0.00597224   -0.0030313        0.00471978   -0.0236748
+  ⋮                                        ⋱
+ -0.00769785   -0.0406075    -0.012298         0.0234008    -0.0161668
+ -0.0133522    -0.0122384    -0.0861289        0.094258     -0.0224369
+ -0.00838137    0.00335528    0.00422692       0.0168207     0.0502388
+ -0.000645754  -0.00632483   -0.00175497      -0.0798787     0.0198685
+ -0.00102918   -0.00637199   -0.00825901   …  -0.187875      0.0275525
+ -0.000640514  -0.00315513   -0.000651312      0.140072     -0.0624593
+ -0.00183032   -0.00321512   -0.0232591       -0.36789       0.0382077
+ -0.00114518   -0.000510353  -0.00663509       0.426481     -0.0867558
+ -0.000725468   0.000627446   0.000814462     -0.0867558     0.189954
julia> model.Bcov40×40 Matrix{Float64}:
+ 157.174     -19.3776    -19.4223    …   -0.257223   -1.24371    -1.90444
+ -19.3776    161.611     -22.5065        -1.3117     -1.42903    -1.33411
+ -19.4223    -22.5065    150.341         -1.38577    -2.56388    -1.25275
+ -14.6304    -12.8425    -19.2089        -2.29579    -2.03511    -1.00321
+ -11.6747    -13.9337    -12.4649        -0.198984   -1.44227    -2.31342
+ -24.2206    -18.4552     -7.96731   …   -3.00702    -1.14626    -0.923721
+ -13.3431    -17.4951    -15.8672        -1.57057    -0.945035   -1.77244
+ -12.7248    -19.4206    -11.7579        13.2375     -1.25172    -1.08592
+ -17.2277    -19.1384    -19.5393        -1.22189    12.8571     -0.396249
+ -15.0949    -12.6969    -15.9363        -1.23333    -0.422978   12.5215
+   ⋮                                 ⋱
+  -2.12779    13.4419     -1.25447       -9.26816    -9.00194    -6.01701
+  -0.752161   -1.49914    13.5685        -5.62018    -9.24482    -8.16962
+  -0.737933   -0.925351   -1.71971      -11.7708     -8.59325   -10.0889
+  -1.49387    -1.2578     -1.04615       -5.98367   -12.911     -13.1438
+  -1.72094    -2.33376    -0.495492  …   -9.81655    -5.37389    -8.16594
+  -1.53932    -0.763366   -2.22368       -9.36946    -7.53599   -10.4206
+  -0.257223   -1.3117     -1.38577       79.0721     -7.3782     -9.48116
+  -1.24371    -1.42903    -2.56388       -7.3782     75.6156     -0.921828
+  -1.90444    -1.33411    -1.25275       -9.48116    -0.921828   78.0356

Corresponding standard error of these estimates are

julia> sqrt.(diag(model.Σcov))50-element Vector{Float64}:
+ 0.6885637727726415
+ 0.529495666135442
+ 0.6956949396229591
+ 0.33645040355906397
+ 0.8084217056989712
+ 0.7442584972630495
+ 0.3639694672208399
+ 1.3688056422005268
+ 0.4677057749056628
+ 0.3179680333750869
+ ⋮
+ 0.5524130983848243
+ 0.8823995615262973
+ 0.43111265513125097
+ 0.7402837878083752
+ 0.9231964895406697
+ 0.41384022887389565
+ 1.8662336914254827
+ 0.6530553700433331
+ 0.43583743574349515
julia> sqrt.(diag(model.Bcov))40-element Vector{Float64}:
+ 12.536904651555659
+ 12.712633608569577
+ 12.26134556777767
+ 12.169443903108293
+ 12.663455285629615
+ 11.870615133041053
+ 12.528437624570504
+ 12.51912728097892
+ 12.39394332241073
+ 12.510940364508672
+  ⋮
+  8.869558232699935
+  8.568572559479021
+  8.594848108237025
+  8.973590296072445
+  8.287660643177881
+  8.80370479520269
+  8.892247260593642
+  8.695724537824539
+  8.833773378129575

Residual maximum likelihood estimation

For REML estimation, you can instead:

julia> model = MRVCModel(Y, X, V; reml = true)A multivariate response variance component model
+   * number of responses: 4
+   * number of observations: 1000
+   * number of fixed effects: 10
+   * number of variance components: 5
julia> @timev fit!(model)[ Info: Running MM algorithm for REML estimation
+iter = 0, logl = -26597.537258425444
+iter = 1, logl = -25455.46070096719
+iter = 2, logl = -25062.15815081145
+iter = 3, logl = -24929.900271068564
+iter = 4, logl = -24878.96910611469
+iter = 5, logl = -24853.289345291265
+iter = 6, logl = -24836.614727024702
+iter = 7, logl = -24824.192330128142
+iter = 8, logl = -24814.4419685779
+iter = 9, logl = -24806.666946417736
+iter = 10, logl = -24800.441634611918
+iter = 11, logl = -24795.448814457188
+iter = 12, logl = -24791.434959065882
+iter = 13, logl = -24788.19527844777
+iter = 14, logl = -24785.565671601264
+iter = 15, logl = -24783.416144877603
+iter = 16, logl = -24781.644754384706
+iter = 17, logl = -24780.17213209585
+iter = 18, logl = -24778.936749268694
+iter = 19, logl = -24777.890976339873
+iter = 20, logl = -24776.997905413984
+iter = 21, logl = -24776.228846668713
+iter = 22, logl = -24775.561388912047
+iter = 23, logl = -24774.977914973028
+iter = 24, logl = -24774.46447416872
+iter = 25, logl = -24774.009929862073
+iter = 26, logl = -24773.605316168836
+iter = 27, logl = -24773.243352268248
+iter = 28, logl = -24772.918074830915
+iter = 29, logl = -24772.62455872271
+iter = 30, logl = -24772.358703657163
+iter = 31, logl = -24772.117070189604
+iter = 32, logl = -24771.896752736797
+iter = 33, logl = -24771.695280510114
+iter = 34, logl = -24771.51053960551
+iter = 35, logl = -24771.340711232333
+iter = 36, logl = -24771.18422234522
+iter = 37, logl = -24771.03970587292
+iter = 38, logl = -24770.90596843845
+iter = 39, logl = -24770.781963965892
+iter = 40, logl = -24770.66677195723
+iter = 41, logl = -24770.55957949347
+iter = 42, logl = -24770.459666236435
+iter = 43, logl = -24770.36639185783
+iter = 44, logl = -24770.27918545024
+iter = 45, logl = -24770.19753656323
+iter = 46, logl = -24770.120987582057
+iter = 47, logl = -24770.049127218048
+iter = 48, logl = -24769.981584929763
+iter = 49, logl = -24769.918026121268
+iter = 50, logl = -24769.85814799603
+iter = 51, logl = -24769.80167596465
+iter = 52, logl = -24769.74836052275
+iter = 53, logl = -24769.69797452873
+iter = 54, logl = -24769.65031082431
+iter = 55, logl = -24769.60518014765
+iter = 56, logl = -24769.56240929961
+iter = 57, logl = -24769.521839527268
+iter = 58, logl = -24769.48332509584
+iter = 59, logl = -24769.446732024946
+iter = 60, logl = -24769.41193696552
+iter = 61, logl = -24769.378826202505
+iter = 62, logl = -24769.34729476516
+iter = 63, logl = -24769.317245631824
+iter = 64, logl = -24769.288589020118
+iter = 65, logl = -24769.261241748998
+iter = 66, logl = -24769.235126666277
+iter = 67, logl = -24769.210172133422
+iter = 68, logl = -24769.186311562113
+[ Info: Updates converged!
+[ Info: Calculating standard errors
+ 54.436487 seconds (5.65 k allocations: 15.296 MiB, 0.01% compilation time)
+elapsed time (ns):  54436486875
+gc time (ns):       0
+bytes allocated:    16039272
+pool allocs:        5451
+non-pool GC allocs: 196
+malloc() calls:     7
+free() calls:       0
+minor collections:  0
+full collections:   0
+Converged after 69 iterations.

Variance components and mean effects estimates and their standard errors can be accessed through:

julia> model.Σ5-element Vector{Matrix{Float64}}:
+ [2.9338500446816433 -0.9556577622339366 1.681944315565907 0.82097923904986; -0.9556577622339366 7.523441348688346 -4.724626084973893 -0.6045832212502729; 1.6819443155659073 -4.724626084973893 8.695787597891757 0.3975584198892441; 0.8209792390498599 -0.6045832212502729 0.397558419889244 1.0347822362989731]
+ [3.8129402322306207 2.3985158343212087 -0.3501640293021632 -1.361452145970671; 2.3985158343212087 2.0221274445059243 -0.26055606213368854 -1.595835001991699; -0.3501640293021632 -0.26055606213368826 4.164419136795955 -2.293103534511798; -1.3614521459706712 -1.595835001991699 -2.293103534511798 4.545249416292042]
+ [3.1531618876945027 -0.08093870739430353 -0.27534179496140915 0.4229588473703198; -0.08093870739430353 3.4956494601351706 -0.2073584361891652 0.6627085784490545; -0.27534179496140915 -0.20735843618916527 1.976397882530889 0.5036124449893044; 0.4229588473703198 0.6627085784490545 0.5036124449893044 0.5585662911372018]
+ [3.635801209038145 1.250204732685735 0.3660670881565596 0.07071405261549359; 1.2502047326857344 0.983771630227471 1.65237696435348 0.6004734046277869; 0.3660670881565597 1.6523769643534798 5.891632852330085 0.9668222589182407; 0.07071405261549359 0.6004734046277869 0.9668222589182407 0.9633632310978463]
+ [5.704238993549306 -2.02059232700069 1.131104825718486 1.4423754007372802; -2.0205923270006902 6.0631121455622194 9.453887154496428 -2.1594622671919748; 1.131104825718486 9.453887154496428 19.22805021076556 -3.065314862756238; 1.4423754007372802 -2.1594622671919748 -3.065314862756238 3.4673781742053422]
julia> model.B_reml10×4 Matrix{Float64}:
+   7.73978     0.0626092   29.9317     10.8401
+  -1.72827    17.0507      26.4331      5.24119
+  -4.99965    28.2624     -20.984     -13.0466
+ -10.2505     -5.98131     12.3119    -16.3662
+  -3.06092     0.788225    15.9509      1.1041
+  14.1047     -3.98098     11.5646     11.254
+  -1.45427    -8.264        5.3133     -0.233005
+   6.05113     5.20727      0.224061    3.95755
+  -0.679987  -14.213      -29.2147      4.91466
+   6.17329   -13.234      -23.6101     -3.48756
julia> hcat(vec(B), vec(model.B_reml))40×2 Matrix{Float64}:
+ 0.866192    7.73978
+ 0.157873   -1.72827
+ 0.818581   -4.99965
+ 0.707496  -10.2505
+ 0.357756   -3.06092
+ 0.722163   14.1047
+ 0.441486   -1.45427
+ 0.812486    6.05113
+ 0.889787   -0.679987
+ 0.331484    6.17329
+ ⋮
+ 0.068644    5.24119
+ 0.451072  -13.0466
+ 0.618679  -16.3662
+ 0.54638     1.1041
+ 0.616161   11.254
+ 0.400984   -0.233005
+ 0.482087    3.95755
+ 0.582507    4.91466
+ 0.456195   -3.48756
julia> reduce(hcat, [hcat(vec(Σ[i]), vec(model.Σ[i])) for i in 1:m])16×10 Matrix{Float64}:
+  2.59353    2.93385    3.32588    …  3.6358      5.37835    5.70424
+ -0.908817  -0.955658   2.21659       1.2502     -1.43756   -2.02059
+  1.58682    1.68194   -0.0345923     0.366067    1.65263    1.1311
+  0.365451   0.820979  -1.36959       0.0707141   0.983537   1.44238
+ -0.908817  -0.955658   2.21659       1.2502     -1.43756   -2.02059
+  7.12022    7.52344    1.75764    …  0.983772    6.84955    6.06311
+ -4.72969   -4.72463   -0.0950186     1.65238     9.98583    9.45389
+ -0.430129  -0.604583  -1.55052       0.600473   -1.98954   -2.15946
+  1.58682    1.68194   -0.0345923     0.366067    1.65263    1.1311
+ -4.72969   -4.72463   -0.0950186     1.65238     9.98583    9.45389
+  6.22778    8.69579    4.06617    …  5.89163    19.2083    19.2281
+  0.969228   0.397558  -2.51336       0.966822   -3.5885    -3.06531
+  0.365451   0.820979  -1.36959       0.0707141   0.983537   1.44238
+ -0.430129  -0.604583  -1.55052       0.600473   -1.98954   -2.15946
+  0.969228   0.397558  -2.51336       0.966822   -3.5885    -3.06531
+  1.2446     1.03478    5.02933    …  0.963363    2.85342    3.46738
julia> model.Σcov50×50 Matrix{Float64}:
+  0.49577      -0.00244778    0.111934     …  -0.00120099   -0.000758647
+ -0.00244778    0.29227      -0.0343723       -0.000533875   0.00067098
+  0.111934     -0.0343723     0.505528        -0.00697209    0.000865612
+  0.061864     -0.0253519    -0.00568503      -0.00302124   -0.00430738
+ -0.000533068  -0.000402843  -0.00233763       0.00189715   -0.000601019
+  5.60305e-5    0.0635175    -0.00787928   …   0.00809095   -0.000772694
+ -4.0096e-5     0.0359266    -0.00341432      -0.00646355    0.00386484
+  0.0245991    -0.0126117     0.220813         0.0177754    -0.000996935
+  0.0136707    -0.00925492    0.0581185       -0.0446667     0.0050127
+  0.00759217   -0.00622591   -0.00328015       0.00502603   -0.0249292
+  ⋮                                        ⋱
+ -0.00795163   -0.0428182    -0.0129844        0.0236794    -0.0165915
+ -0.0140199    -0.012921     -0.0907524        0.0966537    -0.0229699
+ -0.00877942    0.00359818    0.00449712       0.0176113     0.0517196
+ -0.000667567  -0.00656994   -0.0018067       -0.0818833     0.0205321
+ -0.00106014   -0.00669857   -0.00860587   …  -0.193476      0.0283973
+ -0.000657675  -0.0033219    -0.000694922      0.144019     -0.0647023
+ -0.0019256    -0.00341227   -0.024527        -0.379053      0.0392726
+ -0.00120099   -0.000533875  -0.00697209       0.441745     -0.089645
+ -0.000758647   0.00067098    0.000865612     -0.089645      0.197463
julia> model.Bcov_reml40×40 Matrix{Float64}:
+ 158.891     -19.5994    -19.6371   …   -0.270249   -1.25569    -1.91414
+ -19.5994    163.384     -22.755        -1.32842    -1.44256    -1.34338
+ -19.6371    -22.755     151.987        -1.39442    -2.57948    -1.26777
+ -14.7913    -12.9925    -19.412        -2.31154    -2.05524    -1.01497
+ -11.8093    -14.0871    -12.5893       -0.204628   -1.46038    -2.33716
+ -24.4932    -18.6219     -8.06361  …   -3.02449    -1.15581    -0.939094
+ -13.4737    -17.7017    -16.0374       -1.58474    -0.955878   -1.787
+ -12.8751    -19.6214    -11.8827       13.362      -1.26469    -1.10389
+ -17.4026    -19.3504    -19.7707       -1.23472    12.9793     -0.399421
+ -15.2551    -12.8481    -16.103        -1.24913    -0.426619   12.6487
+   ⋮                                ⋱
+  -2.14636    13.5663     -1.27382      -9.38304    -9.11398    -6.09505
+  -0.765165   -1.51854    13.6867       -5.69139    -9.3694     -8.2553
+  -0.7517     -0.930899   -1.73573     -11.9126     -8.7008    -10.1908
+  -1.50467    -1.26692    -1.05451      -6.03856   -13.0461    -13.3113
+  -1.74002    -2.34396    -0.50063  …   -9.94374    -5.4518     -8.24799
+  -1.54677    -0.78006    -2.23457      -9.47418    -7.6325    -10.5424
+  -0.270249   -1.32842    -1.39442      80.0047     -7.46137    -9.59586
+  -1.25569    -1.44256    -2.57948      -7.46137    76.5166     -0.942239
+  -1.91414    -1.34338    -1.26777      -9.59586    -0.942239   78.9404
julia> sqrt.(diag(model.Σcov))50-element Vector{Float64}:
+ 0.7041093807634108
+ 0.5406201064474182
+ 0.7110051601946071
+ 0.3444911257855926
+ 0.8243533172952108
+ 0.7594961262777526
+ 0.37223105634402
+ 1.399043387180137
+ 0.47895976215151104
+ 0.32621217092743393
+ ⋮
+ 0.5632649585154991
+ 0.8978995129849152
+ 0.4392731438996924
+ 0.7554591568000877
+ 0.9384463145136606
+ 0.4220581137074655
+ 1.8966703307591692
+ 0.6646389147784445
+ 0.44436833540465226
julia> sqrt.(diag(model.Bcov_reml))40-element Vector{Float64}:
+ 12.60518354143135
+ 12.782193317524683
+ 12.328308752465436
+ 12.23646139990698
+ 12.732636509969742
+ 11.934986834081903
+ 12.597350995825984
+ 12.586984559735523
+ 12.461643282682688
+ 12.578545874909784
+  ⋮
+  8.923342209927311
+  8.620236102433564
+  8.645876453902996
+  9.026071287093929
+  8.337457783234893
+  8.856449014354963
+  8.944535606024061
+  8.747376305559317
+  8.884843293186195

Estimation only

Calculating standard errors can be memory-consuming, so you could instead forego such calculation via:

model = MRVCModel(Y, X, V; se = false) # or model = MRVCModel(Y, X, V; se = false, reml = true)
+@timev fit!(model)

Special case: missing response

You can also fit data with missing response. For example:

Y_miss = Matrix{Union{eltype(Y), Missing}}(missing, size(Y))
+copy!(Y_miss, Y)
+Y_miss[rand(1:length(Y_miss), n)] .= missing
+
+model = MRVCModel(Y_miss, X, V; se = false)
+@timev fit!(model)

Special case: $m = 2$

When there are two variance components, you can accelerate fitting by avoiding large matrix inversion per iteration. To illustrate this, you can first simulate data as we did previously but with larger $n \cdot d$ and $m = 2$.

julia> function simulate(n, d, p, m)
+           X = rand(n, p)
+           B = rand(p, d)
+           V = [zeros(n, n) for _ in 1:m]
+           Σ = [zeros(d, d) for _ in 1:m]
+           Ω = zeros(n * d, n * d)
+           for i in 1:m
+               Vi = randn(n, n)
+               mul!(V[i], transpose(Vi), Vi)
+               Σi = randn(d, d)
+               mul!(Σ[i], transpose(Σi), Σi)
+               kron_axpy!(Σ[i], V[i], Ω) # Ω = Σ[1]⊗V[1] + ... + Σ[m]⊗V[m]
+           end
+           Ωchol = cholesky(Ω)
+           Y = X * B + reshape(Ωchol.L * randn(n * d), n, d)
+           Y, X, V, B, Σ
+       endsimulate (generic function with 1 method)
julia> Y, X, V, B, Σ = simulate(5000, 4, 10, 2)([349.6925410979359 97.16785228684566 224.42107027741739 -140.34648440718144; -280.08708850905515 121.99696938528912 291.6281581993771 -6.846001768944951; … ; -548.2638152432189 -138.9867854016224 165.14392374211317 -23.060532820908882; -430.5883532324437 -19.716621492057072 326.6227822295293 -58.13550088159137], [0.9221277504721419 0.39518286105049993 … 0.06270138790021751 0.997739716155004; 0.7122939351705581 0.9562437050691619 … 0.36859155672988797 0.9350171343957088; … ; 0.16876259557815643 0.4746184757013582 … 0.6693437348722951 0.8334698814347721; 0.2070536310319615 0.28200930113165246 … 0.2787822379761381 0.6772315839982946], [[5020.237833962983 151.28596690934853 … -105.03459026763699 112.94446090456054; 151.28596690934853 4998.5166994878045 … 1.0427422600613658 -43.50983290951208; … ; -105.03459026763699 1.0427422600613658 … 4869.238277472443 -4.700081453367206; 112.94446090456054 -43.50983290951208 … -4.700081453367206 5045.476681920826], [5014.608235889929 59.03322331129151 … -6.593479325359899 -68.2756587359377; 59.03322331129151 4971.447532001538 … -11.081784740947477 -47.668978305002206; … ; -6.593479325359899 -11.081784740947477 … 5039.79216251604 5.688510795412341; -68.2756587359377 -47.668978305002206 … 5.688510795412341 5025.660239417218]], [0.6255264905567314 0.5777925737265706 0.05168141094146861 0.5596066237597734; 0.17597036525499776 0.26877681752446925 0.4087414296943477 0.593506083228733; … ; 0.16458211947035895 0.7469905226911336 0.5646991244731385 0.34544628060488547; 0.3314750932375198 0.3991125956171754 0.5121170379644673 0.9003522250744518], [[11.818256102835363 2.6219095188820427 -4.874713153490494 0.6218526449017076; 2.6219095188820427 1.722853331225835 -1.2520900781473776 0.1296058205688085; -4.874713153490494 -1.2520900781473776 4.171736419731323 -0.21012586644541056; 0.6218526449017076 0.1296058205688085 -0.21012586644541056 0.05201440663338152], [6.831333870227899 2.195994547246214 -2.0178793576268306 -0.6270388077864709; 2.195994547246214 1.5443677565884601 0.21422986747085898 0.3950251301300546; -2.0178793576268306 0.21422986747085898 1.8725123076931875 0.43011866399546794; -0.6270388077864709 0.3950251301300546 0.43011866399546794 1.167702081443099]])

Then you can fit data as follows:

julia> model = MRTVCModel(Y, X, V)A multivariate response variance component model
+   * number of responses: 4
+   * number of observations: 5000
+   * number of fixed effects: 10
+   * number of variance components: 2
julia> @timev fit!(model)[ Info: Running MM algorithm for ML estimation
+iter = 0, logl = -124717.72364382436
+iter = 1, logl = -122856.59872091345
+iter = 2, logl = -121970.2374785459
+iter = 3, logl = -121483.12235491295
+iter = 4, logl = -121174.53945389927
+iter = 5, logl = -120964.61513730836
+iter = 6, logl = -120818.31544841589
+iter = 7, logl = -120716.4548595128
+iter = 8, logl = -120646.46049426559
+iter = 9, logl = -120599.27053480603
+iter = 10, logl = -120568.14059152281
+iter = 11, logl = -120548.06202828637
+iter = 12, logl = -120535.38956726511
+iter = 13, logl = -120527.5478723189
+iter = 14, logl = -120522.77770458376
+iter = 15, logl = -120519.91676483194
+iter = 16, logl = -120518.2201096205
+iter = 17, logl = -120517.22253342139
+iter = 18, logl = -120516.63963691155
+iter = 19, logl = -120516.30046046266
+iter = 20, logl = -120516.1035635785
+iter = 21, logl = -120515.98934193165
+[ Info: Updates converged!
+[ Info: Calculating standard errors
+  1.610761 seconds (1.49 M allocations: 102.958 MiB, 9.62% gc time, 90.96% compilation time)
+elapsed time (ns):  1610760916
+gc time (ns):       154971333
+bytes allocated:    107958804
+pool allocs:        1490900
+non-pool GC allocs: 2920
+malloc() calls:     56
+realloc() calls:    11
+free() calls:       77
+minor collections:  1
+full collections:   1
+Converged after 22 iterations.
julia> reduce(hcat, [hcat(vec(Σ[i]), vec(model.Σ[i])) for i in 1:2])16×4 Matrix{Float64}:
+ 11.8183     11.6733      6.83133    6.81182
+  2.62191     2.64236     2.19599    2.19778
+ -4.87471    -4.6384     -2.01788   -1.87052
+  0.621853    0.642991   -0.627039  -0.69176
+  2.62191     2.64236     2.19599    2.19778
+  1.72285     1.72298     1.54437    1.61116
+ -1.25209    -1.2187      0.21423    0.32403
+  0.129606    0.138002    0.395025   0.43114
+ -4.87471    -4.6384     -2.01788   -1.87052
+ -1.25209    -1.2187      0.21423    0.32403
+  4.17174     4.12349     1.87251    1.85925
+ -0.210126   -0.215268    0.430119   0.503186
+  0.621853    0.642991   -0.627039  -0.69176
+  0.129606    0.138002    0.395025   0.43114
+ -0.210126   -0.215268    0.430119   0.503186
+  0.0520144   0.0549335   1.1677     1.19821
diff --git a/dev/index.html b/dev/index.html new file mode 100644 index 0000000..853bc3e --- /dev/null +++ b/dev/index.html @@ -0,0 +1,3 @@ + +Home · MRVCModels
GitHub

MRVCModels

MRVCModels.jl is a package for fitting and testing multivariate response variance components linear mixed models of form

\[\text{vec}\ \boldsymbol{Y} \sim \mathcal{N}(\text{vec}(\boldsymbol{X} \boldsymbol{B}), \sum_{i=1}^m \boldsymbol{\Gamma}_i \otimes \boldsymbol{V}_i),\]

where $\boldsymbol{Y}$ and $\boldsymbol{X}$ are $n \times d$ response and $n \times p$ predictor matrices, respectively, and $\boldsymbol{V}_1, \ldots, \boldsymbol{V}_m$ are $m$ known $n \times n$ positive semidefinite matrices. $\text{vec}\ \boldsymbol{Y}$ creates an $nd \times 1$ vector from $\boldsymbol{Y}$ by stacking its columns and $\otimes$ denotes the Kronecker product. The parameters of the model include $p \times d$ mean effects $\boldsymbol{B}$ and $d \times d$ variance components ($\boldsymbol{\Gamma}_1, \dots, \boldsymbol{\Gamma}_m$), which MRVCModels.jl estimates through either minorization-maximization (MM) or expectation–maximization (EM) algorithms.

Info

MRVCModels.jl can also handle data with missing response, which destroys the symmetry of the log-likelihood and complicates maximum likelihood estimation. MM algorithm easily adapts to this challenge.

Warning

MRVCModels.jl is not suitable for biobank-scale data, except in the setting of $m = 2$. For $m > 2$, we recommend using this package for datasets of size up to $n \cdot d \approx 60000$. Further note that large $m$ can affect memory needed when calculating standard errors, since it will require $m(nd)^2$ storage space.

Installation

To use MRVCModels.jl, type:

using Pkg
+Pkg.add(url = "https://github.com/Hua-Zhou/MultiResponseVarianceComponentModels.jl.git")

This documentation was built using Documenter.jl.

Documentation built 2024-06-21T05:08:37.605 with Julia 1.10.4
diff --git a/dev/objects.inv b/dev/objects.inv new file mode 100644 index 0000000..88b503b Binary files /dev/null and b/dev/objects.inv differ diff --git a/dev/search_index.js b/dev/search_index.js new file mode 100644 index 0000000..3ad07ac --- /dev/null +++ b/dev/search_index.js @@ -0,0 +1,3 @@ +var documenterSearchIndex = {"docs": +[{"location":"api/#API","page":"API","title":"API","text":"","category":"section"},{"location":"api/","page":"API","title":"API","text":"","category":"page"},{"location":"api/","page":"API","title":"API","text":"Modules = [MultiResponseVarianceComponentModels]","category":"page"},{"location":"api/#MultiResponseVarianceComponentModels.MultiResponseVarianceComponentModels","page":"API","title":"MultiResponseVarianceComponentModels.MultiResponseVarianceComponentModels","text":"MRVCModels.jl permits maximum likelihood (ML) and residual maximum likelihood (REML) estimation as well as inference for multivariate response variance components linear mixed models.\n\n\n\n\n\n","category":"module"},{"location":"api/#MultiResponseVarianceComponentModels.MRVCModel-Union{Tuple{T}, Tuple{Union{AbstractMatrix{T}, AbstractArray{Union{Missing, T}, 2}}, Union{Nothing, AbstractMatrix{T}}, Vector{<:AbstractMatrix{T}}}} where T<:Union{Float32, Float64}","page":"API","title":"MultiResponseVarianceComponentModels.MRVCModel","text":"MRVCModel(\n Y::AbstractVecOrMat,\n X::Union{Nothing, AbstractVecOrMat},\n V::Union{AbstractMatrix, Vector{<:AbstractMatrix}}\n )\nMRTVCModel(\n Y::AbstractVecOrMat,\n X::Union{Nothing, AbstractVecOrMat},\n V::Vector{<:AbstractMatrix}\n )\n\nCreate a new MRVCModel or MRTVCModel instance from response matrix Y, predictor matrix X, and kernel matrices V. For MRTVCModel instance, the number of variance components must be two.\n\nKeyword arguments\n\nse::Bool calculate standard errors; default true\nreml::Bool pursue REML estimation instead of ML estimation; default false\n\n\n\n\n\n","category":"method"},{"location":"api/#MultiResponseVarianceComponentModels.fisher_B!-Union{Tuple{MRVCModel{T}}, Tuple{T}} where T<:Union{Float32, Float64}","page":"API","title":"MultiResponseVarianceComponentModels.fisher_B!","text":"fisher_B!(model::MRVCModel)\n\nCompute the sampling variance-covariance model.Bcov of regression coefficients model.B, assuming inverse of covariance matrix model.Ω is available at model.storage_nd_nd.\n\n\n\n\n\n","category":"method"},{"location":"api/#MultiResponseVarianceComponentModels.fisher_Σ!-Union{Tuple{MRVCModel{T}}, Tuple{T}} where T<:Union{Float32, Float64}","page":"API","title":"MultiResponseVarianceComponentModels.fisher_Σ!","text":"fisher_Σ!(model::MRVCModel)\n\nCompute the sampling variance-covariance model.Σcov of variance component estimates model.Σ, assuming inverse of covariance matrix model.Ω is available at model.storage_nd_nd.\n\n\n\n\n\n","category":"method"},{"location":"api/#MultiResponseVarianceComponentModels.fit!-Union{Tuple{MRVCModel{T}}, Tuple{T}} where T<:Union{Float32, Float64}","page":"API","title":"MultiResponseVarianceComponentModels.fit!","text":"fit!(model::MRVCModel)\nfit!(model::MRTVCModel)\n\nFit a multivariate response variance components model by MM or EM algorithm.\n\nKeyword arguments\n\nmaxiter::Int maximum number of iterations; default 1000\nreltol::Real relative tolerance for convergence; default 1e-6\nverbose::Bool display algorithmic information; default true\ninit::Symbol initialization strategy; :default initializes by least squares, while\n :user uses user-supplied values at model.B and model.Σ\nalgo::Symbol optimization algorithm; :MM (default) or EM (for MRVCModel)\nlog::Bool record iterate history or not; default false\n\n\n\n\n\n","category":"method"},{"location":"api/#MultiResponseVarianceComponentModels.h2-Union{Tuple{MRVCModel{T}}, Tuple{T}} where T<:Union{Float32, Float64}","page":"API","title":"MultiResponseVarianceComponentModels.h2","text":"h2(model::MRVCModel)\n\nCalculate heritability estimates and their standard errors, assuming that all variance components capture genetic effects except the last term. Also return total heritability from sum of individual contributions and its standard error.\n\n\n\n\n\n","category":"method"},{"location":"api/#MultiResponseVarianceComponentModels.kron_axpy!-Union{Tuple{T}, Tuple{AbstractVecOrMat{T}, AbstractVecOrMat{T}, AbstractVecOrMat{T}}} where T<:Real","page":"API","title":"MultiResponseVarianceComponentModels.kron_axpy!","text":"kron_axpy!(A, X, Y)\n\nOverwrite Y with A ⊗ X + Y. Same as Y += kron(A, X), but more memory efficient.\n\n\n\n\n\n","category":"method"},{"location":"api/#MultiResponseVarianceComponentModels.kron_reduction!-Union{Tuple{T}, Tuple{AbstractMatrix{T}, AbstractMatrix{T}, AbstractMatrix{T}}, Tuple{AbstractMatrix{T}, AbstractMatrix{T}, AbstractMatrix{T}, Bool}} where T<:Real","page":"API","title":"MultiResponseVarianceComponentModels.kron_reduction!","text":"kron_reduction!(A, B, C; sym = false)\n\nOverwrite C with the derivative of tr(A' (X ⊗ B)) wrt X. C[i, j] = dot(Aij, B), where Aij is the (i , j) block of A. sym = true indicates A and B are symmetric.\n\n\n\n\n\n","category":"method"},{"location":"api/#MultiResponseVarianceComponentModels.loglikelihood!-Union{Tuple{MRVCModel{T}}, Tuple{T}} where T<:Union{Float32, Float64}","page":"API","title":"MultiResponseVarianceComponentModels.loglikelihood!","text":"loglikelihood!(model::MRVCModel)\n\nOverwrite model.storage_nd_nd by inverse of the covariance matrix model.Ω, overwrite model.storage_nd by U' \\ vec(model.R), and return the log-likelihood. Assume model.Ω and model.R are already updated according to model.Σ and model.B.\n\n\n\n\n\n","category":"method"},{"location":"api/#MultiResponseVarianceComponentModels.loglikelihood_miss!-Union{Tuple{MRVCModel{T}}, Tuple{T}} where T<:Union{Float32, Float64}","page":"API","title":"MultiResponseVarianceComponentModels.loglikelihood_miss!","text":"loglikelihood_miss!(model::MRVCModel)\n\nOverwrite model.storage_nd_nd by inverse of the covariance matrix model.Ω, overwrite model.storage_nd by U' \\ vec(model.R), overwrite model.storage_n_miss_n_miss_1 by conditional variance, precompute model.storage_n_miss_n_obs_1 for conditional mean, and return the value of surrogate Q-function of log-likelihood. Assume model.Ω and model.R are already updated according to model.Σ and model.B.\n\n\n\n\n\n","category":"method"},{"location":"api/#MultiResponseVarianceComponentModels.lrt-Tuple{MRVCModel, MRVCModel}","page":"API","title":"MultiResponseVarianceComponentModels.lrt","text":"lrt(model1::MRVCModel, model0::MRVCModel)\n\nPerform a variation of the likelihood ratio test for univariate variance components models as in Molenberghs and Verbeke 2007 with model1 and model0 being the full and nested models, respectively.\n\n\n\n\n\n","category":"method"},{"location":"api/#MultiResponseVarianceComponentModels.permute-Union{Tuple{AbstractArray{Union{Missing, T}, 2}}, Tuple{T}} where T<:Union{Float32, Float64}","page":"API","title":"MultiResponseVarianceComponentModels.permute","text":"permute(Y::AbstractVecOrMat)\n\nReturn permutation P such that vec(Y)[P] rearranges vec(Y) with missing values spliced after non-missing values. Also return inverse permutation invP such that vec(Y)[P][invP] = vec(Y).\n\n\n\n\n\n","category":"method"},{"location":"api/#MultiResponseVarianceComponentModels.rg-Union{Tuple{MRVCModel{T}}, Tuple{T}} where T<:Union{Float32, Float64}","page":"API","title":"MultiResponseVarianceComponentModels.rg","text":"rg(model::MRVCModel)\n\nCalculate genetic/residual correlation estimates and their standard errors.\n\n\n\n\n\n","category":"method"},{"location":"api/#MultiResponseVarianceComponentModels.update_B!-Union{Tuple{MRVCModel{T}}, Tuple{T}} where T<:Union{Float32, Float64}","page":"API","title":"MultiResponseVarianceComponentModels.update_B!","text":"update_B!(model::MRVCModel)\n\nUpdate the regression coefficients model.B, assuming inverse of covariance matrix model.Ω is available at model.storage_nd_nd.\n\n\n\n\n\n","category":"method"},{"location":"api/#MultiResponseVarianceComponentModels.update_B_miss!-Union{Tuple{MRVCModel{T}}, Tuple{T}} where T<:Union{Float32, Float64}","page":"API","title":"MultiResponseVarianceComponentModels.update_B_miss!","text":"update_B_miss!(model::MRVCModel)\n\nUpdate the regression coefficients model.B, assuming inverse of covariance matrix model.Ω is available at model.storage_nd_nd and model.storage_n_miss_n_obs_1 for conditional mean is precomputed.\n\n\n\n\n\n","category":"method"},{"location":"api/#MultiResponseVarianceComponentModels.update_Γk!-Union{Tuple{T}, Tuple{MRVCModel{T}, Integer}} where T<:Union{Float32, Float64}","page":"API","title":"MultiResponseVarianceComponentModels.update_Γk!","text":"update_Γk!(model, k)\n\nUpdate the parameters model.Γ[k] and model.Ψ[:,k] assuming a diagonal plus low rank structure for model.Σ[k]. Assumes covariance matrix model.Ω is available at model.storage_nd_nd and model.Ω⁻¹R precomputed.\n\n\n\n\n\n","category":"method"},{"location":"api/#MultiResponseVarianceComponentModels.update_Σ!-Union{Tuple{MRVCModel{T}}, Tuple{T}} where T<:Union{Float32, Float64}","page":"API","title":"MultiResponseVarianceComponentModels.update_Σ!","text":"update_Σ!(model::MRVCModel)\n\nUpdate the variance component parameters model.Σ, assuming inverse of covariance matrix model.Ω is available at model.storage_nd_nd. If missing response, assume conditional variance model.storage_n_miss_n_miss_1 is precomputed.\n\n\n\n\n\n","category":"method"},{"location":"api/#MultiResponseVarianceComponentModels.update_Σk!-Union{Tuple{T}, Tuple{MRVCModel{T}, Integer, Integer}} where T<:Union{Float32, Float64}","page":"API","title":"MultiResponseVarianceComponentModels.update_Σk!","text":"update_Σk!(model, k, rk)\n\nUpdate the model.Σ[k] assuming it has rank rk < d, assuming inverse of covariance matrix model.Ω is available at model.storage_nd_nd and model.Ω⁻¹R precomputed.\n\n\n\n\n\n","category":"method"},{"location":"api/#MultiResponseVarianceComponentModels.update_Σk!-Union{Tuple{T}, Tuple{MRVCModel{T}, Integer, Val{:EM}}} where T<:Union{Float32, Float64}","page":"API","title":"MultiResponseVarianceComponentModels.update_Σk!","text":"update_Σk!(model::MRVCModel, k, Val(:EM))\n\nEM update the model.Σ[k] assuming it has full rank d, inverse of covariance matrix model.Ω is available at model.storage_nd_nd, and model.Ω⁻¹R is precomputed.\n\n\n\n\n\n","category":"method"},{"location":"api/#MultiResponseVarianceComponentModels.update_Σk!-Union{Tuple{T}, Tuple{MRVCModel{T}, Integer, Val{:MM}}} where T<:Union{Float32, Float64}","page":"API","title":"MultiResponseVarianceComponentModels.update_Σk!","text":"update_Σk!(model::MRVCModel, k, Val(:MM))\n\nMM update the model.Σ[k] assuming it has full rank d, inverse of covariance matrix model.Ω is available at model.storage_nd_nd, and model.Ω⁻¹R is precomputed.\n\n\n\n\n\n","category":"method"},{"location":"api/#MultiResponseVarianceComponentModels.update_Σk_miss!-Union{Tuple{T}, Tuple{MRVCModel{T}, Integer, Val{:MM}}} where T<:Union{Float32, Float64}","page":"API","title":"MultiResponseVarianceComponentModels.update_Σk_miss!","text":"update_Σk_miss!(model::MRVCModel, k, Val(:MM))\n\nMM update the model.Σ[k] assuming it has full rank d, inverse of covariance matrix model.Ω is available at model.storage_nd_nd, model.Ω⁻¹R is precomputed, and Ω⁻¹P'CPΩ⁻¹ is available at model.storage_nd_nd_miss.\n\n\n\n\n\n","category":"method"},{"location":"examples/#Examples","page":"Examples","title":"Examples","text":"","category":"section"},{"location":"examples/#Simulate-data","page":"Examples","title":"Simulate data","text":"","category":"section"},{"location":"examples/","page":"Examples","title":"Examples","text":"using MultiResponseVarianceComponentModels, LinearAlgebra, Random\nRandom.seed!(6789)\nn = 1_000; # n of observations\nd = 4; # n of responses\np = 10; # n of covariates\nm = 5; # n of variance components\nX = rand(n, p);\nB = rand(p, d)\nV = [zeros(n, n) for _ in 1:m]; # kernel matrices\nΣ = [zeros(d, d) for _ in 1:m]; # variance components\nfor i in 1:m\n Vi = randn(n, n)\n copy!(V[i], Vi' * Vi)\n Σi = randn(d, d)\n copy!(Σ[i], Σi' * Σi)\nend\nΩ = zeros(n * d, n * d); # overall nd-by-nd covariance matrix Ω\nfor i = 1:m\n Ω += kron(Σ[i], V[i])\nend\nΩchol = cholesky(Ω);\nY = X * B + reshape(Ωchol.L * randn(n * d), n, d)","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"note: Note\nIn the case of heritability and genetic correlation analyses, one can use classic genetic relationship matrices (GRMs) for boldsymbolV_i's, which in turn can be constructed using SnpArrays.jl.","category":"page"},{"location":"examples/#Maximum-likelihood-estimation","page":"Examples","title":"Maximum likelihood estimation","text":"","category":"section"},{"location":"examples/","page":"Examples","title":"Examples","text":"model = MRVCModel(Y, X, V)\n@timev fit!(model)","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"Then variance components and mean effects estimates can be accessed through","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"model.Σ\nmodel.B\nhcat(vec(B), vec(model.B))\nreduce(hcat, [hcat(vec(Σ[i]), vec(model.Σ[i])) for i in 1:m])","category":"page"},{"location":"examples/#Standard-errors","page":"Examples","title":"Standard errors","text":"","category":"section"},{"location":"examples/","page":"Examples","title":"Examples","text":"Sampling variance and covariance of these estimates are","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"model.Σcov\nmodel.Bcov","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"Corresponding standard error of these estimates are","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"sqrt.(diag(model.Σcov))\nsqrt.(diag(model.Bcov))","category":"page"},{"location":"examples/#Residual-maximum-likelihood-estimation","page":"Examples","title":"Residual maximum likelihood estimation","text":"","category":"section"},{"location":"examples/","page":"Examples","title":"Examples","text":"For REML estimation, you can instead:","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"model = MRVCModel(Y, X, V; reml = true)\n@timev fit!(model)","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"Variance components and mean effects estimates and their standard errors can be accessed through:","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"model.Σ\nmodel.B_reml\nhcat(vec(B), vec(model.B_reml))\nreduce(hcat, [hcat(vec(Σ[i]), vec(model.Σ[i])) for i in 1:m])\nmodel.Σcov\nmodel.Bcov_reml\nsqrt.(diag(model.Σcov))\nsqrt.(diag(model.Bcov_reml))","category":"page"},{"location":"examples/#Estimation-only","page":"Examples","title":"Estimation only","text":"","category":"section"},{"location":"examples/","page":"Examples","title":"Examples","text":"Calculating standard errors can be memory-consuming, so you could instead forego such calculation via:","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"model = MRVCModel(Y, X, V; se = false) # or model = MRVCModel(Y, X, V; se = false, reml = true)\n@timev fit!(model)","category":"page"},{"location":"examples/#Special-case:-missing-response","page":"Examples","title":"Special case: missing response","text":"","category":"section"},{"location":"examples/","page":"Examples","title":"Examples","text":"You can also fit data with missing response. For example:","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"Y_miss = Matrix{Union{eltype(Y), Missing}}(missing, size(Y))\ncopy!(Y_miss, Y)\nY_miss[rand(1:length(Y_miss), n)] .= missing\n\nmodel = MRVCModel(Y_miss, X, V; se = false)\n@timev fit!(model)","category":"page"},{"location":"examples/#Special-case:-m-2","page":"Examples","title":"Special case: m = 2","text":"","category":"section"},{"location":"examples/","page":"Examples","title":"Examples","text":"When there are two variance components, you can accelerate fitting by avoiding large matrix inversion per iteration. To illustrate this, you can first simulate data as we did previously but with larger n cdot d and m = 2.","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"function simulate(n, d, p, m)\n X = rand(n, p)\n B = rand(p, d)\n V = [zeros(n, n) for _ in 1:m]\n Σ = [zeros(d, d) for _ in 1:m]\n Ω = zeros(n * d, n * d)\n for i in 1:m\n Vi = randn(n, n)\n mul!(V[i], transpose(Vi), Vi)\n Σi = randn(d, d)\n mul!(Σ[i], transpose(Σi), Σi)\n kron_axpy!(Σ[i], V[i], Ω) # Ω = Σ[1]⊗V[1] + ... + Σ[m]⊗V[m]\n end\n Ωchol = cholesky(Ω)\n Y = X * B + reshape(Ωchol.L * randn(n * d), n, d)\n Y, X, V, B, Σ\nend\nY, X, V, B, Σ = simulate(5000, 4, 10, 2)","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"Then you can fit data as follows:","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"model = MRTVCModel(Y, X, V)\n@timev fit!(model)\nreduce(hcat, [hcat(vec(Σ[i]), vec(model.Σ[i])) for i in 1:2])","category":"page"},{"location":"advanced/#Advanced-details","page":"Advanced details","title":"Advanced details","text":"","category":"section"},{"location":"advanced/#Estimation","page":"Advanced details","title":"Estimation","text":"","category":"section"},{"location":"advanced/","page":"Advanced details","title":"Advanced details","text":"For the MM algorithm, the updates in each iteration are","category":"page"},{"location":"advanced/","page":"Advanced details","title":"Advanced details","text":"beginaligned\ntextvec boldsymbolB^(t) = (boldsymbolI_d otimes boldsymbolX^T) boldsymbolOmega^-(t) (boldsymbolI_d otimes boldsymbolX)^-1 (boldsymbolI_d otimes boldsymbolX^T) boldsymbolOmega^-(t) textvec boldsymbolY \nboldsymbolGamma_i^(t + 1) = boldsymbolL_i^-(t)TboldsymbolL_i^(t)TboldsymbolGamma_i^(t)(boldsymbolR^(t)TboldsymbolV_iboldsymbolR^(t))boldsymbolGamma_i^(t)boldsymbolL_i^(t)^12 boldsymbolL_i^-(t)\nendaligned","category":"page"},{"location":"advanced/","page":"Advanced details","title":"Advanced details","text":"where boldsymbolOmega^(t) = sum_i=1^m boldsymbolGamma_i^(t) otimes boldsymbolV_i and boldsymbolL_i^(t) is the Cholesky factor of boldsymbolM_i^(t) = (boldsymbolI_d otimes boldsymbol1_n)^T (boldsymbol1_d boldsymbol1_d^T otimes boldsymbolV_i) odot boldsymbolOmega^-(t) (boldsymbolI_d otimes boldsymbol1_n), while boldsymbolR^(t) is the n times d matrix such that textvec boldsymbolR^(t) = boldsymbolOmega^-(t) textvec(boldsymbolY - boldsymbolX boldsymbolB^(t)). odot denotes the Hadamard product.","category":"page"},{"location":"advanced/","page":"Advanced details","title":"Advanced details","text":"For the EM algorithm, the updates in each iteration are","category":"page"},{"location":"advanced/","page":"Advanced details","title":"Advanced details","text":"beginaligned\ntextvec boldsymbolB^(t) = (boldsymbolI_d otimes boldsymbolX^T) boldsymbolOmega^-(t) (boldsymbolI_d otimes boldsymbolX)^-1 (boldsymbolI_d otimes boldsymbolX^T) boldsymbolOmega^-(t) textvec boldsymbolY \nboldsymbolGamma_i^(t + 1) = frac1r_i boldsymbolGamma_i^(t) ( boldsymbolR^(t)T boldsymbolV_i boldsymbolR^(t) - boldsymbolM_i^(t) ) boldsymbolGamma_i^(t) + boldsymbolGamma_i^(t)\nendaligned","category":"page"},{"location":"advanced/","page":"Advanced details","title":"Advanced details","text":"where r_i = textrank(boldsymbolV_i). As seen, the updates for mean effects boldsymbolB are the same for these two algorithms.","category":"page"},{"location":"advanced/#Inference","page":"Advanced details","title":"Inference","text":"","category":"section"},{"location":"advanced/","page":"Advanced details","title":"Advanced details","text":"Standard errors for our estimates are calculated using the Fisher information matrix, where","category":"page"},{"location":"advanced/","page":"Advanced details","title":"Advanced details","text":"beginaligned\ntextE left- fracpartial^2partial(textvec boldsymbolB)^T partial(textvec boldsymbolB) mathcalL right = (boldsymbolI_d otimes boldsymbolX^T) boldsymbolOmega^-1 (boldsymbolI_d otimes boldsymbolX) \ntextE left - fracpartial^2partial (textvech boldsymbolGamma_i)^T partial (textvec boldsymbolB) mathcalL right = boldsymbol0 \ntextE left - fracpartial^2partial (textvech boldsymbolGamma_j)^T partial (textvech boldsymbolGamma_i) mathcalL right = frac12 boldsymbolU_i^T (boldsymbolOmega^-1 otimes boldsymbolOmega^-1) boldsymbolU_j\nendaligned","category":"page"},{"location":"advanced/","page":"Advanced details","title":"Advanced details","text":"and boldsymbolU_i = (boldsymbolI_d otimes boldsymbolK_nd otimes boldsymbolI_n) (boldsymbolI_d^2 otimes textvec boldsymbolV_i) boldsymbolD_d. Here, boldsymbolK_nd is the nd times nd commutation matrix and boldsymbolD_d the d^2 times fracd(d+1)2 duplication matrix. textvech boldsymbolGamma_i creates an fracd(d+1)2 times 1 vector from boldsymbolGamma_i by stacking its lower triangular part.","category":"page"},{"location":"advanced/#Special-case:-missing-response","page":"Advanced details","title":"Special case: missing response","text":"","category":"section"},{"location":"advanced/","page":"Advanced details","title":"Advanced details","text":"In the setting of missing response, the adjusted MM updates in each interation are","category":"page"},{"location":"advanced/","page":"Advanced details","title":"Advanced details","text":"beginaligned\ntextvec boldsymbolB^(t) = (boldsymbolI_d otimes boldsymbolX^T) boldsymbolOmega^-(t) (boldsymbolI_d otimes boldsymbolX)^-1 (boldsymbolI_d otimes boldsymbolX^T) boldsymbolOmega^-(t) textvec boldsymbolZ^(t) \nboldsymbolGamma_i^(t + 1) = boldsymbolL_i^-(t)TboldsymbolL_i^(t)TboldsymbolGamma_i^(t)(boldsymbolR^*(t)TboldsymbolV_iboldsymbolR^*(t) + boldsymbolM_i^*(t))boldsymbolGamma_i^(t)boldsymbolL_i^(t)^12 boldsymbolL_i^-(t)\nendaligned","category":"page"},{"location":"advanced/","page":"Advanced details","title":"Advanced details","text":"where boldsymbolZ^(t) is the completed response matrix from conditional mean and boldsymbolM_i^*(t) = (boldsymbolI_d otimes boldsymbol1_n)^T (boldsymbol1_d boldsymbol1_d^T otimes boldsymbolV_i) odot (boldsymbolOmega^-(t) boldsymbolP^T boldsymbolC^(t)boldsymbolPboldsymbolOmega^-(t)) (boldsymbolI_d otimes boldsymbol1_n), while boldsymbolR^*(t) is the n times d matrix such that textvec boldsymbolR^*(t) = boldsymbolOmega^-(t) textvec(boldsymbolZ^(t) - boldsymbolX boldsymbolB^(t)). Additionally, boldsymbolP is the nd times nd permutation matrix such that boldsymbolP cdot textvec boldsymbolY = beginbmatrix boldsymboly_textobs boldsymboly_textmis endbmatrix, where boldsymboly_textobs and boldsymboly_textmis are vectors of observed and missing response values, respectively, in column-major order, and the block matrix boldsymbolC^(t) is boldsymbol0 except for a lower-right block consisting of conditional variance. As seen, the two MM updates are of similar form.","category":"page"},{"location":"advanced/#Special-case:-m-2","page":"Advanced details","title":"Special case: m = 2","text":"","category":"section"},{"location":"advanced/","page":"Advanced details","title":"Advanced details","text":"When there are m = 2 variance components such that boldsymbolOmega = boldsymbolGamma_1 otimes boldsymbolV_1 + boldsymbolGamma_2 otimes boldsymbolV_2, repeated inversion of the nd times nd matrix boldsymbolOmega can be avoided and reduced to one d times d generalized eigen-decomposition per iteration. Without loss of generality, if we assume boldsymbolV_2 to be positive definite, the generalized eigen-decomposition of the matrix pair (boldsymbolV_1 boldsymbolV_2) yields generalized eigenvalues boldsymbold = (d_1 dots d_n)^T and generalized eigenvectors boldsymbolU such that boldsymbolU^T boldsymbolV_1 boldsymbolU = boldsymbolD = textdiag(boldsymbold) and boldsymbolU^T boldsymbolV_2 boldsymbolU = boldsymbolI_n. Similarly, if we let the generalized eigen-decomposition of (boldsymbolGamma_1^(t) boldsymbolGamma_2^(t)) be (boldsymbolLambda^(t) boldsymbolPhi^(t)) such that boldsymbolPhi^(t)T boldsymbolGamma_1^(t) boldsymbolPhi^(t) = boldsymbolLambda^(t) = textdiag(boldsymbollambda^(t)) and boldsymbolPhi^(t)T boldsymbolGamma_2^(t) boldsymbolPhi^(t) = boldsymbolI_d, then the MM updates in each iteration become","category":"page"},{"location":"advanced/","page":"Advanced details","title":"Advanced details","text":"beginaligned\ntextvec boldsymbolB^(t) = (boldsymbolPhi^(t)Totimes tildeboldsymbolX)^T (boldsymbolLambda^(t) otimes boldsymbolD + boldsymbolI_d otimes boldsymbolI_n)^-1 (boldsymbolPhi^(t)Totimes tildeboldsymbolX)^-1 \nquad cdot (boldsymbolPhi^(t)Totimes tildeboldsymbolX)^T (boldsymbolLambda^(t) otimes boldsymbolD + boldsymbolI_d otimes boldsymbolI_n)^-1 textvec(tildeboldsymbolY boldsymbolPhi^(t)) \nboldsymbolGamma_i^(t + 1) = boldsymbolL_i^-(t)TboldsymbolL_i^(t)TboldsymbolN_i^(t)TboldsymbolN_i^(t)boldsymbolL_i^(t)^12 boldsymbolL_i^-(t)\nendaligned","category":"page"},{"location":"advanced/","page":"Advanced details","title":"Advanced details","text":"where tildeboldsymbolX = boldsymbolU^T boldsymbolX, tildeboldsymbolY = boldsymbolU^T boldsymbolY, boldsymbolL_1^(t) is the Cholesky factor of boldsymbolPhi^(t)textdiag(texttr(boldsymbolD(lambda_k^(t)boldsymbolD + boldsymbolI_n)^-1) k = 1dots d)boldsymbolPhi^(t)T, boldsymbolL_2^(t) is the Cholesky factor of boldsymbolPhi^(t)textdiag(texttr((lambda_k^(t)boldsymbolD + boldsymbolI_n)^-1) k = 1dots d)boldsymbolPhi^(t)T, boldsymbolN_1^(t) = boldsymbolD^12(tildeboldsymbolY - tildeboldsymbolXboldsymbolB)boldsymbolPhi^(t)oslash(boldsymboldboldsymbollambda^(t)T + boldsymbol1_nboldsymbol1_d^T) boldsymbolLambda^(t)boldsymbolPhi^-(t), and boldsymbolN_2^(t) = (tildeboldsymbolY - tildeboldsymbolXboldsymbolB)boldsymbolPhi^(t)oslash(boldsymboldboldsymbollambda^(t)T + boldsymbol1_nboldsymbol1_d^T) boldsymbolPhi^-(t). oslash denotes the Hadamard quotient.","category":"page"},{"location":"advanced/","page":"Advanced details","title":"Advanced details","text":"In this setting, the Fisher information matrix is equivalent to","category":"page"},{"location":"advanced/","page":"Advanced details","title":"Advanced details","text":"beginaligned\ntextE left- fracpartial^2partial(textvec boldsymbolB)^T partial(textvec boldsymbolB) mathcalL right = (boldsymbolPhi^Totimes tildeboldsymbolX)^T (boldsymbolLambda otimes boldsymbolD + boldsymbolI_d otimes boldsymbolI_n)^-1 (boldsymbolPhi^Totimes tildeboldsymbolX) \ntextE left - fracpartial^2partial (textvech boldsymbolGamma_i)^T partial (textvec boldsymbolB) mathcalL right = boldsymbol0 \ntextE left - fracpartial^2partial (textvech boldsymbolGamma_j)^T partial (textvech boldsymbolGamma_i) mathcalL right = frac12 boldsymbolD_d^T(boldsymbolPhiotimes boldsymbolPhi) textdiag(textvec boldsymbolW_ij) (boldsymbolPhi otimes boldsymbolPhi)^TboldsymbolD_d\nendaligned","category":"page"},{"location":"advanced/","page":"Advanced details","title":"Advanced details","text":"where boldsymbolW_ij is the d times d matrix that has entries","category":"page"},{"location":"advanced/","page":"Advanced details","title":"Advanced details","text":"beginaligned\n(boldsymbolW_11)_kl = texttr(boldsymbolD^2(boldsymbollambda_k boldsymbolD + boldsymbolI_n)^-1(boldsymbollambda_l boldsymbolD + boldsymbolI_n)^-1) \n(boldsymbolW_12)_kl = texttr(boldsymbolD(boldsymbollambda_k boldsymbolD + boldsymbolI_n)^-1(boldsymbollambda_l boldsymbolD + boldsymbolI_n)^-1) \n(boldsymbolW_22)_kl = texttr((boldsymbollambda_k boldsymbolD + boldsymbolI_n)^-1(boldsymbollambda_l boldsymbolD + boldsymbolI_n)^-1)\nendaligned","category":"page"},{"location":"","page":"Home","title":"Home","text":"CurrentModule = MultiResponseVarianceComponentModels","category":"page"},{"location":"#MRVCModels","page":"Home","title":"MRVCModels","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"MRVCModels.jl is a package for fitting and testing multivariate response variance components linear mixed models of form","category":"page"},{"location":"","page":"Home","title":"Home","text":"textvec boldsymbolY sim mathcalN(textvec(boldsymbolX boldsymbolB) sum_i=1^m boldsymbolGamma_i otimes boldsymbolV_i)","category":"page"},{"location":"","page":"Home","title":"Home","text":"where boldsymbolY and boldsymbolX are n times d response and n times p predictor matrices, respectively, and boldsymbolV_1 ldots boldsymbolV_m are m known n times n positive semidefinite matrices. textvec boldsymbolY creates an nd times 1 vector from boldsymbolY by stacking its columns and otimes denotes the Kronecker product. The parameters of the model include p times d mean effects boldsymbolB and d times d variance components (boldsymbolGamma_1 dots boldsymbolGamma_m), which MRVCModels.jl estimates through either minorization-maximization (MM) or expectation–maximization (EM) algorithms.","category":"page"},{"location":"","page":"Home","title":"Home","text":"info: Info\nMRVCModels.jl can also handle data with missing response, which destroys the symmetry of the log-likelihood and complicates maximum likelihood estimation. MM algorithm easily adapts to this challenge.","category":"page"},{"location":"","page":"Home","title":"Home","text":"warning: Warning\nMRVCModels.jl is not suitable for biobank-scale data, except in the setting of m = 2. For m 2, we recommend using this package for datasets of size up to n cdot d approx 60000. Further note that large m can affect memory needed when calculating standard errors, since it will require m(nd)^2 storage space.","category":"page"},{"location":"#Installation","page":"Home","title":"Installation","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"To use MRVCModels.jl, type:","category":"page"},{"location":"","page":"Home","title":"Home","text":"using Pkg\nPkg.add(url = \"https://github.com/Hua-Zhou/MultiResponseVarianceComponentModels.jl.git\")","category":"page"},{"location":"","page":"Home","title":"Home","text":"This documentation was built using Documenter.jl.","category":"page"},{"location":"","page":"Home","title":"Home","text":"using Dates # hide\nprintln(\"Documentation built $(Dates.now()) with Julia $(VERSION)\") # hide","category":"page"}] +} diff --git a/dev/siteinfo.js b/dev/siteinfo.js new file mode 100644 index 0000000..3343491 --- /dev/null +++ b/dev/siteinfo.js @@ -0,0 +1 @@ +var DOCUMENTER_CURRENT_VERSION = "dev"; diff --git a/index.html b/index.html new file mode 100644 index 0000000..6a5afc3 --- /dev/null +++ b/index.html @@ -0,0 +1,2 @@ + + diff --git a/stable b/stable new file mode 120000 index 0000000..1e66a61 --- /dev/null +++ b/stable @@ -0,0 +1 @@ +v0.3.1 \ No newline at end of file diff --git a/v0.2 b/v0.2 new file mode 120000 index 0000000..eac0a14 --- /dev/null +++ b/v0.2 @@ -0,0 +1 @@ +v0.2.1 \ No newline at end of file diff --git a/v0.2.0/.documenter-siteinfo.json b/v0.2.0/.documenter-siteinfo.json new file mode 100644 index 0000000..ca82a61 --- /dev/null +++ b/v0.2.0/.documenter-siteinfo.json @@ -0,0 +1 @@ +{"documenter":{"julia_version":"1.10.3","generation_timestamp":"2024-05-24T07:36:54","documenter_version":"1.4.1"}} \ No newline at end of file diff --git a/v0.2.0/advanced/index.html b/v0.2.0/advanced/index.html new file mode 100644 index 0000000..1f461aa --- /dev/null +++ b/v0.2.0/advanced/index.html @@ -0,0 +1,12 @@ + +Advanced details · MRVCModels.jl
GitHub

Advanced details

Estimation

For the MM algorithm, the updates in each iteration are

\[\begin{aligned} +\text{vec}\ \boldsymbol{B}^{(t)} &= [(\boldsymbol{I}_d \otimes \boldsymbol{X}^T) \boldsymbol{\Omega}^{-(t)} (\boldsymbol{I}_d \otimes \boldsymbol{X})]^{-1} (\boldsymbol{I}_d \otimes \boldsymbol{X}^T) \boldsymbol{\Omega}^{-(t)} \text{vec}\ \boldsymbol{Y} \\ +\boldsymbol{\Gamma}_i^{(t + 1)} &= \boldsymbol{L}_i^{-(t)T}[\boldsymbol{L}_i^{(t)T}(\boldsymbol{\Gamma}_i^{(t)}\boldsymbol{R}^{(t)T}\boldsymbol{V}_i\boldsymbol{R}^{(t)}\boldsymbol{\Gamma}_i^{(t)})\boldsymbol{L}_i^{(t)}]^{1/2} \boldsymbol{L}_i^{-(t)}, +\end{aligned}\]

where $\boldsymbol{\Omega}^{(t)} = \sum_{i=1}^m \boldsymbol{\Gamma}_i^{(t)} \otimes \boldsymbol{V}_i$ and $\boldsymbol{L}_i^{(t)}$ is the Cholesky factor of $(\boldsymbol{I}_d \otimes \boldsymbol{1}_n)^T [(\boldsymbol{1}_d \boldsymbol{1}_d^T \otimes \boldsymbol{V}_i) \odot \boldsymbol{\Omega}^{-(t)}] (\boldsymbol{I}_d \otimes \boldsymbol{1}_n)$, while $\boldsymbol{R}^{(t)}$ is the $n \times d$ matrix such that $\text{vec}\ \boldsymbol{R}^{(t)} = \boldsymbol{\Omega}^{-(t)} \text{vec}(\boldsymbol{Y} - \boldsymbol{X} \boldsymbol{B}^{(t)})$.

For the EM algorithm, the updates in each iteration are

\[\begin{aligned} +\text{vec}\ \boldsymbol{B}^{(t)} &= [(\boldsymbol{I}_d \otimes \boldsymbol{X}^T) \boldsymbol{\Omega}^{-(t)} (\boldsymbol{I}_d \otimes \boldsymbol{X})]^{-1} (\boldsymbol{I}_d \otimes \boldsymbol{X}^T) \boldsymbol{\Omega}^{-(t)} \text{vec}\ \boldsymbol{Y} \\ +\boldsymbol{\Gamma}_i^{(t + 1)} &= \frac{1}{r_i} \boldsymbol{\Gamma}_i^{(t)} \{ \boldsymbol{R}^{(t)T} \boldsymbol{V}_i \boldsymbol{R}^{(t)} - (\boldsymbol{I}_d \otimes \boldsymbol{1}_n)^T [(\boldsymbol{1}_d \boldsymbol{1}_d^T \otimes \boldsymbol{V}_i) \odot \boldsymbol{\Omega}^{-(t)}] (\boldsymbol{I}_d \otimes \boldsymbol{1}_n)\} \boldsymbol{\Gamma}_i^{(t)} + \boldsymbol{\Gamma}_i^{(t)}, +\end{aligned}\]

where $r_i = \text{rank}(\boldsymbol{V}_i)$. As seen, the updates for mean effects $\boldsymbol{B}$ are the same for these two algorithms.

Inference

Standard errors for our estimates were calculated using the Fisher information matrix, where

\[\begin{aligned} +\text{E} \left[- \frac{\partial^2}{\partial(\text{vec}\ \boldsymbol{B})^T \partial(\text{vec}\ \boldsymbol{B})} \mathcal{L} \right] &= (\boldsymbol{I}_d \otimes \boldsymbol{X}^T) \boldsymbol{\Omega}^{-1} (\boldsymbol{I}_d \otimes \boldsymbol{X}) \\ +\text{E} \left[ - \frac{\partial^2}{\partial (\text{vech}\ \boldsymbol{\Gamma}_i)^T \partial (\text{vec}\ \boldsymbol{B})} \mathcal{L} \right] &= \boldsymbol{0} \\ +\text{E} \left[ - \frac{\partial^2}{\partial (\text{vech}\ \boldsymbol{\Gamma}_j)^T \partial (\text{vech}\ \boldsymbol{\Gamma}_i)} \mathcal{L} \right] &= \frac{1}{2} \boldsymbol{U}_i^T (\boldsymbol{\Omega}^{-1} \otimes \boldsymbol{\Omega}^{-1}) \boldsymbol{U}_j +\end{aligned}\]

and $\boldsymbol{U}_i = (\boldsymbol{I}_d \otimes \boldsymbol{K}_{nd} \otimes \boldsymbol{I}_n) (\boldsymbol{I}_{d^2} \otimes \text{vec}\ \boldsymbol{V}_i) \boldsymbol{D}_{d}$. Here, $\boldsymbol{K}_{nd}$ is the $nd \times nd$ commutation matrix and $\boldsymbol{D}_{d}$ the $d^2 \times \frac{d(d+1)}{2}$ duplication matrix. $\text{vech}\ \boldsymbol{\Gamma}_i$ creates an $\frac{d(d+1)}{2} \times 1$ vector from $\boldsymbol{\Gamma}_i$ by stacking its lower triangular part.

diff --git a/v0.2.0/api/index.html b/v0.2.0/api/index.html new file mode 100644 index 0000000..926feb8 --- /dev/null +++ b/v0.2.0/api/index.html @@ -0,0 +1,2 @@ + +API · MRVCModels.jl
GitHub

API

MultiResponseVarianceComponentModels.fisher_B! — Method
fisher_B!(model::MultiResponseVarianceComponentModel)

Compute the sampling variance-covariance of regression coefficients model.B, assuming inverse of covariance matrix model.Ω is available at model.storage_nd_nd.

source
MultiResponseVarianceComponentModels.fisher_Σ! — Method
fisher_Σ!(model::MultiResponseVarianceComponentModel)

Compute the sampling variance-covariance of variance component estimates model.Σ, assuming inverse of covariance matrix model.Ω is available at model.storage_nd_nd.

source
MultiResponseVarianceComponentModels.fit! — Method
fit!(model::MultiResponseVarianceComponentModel)

Fit a multivariate response variance component model by an MM or EM algorithm.

Positional arguments

  • model : a MultiResponseVarianceComponentModel instance.

Keyword arguments

  • maxiter::Integer : maximum number of iterations. Default is 1000.
  • reltol::Real : relative tolerance for convergence. Default is 1e-6.
  • verbose::Bool : display algorithmic information. Default is false.
  • init::Symbol : initialization strategy. :default initialize by least squares. :user uses user supplied value at model.B and model.Σ.
  • algo::Symbol : optimization algorithm. :MM (default) or EM.
  • log::Bool : record iterate history or not. Defaut is false.
  • reml::Bool : REML instead of ML estimation. Default is false.
  • se::Bool : calculate standard errors. Default is true.

Output

  • history : iterate history.
source
MultiResponseVarianceComponentModels.h2 — Method
h2(model::MultiResponseVarianceComponentModel)

Calculate heritability estimates and their standard errors, assuming that all variance components capture genetic effects except the last term. Also returns total heritability and its standard error from sum of individual contributions.

source
MultiResponseVarianceComponentModels.loglikelihood! — Method
loglikelihood!(model::MultiResponseVarianceComponentModel)

Overwrite model.storage_nd_nd by inverse of the covariance matrix model.Ω, overwrite model.storage_nd by U' \ vec(model.R), and return the log-likelihood. This function assumes model.Ω and model.R are already updated according to model.Σ and model.B.

source
MultiResponseVarianceComponentModels.lrt — Method
lrt(model1::MultiResponseVarianceComponentModel, model0::MultiResponseVarianceComponentModel)

Perform a variation of the likelihood ratio test for univariate variance components models as in Molenberghs and Verbeke 2007 with model1 and model0 being the full and nested models, respectively.

source
MultiResponseVarianceComponentModels.update_Γk! — Method
update_Γk!(model, k)

Update the parameters model.Γ[k] and model.Ψ[:,k] assuming a diagonal plus low rank structure for model.Σ[k]. Assumes covariance matrix model.Ω is available at model.storage_nd_nd and model.Ω⁻¹R precomputed.

source
MultiResponseVarianceComponentModels.update_Σk! — Method
update_Σk!(model::MultiResponseVarianceComponentModel, k, Val(:EM))

EM update the model.Σ[k] assuming it has full rank d, inverse of covariance matrix model.Ω is available at model.storage_nd_nd, and model.Ω⁻¹R precomputed.

source
MultiResponseVarianceComponentModels.update_Σk! — Method
update_Σk!(model::MultiResponseVarianceComponentModel, k, Val(:MM))

MM update the model.Σ[k] assuming it has full rank d, inverse of covariance matrix model.Ω is available at model.storage_nd_nd, and model.Ω⁻¹R precomputed.

source
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+$(document).ready(function () { + $("#documenter .docs-navbar").headroom({ + tolerance: { up: 10, down: 10 }, + }); +}); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +$(document).ready(function () { + let meta = $("div[data-docstringscollapsed]").data(); + + if (meta?.docstringscollapsed) { + $("#documenter-article-toggle-button").trigger({ + type: "click", + noToggleAnimation: true, + }); + } +}); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +/* +To get an in-depth about the thought process you can refer: https://hetarth02.hashnode.dev/series/gsoc + +PSEUDOCODE: + +Searching happens automatically as the user types or adjusts the selected filters. +To preserve responsiveness, as much as possible of the slow parts of the search are done +in a web worker. Searching and result generation are done in the worker, and filtering and +DOM updates are done in the main thread. The filters are in the main thread as they should +be very quick to apply. This lets filters be changed without re-searching with minisearch +(which is possible even if filtering is on the worker thread) and also lets filters be +changed _while_ the worker is searching and without message passing (neither of which are +possible if filtering is on the worker thread) + +SEARCH WORKER: + +Import minisearch + +Build index + +On message from main thread + run search + find the first 200 unique results from each category, and compute their divs for display + note that this is necessary and sufficient information for the main thread to find the + first 200 unique results from any given filter set + post results to main thread + +MAIN: + +Launch worker + +Declare nonconstant globals (worker_is_running, last_search_text, unfiltered_results) + +On text update + if worker is not running, launch_search() + +launch_search + set worker_is_running to true, set last_search_text to the search text + post the search query to worker + +on message from worker + if last_search_text is not the same as the text in the search field, + the latest search result is not reflective of the latest search query, so update again + launch_search() + otherwise + set worker_is_running to false + + regardless, display the new search results to the user + save the unfiltered_results as a global + update_search() + +on filter click + adjust the filter selection + update_search() + +update_search + apply search filters by looping through the unfiltered_results and finding the first 200 + unique results that match the filters + + Update the DOM +*/ + +/////// SEARCH WORKER /////// + +function worker_function(documenterSearchIndex, documenterBaseURL, filters) { + importScripts( + "https://cdn.jsdelivr.net/npm/minisearch@6.1.0/dist/umd/index.min.js" + ); + + let data = documenterSearchIndex.map((x, key) => { + x["id"] = key; // minisearch requires a unique for each object + return x; + }); + + // list below is the lunr 2.1.3 list minus the intersect with names(Base) + // (all, any, get, in, is, only, which) and (do, else, for, let, where, while, with) + // ideally we'd just filter the original list but it's not available as a variable + const stopWords = new Set([ + "a", + "able", + "about", + "across", + "after", + "almost", + "also", + "am", + "among", + "an", + "and", + "are", + "as", + "at", + "be", + "because", + "been", + "but", + "by", + "can", + "cannot", + "could", + "dear", + "did", + "does", + "either", + "ever", + "every", + "from", + "got", + "had", + "has", + "have", + "he", + "her", + "hers", + "him", + "his", + "how", + "however", + "i", + "if", + "into", + "it", + "its", + "just", + "least", + "like", + "likely", + "may", + "me", + "might", + "most", + "must", + "my", + "neither", + "no", + "nor", + "not", + "of", + "off", + "often", + "on", + "or", + "other", + "our", + "own", + "rather", + "said", + "say", + "says", + "she", + "should", + "since", + "so", + "some", + "than", + "that", + "the", + "their", + "them", + "then", + "there", + "these", + "they", + "this", + "tis", + "to", + "too", + "twas", + "us", + "wants", + "was", + "we", + "were", + "what", + "when", + "who", + "whom", + "why", + "will", + "would", + "yet", + "you", + "your", + ]); + + let index = new MiniSearch({ + fields: ["title", "text"], // fields to index for full-text search + storeFields: ["location", "title", "text", "category", "page"], // fields to return with results + processTerm: (term) => { + let word = stopWords.has(term) ? null : term; + if (word) { + // custom trimmer that doesn't strip @ and !, which are used in julia macro and function names + word = word + .replace(/^[^a-zA-Z0-9@!]+/, "") + .replace(/[^a-zA-Z0-9@!]+$/, ""); + + word = word.toLowerCase(); + } + + return word ?? null; + }, + // add . as a separator, because otherwise "title": "Documenter.Anchors.add!", would not + // find anything if searching for "add!", only for the entire qualification + tokenize: (string) => string.split(/[\s\-\.]+/), + // options which will be applied during the search + searchOptions: { + prefix: true, + boost: { title: 100 }, + fuzzy: 2, + }, + }); + + index.addAll(data); + + /** + * Used to map characters to HTML entities. + * Refer: https://github.com/lodash/lodash/blob/main/src/escape.ts + */ + const htmlEscapes = { + "&": "&", + "<": "<", + ">": ">", + '"': """, + "'": "'", + }; + + /** + * Used to match HTML entities and HTML characters. + * Refer: https://github.com/lodash/lodash/blob/main/src/escape.ts + */ + const reUnescapedHtml = /[&<>"']/g; + const reHasUnescapedHtml = RegExp(reUnescapedHtml.source); + + /** + * Escape function from lodash + * Refer: https://github.com/lodash/lodash/blob/main/src/escape.ts + */ + function escape(string) { + return string && reHasUnescapedHtml.test(string) + ? string.replace(reUnescapedHtml, (chr) => htmlEscapes[chr]) + : string || ""; + } + + /** + * Make the result component given a minisearch result data object and the value + * of the search input as queryString. To view the result object structure, refer: + * https://lucaong.github.io/minisearch/modules/_minisearch_.html#searchresult + * + * @param {object} result + * @param {string} querystring + * @returns string + */ + function make_search_result(result, querystring) { + let search_divider = `
`; + let display_link = + result.location.slice(Math.max(0), Math.min(50, result.location.length)) + + (result.location.length > 30 ? "..." : ""); // To cut-off the link because it messes with the overflow of the whole div + + if (result.page !== "") { + display_link += ` (${result.page})`; + } + + let textindex = new RegExp(`${querystring}`, "i").exec(result.text); + let text = + textindex !== null + ? result.text.slice( + Math.max(textindex.index - 100, 0), + Math.min( + textindex.index + querystring.length + 100, + result.text.length + ) + ) + : ""; // cut-off text before and after from the match + + text = text.length ? escape(text) : ""; + + let display_result = text.length + ? "..." + + text.replace( + new RegExp(`${escape(querystring)}`, "i"), // For first occurrence + '$&' + ) + + "..." + : ""; // highlights the match + + let in_code = false; + if (!["page", "section"].includes(result.category.toLowerCase())) { + in_code = true; + } + + // We encode the full url to escape some special characters which can lead to broken links + let result_div = ` + +
+
${escape(result.title)}
+
${result.category}
+
+

+ ${display_result} +

+
+ ${display_link} +
+ + ${search_divider} + `; + + return result_div; + } + + self.onmessage = function (e) { + let query = e.data; + let results = index.search(query, { + filter: (result) => { + // Only return relevant results + return result.score >= 1; + }, + }); + + // Pre-filter to deduplicate and limit to 200 per category to the extent + // possible without knowing what the filters are. + let filtered_results = []; + let counts = {}; + for (let filter of filters) { + counts[filter] = 0; + } + let present = {}; + + for (let result of results) { + cat = result.category; + cnt = counts[cat]; + if (cnt < 200) { + id = cat + "---" + result.location; + if (present[id]) { + continue; + } + present[id] = true; + filtered_results.push({ + location: result.location, + category: cat, + div: make_search_result(result, query), + }); + } + } + + postMessage(filtered_results); + }; +} + +// `worker = Threads.@spawn worker_function(documenterSearchIndex)`, but in JavaScript! +const filters = [ + ...new Set(documenterSearchIndex["docs"].map((x) => x.category)), +]; +const worker_str = + "(" + + worker_function.toString() + + ")(" + + JSON.stringify(documenterSearchIndex["docs"]) + + "," + + JSON.stringify(documenterBaseURL) + + "," + + JSON.stringify(filters) + + ")"; +const worker_blob = new Blob([worker_str], { type: "text/javascript" }); +const worker = new Worker(URL.createObjectURL(worker_blob)); + +/////// SEARCH MAIN /////// + +// Whether the worker is currently handling a search. This is a boolean +// as the worker only ever handles 1 or 0 searches at a time. +var worker_is_running = false; + +// The last search text that was sent to the worker. This is used to determine +// if the worker should be launched again when it reports back results. +var last_search_text = ""; + +// The results of the last search. This, in combination with the state of the filters +// in the DOM, is used compute the results to display on calls to update_search. +var unfiltered_results = []; + +// Which filter is currently selected +var selected_filter = ""; + +$(document).on("input", ".documenter-search-input", function (event) { + if (!worker_is_running) { + launch_search(); + } +}); + +function launch_search() { + worker_is_running = true; + last_search_text = $(".documenter-search-input").val(); + worker.postMessage(last_search_text); +} + +worker.onmessage = function (e) { + if (last_search_text !== $(".documenter-search-input").val()) { + launch_search(); + } else { + worker_is_running = false; + } + + unfiltered_results = e.data; + update_search(); +}; + +$(document).on("click", ".search-filter", function () { + if ($(this).hasClass("search-filter-selected")) { + selected_filter = ""; + } else { + selected_filter = $(this).text().toLowerCase(); + } + + // This updates search results and toggles classes for UI: + update_search(); +}); + +/** + * Make/Update the search component + */ +function update_search() { + let querystring = $(".documenter-search-input").val(); + + if (querystring.trim()) { + if (selected_filter == "") { + results = unfiltered_results; + } else { + results = unfiltered_results.filter((result) => { + return selected_filter == result.category.toLowerCase(); + }); + } + + let search_result_container = ``; + let modal_filters = make_modal_body_filters(); + let search_divider = `
`; + + if (results.length) { + let links = []; + let count = 0; + let search_results = ""; + + for (var i = 0, n = results.length; i < n && count < 200; ++i) { + let result = results[i]; + if (result.location && !links.includes(result.location)) { + search_results += result.div; + count++; + links.push(result.location); + } + } + + if (count == 1) { + count_str = "1 result"; + } else if (count == 200) { + count_str = "200+ results"; + } else { + count_str = count + " results"; + } + let result_count = `
${count_str}
`; + + search_result_container = ` +
+ ${modal_filters} + ${search_divider} + ${result_count} +
+ ${search_results} +
+
+ `; + } else { + search_result_container = ` +
+ ${modal_filters} + ${search_divider} +
0 result(s)
+
+
No result found!
+ `; + } + + if ($(".search-modal-card-body").hasClass("is-justify-content-center")) { + $(".search-modal-card-body").removeClass("is-justify-content-center"); + } + + $(".search-modal-card-body").html(search_result_container); + } else { + if (!$(".search-modal-card-body").hasClass("is-justify-content-center")) { + $(".search-modal-card-body").addClass("is-justify-content-center"); + } + + $(".search-modal-card-body").html(` +
Type something to get started!
+ `); + } +} + +/** + * Make the modal filter html + * + * @returns string + */ +function make_modal_body_filters() { + let str = filters + .map((val) => { + if (selected_filter == val.toLowerCase()) { + return `${val}`; + } else { + return `${val}`; + } + }) + .join(""); + + return ` +
+ Filters: + ${str} +
`; +} + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +// Modal settings dialog +$(document).ready(function () { + var settings = $("#documenter-settings"); + $("#documenter-settings-button").click(function () { + settings.toggleClass("is-active"); + }); + // Close the dialog if X is clicked + $("#documenter-settings button.delete").click(function () { + settings.removeClass("is-active"); + }); + // Close dialog if ESC is pressed + $(document).keyup(function (e) { + if (e.keyCode == 27) settings.removeClass("is-active"); + }); +}); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +$(document).ready(function () { + let search_modal_header = ` +
+
+

+ + + + +

+
+
+ +
+
+ `; + + let initial_search_body = ` +
Type something to get started!
+ `; + + let search_modal_footer = ` + + `; + + $(document.body).append( + ` + + ` + ); + + document.querySelector(".docs-search-query").addEventListener("click", () => { + openModal(); + }); + + document + .querySelector(".close-search-modal") + .addEventListener("click", () => { + closeModal(); + }); + + $(document).on("click", ".search-result-link", function () { + closeModal(); + }); + + document.addEventListener("keydown", (event) => { + if ((event.ctrlKey || event.metaKey) && event.key === "/") { + openModal(); + } else if (event.key === "Escape") { + closeModal(); + } + + return false; + }); + + // Functions to open and close a modal + function openModal() { + let searchModal = document.querySelector("#search-modal"); + + searchModal.classList.add("is-active"); + document.querySelector(".documenter-search-input").focus(); + } + + function closeModal() { + let searchModal = document.querySelector("#search-modal"); + let initial_search_body = ` +
Type something to get started!
+ `; + + searchModal.classList.remove("is-active"); + document.querySelector(".documenter-search-input").blur(); + + if (!$(".search-modal-card-body").hasClass("is-justify-content-center")) { + $(".search-modal-card-body").addClass("is-justify-content-center"); + } + + $(".documenter-search-input").val(""); + $(".search-modal-card-body").html(initial_search_body); + } + + document + .querySelector("#search-modal .modal-background") + .addEventListener("click", () => { + closeModal(); + }); +}); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +// Manages the showing and hiding of the sidebar. +$(document).ready(function () { + var sidebar = $("#documenter > .docs-sidebar"); + var sidebar_button = $("#documenter-sidebar-button"); + sidebar_button.click(function (ev) { + ev.preventDefault(); + sidebar.toggleClass("visible"); + if (sidebar.hasClass("visible")) { + // Makes sure that the current menu item is visible in the sidebar. + $("#documenter .docs-menu a.is-active").focus(); + } + }); + $("#documenter > .docs-main").bind("click", function (ev) { + if ($(ev.target).is(sidebar_button)) { + return; + } + if (sidebar.hasClass("visible")) { + sidebar.removeClass("visible"); + } + }); +}); + +// Resizes the package name / sitename in the sidebar if it is too wide. +// Inspired by: https://github.com/davatron5000/FitText.js +$(document).ready(function () { + e = $("#documenter .docs-autofit"); + function resize() { + var L = parseInt(e.css("max-width"), 10); + var L0 = e.width(); + if (L0 > L) { + var h0 = parseInt(e.css("font-size"), 10); + e.css("font-size", (L * h0) / L0); + // TODO: make sure it survives resizes? + } + } + // call once and then register events + resize(); + $(window).resize(resize); + $(window).on("orientationchange", resize); +}); + +// Scroll the navigation bar to the currently selected menu item +$(document).ready(function () { + var sidebar = $("#documenter .docs-menu").get(0); + var active = $("#documenter .docs-menu .is-active").get(0); + if (typeof active !== "undefined") { + sidebar.scrollTop = active.offsetTop - sidebar.offsetTop - 15; + } +}); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +// Theme picker setup +$(document).ready(function () { + // onchange callback + $("#documenter-themepicker").change(function themepick_callback(ev) { + var themename = $("#documenter-themepicker option:selected").attr("value"); + if (themename === "auto") { + // set_theme(window.matchMedia('(prefers-color-scheme: dark)').matches ? 'dark' : 'light'); + window.localStorage.removeItem("documenter-theme"); + } else { + // set_theme(themename); + window.localStorage.setItem("documenter-theme", themename); + } + // We re-use the global function from themeswap.js to actually do the swapping. + set_theme_from_local_storage(); + }); + + // Make sure that the themepicker displays the correct theme when the theme is retrieved + // from localStorage + if (typeof window.localStorage !== "undefined") { + var theme = window.localStorage.getItem("documenter-theme"); + if (theme !== null) { + $("#documenter-themepicker option").each(function (i, e) { + e.selected = e.value === theme; + }); + } + } +}); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +// update the version selector with info from the siteinfo.js and ../versions.js files +$(document).ready(function () { + // If the version selector is disabled with DOCUMENTER_VERSION_SELECTOR_DISABLED in the + // siteinfo.js file, we just return immediately and not display the version selector. + if ( + typeof DOCUMENTER_VERSION_SELECTOR_DISABLED === "boolean" && + DOCUMENTER_VERSION_SELECTOR_DISABLED + ) { + return; + } + + var version_selector = $("#documenter .docs-version-selector"); + var version_selector_select = $("#documenter .docs-version-selector select"); + + version_selector_select.change(function (x) { + target_href = version_selector_select + .children("option:selected") + .get(0).value; + window.location.href = target_href; + }); + + // add the current version to the selector based on siteinfo.js, but only if the selector is empty + if ( + typeof DOCUMENTER_CURRENT_VERSION !== "undefined" && + $("#version-selector > option").length == 0 + ) { + var option = $( + "" + ); + version_selector_select.append(option); + } + + if (typeof DOC_VERSIONS !== "undefined") { + var existing_versions = version_selector_select.children("option"); + var existing_versions_texts = existing_versions.map(function (i, x) { + return x.text; + }); + DOC_VERSIONS.forEach(function (each) { + var version_url = documenterBaseURL + "/../" + each + "/"; + var existing_id = $.inArray(each, existing_versions_texts); + // if not already in the version selector, add it as a new option, + // otherwise update the old option with the URL and enable it + if (existing_id == -1) { + var option = $( + "" + ); + version_selector_select.append(option); + } else { + var option = existing_versions[existing_id]; + option.value = version_url; + option.disabled = false; + } + }); + } + + // only show the version selector if the selector has been populated + if (version_selector_select.children("option").length > 0) { + version_selector.toggleClass("visible"); + } +}); + +}) diff --git a/v0.2.0/assets/logo.svg b/v0.2.0/assets/logo.svg new file mode 100644 index 0000000..df89daf --- /dev/null +++ b/v0.2.0/assets/logo.svg @@ -0,0 +1,169 @@ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +≈ ++ ++…+ + diff --git a/v0.2.0/assets/themes/documenter-dark.css b/v0.2.0/assets/themes/documenter-dark.css new file mode 100644 index 0000000..1d71701 --- /dev/null +++ b/v0.2.0/assets/themes/documenter-dark.css @@ -0,0 +1,7 @@ +html.theme--documenter-dark .pagination-previous,html.theme--documenter-dark .pagination-next,html.theme--documenter-dark .pagination-link,html.theme--documenter-dark .pagination-ellipsis,html.theme--documenter-dark .file-cta,html.theme--documenter-dark .file-name,html.theme--documenter-dark .select select,html.theme--documenter-dark .textarea,html.theme--documenter-dark .input,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input,html.theme--documenter-dark .button{-moz-appearance:none;-webkit-appearance:none;align-items:center;border:1px solid transparent;border-radius:.4em;box-shadow:none;display:inline-flex;font-size:1rem;height:2.5em;justify-content:flex-start;line-height:1.5;padding-bottom:calc(0.5em - 1px);padding-left:calc(0.75em - 1px);padding-right:calc(0.75em - 1px);padding-top:calc(0.5em - 1px);position:relative;vertical-align:top}html.theme--documenter-dark .pagination-previous:focus,html.theme--documenter-dark .pagination-next:focus,html.theme--documenter-dark .pagination-link:focus,html.theme--documenter-dark .pagination-ellipsis:focus,html.theme--documenter-dark .file-cta:focus,html.theme--documenter-dark .file-name:focus,html.theme--documenter-dark .select select:focus,html.theme--documenter-dark .textarea:focus,html.theme--documenter-dark .input:focus,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input:focus,html.theme--documenter-dark .button:focus,html.theme--documenter-dark .is-focused.pagination-previous,html.theme--documenter-dark .is-focused.pagination-next,html.theme--documenter-dark .is-focused.pagination-link,html.theme--documenter-dark .is-focused.pagination-ellipsis,html.theme--documenter-dark .is-focused.file-cta,html.theme--documenter-dark .is-focused.file-name,html.theme--documenter-dark .select select.is-focused,html.theme--documenter-dark .is-focused.textarea,html.theme--documenter-dark .is-focused.input,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.is-focused,html.theme--documenter-dark .is-focused.button,html.theme--documenter-dark .pagination-previous:active,html.theme--documenter-dark .pagination-next:active,html.theme--documenter-dark .pagination-link:active,html.theme--documenter-dark .pagination-ellipsis:active,html.theme--documenter-dark .file-cta:active,html.theme--documenter-dark .file-name:active,html.theme--documenter-dark .select select:active,html.theme--documenter-dark .textarea:active,html.theme--documenter-dark .input:active,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input:active,html.theme--documenter-dark .button:active,html.theme--documenter-dark .is-active.pagination-previous,html.theme--documenter-dark .is-active.pagination-next,html.theme--documenter-dark .is-active.pagination-link,html.theme--documenter-dark .is-active.pagination-ellipsis,html.theme--documenter-dark .is-active.file-cta,html.theme--documenter-dark .is-active.file-name,html.theme--documenter-dark .select select.is-active,html.theme--documenter-dark .is-active.textarea,html.theme--documenter-dark .is-active.input,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.is-active,html.theme--documenter-dark .is-active.button{outline:none}html.theme--documenter-dark .pagination-previous[disabled],html.theme--documenter-dark .pagination-next[disabled],html.theme--documenter-dark .pagination-link[disabled],html.theme--documenter-dark .pagination-ellipsis[disabled],html.theme--documenter-dark 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.button.is-white{background-color:#fff;border-color:#fff;box-shadow:none}html.theme--documenter-dark .button.is-white.is-inverted{background-color:#0a0a0a;color:#fff}html.theme--documenter-dark .button.is-white.is-inverted:hover,html.theme--documenter-dark .button.is-white.is-inverted.is-hovered{background-color:#000}html.theme--documenter-dark .button.is-white.is-inverted[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-white.is-inverted{background-color:#0a0a0a;border-color:transparent;box-shadow:none;color:#fff}html.theme--documenter-dark .button.is-white.is-loading::after{border-color:transparent transparent #0a0a0a #0a0a0a !important}html.theme--documenter-dark .button.is-white.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--documenter-dark .button.is-white.is-outlined:hover,html.theme--documenter-dark .button.is-white.is-outlined.is-hovered,html.theme--documenter-dark .button.is-white.is-outlined:focus,html.theme--documenter-dark .button.is-white.is-outlined.is-focused{background-color:#fff;border-color:#fff;color:#0a0a0a}html.theme--documenter-dark .button.is-white.is-outlined.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--documenter-dark .button.is-white.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-white.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-white.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-white.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #0a0a0a #0a0a0a !important}html.theme--documenter-dark .button.is-white.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-white.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--documenter-dark 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html.theme--documenter-dark .button.is-white.is-inverted.is-outlined{background-color:transparent;border-color:#0a0a0a;box-shadow:none;color:#0a0a0a}html.theme--documenter-dark .button.is-black{background-color:#0a0a0a;border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-black:hover,html.theme--documenter-dark .button.is-black.is-hovered{background-color:#040404;border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-black:focus,html.theme--documenter-dark .button.is-black.is-focused{border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-black:focus:not(:active),html.theme--documenter-dark .button.is-black.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(10,10,10,0.25)}html.theme--documenter-dark .button.is-black:active,html.theme--documenter-dark .button.is-black.is-active{background-color:#000;border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-black[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-black{background-color:#0a0a0a;border-color:#0a0a0a;box-shadow:none}html.theme--documenter-dark .button.is-black.is-inverted{background-color:#fff;color:#0a0a0a}html.theme--documenter-dark .button.is-black.is-inverted:hover,html.theme--documenter-dark .button.is-black.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--documenter-dark .button.is-black.is-inverted[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-black.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#0a0a0a}html.theme--documenter-dark .button.is-black.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--documenter-dark .button.is-black.is-outlined{background-color:transparent;border-color:#0a0a0a;color:#0a0a0a}html.theme--documenter-dark .button.is-black.is-outlined:hover,html.theme--documenter-dark .button.is-black.is-outlined.is-hovered,html.theme--documenter-dark .button.is-black.is-outlined:focus,html.theme--documenter-dark .button.is-black.is-outlined.is-focused{background-color:#0a0a0a;border-color:#0a0a0a;color:#fff}html.theme--documenter-dark .button.is-black.is-outlined.is-loading::after{border-color:transparent transparent #0a0a0a #0a0a0a !important}html.theme--documenter-dark .button.is-black.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-black.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-black.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-black.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--documenter-dark .button.is-black.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-black.is-outlined{background-color:transparent;border-color:#0a0a0a;box-shadow:none;color:#0a0a0a}html.theme--documenter-dark .button.is-black.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--documenter-dark .button.is-black.is-inverted.is-outlined:hover,html.theme--documenter-dark .button.is-black.is-inverted.is-outlined.is-hovered,html.theme--documenter-dark .button.is-black.is-inverted.is-outlined:focus,html.theme--documenter-dark .button.is-black.is-inverted.is-outlined.is-focused{background-color:#fff;color:#0a0a0a}html.theme--documenter-dark .button.is-black.is-inverted.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-black.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-black.is-inverted.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-black.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #0a0a0a #0a0a0a !important}html.theme--documenter-dark .button.is-black.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-black.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--documenter-dark .button.is-light{background-color:#ecf0f1;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--documenter-dark .button.is-light:hover,html.theme--documenter-dark .button.is-light.is-hovered{background-color:#e5eaec;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--documenter-dark .button.is-light:focus,html.theme--documenter-dark .button.is-light.is-focused{border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--documenter-dark .button.is-light:focus:not(:active),html.theme--documenter-dark .button.is-light.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(236,240,241,0.25)}html.theme--documenter-dark .button.is-light:active,html.theme--documenter-dark .button.is-light.is-active{background-color:#dde4e6;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--documenter-dark .button.is-light[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-light{background-color:#ecf0f1;border-color:#ecf0f1;box-shadow:none}html.theme--documenter-dark .button.is-light.is-inverted{background-color:rgba(0,0,0,0.7);color:#ecf0f1}html.theme--documenter-dark .button.is-light.is-inverted:hover,html.theme--documenter-dark .button.is-light.is-inverted.is-hovered{background-color:rgba(0,0,0,0.7)}html.theme--documenter-dark .button.is-light.is-inverted[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-light.is-inverted{background-color:rgba(0,0,0,0.7);border-color:transparent;box-shadow:none;color:#ecf0f1}html.theme--documenter-dark .button.is-light.is-loading::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--documenter-dark .button.is-light.is-outlined{background-color:transparent;border-color:#ecf0f1;color:#ecf0f1}html.theme--documenter-dark .button.is-light.is-outlined:hover,html.theme--documenter-dark .button.is-light.is-outlined.is-hovered,html.theme--documenter-dark .button.is-light.is-outlined:focus,html.theme--documenter-dark .button.is-light.is-outlined.is-focused{background-color:#ecf0f1;border-color:#ecf0f1;color:rgba(0,0,0,0.7)}html.theme--documenter-dark .button.is-light.is-outlined.is-loading::after{border-color:transparent transparent #ecf0f1 #ecf0f1 !important}html.theme--documenter-dark .button.is-light.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-light.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-light.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-light.is-outlined.is-loading.is-focused::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--documenter-dark .button.is-light.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-light.is-outlined{background-color:transparent;border-color:#ecf0f1;box-shadow:none;color:#ecf0f1}html.theme--documenter-dark .button.is-light.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);color:rgba(0,0,0,0.7)}html.theme--documenter-dark .button.is-light.is-inverted.is-outlined:hover,html.theme--documenter-dark .button.is-light.is-inverted.is-outlined.is-hovered,html.theme--documenter-dark .button.is-light.is-inverted.is-outlined:focus,html.theme--documenter-dark .button.is-light.is-inverted.is-outlined.is-focused{background-color:rgba(0,0,0,0.7);color:#ecf0f1}html.theme--documenter-dark .button.is-light.is-inverted.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-light.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-light.is-inverted.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-light.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #ecf0f1 #ecf0f1 !important}html.theme--documenter-dark .button.is-light.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-light.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--documenter-dark .button.is-dark,html.theme--documenter-dark .content kbd.button{background-color:#282f2f;border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-dark:hover,html.theme--documenter-dark .content kbd.button:hover,html.theme--documenter-dark .button.is-dark.is-hovered,html.theme--documenter-dark .content kbd.button.is-hovered{background-color:#232829;border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-dark:focus,html.theme--documenter-dark .content kbd.button:focus,html.theme--documenter-dark .button.is-dark.is-focused,html.theme--documenter-dark .content kbd.button.is-focused{border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-dark:focus:not(:active),html.theme--documenter-dark .content kbd.button:focus:not(:active),html.theme--documenter-dark .button.is-dark.is-focused:not(:active),html.theme--documenter-dark .content kbd.button.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(40,47,47,0.25)}html.theme--documenter-dark .button.is-dark:active,html.theme--documenter-dark .content kbd.button:active,html.theme--documenter-dark .button.is-dark.is-active,html.theme--documenter-dark .content kbd.button.is-active{background-color:#1d2122;border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-dark[disabled],html.theme--documenter-dark .content kbd.button[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-dark,fieldset[disabled] html.theme--documenter-dark .content kbd.button{background-color:#282f2f;border-color:#282f2f;box-shadow:none}html.theme--documenter-dark .button.is-dark.is-inverted,html.theme--documenter-dark .content kbd.button.is-inverted{background-color:#fff;color:#282f2f}html.theme--documenter-dark .button.is-dark.is-inverted:hover,html.theme--documenter-dark .content kbd.button.is-inverted:hover,html.theme--documenter-dark .button.is-dark.is-inverted.is-hovered,html.theme--documenter-dark .content kbd.button.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--documenter-dark .button.is-dark.is-inverted[disabled],html.theme--documenter-dark .content kbd.button.is-inverted[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-dark.is-inverted,fieldset[disabled] html.theme--documenter-dark .content kbd.button.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#282f2f}html.theme--documenter-dark .button.is-dark.is-loading::after,html.theme--documenter-dark .content kbd.button.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--documenter-dark .button.is-dark.is-outlined,html.theme--documenter-dark .content kbd.button.is-outlined{background-color:transparent;border-color:#282f2f;color:#282f2f}html.theme--documenter-dark .button.is-dark.is-outlined:hover,html.theme--documenter-dark .content kbd.button.is-outlined:hover,html.theme--documenter-dark .button.is-dark.is-outlined.is-hovered,html.theme--documenter-dark .content kbd.button.is-outlined.is-hovered,html.theme--documenter-dark .button.is-dark.is-outlined:focus,html.theme--documenter-dark .content kbd.button.is-outlined:focus,html.theme--documenter-dark .button.is-dark.is-outlined.is-focused,html.theme--documenter-dark .content kbd.button.is-outlined.is-focused{background-color:#282f2f;border-color:#282f2f;color:#fff}html.theme--documenter-dark .button.is-dark.is-outlined.is-loading::after,html.theme--documenter-dark .content kbd.button.is-outlined.is-loading::after{border-color:transparent transparent #282f2f #282f2f !important}html.theme--documenter-dark .button.is-dark.is-outlined.is-loading:hover::after,html.theme--documenter-dark .content kbd.button.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-dark.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .content kbd.button.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-dark.is-outlined.is-loading:focus::after,html.theme--documenter-dark .content kbd.button.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-dark.is-outlined.is-loading.is-focused::after,html.theme--documenter-dark .content kbd.button.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--documenter-dark .button.is-dark.is-outlined[disabled],html.theme--documenter-dark .content kbd.button.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-dark.is-outlined,fieldset[disabled] html.theme--documenter-dark .content kbd.button.is-outlined{background-color:transparent;border-color:#282f2f;box-shadow:none;color:#282f2f}html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined:hover,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined:hover,html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-hovered,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-hovered,html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined:focus,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined:focus,html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-focused,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-focused{background-color:#fff;color:#282f2f}html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-loading:hover::after,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-loading:focus::after,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-loading.is-focused::after,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #282f2f #282f2f !important}html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined[disabled],html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined,fieldset[disabled] html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--documenter-dark .button.is-primary,html.theme--documenter-dark .docstring>section>a.button.docs-sourcelink{background-color:#375a7f;border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-primary:hover,html.theme--documenter-dark .docstring>section>a.button.docs-sourcelink:hover,html.theme--documenter-dark .button.is-primary.is-hovered,html.theme--documenter-dark .docstring>section>a.button.is-hovered.docs-sourcelink{background-color:#335476;border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-primary:focus,html.theme--documenter-dark 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.button.is-warning.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-warning.is-outlined{background-color:transparent;border-color:#ad8100;box-shadow:none;color:#ad8100}html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined:hover,html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined.is-hovered,html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined:focus,html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined.is-focused{background-color:#fff;color:#ad8100}html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #ad8100 #ad8100 !important}html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--documenter-dark .button.is-warning.is-light{background-color:#fffaeb;color:#d19c00}html.theme--documenter-dark .button.is-warning.is-light:hover,html.theme--documenter-dark .button.is-warning.is-light.is-hovered{background-color:#fff7de;border-color:transparent;color:#d19c00}html.theme--documenter-dark .button.is-warning.is-light:active,html.theme--documenter-dark .button.is-warning.is-light.is-active{background-color:#fff3d1;border-color:transparent;color:#d19c00}html.theme--documenter-dark .button.is-danger{background-color:#9e1b0d;border-color:transparent;color:#fff}html.theme--documenter-dark 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.button.is-danger.is-inverted:hover,html.theme--documenter-dark .button.is-danger.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--documenter-dark .button.is-danger.is-inverted[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-danger.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#9e1b0d}html.theme--documenter-dark .button.is-danger.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--documenter-dark .button.is-danger.is-outlined{background-color:transparent;border-color:#9e1b0d;color:#9e1b0d}html.theme--documenter-dark .button.is-danger.is-outlined:hover,html.theme--documenter-dark .button.is-danger.is-outlined.is-hovered,html.theme--documenter-dark .button.is-danger.is-outlined:focus,html.theme--documenter-dark .button.is-danger.is-outlined.is-focused{background-color:#9e1b0d;border-color:#9e1b0d;color:#fff}html.theme--documenter-dark .button.is-danger.is-outlined.is-loading::after{border-color:transparent transparent #9e1b0d #9e1b0d !important}html.theme--documenter-dark .button.is-danger.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-danger.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-danger.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-danger.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--documenter-dark .button.is-danger.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-danger.is-outlined{background-color:transparent;border-color:#9e1b0d;box-shadow:none;color:#9e1b0d}html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined:hover,html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined.is-hovered,html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined:focus,html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined.is-focused{background-color:#fff;color:#9e1b0d}html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #9e1b0d #9e1b0d !important}html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--documenter-dark .button.is-danger.is-light{background-color:#fdeeec;color:#ec311d}html.theme--documenter-dark .button.is-danger.is-light:hover,html.theme--documenter-dark .button.is-danger.is-light.is-hovered{background-color:#fce3e0;border-color:transparent;color:#ec311d}html.theme--documenter-dark .button.is-danger.is-light:active,html.theme--documenter-dark .button.is-danger.is-light.is-active{background-color:#fcd8d5;border-color:transparent;color:#ec311d}html.theme--documenter-dark .button.is-small,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.button{font-size:.75rem}html.theme--documenter-dark .button.is-small:not(.is-rounded),html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.button:not(.is-rounded){border-radius:3px}html.theme--documenter-dark .button.is-normal{font-size:1rem}html.theme--documenter-dark .button.is-medium{font-size:1.25rem}html.theme--documenter-dark .button.is-large{font-size:1.5rem}html.theme--documenter-dark .button[disabled],fieldset[disabled] html.theme--documenter-dark .button{background-color:#8c9b9d;border-color:#5e6d6f;box-shadow:none;opacity:.5}html.theme--documenter-dark .button.is-fullwidth{display:flex;width:100%}html.theme--documenter-dark .button.is-loading{color:transparent !important;pointer-events:none}html.theme--documenter-dark .button.is-loading::after{position:absolute;left:calc(50% - (1em * 0.5));top:calc(50% - (1em * 0.5));position:absolute !important}html.theme--documenter-dark .button.is-static{background-color:#282f2f;border-color:#5e6d6f;color:#dbdee0;box-shadow:none;pointer-events:none}html.theme--documenter-dark .button.is-rounded,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.button{border-radius:9999px;padding-left:calc(1em + 0.25em);padding-right:calc(1em + 0.25em)}html.theme--documenter-dark .buttons{align-items:center;display:flex;flex-wrap:wrap;justify-content:flex-start}html.theme--documenter-dark .buttons .button{margin-bottom:0.5rem}html.theme--documenter-dark .buttons .button:not(:last-child):not(.is-fullwidth){margin-right:.5rem}html.theme--documenter-dark .buttons:last-child{margin-bottom:-0.5rem}html.theme--documenter-dark .buttons:not(:last-child){margin-bottom:1rem}html.theme--documenter-dark .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large){font-size:.75rem}html.theme--documenter-dark .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large):not(.is-rounded){border-radius:3px}html.theme--documenter-dark .buttons.are-medium .button:not(.is-small):not(.is-normal):not(.is-large){font-size:1.25rem}html.theme--documenter-dark .buttons.are-large .button:not(.is-small):not(.is-normal):not(.is-medium){font-size:1.5rem}html.theme--documenter-dark .buttons.has-addons .button:not(:first-child){border-bottom-left-radius:0;border-top-left-radius:0}html.theme--documenter-dark .buttons.has-addons .button:not(:last-child){border-bottom-right-radius:0;border-top-right-radius:0;margin-right:-1px}html.theme--documenter-dark .buttons.has-addons .button:last-child{margin-right:0}html.theme--documenter-dark .buttons.has-addons .button:hover,html.theme--documenter-dark .buttons.has-addons .button.is-hovered{z-index:2}html.theme--documenter-dark .buttons.has-addons .button:focus,html.theme--documenter-dark .buttons.has-addons .button.is-focused,html.theme--documenter-dark .buttons.has-addons .button:active,html.theme--documenter-dark .buttons.has-addons .button.is-active,html.theme--documenter-dark .buttons.has-addons .button.is-selected{z-index:3}html.theme--documenter-dark .buttons.has-addons .button:focus:hover,html.theme--documenter-dark .buttons.has-addons .button.is-focused:hover,html.theme--documenter-dark .buttons.has-addons .button:active:hover,html.theme--documenter-dark .buttons.has-addons .button.is-active:hover,html.theme--documenter-dark .buttons.has-addons .button.is-selected:hover{z-index:4}html.theme--documenter-dark .buttons.has-addons .button.is-expanded{flex-grow:1;flex-shrink:1}html.theme--documenter-dark .buttons.is-centered{justify-content:center}html.theme--documenter-dark .buttons.is-centered:not(.has-addons) .button:not(.is-fullwidth){margin-left:0.25rem;margin-right:0.25rem}html.theme--documenter-dark .buttons.is-right{justify-content:flex-end}html.theme--documenter-dark .buttons.is-right:not(.has-addons) .button:not(.is-fullwidth){margin-left:0.25rem;margin-right:0.25rem}@media screen and (max-width: 768px){html.theme--documenter-dark .button.is-responsive.is-small,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.is-responsive{font-size:.5625rem}html.theme--documenter-dark .button.is-responsive,html.theme--documenter-dark .button.is-responsive.is-normal{font-size:.65625rem}html.theme--documenter-dark .button.is-responsive.is-medium{font-size:.75rem}html.theme--documenter-dark .button.is-responsive.is-large{font-size:1rem}}@media screen and (min-width: 769px) and (max-width: 1055px){html.theme--documenter-dark .button.is-responsive.is-small,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.is-responsive{font-size:.65625rem}html.theme--documenter-dark .button.is-responsive,html.theme--documenter-dark .button.is-responsive.is-normal{font-size:.75rem}html.theme--documenter-dark .button.is-responsive.is-medium{font-size:1rem}html.theme--documenter-dark .button.is-responsive.is-large{font-size:1.25rem}}html.theme--documenter-dark .container{flex-grow:1;margin:0 auto;position:relative;width:auto}html.theme--documenter-dark .container.is-fluid{max-width:none !important;padding-left:32px;padding-right:32px;width:100%}@media screen and (min-width: 1056px){html.theme--documenter-dark .container{max-width:992px}}@media screen and (max-width: 1215px){html.theme--documenter-dark .container.is-widescreen:not(.is-max-desktop){max-width:1152px}}@media screen and (max-width: 1407px){html.theme--documenter-dark .container.is-fullhd:not(.is-max-desktop):not(.is-max-widescreen){max-width:1344px}}@media screen and (min-width: 1216px){html.theme--documenter-dark .container:not(.is-max-desktop){max-width:1152px}}@media screen and (min-width: 1408px){html.theme--documenter-dark .container:not(.is-max-desktop):not(.is-max-widescreen){max-width:1344px}}html.theme--documenter-dark .content li+li{margin-top:0.25em}html.theme--documenter-dark .content p:not(:last-child),html.theme--documenter-dark .content dl:not(:last-child),html.theme--documenter-dark .content ol:not(:last-child),html.theme--documenter-dark .content ul:not(:last-child),html.theme--documenter-dark .content blockquote:not(:last-child),html.theme--documenter-dark .content pre:not(:last-child),html.theme--documenter-dark .content table:not(:last-child){margin-bottom:1em}html.theme--documenter-dark .content h1,html.theme--documenter-dark .content h2,html.theme--documenter-dark .content h3,html.theme--documenter-dark .content h4,html.theme--documenter-dark .content h5,html.theme--documenter-dark .content h6{color:#f2f2f2;font-weight:600;line-height:1.125}html.theme--documenter-dark .content h1{font-size:2em;margin-bottom:0.5em}html.theme--documenter-dark .content h1:not(:first-child){margin-top:1em}html.theme--documenter-dark .content h2{font-size:1.75em;margin-bottom:0.5714em}html.theme--documenter-dark .content h2:not(:first-child){margin-top:1.1428em}html.theme--documenter-dark .content h3{font-size:1.5em;margin-bottom:0.6666em}html.theme--documenter-dark .content h3:not(:first-child){margin-top:1.3333em}html.theme--documenter-dark .content h4{font-size:1.25em;margin-bottom:0.8em}html.theme--documenter-dark .content h5{font-size:1.125em;margin-bottom:0.8888em}html.theme--documenter-dark .content h6{font-size:1em;margin-bottom:1em}html.theme--documenter-dark .content blockquote{background-color:#282f2f;border-left:5px solid #5e6d6f;padding:1.25em 1.5em}html.theme--documenter-dark .content ol{list-style-position:outside;margin-left:2em;margin-top:1em}html.theme--documenter-dark .content ol:not([type]){list-style-type:decimal}html.theme--documenter-dark .content ol.is-lower-alpha:not([type]){list-style-type:lower-alpha}html.theme--documenter-dark .content ol.is-lower-roman:not([type]){list-style-type:lower-roman}html.theme--documenter-dark .content ol.is-upper-alpha:not([type]){list-style-type:upper-alpha}html.theme--documenter-dark .content ol.is-upper-roman:not([type]){list-style-type:upper-roman}html.theme--documenter-dark .content ul{list-style:disc outside;margin-left:2em;margin-top:1em}html.theme--documenter-dark .content ul ul{list-style-type:circle;margin-top:0.5em}html.theme--documenter-dark .content ul ul ul{list-style-type:square}html.theme--documenter-dark .content dd{margin-left:2em}html.theme--documenter-dark .content figure{margin-left:2em;margin-right:2em;text-align:center}html.theme--documenter-dark .content figure:not(:first-child){margin-top:2em}html.theme--documenter-dark .content figure:not(:last-child){margin-bottom:2em}html.theme--documenter-dark .content figure img{display:inline-block}html.theme--documenter-dark .content figure figcaption{font-style:italic}html.theme--documenter-dark .content pre{-webkit-overflow-scrolling:touch;overflow-x:auto;padding:0;white-space:pre;word-wrap:normal}html.theme--documenter-dark .content sup,html.theme--documenter-dark .content sub{font-size:75%}html.theme--documenter-dark .content table{width:100%}html.theme--documenter-dark .content table td,html.theme--documenter-dark .content table th{border:1px solid #5e6d6f;border-width:0 0 1px;padding:0.5em 0.75em;vertical-align:top}html.theme--documenter-dark .content table th{color:#f2f2f2}html.theme--documenter-dark .content table th:not([align]){text-align:inherit}html.theme--documenter-dark .content table thead td,html.theme--documenter-dark .content table thead th{border-width:0 0 2px;color:#f2f2f2}html.theme--documenter-dark .content table tfoot td,html.theme--documenter-dark .content table tfoot th{border-width:2px 0 0;color:#f2f2f2}html.theme--documenter-dark .content table tbody tr:last-child td,html.theme--documenter-dark .content table tbody tr:last-child th{border-bottom-width:0}html.theme--documenter-dark .content .tabs li+li{margin-top:0}html.theme--documenter-dark .content.is-small,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.content{font-size:.75rem}html.theme--documenter-dark .content.is-normal{font-size:1rem}html.theme--documenter-dark .content.is-medium{font-size:1.25rem}html.theme--documenter-dark .content.is-large{font-size:1.5rem}html.theme--documenter-dark .icon{align-items:center;display:inline-flex;justify-content:center;height:1.5rem;width:1.5rem}html.theme--documenter-dark .icon.is-small,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.icon{height:1rem;width:1rem}html.theme--documenter-dark .icon.is-medium{height:2rem;width:2rem}html.theme--documenter-dark .icon.is-large{height:3rem;width:3rem}html.theme--documenter-dark .icon-text{align-items:flex-start;color:inherit;display:inline-flex;flex-wrap:wrap;line-height:1.5rem;vertical-align:top}html.theme--documenter-dark .icon-text .icon{flex-grow:0;flex-shrink:0}html.theme--documenter-dark .icon-text .icon:not(:last-child){margin-right:.25em}html.theme--documenter-dark .icon-text .icon:not(:first-child){margin-left:.25em}html.theme--documenter-dark div.icon-text{display:flex}html.theme--documenter-dark .image,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img{display:block;position:relative}html.theme--documenter-dark .image img,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img img{display:block;height:auto;width:100%}html.theme--documenter-dark .image img.is-rounded,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img img.is-rounded{border-radius:9999px}html.theme--documenter-dark .image.is-fullwidth,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-fullwidth{width:100%}html.theme--documenter-dark .image.is-square img,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-square img,html.theme--documenter-dark .image.is-square .has-ratio,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-square .has-ratio,html.theme--documenter-dark .image.is-1by1 img,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-1by1 img,html.theme--documenter-dark .image.is-1by1 .has-ratio,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-1by1 .has-ratio,html.theme--documenter-dark .image.is-5by4 img,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-5by4 img,html.theme--documenter-dark .image.is-5by4 .has-ratio,html.theme--documenter-dark #documenter 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+ Maintainer: @highlightjs/core-team + Website: https://highlightjs.org/ + License: see project LICENSE + Touched: 2021 +*/pre code.hljs{display:block;overflow-x:auto;padding:1em}code.hljs{padding:3px 5px}.hljs{background:#F3F3F3;color:#444}.hljs-comment{color:#697070}.hljs-tag,.hljs-punctuation{color:#444a}.hljs-tag .hljs-name,.hljs-tag .hljs-attr{color:#444}.hljs-keyword,.hljs-attribute,.hljs-selector-tag,.hljs-meta .hljs-keyword,.hljs-doctag,.hljs-name{font-weight:bold}.hljs-type,.hljs-string,.hljs-number,.hljs-selector-id,.hljs-selector-class,.hljs-quote,.hljs-template-tag,.hljs-deletion{color:#880000}.hljs-title,.hljs-section{color:#880000;font-weight:bold}.hljs-regexp,.hljs-symbol,.hljs-variable,.hljs-template-variable,.hljs-link,.hljs-selector-attr,.hljs-operator,.hljs-selector-pseudo{color:#ab5656}.hljs-literal{color:#695}.hljs-built_in,.hljs-bullet,.hljs-code,.hljs-addition{color:#397300}.hljs-meta{color:#1f7199}.hljs-meta .hljs-string{color:#38a}.hljs-emphasis{font-style:italic}.hljs-strong{font-weight:bold}.gap-4{gap:1rem} diff --git a/v0.2.0/assets/themeswap.js b/v0.2.0/assets/themeswap.js new file mode 100644 index 0000000..9f5eebe --- /dev/null +++ b/v0.2.0/assets/themeswap.js @@ -0,0 +1,84 @@ +// Small function to quickly swap out themes. Gets put into the tag.. +function set_theme_from_local_storage() { + // Initialize the theme to null, which means default + var theme = null; + // If the browser supports the localstorage and is not disabled then try to get the + // documenter theme + if (window.localStorage != null) { + // Get the user-picked theme from localStorage. May be `null`, which means the default + // theme. + theme = window.localStorage.getItem("documenter-theme"); + } + // Check if the users preference is for dark color scheme + var darkPreference = + window.matchMedia("(prefers-color-scheme: dark)").matches === true; + // Initialize a few variables for the loop: + // + // - active: will contain the index of the theme that should be active. Note that there + // is no guarantee that localStorage contains sane values. If `active` stays `null` + // we either could not find the theme or it is the default (primary) theme anyway. + // Either way, we then need to stick to the primary theme. + // + // - disabled: style sheets that should be disabled (i.e. all the theme style sheets + // that are not the currently active theme) + var active = null; + var disabled = []; + var primaryLightTheme = null; + var primaryDarkTheme = null; + for (var i = 0; i < document.styleSheets.length; i++) { + var ss = document.styleSheets[i]; + // The tag of each style sheet is expected to have a data-theme-name attribute + // which must contain the name of the theme. The names in localStorage much match this. + var themename = ss.ownerNode.getAttribute("data-theme-name"); + // attribute not set => non-theme stylesheet => ignore + if (themename === null) continue; + // To distinguish the default (primary) theme, it needs to have the data-theme-primary + // attribute set. + if (ss.ownerNode.getAttribute("data-theme-primary") !== null) { + primaryLightTheme = themename; + } + // Check if the theme is primary dark theme so that we could store its name in darkTheme + if (ss.ownerNode.getAttribute("data-theme-primary-dark") !== null) { + primaryDarkTheme = themename; + } + // If we find a matching theme (and it's not the default), we'll set active to non-null + if (themename === theme) active = i; + // Store the style sheets of inactive themes so that we could disable them + if (themename !== theme) disabled.push(ss); + } + var activeTheme = null; + if (active !== null) { + // If we did find an active theme, we'll (1) add the theme--$(theme) class to + document.getElementsByTagName("html")[0].className = "theme--" + theme; + activeTheme = theme; + } else { + // If we did _not_ find an active theme, then we need to fall back to the primary theme + // which can either be dark or light, depending on the user's OS preference. + var activeTheme = darkPreference ? primaryDarkTheme : primaryLightTheme; + // In case it somehow happens that the relevant primary theme was not found in the + // preceding loop, we abort without doing anything. + if (activeTheme === null) { + console.error("Unable to determine primary theme."); + return; + } + // When switching to the primary light theme, then we must not have a class name + // for the tag. That's only for non-primary or the primary dark theme. + if (darkPreference) { + document.getElementsByTagName("html")[0].className = + "theme--" + activeTheme; + } else { + document.getElementsByTagName("html")[0].className = ""; + } + } + for (var i = 0; i < document.styleSheets.length; i++) { + var ss = document.styleSheets[i]; + // The tag of each style sheet is expected to have a data-theme-name attribute + // which must contain the name of the theme. The names in localStorage much match this. + var themename = ss.ownerNode.getAttribute("data-theme-name"); + // attribute not set => non-theme stylesheet => ignore + if (themename === null) continue; + // we'll disable all the stylesheets, except for the active one + ss.disabled = !(themename == activeTheme); + } +} +set_theme_from_local_storage(); diff --git a/v0.2.0/assets/warner.js b/v0.2.0/assets/warner.js new file mode 100644 index 0000000..3f6f5d0 --- /dev/null +++ b/v0.2.0/assets/warner.js @@ -0,0 +1,52 @@ +function maybeAddWarning() { + // DOCUMENTER_NEWEST is defined in versions.js, DOCUMENTER_CURRENT_VERSION and DOCUMENTER_STABLE + // in siteinfo.js. + // If either of these are undefined something went horribly wrong, so we abort. + if ( + window.DOCUMENTER_NEWEST === undefined || + window.DOCUMENTER_CURRENT_VERSION === undefined || + window.DOCUMENTER_STABLE === undefined + ) { + return; + } + + // Current version is not a version number, so we can't tell if it's the newest version. 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GitHub

Examples

Simulate data

julia> using MultiResponseVarianceComponentModels, LinearAlgebra, Random
julia> Random.seed!(6789)Random.TaskLocalRNG()
julia> n = 1_000;  # n of observations
julia> d = 4;      # n of responses
julia> p = 10;     # n of covariates
julia> m = 5;      # n of variance components
julia> X = rand(n, p);
julia> B = rand(p, d)10×4 Matrix{Float64}:
+ 0.866192  0.739958   0.951693   0.716391
+ 0.157873  0.633772   0.916356   0.068644
+ 0.818581  0.333021   0.0713489  0.451072
+ 0.707496  0.924568   0.0636189  0.618679
+ 0.357756  0.173016   0.982384   0.54638
+ 0.722163  0.0683381  0.343516   0.616161
+ 0.441486  0.64316    0.862809   0.400984
+ 0.812486  0.275565   0.477624   0.482087
+ 0.889787  0.669932   0.103051   0.582507
+ 0.331484  0.989292   0.785077   0.456195
julia> V = [zeros(n, n) for _ in 1:m]; # kernel matrices
julia> Σ = [zeros(d, d) for _ in 1:m]; # variance components
julia> for i in 1:m
+           Vi = randn(n, n)
+           copy!(V[i], Vi' * Vi)
+           Σi = randn(d, d)
+           copy!(Σ[i], Σi' * Σi)
+       end
julia> Ω = zeros(n * d, n * d); # overall nd-by-nd covariance matrix Ω
julia> for i = 1:m
+           Ω += kron(Σ[i], V[i])
+       end
julia> Ωchol = cholesky(Ω);
julia> Y = X * B + reshape(Ωchol.L * randn(n * d), n, d)1000×4 Matrix{Float64}:
+ -270.382    -66.0256   233.653     158.94
+  168.935    354.423     14.8585    -54.193
+   85.8007   117.902   -217.707      65.1394
+ -136.574    -42.6014  -234.087      75.6404
+ -190.312   -210.051    105.117     -10.0791
+  153.414    187.179     20.691     -62.3589
+  164.97     -22.4103   -13.6301    161.682
+   21.9438  -184.643    231.526    -136.723
+ -284.549     79.7798   -37.1967   -136.943
+  -59.2492  -124.37    -160.587     112.235
+    ⋮
+  -82.4606   200.557     -4.16006   -47.2201
+  170.0       95.8103   222.034    -179.011
+   98.1247   -33.0932   -67.4875    -86.9752
+ -167.871    110.395   -184.76      -71.029
+  111.644    -23.6517   113.328    -219.276
+  -77.2094   168.859     65.7808     22.939
+  129.34    -155.123   -251.557     120.656
+ -359.121    -82.2576   -82.0063     10.8301
+  -77.7126   -90.7158   -79.5722     92.9695
Note

In the case of heritability and genetic correlation analyses, one can use classic genetic relationship matrices (GRMs) for $\boldsymbol{V}_i$'s, which in turn can be constructed using SnpArrays.jl.

Maximum likelihood estimation

julia> model = MultiResponseVarianceComponentModel(Y, X, V)A multivariate response variance component model
+   * number of responses: 4
+   * number of observations: 1000
+   * number of fixed effects: 10
+   * number of variance components: 5
julia> @timev fit!(model, verbose = true)[ Info: Running MM algorithm for ML estimation
+iter = 0, logl = -26845.09612499673
+iter = 1, logl = -25695.041826875506
+iter = 2, logl = -25293.37779878975
+iter = 3, logl = -25159.24906847003
+iter = 4, logl = -25107.52993220223
+iter = 5, logl = -25081.38428962063
+iter = 6, logl = -25064.352419795112
+iter = 7, logl = -25051.625744592016
+iter = 8, logl = -25041.61006910993
+iter = 9, logl = -25033.604422863456
+iter = 10, logl = -25027.18042662134
+iter = 11, logl = -25022.017854787307
+iter = 12, logl = -25017.85980986688
+iter = 13, logl = -25014.498019681763
+iter = 14, logl = -25011.76503965001
+iter = 15, logl = -25009.527835987446
+iter = 16, logl = -25007.68180780364
+iter = 17, logl = -25006.145330198324
+iter = 18, logl = -25004.854995838356
+iter = 19, logl = -25003.761633006412
+iter = 20, logl = -25002.827079011357
+iter = 21, logl = -25002.02162797271
+iter = 22, logl = -25001.322047241047
+iter = 23, logl = -25000.710054799467
+iter = 24, logl = -25000.171160151207
+iter = 25, logl = -24999.693786243588
+iter = 26, logl = -24999.26860570048
+iter = 27, logl = -24998.888038955105
+iter = 28, logl = -24998.545873977288
+iter = 29, logl = -24998.236977041215
+iter = 30, logl = -24997.95707160835
+iter = 31, logl = -24997.702568235134
+iter = 32, logl = -24997.470432811737
+iter = 33, logl = -24997.258083718323
+iter = 34, logl = -24997.063310912818
+iter = 35, logl = -24996.884211761542
+iter = 36, logl = -24996.719139735058
+iter = 37, logl = -24996.566663066376
+iter = 38, logl = -24996.425531184308
+iter = 39, logl = -24996.294647261133
+iter = 40, logl = -24996.173045607276
+iter = 41, logl = -24996.05987293569
+iter = 42, logl = -24995.95437274284
+iter = 43, logl = -24995.855872212444
+iter = 44, logl = -24995.7637711789
+iter = 45, logl = -24995.67753277965
+iter = 46, logl = -24995.596675504443
+iter = 47, logl = -24995.520766401234
+iter = 48, logl = -24995.449415250514
+iter = 49, logl = -24995.38226954833
+iter = 50, logl = -24995.31901017304
+iter = 51, logl = -24995.259347628595
+iter = 52, logl = -24995.203018777862
+iter = 53, logl = -24995.14978399378
+iter = 54, logl = -24995.09942466663
+iter = 55, logl = -24995.051741018295
+iter = 56, logl = -24995.006550180144
+iter = 57, logl = -24994.963684497296
+iter = 58, logl = -24994.922990031344
+iter = 59, logl = -24994.88432523246
+iter = 60, logl = -24994.84755976209
+iter = 61, logl = -24994.812573442905
+iter = 62, logl = -24994.779255325782
+iter = 63, logl = -24994.74750285241
+iter = 64, logl = -24994.717221108924
+iter = 65, logl = -24994.688322154096
+iter = 66, logl = -24994.660724417583
+iter = 67, logl = -24994.63435215651
+iter = 68, logl = -24994.609134967097
+iter = 69, logl = -24994.585007342
+[ Info: Updates converged!
+[ Info: Calculating standard errors
+ 59.389879 seconds (14.34 M allocations: 908.116 MiB, 0.26% gc time, 6.94% compilation time)
+elapsed time (ns):  59389879083
+gc time (ns):       152631625
+bytes allocated:    952228260
+pool allocs:        14311455
+non-pool GC allocs: 24882
+malloc() calls:     906
+realloc() calls:    102
+free() calls:       523
+minor collections:  1
+full collections:   0
+Converged after 70 iterations.

For residual maximum likelihood estimation, you can instead type:

@timev fit!(model, reml = true, verbose = true)

Then variance components and mean effects estimates can be accessed through

julia> model.Σ5-element Vector{Matrix{Float64}}:
+ [2.8964235833864374 -0.9547701186106777 1.6856925389781934 0.8185656173741515; -0.9547701186106778 7.474693619948036 -4.744935846247633 -0.601253052759175; 1.6856925389781932 -4.744935846247633 8.639624062398605 0.4049276157435974; 0.8185656173741512 -0.601253052759175 0.4049276157435973 1.0145168589537352]
+ [3.776260289140289 2.3919496497788035 -0.3666324933910243 -1.365148782876448; 2.391949649778804 1.9914356452751707 -0.2630505419731652 -1.5892702567044454; -0.36663249339102433 -0.26305054197316524 4.0912210911875855 -2.287177667967102; -1.365148782876448 -1.5892702567044454 -2.287177667967102 4.526163931034507]
+ [3.1157472905367367 -0.08262940589305556 -0.27318718635822953 0.4179078660737166; -0.08262940589305548 3.4556143634179275 -0.20341922639890309 0.6648870938984194; -0.27318718635822953 -0.20341922639890314 1.9155083377403541 0.49426216391703903; 0.41790786607371666 0.6648870938984194 0.49426216391703903 0.5435108874298807]
+ [3.6027030953820782 1.2366936673894326 0.3637181414535267 0.06851794976775202; 1.2366936673894326 0.9715515927327851 1.6405295499121633 0.5971896668720472; 0.3637181414535266 1.6405295499121637 5.828012183031251 0.9752007240509573; 0.06851794976775206 0.5971896668720472 0.9752007240509573 0.9502953338899155]
+ [5.669808082536949 -2.01083534935516 1.1222606709505223 1.4405860817195557; -2.0108353493551605 6.032248061565413 9.439926228569771 -2.150730923690082; 1.1222606709505227 9.439926228569773 19.179102992380216 -3.054827357292586; 1.4405860817195555 -2.150730923690082 -3.054827357292586 3.445429926626015]
julia> model.B10×4 Matrix{Float64}:
+   7.76641     0.0581691   30.0176     10.8424
+  -1.70871    17.0653      26.3436      5.24079
+  -4.99575    28.325      -20.9966    -13.1088
+ -10.2939     -6.00867     12.3106    -16.3959
+  -3.07093     0.733837    15.8845      1.17996
+  14.1045     -3.92946     11.5408     11.2497
+  -1.47949    -8.30862      5.49722    -0.213233
+   6.03975     5.27422      0.189634    3.90586
+  -0.632607  -14.288      -29.194       4.9749
+   6.17054   -13.2489     -23.6247     -3.52297
julia> hcat(vec(B), vec(model.B))40×2 Matrix{Float64}:
+ 0.866192    7.76641
+ 0.157873   -1.70871
+ 0.818581   -4.99575
+ 0.707496  -10.2939
+ 0.357756   -3.07093
+ 0.722163   14.1045
+ 0.441486   -1.47949
+ 0.812486    6.03975
+ 0.889787   -0.632607
+ 0.331484    6.17054
+ ⋮
+ 0.068644    5.24079
+ 0.451072  -13.1088
+ 0.618679  -16.3959
+ 0.54638     1.17996
+ 0.616161   11.2497
+ 0.400984   -0.213233
+ 0.482087    3.90586
+ 0.582507    4.9749
+ 0.456195   -3.52297
julia> reduce(hcat, [hcat(vec(Σ[i]), vec(model.Σ[i])) for i in 1:m])16×10 Matrix{Float64}:
+  2.59353    2.89642    3.32588    …  3.6027      5.37835    5.66981
+ -0.908817  -0.95477    2.21659       1.23669    -1.43756   -2.01084
+  1.58682    1.68569   -0.0345923     0.363718    1.65263    1.12226
+  0.365451   0.818566  -1.36959       0.0685179   0.983537   1.44059
+ -0.908817  -0.95477    2.21659       1.23669    -1.43756   -2.01084
+  7.12022    7.47469    1.75764    …  0.971552    6.84955    6.03225
+ -4.72969   -4.74494   -0.0950186     1.64053     9.98583    9.43993
+ -0.430129  -0.601253  -1.55052       0.59719    -1.98954   -2.15073
+  1.58682    1.68569   -0.0345923     0.363718    1.65263    1.12226
+ -4.72969   -4.74494   -0.0950186     1.64053     9.98583    9.43993
+  6.22778    8.63962    4.06617    …  5.82801    19.2083    19.1791
+  0.969228   0.404928  -2.51336       0.975201   -3.5885    -3.05483
+  0.365451   0.818566  -1.36959       0.0685179   0.983537   1.44059
+ -0.430129  -0.601253  -1.55052       0.59719    -1.98954   -2.15073
+  0.969228   0.404928  -2.51336       0.975201   -3.5885    -3.05483
+  1.2446     1.01452    5.02933    …  0.950295    2.85342    3.44543

Standard errors

Sampling variance and covariance of these estimates are

julia> model.Σcov50×50 Matrix{Float64}:
+  0.47412      -0.00272341    0.108306     …  -0.00114518   -0.000725468
+ -0.00272341    0.280366     -0.0353382       -0.000510353   0.000627446
+  0.108306     -0.0353382     0.483991        -0.00663509    0.000814462
+  0.0597015    -0.0241462    -0.00476179      -0.00287831   -0.00410306
+ -0.000531375  -0.000740352  -0.00227336       0.00175773   -0.000550985
+ -5.46266e-6    0.0615992    -0.0084444    …   0.00754037   -0.000713865
+ -7.80744e-5    0.0347709    -0.00356761      -0.00609519    0.00360991
+  0.0240561    -0.0132044     0.213683         0.0166641    -0.000929684
+  0.0133385    -0.00919492    0.0562126       -0.0423021     0.00471451
+  0.00739022   -0.00597224   -0.0030313        0.00471978   -0.0236748
+  ⋮                                        ⋱
+ -0.00769785   -0.0406075    -0.012298         0.0234008    -0.0161668
+ -0.0133522    -0.0122384    -0.0861289        0.094258     -0.0224369
+ -0.00838137    0.00335528    0.00422692       0.0168207     0.0502388
+ -0.000645754  -0.00632483   -0.00175497      -0.0798787     0.0198685
+ -0.00102918   -0.00637199   -0.00825901   …  -0.187875      0.0275525
+ -0.000640514  -0.00315513   -0.000651312      0.140072     -0.0624593
+ -0.00183032   -0.00321512   -0.0232591       -0.36789       0.0382077
+ -0.00114518   -0.000510353  -0.00663509       0.426481     -0.0867558
+ -0.000725468   0.000627446   0.000814462     -0.0867558     0.189954
julia> model.Bcov40×40 Matrix{Float64}:
+ 157.174     -19.3776    -19.4223    …   -0.257223   -1.24371    -1.90444
+ -19.3776    161.611     -22.5065        -1.3117     -1.42903    -1.33411
+ -19.4223    -22.5065    150.341         -1.38577    -2.56388    -1.25275
+ -14.6304    -12.8425    -19.2089        -2.29579    -2.03511    -1.00321
+ -11.6747    -13.9337    -12.4649        -0.198984   -1.44227    -2.31342
+ -24.2206    -18.4552     -7.96731   …   -3.00702    -1.14626    -0.923721
+ -13.3431    -17.4951    -15.8672        -1.57057    -0.945035   -1.77244
+ -12.7248    -19.4206    -11.7579        13.2375     -1.25172    -1.08592
+ -17.2277    -19.1384    -19.5393        -1.22189    12.8571     -0.396249
+ -15.0949    -12.6969    -15.9363        -1.23333    -0.422978   12.5215
+   ⋮                                 ⋱
+  -2.12779    13.4419     -1.25447       -9.26816    -9.00194    -6.01701
+  -0.752161   -1.49914    13.5685        -5.62018    -9.24482    -8.16962
+  -0.737933   -0.925351   -1.71971      -11.7708     -8.59325   -10.0889
+  -1.49387    -1.2578     -1.04615       -5.98367   -12.911     -13.1438
+  -1.72094    -2.33376    -0.495492  …   -9.81655    -5.37389    -8.16594
+  -1.53932    -0.763366   -2.22368       -9.36946    -7.53599   -10.4206
+  -0.257223   -1.3117     -1.38577       79.0721     -7.3782     -9.48116
+  -1.24371    -1.42903    -2.56388       -7.3782     75.6156     -0.921828
+  -1.90444    -1.33411    -1.25275       -9.48116    -0.921828   78.0356

Corresponding standard error of these estimates are

julia> sqrt.(diag(model.Σcov))50-element Vector{Float64}:
+ 0.6885637727726415
+ 0.529495666135442
+ 0.6956949396229591
+ 0.33645040355906397
+ 0.8084217056989712
+ 0.7442584972630495
+ 0.3639694672208399
+ 1.3688056422005268
+ 0.4677057749056628
+ 0.3179680333750869
+ ⋮
+ 0.5524130983848243
+ 0.8823995615262973
+ 0.43111265513125097
+ 0.7402837878083752
+ 0.9231964895406697
+ 0.41384022887389565
+ 1.8662336914254827
+ 0.6530553700433331
+ 0.43583743574349515
julia> sqrt.(diag(model.Bcov))40-element Vector{Float64}:
+ 12.536904651555659
+ 12.712633608569577
+ 12.26134556777767
+ 12.169443903108293
+ 12.663455285629615
+ 11.870615133041053
+ 12.528437624570504
+ 12.51912728097892
+ 12.39394332241073
+ 12.510940364508672
+  ⋮
+  8.869558232699935
+  8.568572559479021
+  8.594848108237025
+  8.973590296072445
+  8.287660643177881
+  8.80370479520269
+  8.892247260593642
+  8.695724537824539
+  8.833773378129575
diff --git a/v0.2.0/index.html b/v0.2.0/index.html new file mode 100644 index 0000000..8097909 --- /dev/null +++ b/v0.2.0/index.html @@ -0,0 +1,3 @@ + +Home · MRVCModels.jl
GitHub

MultiResponseVarianceComponentModels

MultiResponseVarianceComponentModels.jl is a package for fitting and testing multivariate response variance components linear mixed models of form

\[\text{vec}\ \boldsymbol{Y} \sim \mathcal{N}(\text{vec}(\boldsymbol{X} \boldsymbol{B}), \sum_{i=1}^m \boldsymbol{\Gamma}_i \otimes \boldsymbol{V}_i),\]

where $\boldsymbol{Y}$ and $\boldsymbol{X}$ are $n \times d$ response and $n \times p$ predictor matrices, respectively, and $\boldsymbol{V}_1, \ldots, \boldsymbol{V}_m$ are $m$ known $n \times n$ positive semidefinite matrices. $\text{vec}\ \boldsymbol{Y}$ creates an $nd \times 1$ vector from $\boldsymbol{Y}$ by stacking its columns and $\otimes$ denotes the Kronecker product. The parameters of the model include $p \times d$ mean effects $\boldsymbol{B}$ and $d \times d$ variance components ($\boldsymbol{\Gamma}_1, \dots, \boldsymbol{\Gamma}_m$), which MultiResponseVarianceComponentModels.jl estimates through either minorization-maximization (MM) or expectation–maximization (EM) algorithms.

Note

MultiResponseVarianceComponentModels.jl is not suitable for biobank-scale data. We recommend using this package for datasets of size up to $n \cdot d \approx 50000$. This package also currently works for balanced data without any missing data.

Installation

To use MultiResponseVarianceComponentModels.jl, type:

using Pkg
+Pkg.add(url = "https://github.com/Hua-Zhou/MultiResponseVarianceComponentModels.jl.git")

This documentation was built using Documenter.jl.

Documentation built 2024-05-24T07:36:51.884 with Julia 1.10.3
diff --git a/v0.2.0/objects.inv b/v0.2.0/objects.inv new file mode 100644 index 0000000..e920b2c --- /dev/null +++ b/v0.2.0/objects.inv @@ -0,0 +1,9 @@ +# Sphinx inventory version 2 +# Project: MRVCModels.jl +# Version: 0.2.0 +# The remainder of this file is compressed using zlib. +xێ0^KPuUEj7v%8q䘖 +}Ospa! +U%$g~ 3 o0`BǮ=i5jQ&TGg<6px(O@K˾xD&Ϊ̭ĂX +*3L P2fLQ:}VehYf!p[YX K,i$˳3~g UQ,{xfi(I3)∅11'ɜ] +A셱n!՘ոa eܛJmsXX(MXF޻V:ʰh6,޼^$̼먵F]m޽O^ٸK%7ͮlkdӪL,8,|ijU Wj?łwP!UH fvl]H)3P[=$,~\!FA}._5jvݼ0C"UJ8of/m`qP/0n>.*j =pl;%wY8<ՑTro}dtdUƑYzn:SZbTVC:*:?WktV v_ [4V(6EV-,o146մg?C \ No newline at end of file diff --git a/v0.2.0/search/index.html b/v0.2.0/search/index.html new file mode 100644 index 0000000..4e4a97f --- /dev/null +++ b/v0.2.0/search/index.html @@ -0,0 +1,2 @@ + +Search · MRVCs.jl

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    diff --git a/v0.2.0/search_index.js b/v0.2.0/search_index.js new file mode 100644 index 0000000..a77f4c9 --- /dev/null +++ b/v0.2.0/search_index.js @@ -0,0 +1,3 @@ +var documenterSearchIndex = {"docs": +[{"location":"api/#API","page":"API","title":"API","text":"","category":"section"},{"location":"api/","page":"API","title":"API","text":"","category":"page"},{"location":"api/","page":"API","title":"API","text":"Modules = [MultiResponseVarianceComponentModels]","category":"page"},{"location":"api/#MultiResponseVarianceComponentModels.fisher_B!-Union{Tuple{MultiResponseVarianceComponentModel{T}}, Tuple{T}} where T<:Union{Float32, Float64}","page":"API","title":"MultiResponseVarianceComponentModels.fisher_B!","text":"fisher_B!(model::MultiResponseVarianceComponentModel)\n\nCompute the sampling variance-covariance of regression coefficients model.B, assuming inverse of covariance matrix model.Ω is available at model.storage_nd_nd.\n\n\n\n\n\n","category":"method"},{"location":"api/#MultiResponseVarianceComponentModels.fisher_Σ!-Union{Tuple{MultiResponseVarianceComponentModel{T}}, Tuple{T}} where T<:Union{Float32, Float64}","page":"API","title":"MultiResponseVarianceComponentModels.fisher_Σ!","text":"fisher_Σ!(model::MultiResponseVarianceComponentModel)\n\nCompute the sampling variance-covariance of variance component estimates model.Σ, assuming inverse of covariance matrix model.Ω is available at model.storage_nd_nd.\n\n\n\n\n\n","category":"method"},{"location":"api/#MultiResponseVarianceComponentModels.fit!-Union{Tuple{MultiResponseVarianceComponentModel{T}}, Tuple{T}} where T<:Union{Float32, Float64}","page":"API","title":"MultiResponseVarianceComponentModels.fit!","text":"fit!(model::MultiResponseVarianceComponentModel)\n\nFit a multivariate response variance component model by an MM or EM algorithm.\n\nPositional arguments\n\nmodel : a MultiResponseVarianceComponentModel instance. \n\nKeyword arguments\n\nmaxiter::Integer : maximum number of iterations. Default is 1000.\nreltol::Real : relative tolerance for convergence. Default is 1e-6.\nverbose::Bool : display algorithmic information. Default is false.\ninit::Symbol : initialization strategy. :default initialize by least squares. :user uses user supplied value at model.B and model.Σ.\nalgo::Symbol : optimization algorithm. :MM (default) or EM.\nlog::Bool : record iterate history or not. Defaut is false.\nreml::Bool : REML instead of ML estimation. Default is false.\nse::Bool : calculate standard errors. Default is true.\n\nOutput\n\nhistory : iterate history.\n\n\n\n\n\n","category":"method"},{"location":"api/#MultiResponseVarianceComponentModels.h2-Union{Tuple{MultiResponseVarianceComponentModel{T}}, Tuple{T}} where T<:Union{Float32, Float64}","page":"API","title":"MultiResponseVarianceComponentModels.h2","text":"h2(model::MultiResponseVarianceComponentModel)\n\nCalculate heritability estimates and their standard errors, assuming that all variance components capture genetic effects except the last term. Also returns total heritability and its standard error from sum of individual contributions.\n\n\n\n\n\n","category":"method"},{"location":"api/#MultiResponseVarianceComponentModels.kron_axpy!-Union{Tuple{T}, Tuple{AbstractVecOrMat{T}, AbstractVecOrMat{T}, AbstractVecOrMat{T}}} where T<:Real","page":"API","title":"MultiResponseVarianceComponentModels.kron_axpy!","text":"kron_axpy!(A, X, Y)\n\nOverwrite Y with A ⊗ X + Y. Same as Y += kron(A, X), but more memory efficient.\n\n\n\n\n\n","category":"method"},{"location":"api/#MultiResponseVarianceComponentModels.kron_reduction!-Union{Tuple{T}, Tuple{AbstractMatrix{T}, AbstractMatrix{T}, AbstractMatrix{T}}, Tuple{AbstractMatrix{T}, AbstractMatrix{T}, AbstractMatrix{T}, Bool}} where T<:Real","page":"API","title":"MultiResponseVarianceComponentModels.kron_reduction!","text":"kron_reduction!(A, B, C; sym = false)\n\nOverwrite C with the derivative of tr(A' (X ⊗ B)) wrt X. C[i, j] = dot(Aij, B), where Aij is the (i , j) block of A. sym = true indicates A and B are symmetric.\n\n\n\n\n\n","category":"method"},{"location":"api/#MultiResponseVarianceComponentModels.loglikelihood!-Union{Tuple{MultiResponseVarianceComponentModel{T}}, Tuple{T}} where T<:Union{Float32, Float64}","page":"API","title":"MultiResponseVarianceComponentModels.loglikelihood!","text":"loglikelihood!(model::MultiResponseVarianceComponentModel)\n\nOverwrite model.storage_nd_nd by inverse of the covariance matrix model.Ω, overwrite model.storage_nd by U' \\ vec(model.R), and return the log-likelihood. This function assumes model.Ω and model.R are already updated according to model.Σ and model.B.\n\n\n\n\n\n","category":"method"},{"location":"api/#MultiResponseVarianceComponentModels.lrt-Tuple{MultiResponseVarianceComponentModel, MultiResponseVarianceComponentModel}","page":"API","title":"MultiResponseVarianceComponentModels.lrt","text":"lrt(model1::MultiResponseVarianceComponentModel, model0::MultiResponseVarianceComponentModel)\n\nPerform a variation of the likelihood ratio test for univariate variance components models as in Molenberghs and Verbeke 2007 with model1 and model0 being the full and nested models, respectively.\n\n\n\n\n\n","category":"method"},{"location":"api/#MultiResponseVarianceComponentModels.rg-Union{Tuple{MultiResponseVarianceComponentModel{T}}, Tuple{T}} where T<:Union{Float32, Float64}","page":"API","title":"MultiResponseVarianceComponentModels.rg","text":"rg(model::MultiResponseVarianceComponentModel)\n\nCalculate genetic/residual correlation estimates and their standard errors.\n\n\n\n\n\n","category":"method"},{"location":"api/#MultiResponseVarianceComponentModels.update_B!-Union{Tuple{MultiResponseVarianceComponentModel{T}}, Tuple{T}} where T<:Union{Float32, Float64}","page":"API","title":"MultiResponseVarianceComponentModels.update_B!","text":"update_B!(model::MultiResponseVarianceComponentModel)\n\nUpdate the regression coefficients model.B, assuming inverse of covariance matrix model.Ω is available at model.storage_nd_nd.\n\n\n\n\n\n","category":"method"},{"location":"api/#MultiResponseVarianceComponentModels.update_Γk!-Union{Tuple{T}, Tuple{MultiResponseVarianceComponentModel{T}, Integer}} where T<:Union{Float32, Float64}","page":"API","title":"MultiResponseVarianceComponentModels.update_Γk!","text":"update_Γk!(model, k)\n\nUpdate the parameters model.Γ[k] and model.Ψ[:,k] assuming a diagonal plus low rank structure for model.Σ[k]. Assumes covariance matrix model.Ω is available at model.storage_nd_nd and model.Ω⁻¹R precomputed.\n\n\n\n\n\n","category":"method"},{"location":"api/#MultiResponseVarianceComponentModels.update_Σ!-Union{Tuple{MultiResponseVarianceComponentModel{T}}, Tuple{T}} where T<:Union{Float32, Float64}","page":"API","title":"MultiResponseVarianceComponentModels.update_Σ!","text":"update_Σ!(model::MultiResponseVarianceComponentModel)\n\nUpdate the variance component parameters model.Σ, assuming inverse of covariance matrix model.Ω is available at model.storage_nd_nd.\n\n\n\n\n\n","category":"method"},{"location":"api/#MultiResponseVarianceComponentModels.update_Σk!-Union{Tuple{T}, Tuple{MultiResponseVarianceComponentModel{T}, Integer, Integer}} where T<:Union{Float32, Float64}","page":"API","title":"MultiResponseVarianceComponentModels.update_Σk!","text":"update_Σk!(model, k, rk)\n\nUpdate the model.Σ[k] assuming it has rank rk < d, assuming inverse of covariance matrix model.Ω is available at model.storage_nd_nd and model.Ω⁻¹R precomputed.\n\n\n\n\n\n","category":"method"},{"location":"api/#MultiResponseVarianceComponentModels.update_Σk!-Union{Tuple{T}, Tuple{MultiResponseVarianceComponentModel{T}, Integer, Val{:EM}}} where T<:Union{Float32, Float64}","page":"API","title":"MultiResponseVarianceComponentModels.update_Σk!","text":"update_Σk!(model::MultiResponseVarianceComponentModel, k, Val(:EM))\n\nEM update the model.Σ[k] assuming it has full rank d, inverse of covariance matrix model.Ω is available at model.storage_nd_nd, and model.Ω⁻¹R precomputed.\n\n\n\n\n\n","category":"method"},{"location":"api/#MultiResponseVarianceComponentModels.update_Σk!-Union{Tuple{T}, Tuple{MultiResponseVarianceComponentModel{T}, Integer, Val{:MM}}} where T<:Union{Float32, Float64}","page":"API","title":"MultiResponseVarianceComponentModels.update_Σk!","text":"update_Σk!(model::MultiResponseVarianceComponentModel, k, Val(:MM))\n\nMM update the model.Σ[k] assuming it has full rank d, inverse of covariance matrix model.Ω is available at model.storage_nd_nd, and model.Ω⁻¹R precomputed.\n\n\n\n\n\n","category":"method"},{"location":"examples/#Examples","page":"Examples","title":"Examples","text":"","category":"section"},{"location":"examples/#Simulate-data","page":"Examples","title":"Simulate data","text":"","category":"section"},{"location":"examples/","page":"Examples","title":"Examples","text":"using MultiResponseVarianceComponentModels, LinearAlgebra, Random\nRandom.seed!(6789)\nn = 1_000; # n of observations\nd = 4; # n of responses\np = 10; # n of covariates\nm = 5; # n of variance components\nX = rand(n, p);\nB = rand(p, d)\nV = [zeros(n, n) for _ in 1:m]; # kernel matrices\nΣ = [zeros(d, d) for _ in 1:m]; # variance components\nfor i in 1:m\n Vi = randn(n, n)\n copy!(V[i], Vi' * Vi)\n Σi = randn(d, d)\n copy!(Σ[i], Σi' * Σi)\nend\nΩ = zeros(n * d, n * d); # overall nd-by-nd covariance matrix Ω\nfor i = 1:m\n Ω += kron(Σ[i], V[i])\nend\nΩchol = cholesky(Ω);\nY = X * B + reshape(Ωchol.L * randn(n * d), n, d)","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"note: Note\nIn the case of heritability and genetic correlation analyses, one can use classic genetic relationship matrices (GRMs) for boldsymbolV_i's, which in turn can be constructed using SnpArrays.jl.","category":"page"},{"location":"examples/#Maximum-likelihood-estimation","page":"Examples","title":"Maximum likelihood estimation","text":"","category":"section"},{"location":"examples/","page":"Examples","title":"Examples","text":"model = MultiResponseVarianceComponentModel(Y, X, V)\n@timev fit!(model, verbose = true)","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"For residual maximum likelihood estimation, you can instead type:","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"@timev fit!(model, reml = true, verbose = true)","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"Then variance components and mean effects estimates can be accessed through","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"model.Σ\nmodel.B\nhcat(vec(B), vec(model.B))\nreduce(hcat, [hcat(vec(Σ[i]), vec(model.Σ[i])) for i in 1:m])","category":"page"},{"location":"examples/#Standard-errors","page":"Examples","title":"Standard errors","text":"","category":"section"},{"location":"examples/","page":"Examples","title":"Examples","text":"Sampling variance and covariance of these estimates are","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"model.Σcov\nmodel.Bcov","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"Corresponding standard error of these estimates are","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"sqrt.(diag(model.Σcov))\nsqrt.(diag(model.Bcov))","category":"page"},{"location":"advanced/#Advanced-details","page":"Advanced details","title":"Advanced details","text":"","category":"section"},{"location":"advanced/#Estimation","page":"Advanced details","title":"Estimation","text":"","category":"section"},{"location":"advanced/","page":"Advanced details","title":"Advanced details","text":"For the MM algorithm, the updates in each iteration are","category":"page"},{"location":"advanced/","page":"Advanced details","title":"Advanced details","text":"beginaligned\ntextvec boldsymbolB^(t) = (boldsymbolI_d otimes boldsymbolX^T) boldsymbolOmega^-(t) (boldsymbolI_d otimes boldsymbolX)^-1 (boldsymbolI_d otimes boldsymbolX^T) boldsymbolOmega^-(t) textvec boldsymbolY \nboldsymbolGamma_i^(t + 1) = boldsymbolL_i^-(t)TboldsymbolL_i^(t)T(boldsymbolGamma_i^(t)boldsymbolR^(t)TboldsymbolV_iboldsymbolR^(t)boldsymbolGamma_i^(t))boldsymbolL_i^(t)^12 boldsymbolL_i^-(t)\nendaligned","category":"page"},{"location":"advanced/","page":"Advanced details","title":"Advanced details","text":"where boldsymbolOmega^(t) = sum_i=1^m boldsymbolGamma_i^(t) otimes boldsymbolV_i and boldsymbolL_i^(t) is the Cholesky factor of (boldsymbolI_d otimes boldsymbol1_n)^T (boldsymbol1_d boldsymbol1_d^T otimes boldsymbolV_i) odot boldsymbolOmega^-(t) (boldsymbolI_d otimes boldsymbol1_n), while boldsymbolR^(t) is the n times d matrix such that textvec boldsymbolR^(t) = boldsymbolOmega^-(t) textvec(boldsymbolY - boldsymbolX boldsymbolB^(t)).","category":"page"},{"location":"advanced/","page":"Advanced details","title":"Advanced details","text":"For the EM algorithm, the updates in each iteration are","category":"page"},{"location":"advanced/","page":"Advanced details","title":"Advanced details","text":"beginaligned\ntextvec boldsymbolB^(t) = (boldsymbolI_d otimes boldsymbolX^T) boldsymbolOmega^-(t) (boldsymbolI_d otimes boldsymbolX)^-1 (boldsymbolI_d otimes boldsymbolX^T) boldsymbolOmega^-(t) textvec boldsymbolY \nboldsymbolGamma_i^(t + 1) = frac1r_i boldsymbolGamma_i^(t) boldsymbolR^(t)T boldsymbolV_i boldsymbolR^(t) - (boldsymbolI_d otimes boldsymbol1_n)^T (boldsymbol1_d boldsymbol1_d^T otimes boldsymbolV_i) odot boldsymbolOmega^-(t) (boldsymbolI_d otimes boldsymbol1_n) boldsymbolGamma_i^(t) + boldsymbolGamma_i^(t)\nendaligned","category":"page"},{"location":"advanced/","page":"Advanced details","title":"Advanced details","text":"where r_i = textrank(boldsymbolV_i). As seen, the updates for mean effects boldsymbolB are the same for these two algorithms.","category":"page"},{"location":"advanced/#Inference","page":"Advanced details","title":"Inference","text":"","category":"section"},{"location":"advanced/","page":"Advanced details","title":"Advanced details","text":"Standard errors for our estimates were calculated using the Fisher information matrix, where","category":"page"},{"location":"advanced/","page":"Advanced details","title":"Advanced details","text":"beginaligned\ntextE left- fracpartial^2partial(textvec boldsymbolB)^T partial(textvec boldsymbolB) mathcalL right = (boldsymbolI_d otimes boldsymbolX^T) boldsymbolOmega^-1 (boldsymbolI_d otimes boldsymbolX) \ntextE left - fracpartial^2partial (textvech boldsymbolGamma_i)^T partial (textvec boldsymbolB) mathcalL right = boldsymbol0 \ntextE left - fracpartial^2partial (textvech boldsymbolGamma_j)^T partial (textvech boldsymbolGamma_i) mathcalL right = frac12 boldsymbolU_i^T (boldsymbolOmega^-1 otimes boldsymbolOmega^-1) boldsymbolU_j\nendaligned","category":"page"},{"location":"advanced/","page":"Advanced details","title":"Advanced details","text":"and boldsymbolU_i = (boldsymbolI_d otimes boldsymbolK_nd otimes boldsymbolI_n) (boldsymbolI_d^2 otimes textvec boldsymbolV_i) boldsymbolD_d. Here, boldsymbolK_nd is the nd times nd commutation matrix and boldsymbolD_d the d^2 times fracd(d+1)2 duplication matrix. textvech boldsymbolGamma_i creates an fracd(d+1)2 times 1 vector from boldsymbolGamma_i by stacking its lower triangular part.","category":"page"},{"location":"","page":"Home","title":"Home","text":"CurrentModule = MultiResponseVarianceComponentModels","category":"page"},{"location":"#MultiResponseVarianceComponentModels","page":"Home","title":"MultiResponseVarianceComponentModels","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"MultiResponseVarianceComponentModels.jl is a package for fitting and testing multivariate response variance components linear mixed models of form","category":"page"},{"location":"","page":"Home","title":"Home","text":"textvec boldsymbolY sim mathcalN(textvec(boldsymbolX boldsymbolB) sum_i=1^m boldsymbolGamma_i otimes boldsymbolV_i)","category":"page"},{"location":"","page":"Home","title":"Home","text":"where boldsymbolY and boldsymbolX are n times d response and n times p predictor matrices, respectively, and boldsymbolV_1 ldots boldsymbolV_m are m known n times n positive semidefinite matrices. textvec boldsymbolY creates an nd times 1 vector from boldsymbolY by stacking its columns and otimes denotes the Kronecker product. The parameters of the model include p times d mean effects boldsymbolB and d times d variance components (boldsymbolGamma_1 dots boldsymbolGamma_m), which MultiResponseVarianceComponentModels.jl estimates through either minorization-maximization (MM) or expectation–maximization (EM) algorithms.","category":"page"},{"location":"","page":"Home","title":"Home","text":"note: Note\nMultiResponseVarianceComponentModels.jl is not suitable for biobank-scale data. We recommend using this package for datasets of size up to n cdot d approx 50000. This package also currently works for balanced data without any missing data.","category":"page"},{"location":"#Installation","page":"Home","title":"Installation","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"To use MultiResponseVarianceComponentModels.jl, type:","category":"page"},{"location":"","page":"Home","title":"Home","text":"using Pkg\nPkg.add(url = \"https://github.com/Hua-Zhou/MultiResponseVarianceComponentModels.jl.git\")","category":"page"},{"location":"","page":"Home","title":"Home","text":"This documentation was built using Documenter.jl.","category":"page"},{"location":"","page":"Home","title":"Home","text":"using Dates # hide\nprintln(\"Documentation built $(Dates.now()) with Julia $(VERSION)\") # hide","category":"page"}] +} diff --git a/v0.2.0/siteinfo.js b/v0.2.0/siteinfo.js new file mode 100644 index 0000000..63ac675 --- /dev/null +++ b/v0.2.0/siteinfo.js @@ -0,0 +1 @@ +var DOCUMENTER_CURRENT_VERSION = "v0.2"; diff --git a/v0.2.1/.documenter-siteinfo.json b/v0.2.1/.documenter-siteinfo.json new file mode 100644 index 0000000..91e6469 --- /dev/null +++ b/v0.2.1/.documenter-siteinfo.json @@ -0,0 +1 @@ +{"documenter":{"julia_version":"1.10.3","generation_timestamp":"2024-05-24T07:37:01","documenter_version":"1.4.1"}} \ No newline at end of file diff --git a/v0.2.1/advanced/index.html b/v0.2.1/advanced/index.html new file mode 100644 index 0000000..5014f9e --- /dev/null +++ b/v0.2.1/advanced/index.html @@ -0,0 +1,12 @@ + +Advanced details · MRVCModels.jl
    GitHub

    Advanced details

    Estimation

    For the MM algorithm, the updates in each iteration are

    \[\begin{aligned} +\text{vec}\ \boldsymbol{B}^{(t)} &= [(\boldsymbol{I}_d \otimes \boldsymbol{X}^T) \boldsymbol{\Omega}^{-(t)} (\boldsymbol{I}_d \otimes \boldsymbol{X})]^{-1} (\boldsymbol{I}_d \otimes \boldsymbol{X}^T) \boldsymbol{\Omega}^{-(t)} \text{vec}\ \boldsymbol{Y} \\ +\boldsymbol{\Gamma}_i^{(t + 1)} &= \boldsymbol{L}_i^{-(t)T}[\boldsymbol{L}_i^{(t)T}(\boldsymbol{\Gamma}_i^{(t)}\boldsymbol{R}^{(t)T}\boldsymbol{V}_i\boldsymbol{R}^{(t)}\boldsymbol{\Gamma}_i^{(t)})\boldsymbol{L}_i^{(t)}]^{1/2} \boldsymbol{L}_i^{-(t)}, +\end{aligned}\]

    where $\boldsymbol{\Omega}^{(t)} = \sum_{i=1}^m \boldsymbol{\Gamma}_i^{(t)} \otimes \boldsymbol{V}_i$ and $\boldsymbol{L}_i^{(t)}$ is the Cholesky factor of $(\boldsymbol{I}_d \otimes \boldsymbol{1}_n)^T [(\boldsymbol{1}_d \boldsymbol{1}_d^T \otimes \boldsymbol{V}_i) \odot \boldsymbol{\Omega}^{-(t)}] (\boldsymbol{I}_d \otimes \boldsymbol{1}_n)$, while $\boldsymbol{R}^{(t)}$ is the $n \times d$ matrix such that $\text{vec}\ \boldsymbol{R}^{(t)} = \boldsymbol{\Omega}^{-(t)} \text{vec}(\boldsymbol{Y} - \boldsymbol{X} \boldsymbol{B}^{(t)})$.

    For the EM algorithm, the updates in each iteration are

    \[\begin{aligned} +\text{vec}\ \boldsymbol{B}^{(t)} &= [(\boldsymbol{I}_d \otimes \boldsymbol{X}^T) \boldsymbol{\Omega}^{-(t)} (\boldsymbol{I}_d \otimes \boldsymbol{X})]^{-1} (\boldsymbol{I}_d \otimes \boldsymbol{X}^T) \boldsymbol{\Omega}^{-(t)} \text{vec}\ \boldsymbol{Y} \\ +\boldsymbol{\Gamma}_i^{(t + 1)} &= \frac{1}{r_i} \boldsymbol{\Gamma}_i^{(t)} \{ \boldsymbol{R}^{(t)T} \boldsymbol{V}_i \boldsymbol{R}^{(t)} - (\boldsymbol{I}_d \otimes \boldsymbol{1}_n)^T [(\boldsymbol{1}_d \boldsymbol{1}_d^T \otimes \boldsymbol{V}_i) \odot \boldsymbol{\Omega}^{-(t)}] (\boldsymbol{I}_d \otimes \boldsymbol{1}_n)\} \boldsymbol{\Gamma}_i^{(t)} + \boldsymbol{\Gamma}_i^{(t)}, +\end{aligned}\]

    where $r_i = \text{rank}(\boldsymbol{V}_i)$. As seen, the updates for mean effects $\boldsymbol{B}$ are the same for these two algorithms.

    Inference

    Standard errors for our estimates were calculated using the Fisher information matrix, where

    \[\begin{aligned} +\text{E} \left[- \frac{\partial^2}{\partial(\text{vec}\ \boldsymbol{B})^T \partial(\text{vec}\ \boldsymbol{B})} \mathcal{L} \right] &= (\boldsymbol{I}_d \otimes \boldsymbol{X}^T) \boldsymbol{\Omega}^{-1} (\boldsymbol{I}_d \otimes \boldsymbol{X}) \\ +\text{E} \left[ - \frac{\partial^2}{\partial (\text{vech}\ \boldsymbol{\Gamma}_i)^T \partial (\text{vec}\ \boldsymbol{B})} \mathcal{L} \right] &= \boldsymbol{0} \\ +\text{E} \left[ - \frac{\partial^2}{\partial (\text{vech}\ \boldsymbol{\Gamma}_j)^T \partial (\text{vech}\ \boldsymbol{\Gamma}_i)} \mathcal{L} \right] &= \frac{1}{2} \boldsymbol{U}_i^T (\boldsymbol{\Omega}^{-1} \otimes \boldsymbol{\Omega}^{-1}) \boldsymbol{U}_j +\end{aligned}\]

    and $\boldsymbol{U}_i = (\boldsymbol{I}_d \otimes \boldsymbol{K}_{nd} \otimes \boldsymbol{I}_n) (\boldsymbol{I}_{d^2} \otimes \text{vec}\ \boldsymbol{V}_i) \boldsymbol{D}_{d}$. Here, $\boldsymbol{K}_{nd}$ is the $nd \times nd$ commutation matrix and $\boldsymbol{D}_{d}$ the $d^2 \times \frac{d(d+1)}{2}$ duplication matrix. $\text{vech}\ \boldsymbol{\Gamma}_i$ creates an $\frac{d(d+1)}{2} \times 1$ vector from $\boldsymbol{\Gamma}_i$ by stacking its lower triangular part.

    diff --git a/v0.2.1/api/index.html b/v0.2.1/api/index.html new file mode 100644 index 0000000..476c6b2 --- /dev/null +++ b/v0.2.1/api/index.html @@ -0,0 +1,2 @@ + +API · MRVCModels.jl
    GitHub

    API

    MultiResponseVarianceComponentModels.fisher_B! — Method
    fisher_B!(model::MultiResponseVarianceComponentModel)

    Compute the sampling variance-covariance of regression coefficients model.B, assuming inverse of covariance matrix model.Ω is available at model.storage_nd_nd.

    source
    MultiResponseVarianceComponentModels.fisher_Σ! — Method
    fisher_Σ!(model::MultiResponseVarianceComponentModel)

    Compute the sampling variance-covariance of variance component estimates model.Σ, assuming inverse of covariance matrix model.Ω is available at model.storage_nd_nd.

    source
    MultiResponseVarianceComponentModels.fit! — Method
    fit!(model::MultiResponseVarianceComponentModel)

    Fit a multivariate response variance component model by an MM or EM algorithm.

    Positional arguments

    • model : a MultiResponseVarianceComponentModel instance.

    Keyword arguments

    • maxiter::Integer : maximum number of iterations. Default is 1000.
    • reltol::Real : relative tolerance for convergence. Default is 1e-6.
    • verbose::Bool : display algorithmic information. Default is false.
    • init::Symbol : initialization strategy. :default initialize by least squares. :user uses user supplied value at model.B and model.Σ.
    • algo::Symbol : optimization algorithm. :MM (default) or EM.
    • log::Bool : record iterate history or not. Defaut is false.
    • reml::Bool : REML instead of ML estimation. Default is false.
    • se::Bool : calculate standard errors. Default is true.

    Output

    • history : iterate history.
    source
    MultiResponseVarianceComponentModels.h2 — Method
    h2(model::MultiResponseVarianceComponentModel)

    Calculate heritability estimates and their standard errors, assuming that all variance components capture genetic effects except the last term. Also returns total heritability and its standard error from sum of individual contributions.

    source
    MultiResponseVarianceComponentModels.loglikelihood! — Method
    loglikelihood!(model::MultiResponseVarianceComponentModel)

    Overwrite model.storage_nd_nd by inverse of the covariance matrix model.Ω, overwrite model.storage_nd by U' \ vec(model.R), and return the log-likelihood. This function assumes model.Ω and model.R are already updated according to model.Σ and model.B.

    source
    MultiResponseVarianceComponentModels.lrt — Method
    lrt(model1::MultiResponseVarianceComponentModel, model0::MultiResponseVarianceComponentModel)

    Perform a variation of the likelihood ratio test for univariate variance components models as in Molenberghs and Verbeke 2007 with model1 and model0 being the full and nested models, respectively.

    source
    MultiResponseVarianceComponentModels.update_Γk! — Method
    update_Γk!(model, k)

    Update the parameters model.Γ[k] and model.Ψ[:,k] assuming a diagonal plus low rank structure for model.Σ[k]. Assumes covariance matrix model.Ω is available at model.storage_nd_nd and model.Ω⁻¹R precomputed.

    source
    MultiResponseVarianceComponentModels.update_Σk! — Method
    update_Σk!(model::MultiResponseVarianceComponentModel, k, Val(:EM))

    EM update the model.Σ[k] assuming it has full rank d, inverse of covariance matrix model.Ω is available at model.storage_nd_nd, and model.Ω⁻¹R precomputed.

    source
    MultiResponseVarianceComponentModels.update_Σk! — Method
    update_Σk!(model::MultiResponseVarianceComponentModel, k, Val(:MM))

    MM update the model.Σ[k] assuming it has full rank d, inverse of covariance matrix model.Ω is available at model.storage_nd_nd, and model.Ω⁻¹R precomputed.

    source
    diff --git a/v0.2.1/assets/MRVC.png b/v0.2.1/assets/MRVC.png new file mode 100644 index 0000000..85e4add Binary files /dev/null and b/v0.2.1/assets/MRVC.png differ diff --git a/v0.2.1/assets/documenter.js b/v0.2.1/assets/documenter.js new file mode 100644 index 0000000..c6562b5 --- /dev/null +++ b/v0.2.1/assets/documenter.js @@ -0,0 +1,1050 @@ +// Generated by Documenter.jl +requirejs.config({ + paths: { + 'highlight-julia': 'https://cdnjs.cloudflare.com/ajax/libs/highlight.js/11.8.0/languages/julia.min', + 'headroom': 'https://cdnjs.cloudflare.com/ajax/libs/headroom/0.12.0/headroom.min', + 'jqueryui': 'https://cdnjs.cloudflare.com/ajax/libs/jqueryui/1.13.2/jquery-ui.min', + 'katex-auto-render': 'https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.8/contrib/auto-render.min', + 'jquery': 'https://cdnjs.cloudflare.com/ajax/libs/jquery/3.7.0/jquery.min', + 'headroom-jquery': 'https://cdnjs.cloudflare.com/ajax/libs/headroom/0.12.0/jQuery.headroom.min', + 'katex': 'https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.8/katex.min', + 'highlight': 'https://cdnjs.cloudflare.com/ajax/libs/highlight.js/11.8.0/highlight.min', + 'highlight-julia-repl': 'https://cdnjs.cloudflare.com/ajax/libs/highlight.js/11.8.0/languages/julia-repl.min', + }, + shim: { + "highlight-julia": { + "deps": [ + "highlight" + ] + }, + "katex-auto-render": { + "deps": [ + "katex" + ] + }, + "headroom-jquery": { + "deps": [ + "jquery", + "headroom" + ] + }, + "highlight-julia-repl": { + "deps": [ + "highlight" + ] + } +} +}); +//////////////////////////////////////////////////////////////////////////////// +require(['jquery', 'katex', 'katex-auto-render'], function($, katex, renderMathInElement) { +$(document).ready(function() { + renderMathInElement( + document.body, + { + "delimiters": [ + { + "left": "$", + "right": "$", + "display": false + }, + { + "left": "$$", + "right": "$$", + "display": true + }, + { + "left": "\\[", + "right": "\\]", + "display": true + } + ] +} + + ); +}) + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery', 'highlight', 'highlight-julia', 'highlight-julia-repl'], function($) { +$(document).ready(function() { + hljs.highlightAll(); +}) + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +let timer = 0; +var isExpanded = true; + +$(document).on("click", ".docstring header", function () { + let articleToggleTitle = "Expand docstring"; + + debounce(() => { + if ($(this).siblings("section").is(":visible")) { + $(this) + .find(".docstring-article-toggle-button") + .removeClass("fa-chevron-down") + .addClass("fa-chevron-right"); + } else { + $(this) + .find(".docstring-article-toggle-button") + .removeClass("fa-chevron-right") + .addClass("fa-chevron-down"); + + articleToggleTitle = "Collapse docstring"; + } + + $(this) + .find(".docstring-article-toggle-button") + .prop("title", articleToggleTitle); + $(this).siblings("section").slideToggle(); + }); +}); + +$(document).on("click", ".docs-article-toggle-button", function (event) { + let articleToggleTitle = "Expand docstring"; + let navArticleToggleTitle = "Expand all docstrings"; + let animationSpeed = event.noToggleAnimation ? 0 : 400; + + debounce(() => { + if (isExpanded) { + $(this).removeClass("fa-chevron-up").addClass("fa-chevron-down"); + $(".docstring-article-toggle-button") + .removeClass("fa-chevron-down") + .addClass("fa-chevron-right"); + + isExpanded = false; + + $(".docstring section").slideUp(animationSpeed); + } else { + $(this).removeClass("fa-chevron-down").addClass("fa-chevron-up"); + $(".docstring-article-toggle-button") + .removeClass("fa-chevron-right") + .addClass("fa-chevron-down"); + + isExpanded = true; + articleToggleTitle = "Collapse docstring"; + navArticleToggleTitle = "Collapse all docstrings"; + + $(".docstring section").slideDown(animationSpeed); + } + + $(this).prop("title", navArticleToggleTitle); + $(".docstring-article-toggle-button").prop("title", articleToggleTitle); + }); +}); + +function debounce(callback, timeout = 300) { + if (Date.now() - timer > timeout) { + callback(); + } + + clearTimeout(timer); + + timer = Date.now(); +} + +}) +//////////////////////////////////////////////////////////////////////////////// +require([], function() { +function addCopyButtonCallbacks() { + for (const el of document.getElementsByTagName("pre")) { + const button = document.createElement("button"); + button.classList.add("copy-button", "fa-solid", "fa-copy"); + button.setAttribute("aria-label", "Copy this code block"); + button.setAttribute("title", "Copy"); + + el.appendChild(button); + + const success = function () { + button.classList.add("success", "fa-check"); + button.classList.remove("fa-copy"); + }; + + const failure = function () { + button.classList.add("error", "fa-xmark"); + button.classList.remove("fa-copy"); + }; + + button.addEventListener("click", function () { + copyToClipboard(el.innerText).then(success, failure); + + setTimeout(function () { + button.classList.add("fa-copy"); + button.classList.remove("success", "fa-check", "fa-xmark"); + }, 5000); + }); + } +} + +function copyToClipboard(text) { + // clipboard API is only available in secure contexts + if (window.navigator && window.navigator.clipboard) { + return window.navigator.clipboard.writeText(text); + } else { + return new Promise(function (resolve, reject) { + try { + const el = document.createElement("textarea"); + el.textContent = text; + el.style.position = "fixed"; + el.style.opacity = 0; + document.body.appendChild(el); + el.select(); + document.execCommand("copy"); + + resolve(); + } catch (err) { + reject(err); + } finally { + document.body.removeChild(el); + } + }); + } +} + +if (document.readyState === "loading") { + document.addEventListener("DOMContentLoaded", addCopyButtonCallbacks); +} else { + addCopyButtonCallbacks(); +} + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery', 'headroom', 'headroom-jquery'], function($, Headroom) { + +// Manages the top navigation bar (hides it when the user starts scrolling down on the +// mobile). +window.Headroom = Headroom; // work around buggy module loading? +$(document).ready(function () { + $("#documenter .docs-navbar").headroom({ + tolerance: { up: 10, down: 10 }, + }); +}); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +$(document).ready(function () { + let meta = $("div[data-docstringscollapsed]").data(); + + if (meta?.docstringscollapsed) { + $("#documenter-article-toggle-button").trigger({ + type: "click", + noToggleAnimation: true, + }); + } +}); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +/* +To get an in-depth about the thought process you can refer: https://hetarth02.hashnode.dev/series/gsoc + +PSEUDOCODE: + +Searching happens automatically as the user types or adjusts the selected filters. +To preserve responsiveness, as much as possible of the slow parts of the search are done +in a web worker. Searching and result generation are done in the worker, and filtering and +DOM updates are done in the main thread. The filters are in the main thread as they should +be very quick to apply. This lets filters be changed without re-searching with minisearch +(which is possible even if filtering is on the worker thread) and also lets filters be +changed _while_ the worker is searching and without message passing (neither of which are +possible if filtering is on the worker thread) + +SEARCH WORKER: + +Import minisearch + +Build index + +On message from main thread + run search + find the first 200 unique results from each category, and compute their divs for display + note that this is necessary and sufficient information for the main thread to find the + first 200 unique results from any given filter set + post results to main thread + +MAIN: + +Launch worker + +Declare nonconstant globals (worker_is_running, last_search_text, unfiltered_results) + +On text update + if worker is not running, launch_search() + +launch_search + set worker_is_running to true, set last_search_text to the search text + post the search query to worker + +on message from worker + if last_search_text is not the same as the text in the search field, + the latest search result is not reflective of the latest search query, so update again + launch_search() + otherwise + set worker_is_running to false + + regardless, display the new search results to the user + save the unfiltered_results as a global + update_search() + +on filter click + adjust the filter selection + update_search() + +update_search + apply search filters by looping through the unfiltered_results and finding the first 200 + unique results that match the filters + + Update the DOM +*/ + +/////// SEARCH WORKER /////// + +function worker_function(documenterSearchIndex, documenterBaseURL, filters) { + importScripts( + "https://cdn.jsdelivr.net/npm/minisearch@6.1.0/dist/umd/index.min.js" + ); + + let data = documenterSearchIndex.map((x, key) => { + x["id"] = key; // minisearch requires a unique for each object + return x; + }); + + // list below is the lunr 2.1.3 list minus the intersect with names(Base) + // (all, any, get, in, is, only, which) and (do, else, for, let, where, while, with) + // ideally we'd just filter the original list but it's not available as a variable + const stopWords = new Set([ + "a", + "able", + "about", + "across", + "after", + "almost", + "also", + "am", + "among", + "an", + "and", + "are", + "as", + "at", + "be", + "because", + "been", + "but", + "by", + "can", + "cannot", + "could", + "dear", + "did", + "does", + "either", + "ever", + "every", + "from", + "got", + "had", + "has", + "have", + "he", + "her", + "hers", + "him", + "his", + "how", + "however", + "i", + "if", + "into", + "it", + "its", + "just", + "least", + "like", + "likely", + "may", + "me", + "might", + "most", + "must", + "my", + "neither", + "no", + "nor", + "not", + "of", + "off", + "often", + "on", + "or", + "other", + "our", + "own", + "rather", + "said", + "say", + "says", + "she", + "should", + "since", + "so", + "some", + "than", + "that", + "the", + "their", + "them", + "then", + "there", + "these", + "they", + "this", + "tis", + "to", + "too", + "twas", + "us", + "wants", + "was", + "we", + "were", + "what", + "when", + "who", + "whom", + "why", + "will", + "would", + "yet", + "you", + "your", + ]); + + let index = new MiniSearch({ + fields: ["title", "text"], // fields to index for full-text search + storeFields: ["location", "title", "text", "category", "page"], // fields to return with results + processTerm: (term) => { + let word = stopWords.has(term) ? null : term; + if (word) { + // custom trimmer that doesn't strip @ and !, which are used in julia macro and function names + word = word + .replace(/^[^a-zA-Z0-9@!]+/, "") + .replace(/[^a-zA-Z0-9@!]+$/, ""); + + word = word.toLowerCase(); + } + + return word ?? null; + }, + // add . as a separator, because otherwise "title": "Documenter.Anchors.add!", would not + // find anything if searching for "add!", only for the entire qualification + tokenize: (string) => string.split(/[\s\-\.]+/), + // options which will be applied during the search + searchOptions: { + prefix: true, + boost: { title: 100 }, + fuzzy: 2, + }, + }); + + index.addAll(data); + + /** + * Used to map characters to HTML entities. + * Refer: https://github.com/lodash/lodash/blob/main/src/escape.ts + */ + const htmlEscapes = { + "&": "&", + "<": "<", + ">": ">", + '"': """, + "'": "'", + }; + + /** + * Used to match HTML entities and HTML characters. + * Refer: https://github.com/lodash/lodash/blob/main/src/escape.ts + */ + const reUnescapedHtml = /[&<>"']/g; + const reHasUnescapedHtml = RegExp(reUnescapedHtml.source); + + /** + * Escape function from lodash + * Refer: https://github.com/lodash/lodash/blob/main/src/escape.ts + */ + function escape(string) { + return string && reHasUnescapedHtml.test(string) + ? string.replace(reUnescapedHtml, (chr) => htmlEscapes[chr]) + : string || ""; + } + + /** + * Make the result component given a minisearch result data object and the value + * of the search input as queryString. To view the result object structure, refer: + * https://lucaong.github.io/minisearch/modules/_minisearch_.html#searchresult + * + * @param {object} result + * @param {string} querystring + * @returns string + */ + function make_search_result(result, querystring) { + let search_divider = `
    `; + let display_link = + result.location.slice(Math.max(0), Math.min(50, result.location.length)) + + (result.location.length > 30 ? "..." : ""); // To cut-off the link because it messes with the overflow of the whole div + + if (result.page !== "") { + display_link += ` (${result.page})`; + } + + let textindex = new RegExp(`${querystring}`, "i").exec(result.text); + let text = + textindex !== null + ? result.text.slice( + Math.max(textindex.index - 100, 0), + Math.min( + textindex.index + querystring.length + 100, + result.text.length + ) + ) + : ""; // cut-off text before and after from the match + + text = text.length ? escape(text) : ""; + + let display_result = text.length + ? "..." + + text.replace( + new RegExp(`${escape(querystring)}`, "i"), // For first occurrence + '$&' + ) + + "..." + : ""; // highlights the match + + let in_code = false; + if (!["page", "section"].includes(result.category.toLowerCase())) { + in_code = true; + } + + // We encode the full url to escape some special characters which can lead to broken links + let result_div = ` + +
    +
    ${escape(result.title)}
    +
    ${result.category}
    +
    +

    + ${display_result} +

    +
    + ${display_link} +
    +     + ${search_divider} + `; + + return result_div; + } + + self.onmessage = function (e) { + let query = e.data; + let results = index.search(query, { + filter: (result) => { + // Only return relevant results + return result.score >= 1; + }, + }); + + // Pre-filter to deduplicate and limit to 200 per category to the extent + // possible without knowing what the filters are. + let filtered_results = []; + let counts = {}; + for (let filter of filters) { + counts[filter] = 0; + } + let present = {}; + + for (let result of results) { + cat = result.category; + cnt = counts[cat]; + if (cnt < 200) { + id = cat + "---" + result.location; + if (present[id]) { + continue; + } + present[id] = true; + filtered_results.push({ + location: result.location, + category: cat, + div: make_search_result(result, query), + }); + } + } + + postMessage(filtered_results); + }; +} + +// `worker = Threads.@spawn worker_function(documenterSearchIndex)`, but in JavaScript! +const filters = [ + ...new Set(documenterSearchIndex["docs"].map((x) => x.category)), +]; +const worker_str = + "(" + + worker_function.toString() + + ")(" + + JSON.stringify(documenterSearchIndex["docs"]) + + "," + + JSON.stringify(documenterBaseURL) + + "," + + JSON.stringify(filters) + + ")"; +const worker_blob = new Blob([worker_str], { type: "text/javascript" }); +const worker = new Worker(URL.createObjectURL(worker_blob)); + +/////// SEARCH MAIN /////// + +// Whether the worker is currently handling a search. This is a boolean +// as the worker only ever handles 1 or 0 searches at a time. +var worker_is_running = false; + +// The last search text that was sent to the worker. This is used to determine +// if the worker should be launched again when it reports back results. +var last_search_text = ""; + +// The results of the last search. This, in combination with the state of the filters +// in the DOM, is used compute the results to display on calls to update_search. +var unfiltered_results = []; + +// Which filter is currently selected +var selected_filter = ""; + +$(document).on("input", ".documenter-search-input", function (event) { + if (!worker_is_running) { + launch_search(); + } +}); + +function launch_search() { + worker_is_running = true; + last_search_text = $(".documenter-search-input").val(); + worker.postMessage(last_search_text); +} + +worker.onmessage = function (e) { + if (last_search_text !== $(".documenter-search-input").val()) { + launch_search(); + } else { + worker_is_running = false; + } + + unfiltered_results = e.data; + update_search(); +}; + +$(document).on("click", ".search-filter", function () { + if ($(this).hasClass("search-filter-selected")) { + selected_filter = ""; + } else { + selected_filter = $(this).text().toLowerCase(); + } + + // This updates search results and toggles classes for UI: + update_search(); +}); + +/** + * Make/Update the search component + */ +function update_search() { + let querystring = $(".documenter-search-input").val(); + + if (querystring.trim()) { + if (selected_filter == "") { + results = unfiltered_results; + } else { + results = unfiltered_results.filter((result) => { + return selected_filter == result.category.toLowerCase(); + }); + } + + let search_result_container = ``; + let modal_filters = make_modal_body_filters(); + let search_divider = `
    `; + + if (results.length) { + let links = []; + let count = 0; + let search_results = ""; + + for (var i = 0, n = results.length; i < n && count < 200; ++i) { + let result = results[i]; + if (result.location && !links.includes(result.location)) { + search_results += result.div; + count++; + links.push(result.location); + } + } + + if (count == 1) { + count_str = "1 result"; + } else if (count == 200) { + count_str = "200+ results"; + } else { + count_str = count + " results"; + } + let result_count = `
    ${count_str}
    `; + + search_result_container = ` +
    + ${modal_filters} + ${search_divider} + ${result_count} +
    + ${search_results} +
    +
    + `; + } else { + search_result_container = ` +
    + ${modal_filters} + ${search_divider} +
    0 result(s)
    +
    +
    No result found!
    + `; + } + + if ($(".search-modal-card-body").hasClass("is-justify-content-center")) { + $(".search-modal-card-body").removeClass("is-justify-content-center"); + } + + $(".search-modal-card-body").html(search_result_container); + } else { + if (!$(".search-modal-card-body").hasClass("is-justify-content-center")) { + $(".search-modal-card-body").addClass("is-justify-content-center"); + } + + $(".search-modal-card-body").html(` +
    Type something to get started!
    + `); + } +} + +/** + * Make the modal filter html + * + * @returns string + */ +function make_modal_body_filters() { + let str = filters + .map((val) => { + if (selected_filter == val.toLowerCase()) { + return `${val}`; + } else { + return `${val}`; + } + }) + .join(""); + + return ` +
    + Filters: + ${str} +
    `; +} + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +// Modal settings dialog +$(document).ready(function () { + var settings = $("#documenter-settings"); + $("#documenter-settings-button").click(function () { + settings.toggleClass("is-active"); + }); + // Close the dialog if X is clicked + $("#documenter-settings button.delete").click(function () { + settings.removeClass("is-active"); + }); + // Close dialog if ESC is pressed + $(document).keyup(function (e) { + if (e.keyCode == 27) settings.removeClass("is-active"); + }); +}); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +$(document).ready(function () { + let search_modal_header = ` +
    +
    +

    + + + + +

    +
    +
    + +
    +
    + `; + + let initial_search_body = ` +
    Type something to get started!
    + `; + + let search_modal_footer = ` +
    + + Ctrl + + / to search + + esc to close +
    + `; + + $(document.body).append( + ` + + ` + ); + + document.querySelector(".docs-search-query").addEventListener("click", () => { + openModal(); + }); + + document + .querySelector(".close-search-modal") + .addEventListener("click", () => { + closeModal(); + }); + + $(document).on("click", ".search-result-link", function () { + closeModal(); + }); + + document.addEventListener("keydown", (event) => { + if ((event.ctrlKey || event.metaKey) && event.key === "/") { + openModal(); + } else if (event.key === "Escape") { + closeModal(); + } + + return false; + }); + + // Functions to open and close a modal + function openModal() { + let searchModal = document.querySelector("#search-modal"); + + searchModal.classList.add("is-active"); + document.querySelector(".documenter-search-input").focus(); + } + + function closeModal() { + let searchModal = document.querySelector("#search-modal"); + let initial_search_body = ` +
    Type something to get started!
    + `; + + searchModal.classList.remove("is-active"); + document.querySelector(".documenter-search-input").blur(); + + if (!$(".search-modal-card-body").hasClass("is-justify-content-center")) { + $(".search-modal-card-body").addClass("is-justify-content-center"); + } + + $(".documenter-search-input").val(""); + $(".search-modal-card-body").html(initial_search_body); + } + + document + .querySelector("#search-modal .modal-background") + .addEventListener("click", () => { + closeModal(); + }); +}); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +// Manages the showing and hiding of the sidebar. +$(document).ready(function () { + var sidebar = $("#documenter > .docs-sidebar"); + var sidebar_button = $("#documenter-sidebar-button"); + sidebar_button.click(function (ev) { + ev.preventDefault(); + sidebar.toggleClass("visible"); + if (sidebar.hasClass("visible")) { + // Makes sure that the current menu item is visible in the sidebar. + $("#documenter .docs-menu a.is-active").focus(); + } + }); + $("#documenter > .docs-main").bind("click", function (ev) { + if ($(ev.target).is(sidebar_button)) { + return; + } + if (sidebar.hasClass("visible")) { + sidebar.removeClass("visible"); + } + }); +}); + +// Resizes the package name / sitename in the sidebar if it is too wide. +// Inspired by: https://github.com/davatron5000/FitText.js +$(document).ready(function () { + e = $("#documenter .docs-autofit"); + function resize() { + var L = parseInt(e.css("max-width"), 10); + var L0 = e.width(); + if (L0 > L) { + var h0 = parseInt(e.css("font-size"), 10); + e.css("font-size", (L * h0) / L0); + // TODO: make sure it survives resizes? + } + } + // call once and then register events + resize(); + $(window).resize(resize); + $(window).on("orientationchange", resize); +}); + +// Scroll the navigation bar to the currently selected menu item +$(document).ready(function () { + var sidebar = $("#documenter .docs-menu").get(0); + var active = $("#documenter .docs-menu .is-active").get(0); + if (typeof active !== "undefined") { + sidebar.scrollTop = active.offsetTop - sidebar.offsetTop - 15; + } +}); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +// Theme picker setup +$(document).ready(function () { + // onchange callback + $("#documenter-themepicker").change(function themepick_callback(ev) { + var themename = $("#documenter-themepicker option:selected").attr("value"); + if (themename === "auto") { + // set_theme(window.matchMedia('(prefers-color-scheme: dark)').matches ? 'dark' : 'light'); + window.localStorage.removeItem("documenter-theme"); + } else { + // set_theme(themename); + window.localStorage.setItem("documenter-theme", themename); + } + // We re-use the global function from themeswap.js to actually do the swapping. + set_theme_from_local_storage(); + }); + + // Make sure that the themepicker displays the correct theme when the theme is retrieved + // from localStorage + if (typeof window.localStorage !== "undefined") { + var theme = window.localStorage.getItem("documenter-theme"); + if (theme !== null) { + $("#documenter-themepicker option").each(function (i, e) { + e.selected = e.value === theme; + }); + } + } +}); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +// update the version selector with info from the siteinfo.js and ../versions.js files +$(document).ready(function () { + // If the version selector is disabled with DOCUMENTER_VERSION_SELECTOR_DISABLED in the + // siteinfo.js file, we just return immediately and not display the version selector. + if ( + typeof DOCUMENTER_VERSION_SELECTOR_DISABLED === "boolean" && + DOCUMENTER_VERSION_SELECTOR_DISABLED + ) { + return; + } + + var version_selector = $("#documenter .docs-version-selector"); + var version_selector_select = $("#documenter .docs-version-selector select"); + + version_selector_select.change(function (x) { + target_href = version_selector_select + .children("option:selected") + .get(0).value; + window.location.href = target_href; + }); + + // add the current version to the selector based on siteinfo.js, but only if the selector is empty + if ( + typeof DOCUMENTER_CURRENT_VERSION !== "undefined" && + $("#version-selector > option").length == 0 + ) { + var option = $( + "" + ); + version_selector_select.append(option); + } + + if (typeof DOC_VERSIONS !== "undefined") { + var existing_versions = version_selector_select.children("option"); + var existing_versions_texts = existing_versions.map(function (i, x) { + return x.text; + }); + DOC_VERSIONS.forEach(function (each) { + var version_url = documenterBaseURL + "/../" + each + "/"; + var existing_id = $.inArray(each, existing_versions_texts); + // if not already in the version selector, add it as a new option, + // otherwise update the old option with the URL and enable it + if (existing_id == -1) { + var option = $( + "" + ); + version_selector_select.append(option); + } else { + var option = existing_versions[existing_id]; + option.value = version_url; + option.disabled = false; + } + }); + } + + // only show the version selector if the selector has been populated + if (version_selector_select.children("option").length > 0) { + version_selector.toggleClass("visible"); + } +}); + +}) diff --git a/v0.2.1/assets/logo.svg b/v0.2.1/assets/logo.svg new file mode 100644 index 0000000..df89daf --- /dev/null +++ b/v0.2.1/assets/logo.svg @@ -0,0 +1,169 @@ + + + + + + + + + + + + + + + + + + + + + + + + + + + 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html.theme--documenter-dark .button.is-dark.is-outlined,fieldset[disabled] html.theme--documenter-dark .content kbd.button.is-outlined{background-color:transparent;border-color:#282f2f;box-shadow:none;color:#282f2f}html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined:hover,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined:hover,html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-hovered,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-hovered,html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined:focus,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined:focus,html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-focused,html.theme--documenter-dark .content 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.button.is-warning.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-warning.is-outlined{background-color:transparent;border-color:#ad8100;box-shadow:none;color:#ad8100}html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined:hover,html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined.is-hovered,html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined:focus,html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined.is-focused{background-color:#fff;color:#ad8100}html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined.is-loading:focus::after,html.theme--documenter-dark 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.button.is-danger.is-inverted.is-outlined.is-hovered,html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined:focus,html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined.is-focused{background-color:#fff;color:#9e1b0d}html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #9e1b0d #9e1b0d !important}html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--documenter-dark .button.is-danger.is-light{background-color:#fdeeec;color:#ec311d}html.theme--documenter-dark .button.is-danger.is-light:hover,html.theme--documenter-dark .button.is-danger.is-light.is-hovered{background-color:#fce3e0;border-color:transparent;color:#ec311d}html.theme--documenter-dark .button.is-danger.is-light:active,html.theme--documenter-dark .button.is-danger.is-light.is-active{background-color:#fcd8d5;border-color:transparent;color:#ec311d}html.theme--documenter-dark .button.is-small,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.button{font-size:.75rem}html.theme--documenter-dark .button.is-small:not(.is-rounded),html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.button:not(.is-rounded){border-radius:3px}html.theme--documenter-dark .button.is-normal{font-size:1rem}html.theme--documenter-dark .button.is-medium{font-size:1.25rem}html.theme--documenter-dark .button.is-large{font-size:1.5rem}html.theme--documenter-dark .button[disabled],fieldset[disabled] html.theme--documenter-dark .button{background-color:#8c9b9d;border-color:#5e6d6f;box-shadow:none;opacity:.5}html.theme--documenter-dark .button.is-fullwidth{display:flex;width:100%}html.theme--documenter-dark .button.is-loading{color:transparent !important;pointer-events:none}html.theme--documenter-dark .button.is-loading::after{position:absolute;left:calc(50% - (1em * 0.5));top:calc(50% - (1em * 0.5));position:absolute !important}html.theme--documenter-dark .button.is-static{background-color:#282f2f;border-color:#5e6d6f;color:#dbdee0;box-shadow:none;pointer-events:none}html.theme--documenter-dark .button.is-rounded,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.button{border-radius:9999px;padding-left:calc(1em + 0.25em);padding-right:calc(1em + 0.25em)}html.theme--documenter-dark .buttons{align-items:center;display:flex;flex-wrap:wrap;justify-content:flex-start}html.theme--documenter-dark .buttons .button{margin-bottom:0.5rem}html.theme--documenter-dark .buttons .button:not(:last-child):not(.is-fullwidth){margin-right:.5rem}html.theme--documenter-dark .buttons:last-child{margin-bottom:-0.5rem}html.theme--documenter-dark .buttons:not(:last-child){margin-bottom:1rem}html.theme--documenter-dark .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large){font-size:.75rem}html.theme--documenter-dark .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large):not(.is-rounded){border-radius:3px}html.theme--documenter-dark .buttons.are-medium .button:not(.is-small):not(.is-normal):not(.is-large){font-size:1.25rem}html.theme--documenter-dark .buttons.are-large .button:not(.is-small):not(.is-normal):not(.is-medium){font-size:1.5rem}html.theme--documenter-dark .buttons.has-addons .button:not(:first-child){border-bottom-left-radius:0;border-top-left-radius:0}html.theme--documenter-dark .buttons.has-addons .button:not(:last-child){border-bottom-right-radius:0;border-top-right-radius:0;margin-right:-1px}html.theme--documenter-dark .buttons.has-addons .button:last-child{margin-right:0}html.theme--documenter-dark .buttons.has-addons .button:hover,html.theme--documenter-dark .buttons.has-addons .button.is-hovered{z-index:2}html.theme--documenter-dark .buttons.has-addons .button:focus,html.theme--documenter-dark .buttons.has-addons .button.is-focused,html.theme--documenter-dark .buttons.has-addons .button:active,html.theme--documenter-dark .buttons.has-addons .button.is-active,html.theme--documenter-dark .buttons.has-addons .button.is-selected{z-index:3}html.theme--documenter-dark .buttons.has-addons .button:focus:hover,html.theme--documenter-dark .buttons.has-addons .button.is-focused:hover,html.theme--documenter-dark .buttons.has-addons .button:active:hover,html.theme--documenter-dark .buttons.has-addons .button.is-active:hover,html.theme--documenter-dark .buttons.has-addons .button.is-selected:hover{z-index:4}html.theme--documenter-dark .buttons.has-addons .button.is-expanded{flex-grow:1;flex-shrink:1}html.theme--documenter-dark .buttons.is-centered{justify-content:center}html.theme--documenter-dark .buttons.is-centered:not(.has-addons) .button:not(.is-fullwidth){margin-left:0.25rem;margin-right:0.25rem}html.theme--documenter-dark .buttons.is-right{justify-content:flex-end}html.theme--documenter-dark .buttons.is-right:not(.has-addons) .button:not(.is-fullwidth){margin-left:0.25rem;margin-right:0.25rem}@media screen and (max-width: 768px){html.theme--documenter-dark .button.is-responsive.is-small,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.is-responsive{font-size:.5625rem}html.theme--documenter-dark .button.is-responsive,html.theme--documenter-dark .button.is-responsive.is-normal{font-size:.65625rem}html.theme--documenter-dark .button.is-responsive.is-medium{font-size:.75rem}html.theme--documenter-dark .button.is-responsive.is-large{font-size:1rem}}@media screen and (min-width: 769px) and (max-width: 1055px){html.theme--documenter-dark .button.is-responsive.is-small,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.is-responsive{font-size:.65625rem}html.theme--documenter-dark .button.is-responsive,html.theme--documenter-dark .button.is-responsive.is-normal{font-size:.75rem}html.theme--documenter-dark .button.is-responsive.is-medium{font-size:1rem}html.theme--documenter-dark .button.is-responsive.is-large{font-size:1.25rem}}html.theme--documenter-dark .container{flex-grow:1;margin:0 auto;position:relative;width:auto}html.theme--documenter-dark .container.is-fluid{max-width:none !important;padding-left:32px;padding-right:32px;width:100%}@media screen and (min-width: 1056px){html.theme--documenter-dark .container{max-width:992px}}@media screen and (max-width: 1215px){html.theme--documenter-dark .container.is-widescreen:not(.is-max-desktop){max-width:1152px}}@media screen and (max-width: 1407px){html.theme--documenter-dark .container.is-fullhd:not(.is-max-desktop):not(.is-max-widescreen){max-width:1344px}}@media screen and (min-width: 1216px){html.theme--documenter-dark .container:not(.is-max-desktop){max-width:1152px}}@media screen and (min-width: 1408px){html.theme--documenter-dark .container:not(.is-max-desktop):not(.is-max-widescreen){max-width:1344px}}html.theme--documenter-dark .content li+li{margin-top:0.25em}html.theme--documenter-dark .content p:not(:last-child),html.theme--documenter-dark .content dl:not(:last-child),html.theme--documenter-dark .content ol:not(:last-child),html.theme--documenter-dark .content ul:not(:last-child),html.theme--documenter-dark .content blockquote:not(:last-child),html.theme--documenter-dark .content pre:not(:last-child),html.theme--documenter-dark .content table:not(:last-child){margin-bottom:1em}html.theme--documenter-dark .content 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html.theme--documenter-dark .navbar-dropdown,html.theme--documenter-dark .navbar-dropdown.is-boxed{border-radius:8px;border-top:none;box-shadow:0 8px 8px rgba(10,10,10,0.1), 0 0 0 1px rgba(10,10,10,0.1);display:block;opacity:0;pointer-events:none;top:calc(100% + (-4px));transform:translateY(-5px);transition-duration:86ms;transition-property:opacity, transform}html.theme--documenter-dark .navbar-dropdown.is-right{left:auto;right:0}html.theme--documenter-dark .navbar-divider{display:block}html.theme--documenter-dark .navbar>.container .navbar-brand,html.theme--documenter-dark .container>.navbar .navbar-brand{margin-left:-.75rem}html.theme--documenter-dark .navbar>.container .navbar-menu,html.theme--documenter-dark .container>.navbar .navbar-menu{margin-right:-.75rem}html.theme--documenter-dark .navbar.is-fixed-bottom-desktop,html.theme--documenter-dark .navbar.is-fixed-top-desktop{left:0;position:fixed;right:0;z-index:30}html.theme--documenter-dark 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a.navbar-item.is-active:not(:focus):not(:hover),html.theme--documenter-dark .navbar-link.is-active:not(:focus):not(:hover){background-color:rgba(0,0,0,0)}html.theme--documenter-dark .navbar-item.has-dropdown:focus .navbar-link,html.theme--documenter-dark .navbar-item.has-dropdown:hover .navbar-link,html.theme--documenter-dark .navbar-item.has-dropdown.is-active .navbar-link{background-color:rgba(0,0,0,0)}}html.theme--documenter-dark .hero.is-fullheight-with-navbar{min-height:calc(100vh - 4rem)}html.theme--documenter-dark .pagination{font-size:1rem;margin:-.25rem}html.theme--documenter-dark .pagination.is-small,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.pagination{font-size:.75rem}html.theme--documenter-dark .pagination.is-medium{font-size:1.25rem}html.theme--documenter-dark .pagination.is-large{font-size:1.5rem}html.theme--documenter-dark .pagination.is-rounded .pagination-previous,html.theme--documenter-dark #documenter .docs-sidebar 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    GitHub

    Examples

    Simulate data

    julia> using MultiResponseVarianceComponentModels, LinearAlgebra, Random
    julia> Random.seed!(6789)Random.TaskLocalRNG()
    julia> n = 1_000;  # n of observations
    julia> d = 4;      # n of responses
    julia> p = 10;     # n of covariates
    julia> m = 5;      # n of variance components
    julia> X = rand(n, p);
    julia> B = rand(p, d)10×4 Matrix{Float64}:
+ 0.866192  0.739958   0.951693   0.716391
+ 0.157873  0.633772   0.916356   0.068644
+ 0.818581  0.333021   0.0713489  0.451072
+ 0.707496  0.924568   0.0636189  0.618679
+ 0.357756  0.173016   0.982384   0.54638
+ 0.722163  0.0683381  0.343516   0.616161
+ 0.441486  0.64316    0.862809   0.400984
+ 0.812486  0.275565   0.477624   0.482087
+ 0.889787  0.669932   0.103051   0.582507
+ 0.331484  0.989292   0.785077   0.456195
    julia> V = [zeros(n, n) for _ in 1:m]; # kernel matrices
    julia> Σ = [zeros(d, d) for _ in 1:m]; # variance components
    julia> for i in 1:m
+           Vi = randn(n, n)
+           copy!(V[i], Vi' * Vi)
+           Σi = randn(d, d)
+           copy!(Σ[i], Σi' * Σi)
+       end
    julia> Ω = zeros(n * d, n * d); # overall nd-by-nd covariance matrix Ω
    julia> for i = 1:m
+           Ω += kron(Σ[i], V[i])
+       end
    julia> Ωchol = cholesky(Ω);
    julia> Y = X * B + reshape(Ωchol.L * randn(n * d), n, d)1000×4 Matrix{Float64}:
+ -270.382    -66.0256   233.653     158.94
+  168.935    354.423     14.8585    -54.193
+   85.8007   117.902   -217.707      65.1394
+ -136.574    -42.6014  -234.087      75.6404
+ -190.312   -210.051    105.117     -10.0791
+  153.414    187.179     20.691     -62.3589
+  164.97     -22.4103   -13.6301    161.682
+   21.9438  -184.643    231.526    -136.723
+ -284.549     79.7798   -37.1967   -136.943
+  -59.2492  -124.37    -160.587     112.235
+    ⋮
+  -82.4606   200.557     -4.16006   -47.2201
+  170.0       95.8103   222.034    -179.011
+   98.1247   -33.0932   -67.4875    -86.9752
+ -167.871    110.395   -184.76      -71.029
+  111.644    -23.6517   113.328    -219.276
+  -77.2094   168.859     65.7808     22.939
+  129.34    -155.123   -251.557     120.656
+ -359.121    -82.2576   -82.0063     10.8301
+  -77.7126   -90.7158   -79.5722     92.9695
    Note

    In the case of heritability and genetic correlation analyses, one can use classic genetic relationship matrices (GRMs) for $\boldsymbol{V}_i$'s, which in turn can be constructed using SnpArrays.jl.

    Maximum likelihood estimation

    julia> model = MultiResponseVarianceComponentModel(Y, X, V)A multivariate response variance component model
+   * number of responses: 4
+   * number of observations: 1000
+   * number of fixed effects: 10
+   * number of variance components: 5
    julia> @timev fit!(model, verbose = true)[ Info: Running MM algorithm for ML estimation
+iter = 0, logl = -26845.09612499673
+iter = 1, logl = -25695.041826875506
+iter = 2, logl = -25293.37779878975
+iter = 3, logl = -25159.24906847003
+iter = 4, logl = -25107.52993220223
+iter = 5, logl = -25081.38428962063
+iter = 6, logl = -25064.352419795112
+iter = 7, logl = -25051.625744592016
+iter = 8, logl = -25041.61006910993
+iter = 9, logl = -25033.604422863456
+iter = 10, logl = -25027.18042662134
+iter = 11, logl = -25022.017854787307
+iter = 12, logl = -25017.85980986688
+iter = 13, logl = -25014.498019681763
+iter = 14, logl = -25011.76503965001
+iter = 15, logl = -25009.527835987446
+iter = 16, logl = -25007.68180780364
+iter = 17, logl = -25006.145330198324
+iter = 18, logl = -25004.854995838356
+iter = 19, logl = -25003.761633006412
+iter = 20, logl = -25002.827079011357
+iter = 21, logl = -25002.02162797271
+iter = 22, logl = -25001.322047241047
+iter = 23, logl = -25000.710054799467
+iter = 24, logl = -25000.171160151207
+iter = 25, logl = -24999.693786243588
+iter = 26, logl = -24999.26860570048
+iter = 27, logl = -24998.888038955105
+iter = 28, logl = -24998.545873977288
+iter = 29, logl = -24998.236977041215
+iter = 30, logl = -24997.95707160835
+iter = 31, logl = -24997.702568235134
+iter = 32, logl = -24997.470432811737
+iter = 33, logl = -24997.258083718323
+iter = 34, logl = -24997.063310912818
+iter = 35, logl = -24996.884211761542
+iter = 36, logl = -24996.719139735058
+iter = 37, logl = -24996.566663066376
+iter = 38, logl = -24996.425531184308
+iter = 39, logl = -24996.294647261133
+iter = 40, logl = -24996.173045607276
+iter = 41, logl = -24996.05987293569
+iter = 42, logl = -24995.95437274284
+iter = 43, logl = -24995.855872212444
+iter = 44, logl = -24995.7637711789
+iter = 45, logl = -24995.67753277965
+iter = 46, logl = -24995.596675504443
+iter = 47, logl = -24995.520766401234
+iter = 48, logl = -24995.449415250514
+iter = 49, logl = -24995.38226954833
+iter = 50, logl = -24995.31901017304
+iter = 51, logl = -24995.259347628595
+iter = 52, logl = -24995.203018777862
+iter = 53, logl = -24995.14978399378
+iter = 54, logl = -24995.09942466663
+iter = 55, logl = -24995.051741018295
+iter = 56, logl = -24995.006550180144
+iter = 57, logl = -24994.963684497296
+iter = 58, logl = -24994.922990031344
+iter = 59, logl = -24994.88432523246
+iter = 60, logl = -24994.84755976209
+iter = 61, logl = -24994.812573442905
+iter = 62, logl = -24994.779255325782
+iter = 63, logl = -24994.74750285241
+iter = 64, logl = -24994.717221108924
+iter = 65, logl = -24994.688322154096
+iter = 66, logl = -24994.660724417583
+iter = 67, logl = -24994.63435215651
+iter = 68, logl = -24994.609134967097
+iter = 69, logl = -24994.585007342
+[ Info: Updates converged!
+[ Info: Calculating standard errors
+ 73.296211 seconds (14.35 M allocations: 908.765 MiB, 0.20% gc time, 7.67% compilation time)
+elapsed time (ns):  73296211375
+gc time (ns):       149341750
+bytes allocated:    952908860
+pool allocs:        14320613
+non-pool GC allocs: 24881
+malloc() calls:     906
+realloc() calls:    102
+free() calls:       523
+minor collections:  1
+full collections:   0
+Converged after 70 iterations.

    For residual maximum likelihood estimation, you can instead type:

    @timev fit!(model, reml = true, verbose = true)

    Then variance components and mean effects estimates can be accessed through

    julia> model.Σ5-element Vector{Matrix{Float64}}:
+ [2.8964235833864374 -0.9547701186106777 1.6856925389781934 0.8185656173741515; -0.9547701186106778 7.474693619948036 -4.744935846247633 -0.601253052759175; 1.6856925389781932 -4.744935846247633 8.639624062398605 0.4049276157435974; 0.8185656173741512 -0.601253052759175 0.4049276157435973 1.0145168589537352]
+ [3.776260289140289 2.3919496497788035 -0.3666324933910243 -1.365148782876448; 2.391949649778804 1.9914356452751707 -0.2630505419731652 -1.5892702567044454; -0.36663249339102433 -0.26305054197316524 4.0912210911875855 -2.287177667967102; -1.365148782876448 -1.5892702567044454 -2.287177667967102 4.526163931034507]
+ [3.1157472905367367 -0.08262940589305556 -0.27318718635822953 0.4179078660737166; -0.08262940589305548 3.4556143634179275 -0.20341922639890309 0.6648870938984194; -0.27318718635822953 -0.20341922639890314 1.9155083377403541 0.49426216391703903; 0.41790786607371666 0.6648870938984194 0.49426216391703903 0.5435108874298807]
+ [3.6027030953820782 1.2366936673894326 0.3637181414535267 0.06851794976775202; 1.2366936673894326 0.9715515927327851 1.6405295499121633 0.5971896668720472; 0.3637181414535266 1.6405295499121637 5.828012183031251 0.9752007240509573; 0.06851794976775206 0.5971896668720472 0.9752007240509573 0.9502953338899155]
+ [5.669808082536949 -2.01083534935516 1.1222606709505223 1.4405860817195557; -2.0108353493551605 6.032248061565413 9.439926228569771 -2.150730923690082; 1.1222606709505227 9.439926228569773 19.179102992380216 -3.054827357292586; 1.4405860817195555 -2.150730923690082 -3.054827357292586 3.445429926626015]
    julia> model.B10×4 Matrix{Float64}:
+   7.76641     0.0581691   30.0176     10.8424
+  -1.70871    17.0653      26.3436      5.24079
+  -4.99575    28.325      -20.9966    -13.1088
+ -10.2939     -6.00867     12.3106    -16.3959
+  -3.07093     0.733837    15.8845      1.17996
+  14.1045     -3.92946     11.5408     11.2497
+  -1.47949    -8.30862      5.49722    -0.213233
+   6.03975     5.27422      0.189634    3.90586
+  -0.632607  -14.288      -29.194       4.9749
+   6.17054   -13.2489     -23.6247     -3.52297
    julia> hcat(vec(B), vec(model.B))40×2 Matrix{Float64}:
+ 0.866192    7.76641
+ 0.157873   -1.70871
+ 0.818581   -4.99575
+ 0.707496  -10.2939
+ 0.357756   -3.07093
+ 0.722163   14.1045
+ 0.441486   -1.47949
+ 0.812486    6.03975
+ 0.889787   -0.632607
+ 0.331484    6.17054
+ ⋮
+ 0.068644    5.24079
+ 0.451072  -13.1088
+ 0.618679  -16.3959
+ 0.54638     1.17996
+ 0.616161   11.2497
+ 0.400984   -0.213233
+ 0.482087    3.90586
+ 0.582507    4.9749
+ 0.456195   -3.52297
    julia> reduce(hcat, [hcat(vec(Σ[i]), vec(model.Σ[i])) for i in 1:m])16×10 Matrix{Float64}:
+  2.59353    2.89642    3.32588    …  3.6027      5.37835    5.66981
+ -0.908817  -0.95477    2.21659       1.23669    -1.43756   -2.01084
+  1.58682    1.68569   -0.0345923     0.363718    1.65263    1.12226
+  0.365451   0.818566  -1.36959       0.0685179   0.983537   1.44059
+ -0.908817  -0.95477    2.21659       1.23669    -1.43756   -2.01084
+  7.12022    7.47469    1.75764    …  0.971552    6.84955    6.03225
+ -4.72969   -4.74494   -0.0950186     1.64053     9.98583    9.43993
+ -0.430129  -0.601253  -1.55052       0.59719    -1.98954   -2.15073
+  1.58682    1.68569   -0.0345923     0.363718    1.65263    1.12226
+ -4.72969   -4.74494   -0.0950186     1.64053     9.98583    9.43993
+  6.22778    8.63962    4.06617    …  5.82801    19.2083    19.1791
+  0.969228   0.404928  -2.51336       0.975201   -3.5885    -3.05483
+  0.365451   0.818566  -1.36959       0.0685179   0.983537   1.44059
+ -0.430129  -0.601253  -1.55052       0.59719    -1.98954   -2.15073
+  0.969228   0.404928  -2.51336       0.975201   -3.5885    -3.05483
+  1.2446     1.01452    5.02933    …  0.950295    2.85342    3.44543

    Standard errors

    Sampling variance and covariance of these estimates are

    julia> model.Σcov50×50 Matrix{Float64}:
+  0.47412      -0.00272341    0.108306     …  -0.00114518   -0.000725468
+ -0.00272341    0.280366     -0.0353382       -0.000510353   0.000627446
+  0.108306     -0.0353382     0.483991        -0.00663509    0.000814462
+  0.0597015    -0.0241462    -0.00476179      -0.00287831   -0.00410306
+ -0.000531375  -0.000740352  -0.00227336       0.00175773   -0.000550985
+ -5.46266e-6    0.0615992    -0.0084444    …   0.00754037   -0.000713865
+ -7.80744e-5    0.0347709    -0.00356761      -0.00609519    0.00360991
+  0.0240561    -0.0132044     0.213683         0.0166641    -0.000929684
+  0.0133385    -0.00919492    0.0562126       -0.0423021     0.00471451
+  0.00739022   -0.00597224   -0.0030313        0.00471978   -0.0236748
+  ⋮                                        ⋱
+ -0.00769785   -0.0406075    -0.012298         0.0234008    -0.0161668
+ -0.0133522    -0.0122384    -0.0861289        0.094258     -0.0224369
+ -0.00838137    0.00335528    0.00422692       0.0168207     0.0502388
+ -0.000645754  -0.00632483   -0.00175497      -0.0798787     0.0198685
+ -0.00102918   -0.00637199   -0.00825901   …  -0.187875      0.0275525
+ -0.000640514  -0.00315513   -0.000651312      0.140072     -0.0624593
+ -0.00183032   -0.00321512   -0.0232591       -0.36789       0.0382077
+ -0.00114518   -0.000510353  -0.00663509       0.426481     -0.0867558
+ -0.000725468   0.000627446   0.000814462     -0.0867558     0.189954
    julia> model.Bcov40×40 Matrix{Float64}:
+ 157.174     -19.3776    -19.4223    …   -0.257223   -1.24371    -1.90444
+ -19.3776    161.611     -22.5065        -1.3117     -1.42903    -1.33411
+ -19.4223    -22.5065    150.341         -1.38577    -2.56388    -1.25275
+ -14.6304    -12.8425    -19.2089        -2.29579    -2.03511    -1.00321
+ -11.6747    -13.9337    -12.4649        -0.198984   -1.44227    -2.31342
+ -24.2206    -18.4552     -7.96731   …   -3.00702    -1.14626    -0.923721
+ -13.3431    -17.4951    -15.8672        -1.57057    -0.945035   -1.77244
+ -12.7248    -19.4206    -11.7579        13.2375     -1.25172    -1.08592
+ -17.2277    -19.1384    -19.5393        -1.22189    12.8571     -0.396249
+ -15.0949    -12.6969    -15.9363        -1.23333    -0.422978   12.5215
+   ⋮                                 ⋱
+  -2.12779    13.4419     -1.25447       -9.26816    -9.00194    -6.01701
+  -0.752161   -1.49914    13.5685        -5.62018    -9.24482    -8.16962
+  -0.737933   -0.925351   -1.71971      -11.7708     -8.59325   -10.0889
+  -1.49387    -1.2578     -1.04615       -5.98367   -12.911     -13.1438
+  -1.72094    -2.33376    -0.495492  …   -9.81655    -5.37389    -8.16594
+  -1.53932    -0.763366   -2.22368       -9.36946    -7.53599   -10.4206
+  -0.257223   -1.3117     -1.38577       79.0721     -7.3782     -9.48116
+  -1.24371    -1.42903    -2.56388       -7.3782     75.6156     -0.921828
+  -1.90444    -1.33411    -1.25275       -9.48116    -0.921828   78.0356

    Corresponding standard error of these estimates are

    julia> sqrt.(diag(model.Σcov))50-element Vector{Float64}:
+ 0.6885637727726415
+ 0.529495666135442
+ 0.6956949396229591
+ 0.33645040355906397
+ 0.8084217056989712
+ 0.7442584972630495
+ 0.3639694672208399
+ 1.3688056422005268
+ 0.4677057749056628
+ 0.3179680333750869
+ ⋮
+ 0.5524130983848243
+ 0.8823995615262973
+ 0.43111265513125097
+ 0.7402837878083752
+ 0.9231964895406697
+ 0.41384022887389565
+ 1.8662336914254827
+ 0.6530553700433331
+ 0.43583743574349515
    julia> sqrt.(diag(model.Bcov))40-element Vector{Float64}:
+ 12.536904651555659
+ 12.712633608569577
+ 12.26134556777767
+ 12.169443903108293
+ 12.663455285629615
+ 11.870615133041053
+ 12.528437624570504
+ 12.51912728097892
+ 12.39394332241073
+ 12.510940364508672
+  ⋮
+  8.869558232699935
+  8.568572559479021
+  8.594848108237025
+  8.973590296072445
+  8.287660643177881
+  8.80370479520269
+  8.892247260593642
+  8.695724537824539
+  8.833773378129575
    diff --git a/v0.2.1/index.html b/v0.2.1/index.html new file mode 100644 index 0000000..582a299 --- /dev/null +++ b/v0.2.1/index.html @@ -0,0 +1,3 @@ + +Home · MRVCModels.jl
    GitHub

    MultiResponseVarianceComponentModels

    MultiResponseVarianceComponentModels.jl is a package for fitting and testing multivariate response variance components linear mixed models of form

    \[\text{vec}\ \boldsymbol{Y} \sim \mathcal{N}(\text{vec}(\boldsymbol{X} \boldsymbol{B}), \sum_{i=1}^m \boldsymbol{\Gamma}_i \otimes \boldsymbol{V}_i),\]

    where $\boldsymbol{Y}$ and $\boldsymbol{X}$ are $n \times d$ response and $n \times p$ predictor matrices, respectively, and $\boldsymbol{V}_1, \ldots, \boldsymbol{V}_m$ are $m$ known $n \times n$ positive semidefinite matrices. $\text{vec}\ \boldsymbol{Y}$ creates an $nd \times 1$ vector from $\boldsymbol{Y}$ by stacking its columns and $\otimes$ denotes the Kronecker product. The parameters of the model include $p \times d$ mean effects $\boldsymbol{B}$ and $d \times d$ variance components ($\boldsymbol{\Gamma}_1, \dots, \boldsymbol{\Gamma}_m$), which MultiResponseVarianceComponentModels.jl estimates through either minorization-maximization (MM) or expectation–maximization (EM) algorithms.

    Note

    MultiResponseVarianceComponentModels.jl is not suitable for biobank-scale data. We recommend using this package for datasets of size up to $n \cdot d \approx 50000$. This package also currently works for balanced data without any missing data.

    Installation

    To use MultiResponseVarianceComponentModels.jl, type:

    using Pkg
+Pkg.add(url = "https://github.com/Hua-Zhou/MultiResponseVarianceComponentModels.jl.git")

    This documentation was built using Documenter.jl.

    Documentation built 2024-05-24T07:36:58.741 with Julia 1.10.3
    diff --git a/v0.2.1/objects.inv b/v0.2.1/objects.inv new file mode 100644 index 0000000..d230a89 --- /dev/null +++ b/v0.2.1/objects.inv @@ -0,0 +1,9 @@ +# Sphinx inventory version 2 +# Project: MRVCModels.jl +# Version: 0.2.1 +# The remainder of this file is compressed using zlib. +xێ0^KPuUEj7v%8q䘖 +}Ospa! +U%$g~ 3 o0`BǮ=i5jQ&TGg<6px(O@K˾xD&Ϊ̭ĂX +*3L P2fLQ:}VehYf!p[YX K,i$˳3~g UQ,{xfi(I3)∅11'ɜ] +A셱n!՘ոa eܛJmsXX(MXF޻V:ʰh6,޼^$̼먵F]m޽O^ٸK%7ͮlkdӪL,8,|ijU Wj?łwP!UH fvl]H)3P[=$,~\!FA}._5jvݼ0C"UJ8of/m`qP/0n>.*j =pl;%wY8<ՑTro}dtdUƑYzn:SZbTVC:*:?WktV v_ [4V(6EV-,o146մg?C \ No newline at end of file diff --git a/v0.2.1/search_index.js b/v0.2.1/search_index.js new file mode 100644 index 0000000..a77f4c9 --- /dev/null +++ b/v0.2.1/search_index.js @@ -0,0 +1,3 @@ +var documenterSearchIndex = {"docs": +[{"location":"api/#API","page":"API","title":"API","text":"","category":"section"},{"location":"api/","page":"API","title":"API","text":"","category":"page"},{"location":"api/","page":"API","title":"API","text":"Modules = [MultiResponseVarianceComponentModels]","category":"page"},{"location":"api/#MultiResponseVarianceComponentModels.fisher_B!-Union{Tuple{MultiResponseVarianceComponentModel{T}}, Tuple{T}} where T<:Union{Float32, Float64}","page":"API","title":"MultiResponseVarianceComponentModels.fisher_B!","text":"fisher_B!(model::MultiResponseVarianceComponentModel)\n\nCompute the sampling variance-covariance of regression coefficients model.B, assuming inverse of covariance matrix model.Ω is available at model.storage_nd_nd.\n\n\n\n\n\n","category":"method"},{"location":"api/#MultiResponseVarianceComponentModels.fisher_Σ!-Union{Tuple{MultiResponseVarianceComponentModel{T}}, Tuple{T}} where T<:Union{Float32, Float64}","page":"API","title":"MultiResponseVarianceComponentModels.fisher_Σ!","text":"fisher_Σ!(model::MultiResponseVarianceComponentModel)\n\nCompute the sampling variance-covariance of variance component estimates model.Σ, assuming inverse of covariance matrix model.Ω is available at model.storage_nd_nd.\n\n\n\n\n\n","category":"method"},{"location":"api/#MultiResponseVarianceComponentModels.fit!-Union{Tuple{MultiResponseVarianceComponentModel{T}}, Tuple{T}} where T<:Union{Float32, Float64}","page":"API","title":"MultiResponseVarianceComponentModels.fit!","text":"fit!(model::MultiResponseVarianceComponentModel)\n\nFit a multivariate response variance component model by an MM or EM algorithm.\n\nPositional arguments\n\nmodel : a MultiResponseVarianceComponentModel instance. \n\nKeyword arguments\n\nmaxiter::Integer : maximum number of iterations. Default is 1000.\nreltol::Real : relative tolerance for convergence. Default is 1e-6.\nverbose::Bool : display algorithmic information. Default is false.\ninit::Symbol : initialization strategy. :default initialize by least squares. :user uses user supplied value at model.B and model.Σ.\nalgo::Symbol : optimization algorithm. :MM (default) or EM.\nlog::Bool : record iterate history or not. Defaut is false.\nreml::Bool : REML instead of ML estimation. Default is false.\nse::Bool : calculate standard errors. Default is true.\n\nOutput\n\nhistory : iterate history.\n\n\n\n\n\n","category":"method"},{"location":"api/#MultiResponseVarianceComponentModels.h2-Union{Tuple{MultiResponseVarianceComponentModel{T}}, Tuple{T}} where T<:Union{Float32, Float64}","page":"API","title":"MultiResponseVarianceComponentModels.h2","text":"h2(model::MultiResponseVarianceComponentModel)\n\nCalculate heritability estimates and their standard errors, assuming that all variance components capture genetic effects except the last term. Also returns total heritability and its standard error from sum of individual contributions.\n\n\n\n\n\n","category":"method"},{"location":"api/#MultiResponseVarianceComponentModels.kron_axpy!-Union{Tuple{T}, Tuple{AbstractVecOrMat{T}, AbstractVecOrMat{T}, AbstractVecOrMat{T}}} where T<:Real","page":"API","title":"MultiResponseVarianceComponentModels.kron_axpy!","text":"kron_axpy!(A, X, Y)\n\nOverwrite Y with A ⊗ X + Y. Same as Y += kron(A, X), but more memory efficient.\n\n\n\n\n\n","category":"method"},{"location":"api/#MultiResponseVarianceComponentModels.kron_reduction!-Union{Tuple{T}, Tuple{AbstractMatrix{T}, AbstractMatrix{T}, AbstractMatrix{T}}, Tuple{AbstractMatrix{T}, AbstractMatrix{T}, AbstractMatrix{T}, Bool}} where T<:Real","page":"API","title":"MultiResponseVarianceComponentModels.kron_reduction!","text":"kron_reduction!(A, B, C; sym = false)\n\nOverwrite C with the derivative of tr(A' (X ⊗ B)) wrt X. C[i, j] = dot(Aij, B), where Aij is the (i , j) block of A. sym = true indicates A and B are symmetric.\n\n\n\n\n\n","category":"method"},{"location":"api/#MultiResponseVarianceComponentModels.loglikelihood!-Union{Tuple{MultiResponseVarianceComponentModel{T}}, Tuple{T}} where T<:Union{Float32, Float64}","page":"API","title":"MultiResponseVarianceComponentModels.loglikelihood!","text":"loglikelihood!(model::MultiResponseVarianceComponentModel)\n\nOverwrite model.storage_nd_nd by inverse of the covariance matrix model.Ω, overwrite model.storage_nd by U' \\ vec(model.R), and return the log-likelihood. This function assumes model.Ω and model.R are already updated according to model.Σ and model.B.\n\n\n\n\n\n","category":"method"},{"location":"api/#MultiResponseVarianceComponentModels.lrt-Tuple{MultiResponseVarianceComponentModel, MultiResponseVarianceComponentModel}","page":"API","title":"MultiResponseVarianceComponentModels.lrt","text":"lrt(model1::MultiResponseVarianceComponentModel, model0::MultiResponseVarianceComponentModel)\n\nPerform a variation of the likelihood ratio test for univariate variance components models as in Molenberghs and Verbeke 2007 with model1 and model0 being the full and nested models, respectively.\n\n\n\n\n\n","category":"method"},{"location":"api/#MultiResponseVarianceComponentModels.rg-Union{Tuple{MultiResponseVarianceComponentModel{T}}, Tuple{T}} where T<:Union{Float32, Float64}","page":"API","title":"MultiResponseVarianceComponentModels.rg","text":"rg(model::MultiResponseVarianceComponentModel)\n\nCalculate genetic/residual correlation estimates and their standard errors.\n\n\n\n\n\n","category":"method"},{"location":"api/#MultiResponseVarianceComponentModels.update_B!-Union{Tuple{MultiResponseVarianceComponentModel{T}}, Tuple{T}} where T<:Union{Float32, Float64}","page":"API","title":"MultiResponseVarianceComponentModels.update_B!","text":"update_B!(model::MultiResponseVarianceComponentModel)\n\nUpdate the regression coefficients model.B, assuming inverse of covariance matrix model.Ω is available at model.storage_nd_nd.\n\n\n\n\n\n","category":"method"},{"location":"api/#MultiResponseVarianceComponentModels.update_Γk!-Union{Tuple{T}, Tuple{MultiResponseVarianceComponentModel{T}, Integer}} where T<:Union{Float32, Float64}","page":"API","title":"MultiResponseVarianceComponentModels.update_Γk!","text":"update_Γk!(model, k)\n\nUpdate the parameters model.Γ[k] and model.Ψ[:,k] assuming a diagonal plus low rank structure for model.Σ[k]. Assumes covariance matrix model.Ω is available at model.storage_nd_nd and model.Ω⁻¹R precomputed.\n\n\n\n\n\n","category":"method"},{"location":"api/#MultiResponseVarianceComponentModels.update_Σ!-Union{Tuple{MultiResponseVarianceComponentModel{T}}, Tuple{T}} where T<:Union{Float32, Float64}","page":"API","title":"MultiResponseVarianceComponentModels.update_Σ!","text":"update_Σ!(model::MultiResponseVarianceComponentModel)\n\nUpdate the variance component parameters model.Σ, assuming inverse of covariance matrix model.Ω is available at model.storage_nd_nd.\n\n\n\n\n\n","category":"method"},{"location":"api/#MultiResponseVarianceComponentModels.update_Σk!-Union{Tuple{T}, Tuple{MultiResponseVarianceComponentModel{T}, Integer, Integer}} where T<:Union{Float32, Float64}","page":"API","title":"MultiResponseVarianceComponentModels.update_Σk!","text":"update_Σk!(model, k, rk)\n\nUpdate the model.Σ[k] assuming it has rank rk < d, assuming inverse of covariance matrix model.Ω is available at model.storage_nd_nd and model.Ω⁻¹R precomputed.\n\n\n\n\n\n","category":"method"},{"location":"api/#MultiResponseVarianceComponentModels.update_Σk!-Union{Tuple{T}, Tuple{MultiResponseVarianceComponentModel{T}, Integer, Val{:EM}}} where T<:Union{Float32, Float64}","page":"API","title":"MultiResponseVarianceComponentModels.update_Σk!","text":"update_Σk!(model::MultiResponseVarianceComponentModel, k, Val(:EM))\n\nEM update the model.Σ[k] assuming it has full rank d, inverse of covariance matrix model.Ω is available at model.storage_nd_nd, and model.Ω⁻¹R precomputed.\n\n\n\n\n\n","category":"method"},{"location":"api/#MultiResponseVarianceComponentModels.update_Σk!-Union{Tuple{T}, Tuple{MultiResponseVarianceComponentModel{T}, Integer, Val{:MM}}} where T<:Union{Float32, Float64}","page":"API","title":"MultiResponseVarianceComponentModels.update_Σk!","text":"update_Σk!(model::MultiResponseVarianceComponentModel, k, Val(:MM))\n\nMM update the model.Σ[k] assuming it has full rank d, inverse of covariance matrix model.Ω is available at model.storage_nd_nd, and model.Ω⁻¹R precomputed.\n\n\n\n\n\n","category":"method"},{"location":"examples/#Examples","page":"Examples","title":"Examples","text":"","category":"section"},{"location":"examples/#Simulate-data","page":"Examples","title":"Simulate data","text":"","category":"section"},{"location":"examples/","page":"Examples","title":"Examples","text":"using MultiResponseVarianceComponentModels, LinearAlgebra, Random\nRandom.seed!(6789)\nn = 1_000; # n of observations\nd = 4; # n of responses\np = 10; # n of covariates\nm = 5; # n of variance components\nX = rand(n, p);\nB = rand(p, d)\nV = [zeros(n, n) for _ in 1:m]; # kernel matrices\nΣ = [zeros(d, d) for _ in 1:m]; # variance components\nfor i in 1:m\n Vi = randn(n, n)\n copy!(V[i], Vi' * Vi)\n Σi = randn(d, d)\n copy!(Σ[i], Σi' * Σi)\nend\nΩ = zeros(n * d, n * d); # overall nd-by-nd covariance matrix Ω\nfor i = 1:m\n Ω += kron(Σ[i], V[i])\nend\nΩchol = cholesky(Ω);\nY = X * B + reshape(Ωchol.L * randn(n * d), n, d)","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"note: Note\nIn the case of heritability and genetic correlation analyses, one can use classic genetic relationship matrices (GRMs) for boldsymbolV_i's, which in turn can be constructed using SnpArrays.jl.","category":"page"},{"location":"examples/#Maximum-likelihood-estimation","page":"Examples","title":"Maximum likelihood estimation","text":"","category":"section"},{"location":"examples/","page":"Examples","title":"Examples","text":"model = MultiResponseVarianceComponentModel(Y, X, V)\n@timev fit!(model, verbose = true)","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"For residual maximum likelihood estimation, you can instead type:","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"@timev fit!(model, reml = true, verbose = true)","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"Then variance components and mean effects estimates can be accessed through","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"model.Σ\nmodel.B\nhcat(vec(B), vec(model.B))\nreduce(hcat, [hcat(vec(Σ[i]), vec(model.Σ[i])) for i in 1:m])","category":"page"},{"location":"examples/#Standard-errors","page":"Examples","title":"Standard errors","text":"","category":"section"},{"location":"examples/","page":"Examples","title":"Examples","text":"Sampling variance and covariance of these estimates are","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"model.Σcov\nmodel.Bcov","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"Corresponding standard error of these estimates are","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"sqrt.(diag(model.Σcov))\nsqrt.(diag(model.Bcov))","category":"page"},{"location":"advanced/#Advanced-details","page":"Advanced details","title":"Advanced details","text":"","category":"section"},{"location":"advanced/#Estimation","page":"Advanced details","title":"Estimation","text":"","category":"section"},{"location":"advanced/","page":"Advanced details","title":"Advanced details","text":"For the MM algorithm, the updates in each iteration are","category":"page"},{"location":"advanced/","page":"Advanced details","title":"Advanced details","text":"beginaligned\ntextvec boldsymbolB^(t) = (boldsymbolI_d otimes boldsymbolX^T) boldsymbolOmega^-(t) (boldsymbolI_d otimes boldsymbolX)^-1 (boldsymbolI_d otimes boldsymbolX^T) boldsymbolOmega^-(t) textvec boldsymbolY \nboldsymbolGamma_i^(t + 1) = boldsymbolL_i^-(t)TboldsymbolL_i^(t)T(boldsymbolGamma_i^(t)boldsymbolR^(t)TboldsymbolV_iboldsymbolR^(t)boldsymbolGamma_i^(t))boldsymbolL_i^(t)^12 boldsymbolL_i^-(t)\nendaligned","category":"page"},{"location":"advanced/","page":"Advanced details","title":"Advanced details","text":"where boldsymbolOmega^(t) = sum_i=1^m boldsymbolGamma_i^(t) otimes boldsymbolV_i and boldsymbolL_i^(t) is the Cholesky factor of (boldsymbolI_d otimes boldsymbol1_n)^T (boldsymbol1_d boldsymbol1_d^T otimes boldsymbolV_i) odot boldsymbolOmega^-(t) (boldsymbolI_d otimes boldsymbol1_n), while boldsymbolR^(t) is the n times d matrix such that textvec boldsymbolR^(t) = boldsymbolOmega^-(t) textvec(boldsymbolY - boldsymbolX boldsymbolB^(t)).","category":"page"},{"location":"advanced/","page":"Advanced details","title":"Advanced details","text":"For the EM algorithm, the updates in each iteration are","category":"page"},{"location":"advanced/","page":"Advanced details","title":"Advanced details","text":"beginaligned\ntextvec boldsymbolB^(t) = (boldsymbolI_d otimes boldsymbolX^T) boldsymbolOmega^-(t) (boldsymbolI_d otimes boldsymbolX)^-1 (boldsymbolI_d otimes boldsymbolX^T) boldsymbolOmega^-(t) textvec boldsymbolY \nboldsymbolGamma_i^(t + 1) = frac1r_i boldsymbolGamma_i^(t) boldsymbolR^(t)T boldsymbolV_i boldsymbolR^(t) - (boldsymbolI_d otimes boldsymbol1_n)^T (boldsymbol1_d boldsymbol1_d^T otimes boldsymbolV_i) odot boldsymbolOmega^-(t) (boldsymbolI_d otimes boldsymbol1_n) boldsymbolGamma_i^(t) + boldsymbolGamma_i^(t)\nendaligned","category":"page"},{"location":"advanced/","page":"Advanced details","title":"Advanced details","text":"where r_i = textrank(boldsymbolV_i). As seen, the updates for mean effects boldsymbolB are the same for these two algorithms.","category":"page"},{"location":"advanced/#Inference","page":"Advanced details","title":"Inference","text":"","category":"section"},{"location":"advanced/","page":"Advanced details","title":"Advanced details","text":"Standard errors for our estimates were calculated using the Fisher information matrix, where","category":"page"},{"location":"advanced/","page":"Advanced details","title":"Advanced details","text":"beginaligned\ntextE left- fracpartial^2partial(textvec boldsymbolB)^T partial(textvec boldsymbolB) mathcalL right = (boldsymbolI_d otimes boldsymbolX^T) boldsymbolOmega^-1 (boldsymbolI_d otimes boldsymbolX) \ntextE left - fracpartial^2partial (textvech boldsymbolGamma_i)^T partial (textvec boldsymbolB) mathcalL right = boldsymbol0 \ntextE left - fracpartial^2partial (textvech boldsymbolGamma_j)^T partial (textvech boldsymbolGamma_i) mathcalL right = frac12 boldsymbolU_i^T (boldsymbolOmega^-1 otimes boldsymbolOmega^-1) boldsymbolU_j\nendaligned","category":"page"},{"location":"advanced/","page":"Advanced details","title":"Advanced details","text":"and boldsymbolU_i = (boldsymbolI_d otimes boldsymbolK_nd otimes boldsymbolI_n) (boldsymbolI_d^2 otimes textvec boldsymbolV_i) boldsymbolD_d. Here, boldsymbolK_nd is the nd times nd commutation matrix and boldsymbolD_d the d^2 times fracd(d+1)2 duplication matrix. textvech boldsymbolGamma_i creates an fracd(d+1)2 times 1 vector from boldsymbolGamma_i by stacking its lower triangular part.","category":"page"},{"location":"","page":"Home","title":"Home","text":"CurrentModule = MultiResponseVarianceComponentModels","category":"page"},{"location":"#MultiResponseVarianceComponentModels","page":"Home","title":"MultiResponseVarianceComponentModels","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"MultiResponseVarianceComponentModels.jl is a package for fitting and testing multivariate response variance components linear mixed models of form","category":"page"},{"location":"","page":"Home","title":"Home","text":"textvec boldsymbolY sim mathcalN(textvec(boldsymbolX boldsymbolB) sum_i=1^m boldsymbolGamma_i otimes boldsymbolV_i)","category":"page"},{"location":"","page":"Home","title":"Home","text":"where boldsymbolY and boldsymbolX are n times d response and n times p predictor matrices, respectively, and boldsymbolV_1 ldots boldsymbolV_m are m known n times n positive semidefinite matrices. textvec boldsymbolY creates an nd times 1 vector from boldsymbolY by stacking its columns and otimes denotes the Kronecker product. The parameters of the model include p times d mean effects boldsymbolB and d times d variance components (boldsymbolGamma_1 dots boldsymbolGamma_m), which MultiResponseVarianceComponentModels.jl estimates through either minorization-maximization (MM) or expectation–maximization (EM) algorithms.","category":"page"},{"location":"","page":"Home","title":"Home","text":"note: Note\nMultiResponseVarianceComponentModels.jl is not suitable for biobank-scale data. We recommend using this package for datasets of size up to n cdot d approx 50000. This package also currently works for balanced data without any missing data.","category":"page"},{"location":"#Installation","page":"Home","title":"Installation","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"To use MultiResponseVarianceComponentModels.jl, type:","category":"page"},{"location":"","page":"Home","title":"Home","text":"using Pkg\nPkg.add(url = \"https://github.com/Hua-Zhou/MultiResponseVarianceComponentModels.jl.git\")","category":"page"},{"location":"","page":"Home","title":"Home","text":"This documentation was built using Documenter.jl.","category":"page"},{"location":"","page":"Home","title":"Home","text":"using Dates # hide\nprintln(\"Documentation built $(Dates.now()) with Julia $(VERSION)\") # hide","category":"page"}] +} diff --git a/v0.2.1/siteinfo.js b/v0.2.1/siteinfo.js new file mode 100644 index 0000000..c32ac74 --- /dev/null +++ b/v0.2.1/siteinfo.js @@ -0,0 +1 @@ +var DOCUMENTER_CURRENT_VERSION = "v0.2.1"; diff --git a/v0.3 b/v0.3 new file mode 120000 index 0000000..1e66a61 --- /dev/null +++ b/v0.3 @@ -0,0 +1 @@ +v0.3.1 \ No newline at end of file diff --git a/v0.3.0/.documenter-siteinfo.json b/v0.3.0/.documenter-siteinfo.json new file mode 100644 index 0000000..1ee011a --- /dev/null +++ b/v0.3.0/.documenter-siteinfo.json @@ -0,0 +1 @@ +{"documenter":{"julia_version":"1.10.4","generation_timestamp":"2024-06-08T07:53:19","documenter_version":"1.4.1"}} \ No newline at end of file diff --git a/v0.3.0/advanced/index.html b/v0.3.0/advanced/index.html new file mode 100644 index 0000000..cc360a4 --- /dev/null +++ b/v0.3.0/advanced/index.html @@ -0,0 +1,15 @@ + +Advanced details · MRVCModels.jl
    GitHub

    Advanced details

    Estimation

    For the MM algorithm, the updates in each iteration are

    \[\begin{aligned} +\text{vec}\ \boldsymbol{B}^{(t)} &= [(\boldsymbol{I}_d \otimes \boldsymbol{X}^T) \boldsymbol{\Omega}^{-(t)} (\boldsymbol{I}_d \otimes \boldsymbol{X})]^{-1} (\boldsymbol{I}_d \otimes \boldsymbol{X}^T) \boldsymbol{\Omega}^{-(t)} \text{vec}\ \boldsymbol{Y} \\ +\boldsymbol{\Gamma}_i^{(t + 1)} &= \boldsymbol{L}_i^{-(t)T}[\boldsymbol{L}_i^{(t)T}(\boldsymbol{\Gamma}_i^{(t)}\boldsymbol{R}^{(t)T}\boldsymbol{V}_i\boldsymbol{R}^{(t)}\boldsymbol{\Gamma}_i^{(t)})\boldsymbol{L}_i^{(t)}]^{1/2} \boldsymbol{L}_i^{-(t)}, +\end{aligned}\]

    where $\boldsymbol{\Omega}^{(t)} = \sum_{i=1}^m \boldsymbol{\Gamma}_i^{(t)} \otimes \boldsymbol{V}_i$ and $\boldsymbol{L}_i^{(t)}$ is the Cholesky factor of $\boldsymbol{M}_i^{(t)} = (\boldsymbol{I}_d \otimes \boldsymbol{1}_n)^T [(\boldsymbol{1}_d \boldsymbol{1}_d^T \otimes \boldsymbol{V}_i) \odot \boldsymbol{\Omega}^{-(t)}] (\boldsymbol{I}_d \otimes \boldsymbol{1}_n)$, while $\boldsymbol{R}^{(t)}$ is the $n \times d$ matrix such that $\text{vec}\ \boldsymbol{R}^{(t)} = \boldsymbol{\Omega}^{-(t)} \text{vec}(\boldsymbol{Y} - \boldsymbol{X} \boldsymbol{B}^{(t)})$.

    For the EM algorithm, the updates in each iteration are

    \[\begin{aligned} +\text{vec}\ \boldsymbol{B}^{(t)} &= [(\boldsymbol{I}_d \otimes \boldsymbol{X}^T) \boldsymbol{\Omega}^{-(t)} (\boldsymbol{I}_d \otimes \boldsymbol{X})]^{-1} (\boldsymbol{I}_d \otimes \boldsymbol{X}^T) \boldsymbol{\Omega}^{-(t)} \text{vec}\ \boldsymbol{Y} \\ +\boldsymbol{\Gamma}_i^{(t + 1)} &= \frac{1}{r_i} \boldsymbol{\Gamma}_i^{(t)} \{ \boldsymbol{R}^{(t)T} \boldsymbol{V}_i \boldsymbol{R}^{(t)} - \boldsymbol{M}_i^{(t)}\} \boldsymbol{\Gamma}_i^{(t)} + \boldsymbol{\Gamma}_i^{(t)}, +\end{aligned}\]

    where $r_i = \text{rank}(\boldsymbol{V}_i)$. As seen, the updates for mean effects $\boldsymbol{B}$ are the same for these two algorithms.

    In the setting of missing response, the adjusted MM updates in each interation are

    \[\begin{aligned} +\text{vec}\ \boldsymbol{B}^{(t)} &= [(\boldsymbol{I}_d \otimes \boldsymbol{X}^T) \boldsymbol{\Omega}^{-(t)} (\boldsymbol{I}_d \otimes \boldsymbol{X})]^{-1} (\boldsymbol{I}_d \otimes \boldsymbol{X}^T) \boldsymbol{\Omega}^{-(t)} \text{vec}\ \boldsymbol{Z}^{(t)} \\ +\boldsymbol{\Gamma}_i^{(t + 1)} &= \boldsymbol{L}_i^{-(t)T}[\boldsymbol{L}_i^{(t)T}\boldsymbol{\Gamma}_i^{(t)}(\boldsymbol{R}^{*(t)T}\boldsymbol{V}_i\boldsymbol{R}^{*(t)} + \boldsymbol{M}_i^{*(t)})\boldsymbol{\Gamma}_i^{(t)}\boldsymbol{L}_i^{(t)}]^{1/2} \boldsymbol{L}_i^{-(t)}, +\end{aligned}\]

    where $\boldsymbol{Z}^{(t)}$ is the completed response matrix from conditional mean and $\boldsymbol{M}_i^{*(t)} = (\boldsymbol{I}_d \otimes \boldsymbol{1}_n)^T [(\boldsymbol{1}_d \boldsymbol{1}_d^T \otimes \boldsymbol{V}_i) \odot (\boldsymbol{\Omega}^{-(t)} \boldsymbol{P}^T \boldsymbol{C}^{(t)}\boldsymbol{P}\boldsymbol{\Omega}^{-(t)})] (\boldsymbol{I}_d \otimes \boldsymbol{1}_n)$, while $\boldsymbol{R}^{*(t)}$ is the $n \times d$ matrix such that $\text{vec}\ \boldsymbol{R}^{*(t)} = \boldsymbol{\Omega}^{-(t)} \text{vec}(\boldsymbol{Z}^{(t)} - \boldsymbol{X} \boldsymbol{B}^{(t)})$. Additionally, $\boldsymbol{P}$ is the permutation matrix such that $\boldsymbol{P} \cdot \text{vec}(\boldsymbol{Y}) = \begin{bmatrix} \boldsymbol{y}_{\text{obs}} \\ \boldsymbol{y}_{\text{mis}} \end{bmatrix}$, where $\boldsymbol{y}_{\text{obs}}$ and $\boldsymbol{y}_{\text{mis}}$ are vectors of observed and missing response values, respectively, in column-major order, and the block matrix $\boldsymbol{C}^{(t)}$ is $\boldsymbol{0}$ except for a lower-right block consisting of conditional variance. As seen, the two MM updates for both mean effects and variance components are of similar form.

    Inference

    Standard errors for our estimates are calculated using the Fisher information matrix, where

    \[\begin{aligned} +\text{E} \left[- \frac{\partial^2}{\partial(\text{vec}\ \boldsymbol{B})^T \partial(\text{vec}\ \boldsymbol{B})} \mathcal{L} \right] &= (\boldsymbol{I}_d \otimes \boldsymbol{X}^T) \boldsymbol{\Omega}^{-1} (\boldsymbol{I}_d \otimes \boldsymbol{X}) \\ +\text{E} \left[ - \frac{\partial^2}{\partial (\text{vech}\ \boldsymbol{\Gamma}_i)^T \partial (\text{vec}\ \boldsymbol{B})} \mathcal{L} \right] &= \boldsymbol{0} \\ +\text{E} \left[ - \frac{\partial^2}{\partial (\text{vech}\ \boldsymbol{\Gamma}_j)^T \partial (\text{vech}\ \boldsymbol{\Gamma}_i)} \mathcal{L} \right] &= \frac{1}{2} \boldsymbol{U}_i^T (\boldsymbol{\Omega}^{-1} \otimes \boldsymbol{\Omega}^{-1}) \boldsymbol{U}_j +\end{aligned}\]

    and $\boldsymbol{U}_i = (\boldsymbol{I}_d \otimes \boldsymbol{K}_{nd} \otimes \boldsymbol{I}_n) (\boldsymbol{I}_{d^2} \otimes \text{vec}\ \boldsymbol{V}_i) \boldsymbol{D}_{d}$. Here, $\boldsymbol{K}_{nd}$ is the $nd \times nd$ commutation matrix and $\boldsymbol{D}_{d}$ the $d^2 \times \frac{d(d+1)}{2}$ duplication matrix. $\text{vech}\ \boldsymbol{\Gamma}_i$ creates an $\frac{d(d+1)}{2} \times 1$ vector from $\boldsymbol{\Gamma}_i$ by stacking its lower triangular part.

    diff --git a/v0.3.0/api/index.html b/v0.3.0/api/index.html new file mode 100644 index 0000000..51d13b9 --- /dev/null +++ b/v0.3.0/api/index.html @@ -0,0 +1,2 @@ + +API · MRVCModels.jl
    GitHub

    API

    MultiResponseVarianceComponentModels.MRVCModel — Method
    MRVCModel(Y, X, V)

    Create a new MRVCModel instance from response matrix Y, predictor matrix X, and kernel matrices V.

    Positional arguments

    • Y::AbstractVecOrMat : response matrix
    • X::Union{Nothing, AbstractVecOrMat} : predictor matrix
    • V::Union{AbstractMatrix, Vector{<:AbstractMatrix}} : kernel matrices

    Keyword arguments

    • se::Bool : calculate standard errors. Default is true.
    • reml::Bool : REML estimation instead of ML estimation. Default is false.
    source
    MultiResponseVarianceComponentModels.fit! — Method
    fit!(model::MRVCModel)

    Fit a multivariate response variance components model by MM or EM algorithm.

    Keyword arguments

    • maxiter::Integer : maximum number of iterations. Default is 1000.
    • reltol::Real : relative tolerance for convergence. Default is 1e-6.
    • verbose::Bool : display algorithmic information. Default is true.
    • init::Symbol : initialization strategy. :default initialize by least squares. :user uses user supplied value at model.B and model.Σ.
    • algo::Symbol : optimization algorithm. :MM (default) or EM.
    • log::Bool : record iterate history or not. Defaut is false.

    Output

    • history : iterate history.
    source
    MultiResponseVarianceComponentModels.h2 — Method
    h2(model::MRVCModel)

    Calculate heritability estimates and their standard errors, assuming that all variance components capture genetic effects except the last term. Also return total heritability from sum of individual contributions and its standard error.

    source
    MultiResponseVarianceComponentModels.loglikelihood! — Method
    loglikelihood!(model::MRVCModel)

    Overwrite model.storage_nd_nd by inverse of the covariance matrix model.Ω, overwrite model.storage_nd by U' \ vec(model.R), and return the log-likelihood. Assume model.Ω and model.R are already updated according to model.Σ and model.B.

    source
    MultiResponseVarianceComponentModels.loglikelihood_miss! — Method
    loglikelihood_miss!(model::MRVCModel)

    Overwrite model.storage_nd_nd by inverse of the covariance matrix model.Ω, overwrite model.storage_nd by U' \ vec(model.R), overwrite model.storage_n_miss_n_miss_1 by conditional variance, and return the surrogate Q-function of log-likelihood. Assume model.Ω and model.R are already updated according to model.Σ and model.B.

    source
    MultiResponseVarianceComponentModels.lrt — Method
    lrt(model1::MRVCModel, model0::MRVCModel)

    Perform a variation of the likelihood ratio test for univariate variance components models as in Molenberghs and Verbeke 2007 with model1 and model0 being the full and nested models, respectively.

    source
    MultiResponseVarianceComponentModels.permute — Method
    permute(Y::AbstractVecOrMat)

    Return permutation P such that vec(Y)[P] rearranges vec(Y) with missing values spliced after non-missing values. Also return inverse permutation invP such that vec(Y)[P][invP] = vec(Y).

    source
    MultiResponseVarianceComponentModels.update_B_miss! — Method
    update_B_miss!(model::MRVCModel)

    Update the regression coefficients model.B, assuming inverse of covariance matrix model.Ω is available at model.storage_nd_nd and conditional covariance model.storage_n_miss_n_obs_1 precomputed.

    source
    MultiResponseVarianceComponentModels.update_Γk! — Method
    update_Γk!(model, k)

    Update the parameters model.Γ[k] and model.Ψ[:,k] assuming a diagonal plus low rank structure for model.Σ[k]. Assumes covariance matrix model.Ω is available at model.storage_nd_nd and model.Ω⁻¹R precomputed.

    source
    MultiResponseVarianceComponentModels.update_Σ! — Method
    update_Σ!(model::MRVCModel)

    Update the variance component parameters model.Σ, assuming inverse of covariance matrix model.Ω is available at model.storage_nd_nd. For missing response, assume conditional variance model.storage_n_miss_n_miss_1 is precomputed.

    source
    MultiResponseVarianceComponentModels.update_Σk_miss! — Method
    update_Σk_miss!(model::MRVCModel, k, Val(:MM))

    MM update the model.Σ[k] assuming it has full rank d, inverse of covariance matrix model.Ω is available at model.storage_nd_nd, and model.Ω⁻¹R, conditional variance model.storage_n_miss_n_miss_1, and Ω⁻¹P'CPΩ⁻¹ precomputed.

    source
    diff --git a/v0.3.0/assets/MRVC.png b/v0.3.0/assets/MRVC.png new file mode 100644 index 0000000..85e4add Binary files /dev/null and b/v0.3.0/assets/MRVC.png differ diff --git a/v0.3.0/assets/documenter.js b/v0.3.0/assets/documenter.js new file mode 100644 index 0000000..c6562b5 --- /dev/null +++ b/v0.3.0/assets/documenter.js @@ -0,0 +1,1050 @@ +// Generated by Documenter.jl +requirejs.config({ + paths: { + 'highlight-julia': 'https://cdnjs.cloudflare.com/ajax/libs/highlight.js/11.8.0/languages/julia.min', + 'headroom': 'https://cdnjs.cloudflare.com/ajax/libs/headroom/0.12.0/headroom.min', + 'jqueryui': 'https://cdnjs.cloudflare.com/ajax/libs/jqueryui/1.13.2/jquery-ui.min', + 'katex-auto-render': 'https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.8/contrib/auto-render.min', + 'jquery': 'https://cdnjs.cloudflare.com/ajax/libs/jquery/3.7.0/jquery.min', + 'headroom-jquery': 'https://cdnjs.cloudflare.com/ajax/libs/headroom/0.12.0/jQuery.headroom.min', + 'katex': 'https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.8/katex.min', + 'highlight': 'https://cdnjs.cloudflare.com/ajax/libs/highlight.js/11.8.0/highlight.min', + 'highlight-julia-repl': 'https://cdnjs.cloudflare.com/ajax/libs/highlight.js/11.8.0/languages/julia-repl.min', + }, + shim: { + "highlight-julia": { + "deps": [ + "highlight" + ] + }, + "katex-auto-render": { + "deps": [ + "katex" + ] + }, + "headroom-jquery": { + "deps": [ + "jquery", + "headroom" + ] + }, + "highlight-julia-repl": { + "deps": [ + "highlight" + ] + } +} +}); +//////////////////////////////////////////////////////////////////////////////// +require(['jquery', 'katex', 'katex-auto-render'], function($, katex, renderMathInElement) { +$(document).ready(function() { + renderMathInElement( + document.body, + { + "delimiters": [ + { + "left": "$", + "right": "$", + "display": false + }, + { + "left": "$$", + "right": "$$", + "display": true + }, + { + "left": "\\[", + "right": "\\]", + "display": true + } + ] +} + + ); +}) + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery', 'highlight', 'highlight-julia', 'highlight-julia-repl'], function($) { +$(document).ready(function() { + hljs.highlightAll(); +}) + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +let timer = 0; +var isExpanded = true; + +$(document).on("click", ".docstring header", function () { + let articleToggleTitle = "Expand docstring"; + + debounce(() => { + if ($(this).siblings("section").is(":visible")) { + $(this) + .find(".docstring-article-toggle-button") + .removeClass("fa-chevron-down") + .addClass("fa-chevron-right"); + } else { + $(this) + .find(".docstring-article-toggle-button") + .removeClass("fa-chevron-right") + .addClass("fa-chevron-down"); + + articleToggleTitle = "Collapse docstring"; + } + + $(this) + .find(".docstring-article-toggle-button") + .prop("title", articleToggleTitle); + $(this).siblings("section").slideToggle(); + }); +}); + +$(document).on("click", ".docs-article-toggle-button", function (event) { + let articleToggleTitle = "Expand docstring"; + let navArticleToggleTitle = "Expand all docstrings"; + let animationSpeed = event.noToggleAnimation ? 0 : 400; + + debounce(() => { + if (isExpanded) { + $(this).removeClass("fa-chevron-up").addClass("fa-chevron-down"); + $(".docstring-article-toggle-button") + .removeClass("fa-chevron-down") + .addClass("fa-chevron-right"); + + isExpanded = false; + + $(".docstring section").slideUp(animationSpeed); + } else { + $(this).removeClass("fa-chevron-down").addClass("fa-chevron-up"); + $(".docstring-article-toggle-button") + .removeClass("fa-chevron-right") + .addClass("fa-chevron-down"); + + isExpanded = true; + articleToggleTitle = "Collapse docstring"; + navArticleToggleTitle = "Collapse all docstrings"; + + $(".docstring section").slideDown(animationSpeed); + } + + $(this).prop("title", navArticleToggleTitle); + $(".docstring-article-toggle-button").prop("title", articleToggleTitle); + }); +}); + +function debounce(callback, timeout = 300) { + if (Date.now() - timer > timeout) { + callback(); + } + + clearTimeout(timer); + + timer = Date.now(); +} + +}) +//////////////////////////////////////////////////////////////////////////////// +require([], function() { +function addCopyButtonCallbacks() { + for (const el of document.getElementsByTagName("pre")) { + const button = document.createElement("button"); + button.classList.add("copy-button", "fa-solid", "fa-copy"); + button.setAttribute("aria-label", "Copy this code block"); + button.setAttribute("title", "Copy"); + + el.appendChild(button); + + const success = function () { + button.classList.add("success", "fa-check"); + button.classList.remove("fa-copy"); + }; + + const failure = function () { + button.classList.add("error", "fa-xmark"); + button.classList.remove("fa-copy"); + }; + + button.addEventListener("click", function () { + copyToClipboard(el.innerText).then(success, failure); + + setTimeout(function () { + button.classList.add("fa-copy"); + button.classList.remove("success", "fa-check", "fa-xmark"); + }, 5000); + }); + } +} + +function copyToClipboard(text) { + // clipboard API is only available in secure contexts + if (window.navigator && window.navigator.clipboard) { + return window.navigator.clipboard.writeText(text); + } else { + return new Promise(function (resolve, reject) { + try { + const el = document.createElement("textarea"); + el.textContent = text; + el.style.position = "fixed"; + el.style.opacity = 0; + document.body.appendChild(el); + el.select(); + document.execCommand("copy"); + + resolve(); + } catch (err) { + reject(err); + } finally { + document.body.removeChild(el); + } + }); + } +} + +if (document.readyState === "loading") { + document.addEventListener("DOMContentLoaded", addCopyButtonCallbacks); +} else { + addCopyButtonCallbacks(); +} + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery', 'headroom', 'headroom-jquery'], function($, Headroom) { + +// Manages the top navigation bar (hides it when the user starts scrolling down on the +// mobile). +window.Headroom = Headroom; // work around buggy module loading? +$(document).ready(function () { + $("#documenter .docs-navbar").headroom({ + tolerance: { up: 10, down: 10 }, + }); +}); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +$(document).ready(function () { + let meta = $("div[data-docstringscollapsed]").data(); + + if (meta?.docstringscollapsed) { + $("#documenter-article-toggle-button").trigger({ + type: "click", + noToggleAnimation: true, + }); + } +}); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +/* +To get an in-depth about the thought process you can refer: https://hetarth02.hashnode.dev/series/gsoc + +PSEUDOCODE: + +Searching happens automatically as the user types or adjusts the selected filters. +To preserve responsiveness, as much as possible of the slow parts of the search are done +in a web worker. Searching and result generation are done in the worker, and filtering and +DOM updates are done in the main thread. The filters are in the main thread as they should +be very quick to apply. This lets filters be changed without re-searching with minisearch +(which is possible even if filtering is on the worker thread) and also lets filters be +changed _while_ the worker is searching and without message passing (neither of which are +possible if filtering is on the worker thread) + +SEARCH WORKER: + +Import minisearch + +Build index + +On message from main thread + run search + find the first 200 unique results from each category, and compute their divs for display + note that this is necessary and sufficient information for the main thread to find the + first 200 unique results from any given filter set + post results to main thread + +MAIN: + +Launch worker + +Declare nonconstant globals (worker_is_running, last_search_text, unfiltered_results) + +On text update + if worker is not running, launch_search() + +launch_search + set worker_is_running to true, set last_search_text to the search text + post the search query to worker + +on message from worker + if last_search_text is not the same as the text in the search field, + the latest search result is not reflective of the latest search query, so update again + launch_search() + otherwise + set worker_is_running to false + + regardless, display the new search results to the user + save the unfiltered_results as a global + update_search() + +on filter click + adjust the filter selection + update_search() + +update_search + apply search filters by looping through the unfiltered_results and finding the first 200 + unique results that match the filters + + Update the DOM +*/ + +/////// SEARCH WORKER /////// + +function worker_function(documenterSearchIndex, documenterBaseURL, filters) { + importScripts( + "https://cdn.jsdelivr.net/npm/minisearch@6.1.0/dist/umd/index.min.js" + ); + + let data = documenterSearchIndex.map((x, key) => { + x["id"] = key; // minisearch requires a unique for each object + return x; + }); + + // list below is the lunr 2.1.3 list minus the intersect with names(Base) + // (all, any, get, in, is, only, which) and (do, else, for, let, where, while, with) + // ideally we'd just filter the original list but it's not available as a variable + const stopWords = new Set([ + "a", + "able", + "about", + "across", + "after", + "almost", + "also", + "am", + "among", + "an", + "and", + "are", + "as", + "at", + "be", + "because", + "been", + "but", + "by", + "can", + "cannot", + "could", + "dear", + "did", + "does", + "either", + "ever", + "every", + "from", + "got", + "had", + "has", + "have", + "he", + "her", + "hers", + "him", + "his", + "how", + "however", + "i", + "if", + "into", + "it", + "its", + "just", + "least", + "like", + "likely", + "may", + "me", + "might", + "most", + "must", + "my", + "neither", + "no", + "nor", + "not", + "of", + "off", + "often", + "on", + "or", + "other", + "our", + "own", + "rather", + "said", + "say", + "says", + "she", + "should", + "since", + "so", + "some", + "than", + "that", + "the", + "their", + "them", + "then", + "there", + "these", + "they", + "this", + "tis", + "to", + "too", + "twas", + "us", + "wants", + "was", + "we", + "were", + "what", + "when", + "who", + "whom", + "why", + "will", + "would", + "yet", + "you", + "your", + ]); + + let index = new MiniSearch({ + fields: ["title", "text"], // fields to index for full-text search + storeFields: ["location", "title", "text", "category", "page"], // fields to return with results + processTerm: (term) => { + let word = stopWords.has(term) ? null : term; + if (word) { + // custom trimmer that doesn't strip @ and !, which are used in julia macro and function names + word = word + .replace(/^[^a-zA-Z0-9@!]+/, "") + .replace(/[^a-zA-Z0-9@!]+$/, ""); + + word = word.toLowerCase(); + } + + return word ?? null; + }, + // add . as a separator, because otherwise "title": "Documenter.Anchors.add!", would not + // find anything if searching for "add!", only for the entire qualification + tokenize: (string) => string.split(/[\s\-\.]+/), + // options which will be applied during the search + searchOptions: { + prefix: true, + boost: { title: 100 }, + fuzzy: 2, + }, + }); + + index.addAll(data); + + /** + * Used to map characters to HTML entities. + * Refer: https://github.com/lodash/lodash/blob/main/src/escape.ts + */ + const htmlEscapes = { + "&": "&", + "<": "<", + ">": ">", + '"': """, + "'": "'", + }; + + /** + * Used to match HTML entities and HTML characters. + * Refer: https://github.com/lodash/lodash/blob/main/src/escape.ts + */ + const reUnescapedHtml = /[&<>"']/g; + const reHasUnescapedHtml = RegExp(reUnescapedHtml.source); + + /** + * Escape function from lodash + * Refer: https://github.com/lodash/lodash/blob/main/src/escape.ts + */ + function escape(string) { + return string && reHasUnescapedHtml.test(string) + ? string.replace(reUnescapedHtml, (chr) => htmlEscapes[chr]) + : string || ""; + } + + /** + * Make the result component given a minisearch result data object and the value + * of the search input as queryString. To view the result object structure, refer: + * https://lucaong.github.io/minisearch/modules/_minisearch_.html#searchresult + * + * @param {object} result + * @param {string} querystring + * @returns string + */ + function make_search_result(result, querystring) { + let search_divider = `
    `; + let display_link = + result.location.slice(Math.max(0), Math.min(50, result.location.length)) + + (result.location.length > 30 ? "..." : ""); // To cut-off the link because it messes with the overflow of the whole div + + if (result.page !== "") { + display_link += ` (${result.page})`; + } + + let textindex = new RegExp(`${querystring}`, "i").exec(result.text); + let text = + textindex !== null + ? result.text.slice( + Math.max(textindex.index - 100, 0), + Math.min( + textindex.index + querystring.length + 100, + result.text.length + ) + ) + : ""; // cut-off text before and after from the match + + text = text.length ? escape(text) : ""; + + let display_result = text.length + ? "..." + + text.replace( + new RegExp(`${escape(querystring)}`, "i"), // For first occurrence + '$&' + ) + + "..." + : ""; // highlights the match + + let in_code = false; + if (!["page", "section"].includes(result.category.toLowerCase())) { + in_code = true; + } + + // We encode the full url to escape some special characters which can lead to broken links + let result_div = ` + +
    +
    ${escape(result.title)}
    +
    ${result.category}
    +
    +

    + ${display_result} +

    +
    + ${display_link} +
    +     + ${search_divider} + `; + + return result_div; + } + + self.onmessage = function (e) { + let query = e.data; + let results = index.search(query, { + filter: (result) => { + // Only return relevant results + return result.score >= 1; + }, + }); + + // Pre-filter to deduplicate and limit to 200 per category to the extent + // possible without knowing what the filters are. + let filtered_results = []; + let counts = {}; + for (let filter of filters) { + counts[filter] = 0; + } + let present = {}; + + for (let result of results) { + cat = result.category; + cnt = counts[cat]; + if (cnt < 200) { + id = cat + "---" + result.location; + if (present[id]) { + continue; + } + present[id] = true; + filtered_results.push({ + location: result.location, + category: cat, + div: make_search_result(result, query), + }); + } + } + + postMessage(filtered_results); + }; +} + +// `worker = Threads.@spawn worker_function(documenterSearchIndex)`, but in JavaScript! +const filters = [ + ...new Set(documenterSearchIndex["docs"].map((x) => x.category)), +]; +const worker_str = + "(" + + worker_function.toString() + + ")(" + + JSON.stringify(documenterSearchIndex["docs"]) + + "," + + JSON.stringify(documenterBaseURL) + + "," + + JSON.stringify(filters) + + ")"; +const worker_blob = new Blob([worker_str], { type: "text/javascript" }); +const worker = new Worker(URL.createObjectURL(worker_blob)); + +/////// SEARCH MAIN /////// + +// Whether the worker is currently handling a search. This is a boolean +// as the worker only ever handles 1 or 0 searches at a time. +var worker_is_running = false; + +// The last search text that was sent to the worker. This is used to determine +// if the worker should be launched again when it reports back results. +var last_search_text = ""; + +// The results of the last search. This, in combination with the state of the filters +// in the DOM, is used compute the results to display on calls to update_search. +var unfiltered_results = []; + +// Which filter is currently selected +var selected_filter = ""; + +$(document).on("input", ".documenter-search-input", function (event) { + if (!worker_is_running) { + launch_search(); + } +}); + +function launch_search() { + worker_is_running = true; + last_search_text = $(".documenter-search-input").val(); + worker.postMessage(last_search_text); +} + +worker.onmessage = function (e) { + if (last_search_text !== $(".documenter-search-input").val()) { + launch_search(); + } else { + worker_is_running = false; + } + + unfiltered_results = e.data; + update_search(); +}; + +$(document).on("click", ".search-filter", function () { + if ($(this).hasClass("search-filter-selected")) { + selected_filter = ""; + } else { + selected_filter = $(this).text().toLowerCase(); + } + + // This updates search results and toggles classes for UI: + update_search(); +}); + +/** + * Make/Update the search component + */ +function update_search() { + let querystring = $(".documenter-search-input").val(); + + if (querystring.trim()) { + if (selected_filter == "") { + results = unfiltered_results; + } else { + results = unfiltered_results.filter((result) => { + return selected_filter == result.category.toLowerCase(); + }); + } + + let search_result_container = ``; + let modal_filters = make_modal_body_filters(); + let search_divider = `
    `; + + if (results.length) { + let links = []; + let count = 0; + let search_results = ""; + + for (var i = 0, n = results.length; i < n && count < 200; ++i) { + let result = results[i]; + if (result.location && !links.includes(result.location)) { + search_results += result.div; + count++; + links.push(result.location); + } + } + + if (count == 1) { + count_str = "1 result"; + } else if (count == 200) { + count_str = "200+ results"; + } else { + count_str = count + " results"; + } + let result_count = `
    ${count_str}
    `; + + search_result_container = ` +
    + ${modal_filters} + ${search_divider} + ${result_count} +
    + ${search_results} +
    +
    + `; + } else { + search_result_container = ` +
    + ${modal_filters} + ${search_divider} +
    0 result(s)
    +
    +
    No result found!
    + `; + } + + if ($(".search-modal-card-body").hasClass("is-justify-content-center")) { + $(".search-modal-card-body").removeClass("is-justify-content-center"); + } + + $(".search-modal-card-body").html(search_result_container); + } else { + if (!$(".search-modal-card-body").hasClass("is-justify-content-center")) { + $(".search-modal-card-body").addClass("is-justify-content-center"); + } + + $(".search-modal-card-body").html(` +
    Type something to get started!
    + `); + } +} + +/** + * Make the modal filter html + * + * @returns string + */ +function make_modal_body_filters() { + let str = filters + .map((val) => { + if (selected_filter == val.toLowerCase()) { + return `${val}`; + } else { + return `${val}`; + } + }) + .join(""); + + return ` +
    + Filters: + ${str} +
    `; +} + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +// Modal settings dialog +$(document).ready(function () { + var settings = $("#documenter-settings"); + $("#documenter-settings-button").click(function () { + settings.toggleClass("is-active"); + }); + // Close the dialog if X is clicked + $("#documenter-settings button.delete").click(function () { + settings.removeClass("is-active"); + }); + // Close dialog if ESC is pressed + $(document).keyup(function (e) { + if (e.keyCode == 27) settings.removeClass("is-active"); + }); +}); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +$(document).ready(function () { + let search_modal_header = ` +
    +
    +

    + + + + +

    +
    +
    + +
    +
    + `; + + let initial_search_body = ` +
    Type something to get started!
    + `; + + let search_modal_footer = ` +
    + + Ctrl + + / to search + + esc to close +
    + `; + + $(document.body).append( + ` + + ` + ); + + document.querySelector(".docs-search-query").addEventListener("click", () => { + openModal(); + }); + + document + .querySelector(".close-search-modal") + .addEventListener("click", () => { + closeModal(); + }); + + $(document).on("click", ".search-result-link", function () { + closeModal(); + }); + + document.addEventListener("keydown", (event) => { + if ((event.ctrlKey || event.metaKey) && event.key === "/") { + openModal(); + } else if (event.key === "Escape") { + closeModal(); + } + + return false; + }); + + // Functions to open and close a modal + function openModal() { + let searchModal = document.querySelector("#search-modal"); + + searchModal.classList.add("is-active"); + document.querySelector(".documenter-search-input").focus(); + } + + function closeModal() { + let searchModal = document.querySelector("#search-modal"); + let initial_search_body = ` +
    Type something to get started!
    + `; + + searchModal.classList.remove("is-active"); + document.querySelector(".documenter-search-input").blur(); + + if (!$(".search-modal-card-body").hasClass("is-justify-content-center")) { + $(".search-modal-card-body").addClass("is-justify-content-center"); + } + + $(".documenter-search-input").val(""); + $(".search-modal-card-body").html(initial_search_body); + } + + document + .querySelector("#search-modal .modal-background") + .addEventListener("click", () => { + closeModal(); + }); +}); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +// Manages the showing and hiding of the sidebar. +$(document).ready(function () { + var sidebar = $("#documenter > .docs-sidebar"); + var sidebar_button = $("#documenter-sidebar-button"); + sidebar_button.click(function (ev) { + ev.preventDefault(); + sidebar.toggleClass("visible"); + if (sidebar.hasClass("visible")) { + // Makes sure that the current menu item is visible in the sidebar. + $("#documenter .docs-menu a.is-active").focus(); + } + }); + $("#documenter > .docs-main").bind("click", function (ev) { + if ($(ev.target).is(sidebar_button)) { + return; + } + if (sidebar.hasClass("visible")) { + sidebar.removeClass("visible"); + } + }); +}); + +// Resizes the package name / sitename in the sidebar if it is too wide. +// Inspired by: https://github.com/davatron5000/FitText.js +$(document).ready(function () { + e = $("#documenter .docs-autofit"); + function resize() { + var L = parseInt(e.css("max-width"), 10); + var L0 = e.width(); + if (L0 > L) { + var h0 = parseInt(e.css("font-size"), 10); + e.css("font-size", (L * h0) / L0); + // TODO: make sure it survives resizes? + } + } + // call once and then register events + resize(); + $(window).resize(resize); + $(window).on("orientationchange", resize); +}); + +// Scroll the navigation bar to the currently selected menu item +$(document).ready(function () { + var sidebar = $("#documenter .docs-menu").get(0); + var active = $("#documenter .docs-menu .is-active").get(0); + if (typeof active !== "undefined") { + sidebar.scrollTop = active.offsetTop - sidebar.offsetTop - 15; + } +}); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +// Theme picker setup +$(document).ready(function () { + // onchange callback + $("#documenter-themepicker").change(function themepick_callback(ev) { + var themename = $("#documenter-themepicker option:selected").attr("value"); + if (themename === "auto") { + // set_theme(window.matchMedia('(prefers-color-scheme: dark)').matches ? 'dark' : 'light'); + window.localStorage.removeItem("documenter-theme"); + } else { + // set_theme(themename); + window.localStorage.setItem("documenter-theme", themename); + } + // We re-use the global function from themeswap.js to actually do the swapping. + set_theme_from_local_storage(); + }); + + // Make sure that the themepicker displays the correct theme when the theme is retrieved + // from localStorage + if (typeof window.localStorage !== "undefined") { + var theme = window.localStorage.getItem("documenter-theme"); + if (theme !== null) { + $("#documenter-themepicker option").each(function (i, e) { + e.selected = e.value === theme; + }); + } + } +}); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +// update the version selector with info from the siteinfo.js and ../versions.js files +$(document).ready(function () { + // If the version selector is disabled with DOCUMENTER_VERSION_SELECTOR_DISABLED in the + // siteinfo.js file, we just return immediately and not display the version selector. + if ( + typeof DOCUMENTER_VERSION_SELECTOR_DISABLED === "boolean" && + DOCUMENTER_VERSION_SELECTOR_DISABLED + ) { + return; + } + + var version_selector = $("#documenter .docs-version-selector"); + var version_selector_select = $("#documenter .docs-version-selector select"); + + version_selector_select.change(function (x) { + target_href = version_selector_select + .children("option:selected") + .get(0).value; + window.location.href = target_href; + }); + + // add the current version to the selector based on siteinfo.js, but only if the selector is empty + if ( + typeof DOCUMENTER_CURRENT_VERSION !== "undefined" && + $("#version-selector > option").length == 0 + ) { + var option = $( + "" + ); + version_selector_select.append(option); + } + + if (typeof DOC_VERSIONS !== "undefined") { + var existing_versions = version_selector_select.children("option"); + var existing_versions_texts = existing_versions.map(function (i, x) { + return x.text; + }); + DOC_VERSIONS.forEach(function (each) { + var version_url = documenterBaseURL + "/../" + each + "/"; + var existing_id = $.inArray(each, existing_versions_texts); + // if not already in the version selector, add it as a new option, + // otherwise update the old option with the URL and enable it + if (existing_id == -1) { + var option = $( + "" + ); + version_selector_select.append(option); + } else { + var option = existing_versions[existing_id]; + option.value = version_url; + option.disabled = false; + } + }); + } + + // only show the version selector if the selector has been populated + if (version_selector_select.children("option").length > 0) { + version_selector.toggleClass("visible"); + } +}); + +}) diff --git a/v0.3.0/assets/logo.svg b/v0.3.0/assets/logo.svg new file mode 100644 index 0000000..df89daf --- /dev/null +++ b/v0.3.0/assets/logo.svg @@ -0,0 +1,169 @@ + + + + + + + + + + + + + + + + + + + + + + + + + + + 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html.theme--documenter-dark .button.is-dark.is-outlined,fieldset[disabled] html.theme--documenter-dark .content kbd.button.is-outlined{background-color:transparent;border-color:#282f2f;box-shadow:none;color:#282f2f}html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined:hover,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined:hover,html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-hovered,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-hovered,html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined:focus,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined:focus,html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-focused,html.theme--documenter-dark .content 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.button.is-warning.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-warning.is-outlined{background-color:transparent;border-color:#ad8100;box-shadow:none;color:#ad8100}html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined:hover,html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined.is-hovered,html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined:focus,html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined.is-focused{background-color:#fff;color:#ad8100}html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined.is-loading:focus::after,html.theme--documenter-dark 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.button.is-danger.is-inverted.is-outlined.is-hovered,html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined:focus,html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined.is-focused{background-color:#fff;color:#9e1b0d}html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #9e1b0d #9e1b0d !important}html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--documenter-dark .button.is-danger.is-light{background-color:#fdeeec;color:#ec311d}html.theme--documenter-dark .button.is-danger.is-light:hover,html.theme--documenter-dark .button.is-danger.is-light.is-hovered{background-color:#fce3e0;border-color:transparent;color:#ec311d}html.theme--documenter-dark .button.is-danger.is-light:active,html.theme--documenter-dark .button.is-danger.is-light.is-active{background-color:#fcd8d5;border-color:transparent;color:#ec311d}html.theme--documenter-dark .button.is-small,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.button{font-size:.75rem}html.theme--documenter-dark .button.is-small:not(.is-rounded),html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.button:not(.is-rounded){border-radius:3px}html.theme--documenter-dark .button.is-normal{font-size:1rem}html.theme--documenter-dark .button.is-medium{font-size:1.25rem}html.theme--documenter-dark .button.is-large{font-size:1.5rem}html.theme--documenter-dark .button[disabled],fieldset[disabled] html.theme--documenter-dark .button{background-color:#8c9b9d;border-color:#5e6d6f;box-shadow:none;opacity:.5}html.theme--documenter-dark .button.is-fullwidth{display:flex;width:100%}html.theme--documenter-dark .button.is-loading{color:transparent !important;pointer-events:none}html.theme--documenter-dark .button.is-loading::after{position:absolute;left:calc(50% - (1em * 0.5));top:calc(50% - (1em * 0.5));position:absolute !important}html.theme--documenter-dark .button.is-static{background-color:#282f2f;border-color:#5e6d6f;color:#dbdee0;box-shadow:none;pointer-events:none}html.theme--documenter-dark .button.is-rounded,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.button{border-radius:9999px;padding-left:calc(1em + 0.25em);padding-right:calc(1em + 0.25em)}html.theme--documenter-dark .buttons{align-items:center;display:flex;flex-wrap:wrap;justify-content:flex-start}html.theme--documenter-dark .buttons .button{margin-bottom:0.5rem}html.theme--documenter-dark .buttons .button:not(:last-child):not(.is-fullwidth){margin-right:.5rem}html.theme--documenter-dark .buttons:last-child{margin-bottom:-0.5rem}html.theme--documenter-dark .buttons:not(:last-child){margin-bottom:1rem}html.theme--documenter-dark .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large){font-size:.75rem}html.theme--documenter-dark .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large):not(.is-rounded){border-radius:3px}html.theme--documenter-dark .buttons.are-medium .button:not(.is-small):not(.is-normal):not(.is-large){font-size:1.25rem}html.theme--documenter-dark .buttons.are-large .button:not(.is-small):not(.is-normal):not(.is-medium){font-size:1.5rem}html.theme--documenter-dark .buttons.has-addons .button:not(:first-child){border-bottom-left-radius:0;border-top-left-radius:0}html.theme--documenter-dark .buttons.has-addons .button:not(:last-child){border-bottom-right-radius:0;border-top-right-radius:0;margin-right:-1px}html.theme--documenter-dark .buttons.has-addons .button:last-child{margin-right:0}html.theme--documenter-dark .buttons.has-addons .button:hover,html.theme--documenter-dark .buttons.has-addons .button.is-hovered{z-index:2}html.theme--documenter-dark .buttons.has-addons .button:focus,html.theme--documenter-dark .buttons.has-addons .button.is-focused,html.theme--documenter-dark .buttons.has-addons .button:active,html.theme--documenter-dark .buttons.has-addons .button.is-active,html.theme--documenter-dark .buttons.has-addons .button.is-selected{z-index:3}html.theme--documenter-dark .buttons.has-addons .button:focus:hover,html.theme--documenter-dark .buttons.has-addons .button.is-focused:hover,html.theme--documenter-dark .buttons.has-addons .button:active:hover,html.theme--documenter-dark .buttons.has-addons .button.is-active:hover,html.theme--documenter-dark .buttons.has-addons .button.is-selected:hover{z-index:4}html.theme--documenter-dark .buttons.has-addons .button.is-expanded{flex-grow:1;flex-shrink:1}html.theme--documenter-dark .buttons.is-centered{justify-content:center}html.theme--documenter-dark .buttons.is-centered:not(.has-addons) .button:not(.is-fullwidth){margin-left:0.25rem;margin-right:0.25rem}html.theme--documenter-dark .buttons.is-right{justify-content:flex-end}html.theme--documenter-dark .buttons.is-right:not(.has-addons) .button:not(.is-fullwidth){margin-left:0.25rem;margin-right:0.25rem}@media screen and (max-width: 768px){html.theme--documenter-dark .button.is-responsive.is-small,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.is-responsive{font-size:.5625rem}html.theme--documenter-dark .button.is-responsive,html.theme--documenter-dark .button.is-responsive.is-normal{font-size:.65625rem}html.theme--documenter-dark .button.is-responsive.is-medium{font-size:.75rem}html.theme--documenter-dark .button.is-responsive.is-large{font-size:1rem}}@media screen and (min-width: 769px) and (max-width: 1055px){html.theme--documenter-dark .button.is-responsive.is-small,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.is-responsive{font-size:.65625rem}html.theme--documenter-dark .button.is-responsive,html.theme--documenter-dark .button.is-responsive.is-normal{font-size:.75rem}html.theme--documenter-dark .button.is-responsive.is-medium{font-size:1rem}html.theme--documenter-dark .button.is-responsive.is-large{font-size:1.25rem}}html.theme--documenter-dark .container{flex-grow:1;margin:0 auto;position:relative;width:auto}html.theme--documenter-dark .container.is-fluid{max-width:none !important;padding-left:32px;padding-right:32px;width:100%}@media screen and (min-width: 1056px){html.theme--documenter-dark .container{max-width:992px}}@media screen and (max-width: 1215px){html.theme--documenter-dark .container.is-widescreen:not(.is-max-desktop){max-width:1152px}}@media screen and (max-width: 1407px){html.theme--documenter-dark .container.is-fullhd:not(.is-max-desktop):not(.is-max-widescreen){max-width:1344px}}@media screen and (min-width: 1216px){html.theme--documenter-dark .container:not(.is-max-desktop){max-width:1152px}}@media screen and (min-width: 1408px){html.theme--documenter-dark .container:not(.is-max-desktop):not(.is-max-widescreen){max-width:1344px}}html.theme--documenter-dark .content li+li{margin-top:0.25em}html.theme--documenter-dark .content p:not(:last-child),html.theme--documenter-dark .content dl:not(:last-child),html.theme--documenter-dark .content ol:not(:last-child),html.theme--documenter-dark .content ul:not(:last-child),html.theme--documenter-dark .content blockquote:not(:last-child),html.theme--documenter-dark .content pre:not(:last-child),html.theme--documenter-dark .content table:not(:last-child){margin-bottom:1em}html.theme--documenter-dark .content 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html.theme--documenter-dark .navbar-dropdown,html.theme--documenter-dark .navbar-dropdown.is-boxed{border-radius:8px;border-top:none;box-shadow:0 8px 8px rgba(10,10,10,0.1), 0 0 0 1px rgba(10,10,10,0.1);display:block;opacity:0;pointer-events:none;top:calc(100% + (-4px));transform:translateY(-5px);transition-duration:86ms;transition-property:opacity, transform}html.theme--documenter-dark .navbar-dropdown.is-right{left:auto;right:0}html.theme--documenter-dark .navbar-divider{display:block}html.theme--documenter-dark .navbar>.container .navbar-brand,html.theme--documenter-dark .container>.navbar .navbar-brand{margin-left:-.75rem}html.theme--documenter-dark .navbar>.container .navbar-menu,html.theme--documenter-dark .container>.navbar .navbar-menu{margin-right:-.75rem}html.theme--documenter-dark .navbar.is-fixed-bottom-desktop,html.theme--documenter-dark .navbar.is-fixed-top-desktop{left:0;position:fixed;right:0;z-index:30}html.theme--documenter-dark 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a.navbar-item.is-active:not(:focus):not(:hover),html.theme--documenter-dark .navbar-link.is-active:not(:focus):not(:hover){background-color:rgba(0,0,0,0)}html.theme--documenter-dark .navbar-item.has-dropdown:focus .navbar-link,html.theme--documenter-dark .navbar-item.has-dropdown:hover .navbar-link,html.theme--documenter-dark .navbar-item.has-dropdown.is-active .navbar-link{background-color:rgba(0,0,0,0)}}html.theme--documenter-dark .hero.is-fullheight-with-navbar{min-height:calc(100vh - 4rem)}html.theme--documenter-dark .pagination{font-size:1rem;margin:-.25rem}html.theme--documenter-dark .pagination.is-small,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.pagination{font-size:.75rem}html.theme--documenter-dark .pagination.is-medium{font-size:1.25rem}html.theme--documenter-dark .pagination.is-large{font-size:1.5rem}html.theme--documenter-dark .pagination.is-rounded .pagination-previous,html.theme--documenter-dark #documenter .docs-sidebar 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    GitHub

    Examples

    Simulate data

    julia> using MultiResponseVarianceComponentModels, LinearAlgebra, Random
    julia> Random.seed!(6789)Random.TaskLocalRNG()
    julia> n = 1_000;  # n of observations
    julia> d = 4;      # n of responses
    julia> p = 10;     # n of covariates
    julia> m = 5;      # n of variance components
    julia> X = rand(n, p);
    julia> B = rand(p, d)10×4 Matrix{Float64}:
+ 0.866192  0.739958   0.951693   0.716391
+ 0.157873  0.633772   0.916356   0.068644
+ 0.818581  0.333021   0.0713489  0.451072
+ 0.707496  0.924568   0.0636189  0.618679
+ 0.357756  0.173016   0.982384   0.54638
+ 0.722163  0.0683381  0.343516   0.616161
+ 0.441486  0.64316    0.862809   0.400984
+ 0.812486  0.275565   0.477624   0.482087
+ 0.889787  0.669932   0.103051   0.582507
+ 0.331484  0.989292   0.785077   0.456195
    julia> V = [zeros(n, n) for _ in 1:m]; # kernel matrices
    julia> Σ = [zeros(d, d) for _ in 1:m]; # variance components
    julia> for i in 1:m
+           Vi = randn(n, n)
+           copy!(V[i], Vi' * Vi)
+           Σi = randn(d, d)
+           copy!(Σ[i], Σi' * Σi)
+       end
    julia> Ω = zeros(n * d, n * d); # overall nd-by-nd covariance matrix Ω
    julia> for i = 1:m
+           Ω += kron(Σ[i], V[i])
+       end
    julia> Ωchol = cholesky(Ω);
    julia> Y = X * B + reshape(Ωchol.L * randn(n * d), n, d)1000×4 Matrix{Float64}:
+ -270.382    -66.0256   233.653     158.94
+  168.935    354.423     14.8585    -54.193
+   85.8007   117.902   -217.707      65.1394
+ -136.574    -42.6014  -234.087      75.6404
+ -190.312   -210.051    105.117     -10.0791
+  153.414    187.179     20.691     -62.3589
+  164.97     -22.4103   -13.6301    161.682
+   21.9438  -184.643    231.526    -136.723
+ -284.549     79.7798   -37.1967   -136.943
+  -59.2492  -124.37    -160.587     112.235
+    ⋮
+  -82.4606   200.557     -4.16006   -47.2201
+  170.0       95.8103   222.034    -179.011
+   98.1247   -33.0932   -67.4875    -86.9752
+ -167.871    110.395   -184.76      -71.029
+  111.644    -23.6517   113.328    -219.276
+  -77.2094   168.859     65.7808     22.939
+  129.34    -155.123   -251.557     120.656
+ -359.121    -82.2576   -82.0063     10.8301
+  -77.7126   -90.7158   -79.5722     92.9695
    Note

    In the case of heritability and genetic correlation analyses, one can use classic genetic relationship matrices (GRMs) for $\boldsymbol{V}_i$'s, which in turn can be constructed using SnpArrays.jl.

    Maximum likelihood estimation

    julia> model = MRVCModel(Y, X, V)A multivariate response variance component model
+   * number of responses: 4
+   * number of observations: 1000
+   * number of fixed effects: 10
+   * number of variance components: 5
    julia> @timev fit!(model)[ Info: Running MM algorithm for ML estimation
+iter = 0, logl = -26845.09612499673
+iter = 1, logl = -25695.041826875506
+iter = 2, logl = -25293.37779878975
+iter = 3, logl = -25159.24906847003
+iter = 4, logl = -25107.52993220223
+iter = 5, logl = -25081.38428962063
+iter = 6, logl = -25064.352419795112
+iter = 7, logl = -25051.625744592016
+iter = 8, logl = -25041.61006910993
+iter = 9, logl = -25033.604422863456
+iter = 10, logl = -25027.18042662134
+iter = 11, logl = -25022.017854787307
+iter = 12, logl = -25017.85980986688
+iter = 13, logl = -25014.498019681763
+iter = 14, logl = -25011.76503965001
+iter = 15, logl = -25009.527835987446
+iter = 16, logl = -25007.68180780364
+iter = 17, logl = -25006.145330198324
+iter = 18, logl = -25004.854995838356
+iter = 19, logl = -25003.761633006412
+iter = 20, logl = -25002.827079011357
+iter = 21, logl = -25002.02162797271
+iter = 22, logl = -25001.322047241047
+iter = 23, logl = -25000.710054799467
+iter = 24, logl = -25000.171160151207
+iter = 25, logl = -24999.693786243588
+iter = 26, logl = -24999.26860570048
+iter = 27, logl = -24998.888038955105
+iter = 28, logl = -24998.545873977288
+iter = 29, logl = -24998.236977041215
+iter = 30, logl = -24997.95707160835
+iter = 31, logl = -24997.702568235134
+iter = 32, logl = -24997.470432811737
+iter = 33, logl = -24997.258083718323
+iter = 34, logl = -24997.063310912818
+iter = 35, logl = -24996.884211761542
+iter = 36, logl = -24996.719139735058
+iter = 37, logl = -24996.566663066376
+iter = 38, logl = -24996.425531184308
+iter = 39, logl = -24996.294647261133
+iter = 40, logl = -24996.173045607276
+iter = 41, logl = -24996.05987293569
+iter = 42, logl = -24995.95437274284
+iter = 43, logl = -24995.855872212444
+iter = 44, logl = -24995.7637711789
+iter = 45, logl = -24995.67753277965
+iter = 46, logl = -24995.596675504443
+iter = 47, logl = -24995.520766401234
+iter = 48, logl = -24995.449415250514
+iter = 49, logl = -24995.38226954833
+iter = 50, logl = -24995.31901017304
+iter = 51, logl = -24995.259347628595
+iter = 52, logl = -24995.203018777862
+iter = 53, logl = -24995.14978399378
+iter = 54, logl = -24995.09942466663
+iter = 55, logl = -24995.051741018295
+iter = 56, logl = -24995.006550180144
+iter = 57, logl = -24994.963684497296
+iter = 58, logl = -24994.922990031344
+iter = 59, logl = -24994.88432523246
+iter = 60, logl = -24994.84755976209
+iter = 61, logl = -24994.812573442905
+iter = 62, logl = -24994.779255325782
+iter = 63, logl = -24994.74750285241
+iter = 64, logl = -24994.717221108924
+iter = 65, logl = -24994.688322154096
+iter = 66, logl = -24994.660724417583
+iter = 67, logl = -24994.63435215651
+iter = 68, logl = -24994.609134967097
+iter = 69, logl = -24994.585007342
+[ Info: Updates converged!
+[ Info: Calculating standard errors
+ 60.387438 seconds (15.64 M allocations: 992.951 MiB, 0.31% gc time, 8.42% compilation time)
+elapsed time (ns):  60387438125
+gc time (ns):       185322875
+bytes allocated:    1041185092
+pool allocs:        15607324
+non-pool GC allocs: 26823
+malloc() calls:     932
+realloc() calls:    107
+free() calls:       335
+minor collections:  1
+full collections:   0
+Converged after 70 iterations.

    Then variance components and mean effects estimates can be accessed through

    julia> model.Σ5-element Vector{Matrix{Float64}}:
+ [2.8964235833864374 -0.9547701186106777 1.6856925389781934 0.8185656173741515; -0.9547701186106778 7.474693619948036 -4.744935846247633 -0.601253052759175; 1.6856925389781932 -4.744935846247633 8.639624062398605 0.4049276157435974; 0.8185656173741512 -0.601253052759175 0.4049276157435973 1.0145168589537352]
+ [3.776260289140289 2.3919496497788035 -0.3666324933910243 -1.365148782876448; 2.391949649778804 1.9914356452751707 -0.2630505419731652 -1.5892702567044454; -0.36663249339102433 -0.26305054197316524 4.0912210911875855 -2.287177667967102; -1.365148782876448 -1.5892702567044454 -2.287177667967102 4.526163931034507]
+ [3.1157472905367367 -0.08262940589305556 -0.27318718635822953 0.4179078660737166; -0.08262940589305548 3.4556143634179275 -0.20341922639890309 0.6648870938984194; -0.27318718635822953 -0.20341922639890314 1.9155083377403541 0.49426216391703903; 0.41790786607371666 0.6648870938984194 0.49426216391703903 0.5435108874298807]
+ [3.6027030953820782 1.2366936673894326 0.3637181414535267 0.06851794976775202; 1.2366936673894326 0.9715515927327851 1.6405295499121633 0.5971896668720472; 0.3637181414535266 1.6405295499121637 5.828012183031251 0.9752007240509573; 0.06851794976775206 0.5971896668720472 0.9752007240509573 0.9502953338899155]
+ [5.669808082536949 -2.01083534935516 1.1222606709505223 1.4405860817195557; -2.0108353493551605 6.032248061565413 9.439926228569771 -2.150730923690082; 1.1222606709505227 9.439926228569773 19.179102992380216 -3.054827357292586; 1.4405860817195555 -2.150730923690082 -3.054827357292586 3.445429926626015]
    julia> model.B10×4 Matrix{Float64}:
+   7.76641     0.0581691   30.0176     10.8424
+  -1.70871    17.0653      26.3436      5.24079
+  -4.99575    28.325      -20.9966    -13.1088
+ -10.2939     -6.00867     12.3106    -16.3959
+  -3.07093     0.733837    15.8845      1.17996
+  14.1045     -3.92946     11.5408     11.2497
+  -1.47949    -8.30862      5.49722    -0.213233
+   6.03975     5.27422      0.189634    3.90586
+  -0.632607  -14.288      -29.194       4.9749
+   6.17054   -13.2489     -23.6247     -3.52297
    julia> hcat(vec(B), vec(model.B))40×2 Matrix{Float64}:
+ 0.866192    7.76641
+ 0.157873   -1.70871
+ 0.818581   -4.99575
+ 0.707496  -10.2939
+ 0.357756   -3.07093
+ 0.722163   14.1045
+ 0.441486   -1.47949
+ 0.812486    6.03975
+ 0.889787   -0.632607
+ 0.331484    6.17054
+ ⋮
+ 0.068644    5.24079
+ 0.451072  -13.1088
+ 0.618679  -16.3959
+ 0.54638     1.17996
+ 0.616161   11.2497
+ 0.400984   -0.213233
+ 0.482087    3.90586
+ 0.582507    4.9749
+ 0.456195   -3.52297
    julia> reduce(hcat, [hcat(vec(Σ[i]), vec(model.Σ[i])) for i in 1:m])16×10 Matrix{Float64}:
+  2.59353    2.89642    3.32588    …  3.6027      5.37835    5.66981
+ -0.908817  -0.95477    2.21659       1.23669    -1.43756   -2.01084
+  1.58682    1.68569   -0.0345923     0.363718    1.65263    1.12226
+  0.365451   0.818566  -1.36959       0.0685179   0.983537   1.44059
+ -0.908817  -0.95477    2.21659       1.23669    -1.43756   -2.01084
+  7.12022    7.47469    1.75764    …  0.971552    6.84955    6.03225
+ -4.72969   -4.74494   -0.0950186     1.64053     9.98583    9.43993
+ -0.430129  -0.601253  -1.55052       0.59719    -1.98954   -2.15073
+  1.58682    1.68569   -0.0345923     0.363718    1.65263    1.12226
+ -4.72969   -4.74494   -0.0950186     1.64053     9.98583    9.43993
+  6.22778    8.63962    4.06617    …  5.82801    19.2083    19.1791
+  0.969228   0.404928  -2.51336       0.975201   -3.5885    -3.05483
+  0.365451   0.818566  -1.36959       0.0685179   0.983537   1.44059
+ -0.430129  -0.601253  -1.55052       0.59719    -1.98954   -2.15073
+  0.969228   0.404928  -2.51336       0.975201   -3.5885    -3.05483
+  1.2446     1.01452    5.02933    …  0.950295    2.85342    3.44543

    Standard errors

    Sampling variance and covariance of these estimates are

    julia> model.Σcov50×50 Matrix{Float64}:
+  0.47412      -0.00272341    0.108306     …  -0.00114518   -0.000725468
+ -0.00272341    0.280366     -0.0353382       -0.000510353   0.000627446
+  0.108306     -0.0353382     0.483991        -0.00663509    0.000814462
+  0.0597015    -0.0241462    -0.00476179      -0.00287831   -0.00410306
+ -0.000531375  -0.000740352  -0.00227336       0.00175773   -0.000550985
+ -5.46266e-6    0.0615992    -0.0084444    …   0.00754037   -0.000713865
+ -7.80744e-5    0.0347709    -0.00356761      -0.00609519    0.00360991
+  0.0240561    -0.0132044     0.213683         0.0166641    -0.000929684
+  0.0133385    -0.00919492    0.0562126       -0.0423021     0.00471451
+  0.00739022   -0.00597224   -0.0030313        0.00471978   -0.0236748
+  ⋮                                        ⋱
+ -0.00769785   -0.0406075    -0.012298         0.0234008    -0.0161668
+ -0.0133522    -0.0122384    -0.0861289        0.094258     -0.0224369
+ -0.00838137    0.00335528    0.00422692       0.0168207     0.0502388
+ -0.000645754  -0.00632483   -0.00175497      -0.0798787     0.0198685
+ -0.00102918   -0.00637199   -0.00825901   …  -0.187875      0.0275525
+ -0.000640514  -0.00315513   -0.000651312      0.140072     -0.0624593
+ -0.00183032   -0.00321512   -0.0232591       -0.36789       0.0382077
+ -0.00114518   -0.000510353  -0.00663509       0.426481     -0.0867558
+ -0.000725468   0.000627446   0.000814462     -0.0867558     0.189954
    julia> model.Bcov40×40 Matrix{Float64}:
+ 157.174     -19.3776    -19.4223    …   -0.257223   -1.24371    -1.90444
+ -19.3776    161.611     -22.5065        -1.3117     -1.42903    -1.33411
+ -19.4223    -22.5065    150.341         -1.38577    -2.56388    -1.25275
+ -14.6304    -12.8425    -19.2089        -2.29579    -2.03511    -1.00321
+ -11.6747    -13.9337    -12.4649        -0.198984   -1.44227    -2.31342
+ -24.2206    -18.4552     -7.96731   …   -3.00702    -1.14626    -0.923721
+ -13.3431    -17.4951    -15.8672        -1.57057    -0.945035   -1.77244
+ -12.7248    -19.4206    -11.7579        13.2375     -1.25172    -1.08592
+ -17.2277    -19.1384    -19.5393        -1.22189    12.8571     -0.396249
+ -15.0949    -12.6969    -15.9363        -1.23333    -0.422978   12.5215
+   ⋮                                 ⋱
+  -2.12779    13.4419     -1.25447       -9.26816    -9.00194    -6.01701
+  -0.752161   -1.49914    13.5685        -5.62018    -9.24482    -8.16962
+  -0.737933   -0.925351   -1.71971      -11.7708     -8.59325   -10.0889
+  -1.49387    -1.2578     -1.04615       -5.98367   -12.911     -13.1438
+  -1.72094    -2.33376    -0.495492  …   -9.81655    -5.37389    -8.16594
+  -1.53932    -0.763366   -2.22368       -9.36946    -7.53599   -10.4206
+  -0.257223   -1.3117     -1.38577       79.0721     -7.3782     -9.48116
+  -1.24371    -1.42903    -2.56388       -7.3782     75.6156     -0.921828
+  -1.90444    -1.33411    -1.25275       -9.48116    -0.921828   78.0356

    Corresponding standard error of these estimates are

    julia> sqrt.(diag(model.Σcov))50-element Vector{Float64}:
+ 0.6885637727726415
+ 0.529495666135442
+ 0.6956949396229591
+ 0.33645040355906397
+ 0.8084217056989712
+ 0.7442584972630495
+ 0.3639694672208399
+ 1.3688056422005268
+ 0.4677057749056628
+ 0.3179680333750869
+ ⋮
+ 0.5524130983848243
+ 0.8823995615262973
+ 0.43111265513125097
+ 0.7402837878083752
+ 0.9231964895406697
+ 0.41384022887389565
+ 1.8662336914254827
+ 0.6530553700433331
+ 0.43583743574349515
    julia> sqrt.(diag(model.Bcov))40-element Vector{Float64}:
+ 12.536904651555659
+ 12.712633608569577
+ 12.26134556777767
+ 12.169443903108293
+ 12.663455285629615
+ 11.870615133041053
+ 12.528437624570504
+ 12.51912728097892
+ 12.39394332241073
+ 12.510940364508672
+  ⋮
+  8.869558232699935
+  8.568572559479021
+  8.594848108237025
+  8.973590296072445
+  8.287660643177881
+  8.80370479520269
+  8.892247260593642
+  8.695724537824539
+  8.833773378129575

    Residual maximum likelihood estimation

    For REML estimation, you can instead:

    julia> model = MRVCModel(Y, X, V; reml = true)A multivariate response variance component model
+   * number of responses: 4
+   * number of observations: 1000
+   * number of fixed effects: 10
+   * number of variance components: 5
    julia> @timev fit!(model)[ Info: Running MM algorithm for REML estimation
+iter = 0, logl = -26597.537258425444
+iter = 1, logl = -25455.46070096719
+iter = 2, logl = -25062.15815081145
+iter = 3, logl = -24929.900271068564
+iter = 4, logl = -24878.96910611469
+iter = 5, logl = -24853.289345291265
+iter = 6, logl = -24836.614727024702
+iter = 7, logl = -24824.192330128142
+iter = 8, logl = -24814.4419685779
+iter = 9, logl = -24806.666946417736
+iter = 10, logl = -24800.441634611918
+iter = 11, logl = -24795.448814457188
+iter = 12, logl = -24791.434959065882
+iter = 13, logl = -24788.19527844777
+iter = 14, logl = -24785.565671601264
+iter = 15, logl = -24783.416144877603
+iter = 16, logl = -24781.644754384706
+iter = 17, logl = -24780.17213209585
+iter = 18, logl = -24778.936749268694
+iter = 19, logl = -24777.890976339873
+iter = 20, logl = -24776.997905413984
+iter = 21, logl = -24776.228846668713
+iter = 22, logl = -24775.561388912047
+iter = 23, logl = -24774.977914973028
+iter = 24, logl = -24774.46447416872
+iter = 25, logl = -24774.009929862073
+iter = 26, logl = -24773.605316168836
+iter = 27, logl = -24773.243352268248
+iter = 28, logl = -24772.918074830915
+iter = 29, logl = -24772.62455872271
+iter = 30, logl = -24772.358703657163
+iter = 31, logl = -24772.117070189604
+iter = 32, logl = -24771.896752736797
+iter = 33, logl = -24771.695280510114
+iter = 34, logl = -24771.51053960551
+iter = 35, logl = -24771.340711232333
+iter = 36, logl = -24771.18422234522
+iter = 37, logl = -24771.03970587292
+iter = 38, logl = -24770.90596843845
+iter = 39, logl = -24770.781963965892
+iter = 40, logl = -24770.66677195723
+iter = 41, logl = -24770.55957949347
+iter = 42, logl = -24770.459666236435
+iter = 43, logl = -24770.36639185783
+iter = 44, logl = -24770.27918545024
+iter = 45, logl = -24770.19753656323
+iter = 46, logl = -24770.120987582057
+iter = 47, logl = -24770.049127218048
+iter = 48, logl = -24769.981584929763
+iter = 49, logl = -24769.918026121268
+iter = 50, logl = -24769.85814799603
+iter = 51, logl = -24769.80167596465
+iter = 52, logl = -24769.74836052275
+iter = 53, logl = -24769.69797452873
+iter = 54, logl = -24769.65031082431
+iter = 55, logl = -24769.60518014765
+iter = 56, logl = -24769.56240929961
+iter = 57, logl = -24769.521839527268
+iter = 58, logl = -24769.48332509584
+iter = 59, logl = -24769.446732024946
+iter = 60, logl = -24769.41193696552
+iter = 61, logl = -24769.378826202505
+iter = 62, logl = -24769.34729476516
+iter = 63, logl = -24769.317245631824
+iter = 64, logl = -24769.288589020118
+iter = 65, logl = -24769.261241748998
+iter = 66, logl = -24769.235126666277
+iter = 67, logl = -24769.210172133422
+iter = 68, logl = -24769.186311562113
+[ Info: Updates converged!
+[ Info: Calculating standard errors
+ 54.412696 seconds (5.65 k allocations: 15.296 MiB, 0.01% compilation time)
+elapsed time (ns):  54412695958
+gc time (ns):       0
+bytes allocated:    16039272
+pool allocs:        5451
+non-pool GC allocs: 196
+malloc() calls:     7
+free() calls:       0
+minor collections:  0
+full collections:   0
+Converged after 69 iterations.

    Variance components and mean effects estimates and their standard errors can be accessed through:

    julia> model.Σ5-element Vector{Matrix{Float64}}:
+ [2.9338500446816433 -0.9556577622339366 1.681944315565907 0.82097923904986; -0.9556577622339366 7.523441348688346 -4.724626084973893 -0.6045832212502729; 1.6819443155659073 -4.724626084973893 8.695787597891757 0.3975584198892441; 0.8209792390498599 -0.6045832212502729 0.397558419889244 1.0347822362989731]
+ [3.8129402322306207 2.3985158343212087 -0.3501640293021632 -1.361452145970671; 2.3985158343212087 2.0221274445059243 -0.26055606213368854 -1.595835001991699; -0.3501640293021632 -0.26055606213368826 4.164419136795955 -2.293103534511798; -1.3614521459706712 -1.595835001991699 -2.293103534511798 4.545249416292042]
+ [3.1531618876945027 -0.08093870739430353 -0.27534179496140915 0.4229588473703198; -0.08093870739430353 3.4956494601351706 -0.2073584361891652 0.6627085784490545; -0.27534179496140915 -0.20735843618916527 1.976397882530889 0.5036124449893044; 0.4229588473703198 0.6627085784490545 0.5036124449893044 0.5585662911372018]
+ [3.635801209038145 1.250204732685735 0.3660670881565596 0.07071405261549359; 1.2502047326857344 0.983771630227471 1.65237696435348 0.6004734046277869; 0.3660670881565597 1.6523769643534798 5.891632852330085 0.9668222589182407; 0.07071405261549359 0.6004734046277869 0.9668222589182407 0.9633632310978463]
+ [5.704238993549306 -2.02059232700069 1.131104825718486 1.4423754007372802; -2.0205923270006902 6.0631121455622194 9.453887154496428 -2.1594622671919748; 1.131104825718486 9.453887154496428 19.22805021076556 -3.065314862756238; 1.4423754007372802 -2.1594622671919748 -3.065314862756238 3.4673781742053422]
    julia> model.B_reml10×4 Matrix{Float64}:
+   7.73978     0.0626092   29.9317     10.8401
+  -1.72827    17.0507      26.4331      5.24119
+  -4.99965    28.2624     -20.984     -13.0466
+ -10.2505     -5.98131     12.3119    -16.3662
+  -3.06092     0.788225    15.9509      1.1041
+  14.1047     -3.98098     11.5646     11.254
+  -1.45427    -8.264        5.3133     -0.233005
+   6.05113     5.20727      0.224061    3.95755
+  -0.679987  -14.213      -29.2147      4.91466
+   6.17329   -13.234      -23.6101     -3.48756
    julia> hcat(vec(B), vec(model.B_reml))40×2 Matrix{Float64}:
+ 0.866192    7.73978
+ 0.157873   -1.72827
+ 0.818581   -4.99965
+ 0.707496  -10.2505
+ 0.357756   -3.06092
+ 0.722163   14.1047
+ 0.441486   -1.45427
+ 0.812486    6.05113
+ 0.889787   -0.679987
+ 0.331484    6.17329
+ ⋮
+ 0.068644    5.24119
+ 0.451072  -13.0466
+ 0.618679  -16.3662
+ 0.54638     1.1041
+ 0.616161   11.254
+ 0.400984   -0.233005
+ 0.482087    3.95755
+ 0.582507    4.91466
+ 0.456195   -3.48756
    julia> reduce(hcat, [hcat(vec(Σ[i]), vec(model.Σ[i])) for i in 1:m])16×10 Matrix{Float64}:
+  2.59353    2.93385    3.32588    …  3.6358      5.37835    5.70424
+ -0.908817  -0.955658   2.21659       1.2502     -1.43756   -2.02059
+  1.58682    1.68194   -0.0345923     0.366067    1.65263    1.1311
+  0.365451   0.820979  -1.36959       0.0707141   0.983537   1.44238
+ -0.908817  -0.955658   2.21659       1.2502     -1.43756   -2.02059
+  7.12022    7.52344    1.75764    …  0.983772    6.84955    6.06311
+ -4.72969   -4.72463   -0.0950186     1.65238     9.98583    9.45389
+ -0.430129  -0.604583  -1.55052       0.600473   -1.98954   -2.15946
+  1.58682    1.68194   -0.0345923     0.366067    1.65263    1.1311
+ -4.72969   -4.72463   -0.0950186     1.65238     9.98583    9.45389
+  6.22778    8.69579    4.06617    …  5.89163    19.2083    19.2281
+  0.969228   0.397558  -2.51336       0.966822   -3.5885    -3.06531
+  0.365451   0.820979  -1.36959       0.0707141   0.983537   1.44238
+ -0.430129  -0.604583  -1.55052       0.600473   -1.98954   -2.15946
+  0.969228   0.397558  -2.51336       0.966822   -3.5885    -3.06531
+  1.2446     1.03478    5.02933    …  0.963363    2.85342    3.46738
    julia> model.Σcov50×50 Matrix{Float64}:
+  0.49577      -0.00244778    0.111934     …  -0.00120099   -0.000758647
+ -0.00244778    0.29227      -0.0343723       -0.000533875   0.00067098
+  0.111934     -0.0343723     0.505528        -0.00697209    0.000865612
+  0.061864     -0.0253519    -0.00568503      -0.00302124   -0.00430738
+ -0.000533068  -0.000402843  -0.00233763       0.00189715   -0.000601019
+  5.60305e-5    0.0635175    -0.00787928   …   0.00809095   -0.000772694
+ -4.0096e-5     0.0359266    -0.00341432      -0.00646355    0.00386484
+  0.0245991    -0.0126117     0.220813         0.0177754    -0.000996935
+  0.0136707    -0.00925492    0.0581185       -0.0446667     0.0050127
+  0.00759217   -0.00622591   -0.00328015       0.00502603   -0.0249292
+  ⋮                                        ⋱
+ -0.00795163   -0.0428182    -0.0129844        0.0236794    -0.0165915
+ -0.0140199    -0.012921     -0.0907524        0.0966537    -0.0229699
+ -0.00877942    0.00359818    0.00449712       0.0176113     0.0517196
+ -0.000667567  -0.00656994   -0.0018067       -0.0818833     0.0205321
+ -0.00106014   -0.00669857   -0.00860587   …  -0.193476      0.0283973
+ -0.000657675  -0.0033219    -0.000694922      0.144019     -0.0647023
+ -0.0019256    -0.00341227   -0.024527        -0.379053      0.0392726
+ -0.00120099   -0.000533875  -0.00697209       0.441745     -0.089645
+ -0.000758647   0.00067098    0.000865612     -0.089645      0.197463
    julia> model.Bcov_reml40×40 Matrix{Float64}:
+ 158.891     -19.5994    -19.6371   …   -0.270249   -1.25569    -1.91414
+ -19.5994    163.384     -22.755        -1.32842    -1.44256    -1.34338
+ -19.6371    -22.755     151.987        -1.39442    -2.57948    -1.26777
+ -14.7913    -12.9925    -19.412        -2.31154    -2.05524    -1.01497
+ -11.8093    -14.0871    -12.5893       -0.204628   -1.46038    -2.33716
+ -24.4932    -18.6219     -8.06361  …   -3.02449    -1.15581    -0.939094
+ -13.4737    -17.7017    -16.0374       -1.58474    -0.955878   -1.787
+ -12.8751    -19.6214    -11.8827       13.362      -1.26469    -1.10389
+ -17.4026    -19.3504    -19.7707       -1.23472    12.9793     -0.399421
+ -15.2551    -12.8481    -16.103        -1.24913    -0.426619   12.6487
+   ⋮                                ⋱
+  -2.14636    13.5663     -1.27382      -9.38304    -9.11398    -6.09505
+  -0.765165   -1.51854    13.6867       -5.69139    -9.3694     -8.2553
+  -0.7517     -0.930899   -1.73573     -11.9126     -8.7008    -10.1908
+  -1.50467    -1.26692    -1.05451      -6.03856   -13.0461    -13.3113
+  -1.74002    -2.34396    -0.50063  …   -9.94374    -5.4518     -8.24799
+  -1.54677    -0.78006    -2.23457      -9.47418    -7.6325    -10.5424
+  -0.270249   -1.32842    -1.39442      80.0047     -7.46137    -9.59586
+  -1.25569    -1.44256    -2.57948      -7.46137    76.5166     -0.942239
+  -1.91414    -1.34338    -1.26777      -9.59586    -0.942239   78.9404
    julia> sqrt.(diag(model.Σcov))50-element Vector{Float64}:
+ 0.7041093807634108
+ 0.5406201064474182
+ 0.7110051601946071
+ 0.3444911257855926
+ 0.8243533172952108
+ 0.7594961262777526
+ 0.37223105634402
+ 1.399043387180137
+ 0.47895976215151104
+ 0.32621217092743393
+ ⋮
+ 0.5632649585154991
+ 0.8978995129849152
+ 0.4392731438996924
+ 0.7554591568000877
+ 0.9384463145136606
+ 0.4220581137074655
+ 1.8966703307591692
+ 0.6646389147784445
+ 0.44436833540465226
    julia> sqrt.(diag(model.Bcov_reml))40-element Vector{Float64}:
+ 12.60518354143135
+ 12.782193317524683
+ 12.328308752465436
+ 12.23646139990698
+ 12.732636509969742
+ 11.934986834081903
+ 12.597350995825984
+ 12.586984559735523
+ 12.461643282682688
+ 12.578545874909784
+  ⋮
+  8.923342209927311
+  8.620236102433564
+  8.645876453902996
+  9.026071287093929
+  8.337457783234893
+  8.856449014354963
+  8.944535606024061
+  8.747376305559317
+  8.884843293186195

    Estimation only

    Calculating standard errors can be memory-consuming, so you could instead forego such calculation via:

    model = MRVCModel(Y, X, V; se = false) # or model = MRVCModel(Y, X, V; se = false, reml = true)
+@timev fit!(model)

    Missing response

    You can also fit data with missing response. For example:

    Y_miss = Matrix{Union{eltype(Y), Missing}}(missing, size(Y))
+copy!(Y_miss, Y)
+Y_miss[rand(1:length(Y_miss), n)] .= missing
+
+model = MRVCModel(Y_miss, X, V, se = false)
+@timev fit!(model)
    diff --git a/v0.3.0/index.html b/v0.3.0/index.html new file mode 100644 index 0000000..8e5fb24 --- /dev/null +++ b/v0.3.0/index.html @@ -0,0 +1,3 @@ + +Home · MRVCModels.jl
    GitHub

    MultiResponseVarianceComponentModels

    MRVCModels.jl is a package for fitting and testing multivariate response variance components linear mixed models of form

    \[\text{vec}\ \boldsymbol{Y} \sim \mathcal{N}(\text{vec}(\boldsymbol{X} \boldsymbol{B}), \sum_{i=1}^m \boldsymbol{\Gamma}_i \otimes \boldsymbol{V}_i),\]

    where $\boldsymbol{Y}$ and $\boldsymbol{X}$ are $n \times d$ response and $n \times p$ predictor matrices, respectively, and $\boldsymbol{V}_1, \ldots, \boldsymbol{V}_m$ are $m$ known $n \times n$ positive semidefinite matrices. $\text{vec}\ \boldsymbol{Y}$ creates an $nd \times 1$ vector from $\boldsymbol{Y}$ by stacking its columns and $\otimes$ denotes the Kronecker product. The parameters of the model include $p \times d$ mean effects $\boldsymbol{B}$ and $d \times d$ variance components ($\boldsymbol{\Gamma}_1, \dots, \boldsymbol{\Gamma}_m$), which MRVCModels.jl estimates through either minorization-maximization (MM) or expectation–maximization (EM) algorithms.

    Note

    MRVCModels.jl is not suitable for biobank-scale data. We recommend using this package for datasets of size up to $n \cdot d \approx 50000$. This package can also handle data with missing response.

    Installation

    To use MRVCModels.jl, type:

    using Pkg
+Pkg.add(url = "https://github.com/Hua-Zhou/MultiResponseVarianceComponentModels.jl.git")

    This documentation was built using Documenter.jl.

    Documentation built 2024-06-08T07:53:17.178 with Julia 1.10.4
    diff --git a/v0.3.0/objects.inv b/v0.3.0/objects.inv new file mode 100644 index 0000000..34460df Binary files /dev/null and b/v0.3.0/objects.inv differ diff --git a/v0.3.0/search_index.js b/v0.3.0/search_index.js new file mode 100644 index 0000000..6b32302 --- /dev/null +++ b/v0.3.0/search_index.js @@ -0,0 +1,3 @@ +var documenterSearchIndex = {"docs": +[{"location":"api/#API","page":"API","title":"API","text":"","category":"section"},{"location":"api/","page":"API","title":"API","text":"","category":"page"},{"location":"api/","page":"API","title":"API","text":"Modules = [MultiResponseVarianceComponentModels]","category":"page"},{"location":"api/#MultiResponseVarianceComponentModels.MultiResponseVarianceComponentModels","page":"API","title":"MultiResponseVarianceComponentModels.MultiResponseVarianceComponentModels","text":"MultiResponseVarianceComponentModels.jl or MRVCModels.jl permits maximum likelihood (ML) and residual maximum likelihood (REML) estimation as well as inference for multivariate response variance components linear mixed models.\n\n\n\n\n\n","category":"module"},{"location":"api/#MultiResponseVarianceComponentModels.MRVCModel-Union{Tuple{T}, Tuple{Union{AbstractMatrix{T}, AbstractArray{Union{Missing, T}, 2}}, Union{Nothing, AbstractMatrix{T}}, Vector{<:AbstractMatrix{T}}}} where T<:Union{Float32, Float64}","page":"API","title":"MultiResponseVarianceComponentModels.MRVCModel","text":"MRVCModel(Y, X, V)\n\nCreate a new MRVCModel instance from response matrix Y, predictor matrix X, and kernel matrices V.\n\nPositional arguments\n\nY::AbstractVecOrMat : response matrix\nX::Union{Nothing, AbstractVecOrMat} : predictor matrix\nV::Union{AbstractMatrix, Vector{<:AbstractMatrix}} : kernel matrices\n\nKeyword arguments\n\nse::Bool : calculate standard errors. Default is true.\nreml::Bool : REML estimation instead of ML estimation. Default is false.\n\n\n\n\n\n","category":"method"},{"location":"api/#MultiResponseVarianceComponentModels.fisher_B!-Union{Tuple{MRVCModel{T}}, Tuple{T}} where T<:Union{Float32, Float64}","page":"API","title":"MultiResponseVarianceComponentModels.fisher_B!","text":"fisher_B!(model::MRVCModel)\n\nCompute the sampling variance-covariance of regression coefficients model.B, assuming inverse of covariance matrix model.Ω is available at model.storage_nd_nd.\n\n\n\n\n\n","category":"method"},{"location":"api/#MultiResponseVarianceComponentModels.fisher_Σ!-Union{Tuple{MRVCModel{T}}, Tuple{T}} where T<:Union{Float32, Float64}","page":"API","title":"MultiResponseVarianceComponentModels.fisher_Σ!","text":"fisher_Σ!(model::MRVCModel)\n\nCompute the sampling variance-covariance of variance component estimates model.Σ, assuming inverse of covariance matrix model.Ω is available at model.storage_nd_nd.\n\n\n\n\n\n","category":"method"},{"location":"api/#MultiResponseVarianceComponentModels.fit!-Union{Tuple{MRVCModel{T}}, Tuple{T}} where T<:Union{Float32, Float64}","page":"API","title":"MultiResponseVarianceComponentModels.fit!","text":"fit!(model::MRVCModel)\n\nFit a multivariate response variance components model by MM or EM algorithm.\n\nKeyword arguments\n\nmaxiter::Integer : maximum number of iterations. Default is 1000.\nreltol::Real : relative tolerance for convergence. Default is 1e-6.\nverbose::Bool : display algorithmic information. Default is true.\ninit::Symbol : initialization strategy. :default initialize by least squares. :user uses user supplied value at model.B and model.Σ.\nalgo::Symbol : optimization algorithm. :MM (default) or EM.\nlog::Bool : record iterate history or not. Defaut is false.\n\nOutput\n\nhistory : iterate history.\n\n\n\n\n\n","category":"method"},{"location":"api/#MultiResponseVarianceComponentModels.h2-Union{Tuple{MRVCModel{T}}, Tuple{T}} where T<:Union{Float32, Float64}","page":"API","title":"MultiResponseVarianceComponentModels.h2","text":"h2(model::MRVCModel)\n\nCalculate heritability estimates and their standard errors, assuming that all variance components capture genetic effects except the last term. Also return total heritability from sum of individual contributions and its standard error.\n\n\n\n\n\n","category":"method"},{"location":"api/#MultiResponseVarianceComponentModels.kron_axpy!-Union{Tuple{T}, Tuple{AbstractVecOrMat{T}, AbstractVecOrMat{T}, AbstractVecOrMat{T}}} where T<:Real","page":"API","title":"MultiResponseVarianceComponentModels.kron_axpy!","text":"kron_axpy!(A, X, Y)\n\nOverwrite Y with A ⊗ X + Y. Same as Y += kron(A, X), but more memory efficient.\n\n\n\n\n\n","category":"method"},{"location":"api/#MultiResponseVarianceComponentModels.kron_reduction!-Union{Tuple{T}, Tuple{AbstractMatrix{T}, AbstractMatrix{T}, AbstractMatrix{T}}, Tuple{AbstractMatrix{T}, AbstractMatrix{T}, AbstractMatrix{T}, Bool}} where T<:Real","page":"API","title":"MultiResponseVarianceComponentModels.kron_reduction!","text":"kron_reduction!(A, B, C; sym = false)\n\nOverwrite C with the derivative of tr(A' (X ⊗ B)) wrt X. C[i, j] = dot(Aij, B), where Aij is the (i , j) block of A. sym = true indicates A and B are symmetric.\n\n\n\n\n\n","category":"method"},{"location":"api/#MultiResponseVarianceComponentModels.loglikelihood!-Union{Tuple{MRVCModel{T}}, Tuple{T}} where T<:Union{Float32, Float64}","page":"API","title":"MultiResponseVarianceComponentModels.loglikelihood!","text":"loglikelihood!(model::MRVCModel)\n\nOverwrite model.storage_nd_nd by inverse of the covariance matrix model.Ω, overwrite model.storage_nd by U' \\ vec(model.R), and return the log-likelihood. Assume model.Ω and model.R are already updated according to model.Σ and model.B.\n\n\n\n\n\n","category":"method"},{"location":"api/#MultiResponseVarianceComponentModels.loglikelihood_miss!-Union{Tuple{MRVCModel{T}}, Tuple{T}} where T<:Union{Float32, Float64}","page":"API","title":"MultiResponseVarianceComponentModels.loglikelihood_miss!","text":"loglikelihood_miss!(model::MRVCModel)\n\nOverwrite model.storage_nd_nd by inverse of the covariance matrix model.Ω, overwrite model.storage_nd by U' \\ vec(model.R), overwrite model.storage_n_miss_n_miss_1 by conditional variance, and return the surrogate Q-function of log-likelihood. Assume model.Ω and model.R are already updated according to model.Σ and model.B.\n\n\n\n\n\n","category":"method"},{"location":"api/#MultiResponseVarianceComponentModels.lrt-Tuple{MRVCModel, MRVCModel}","page":"API","title":"MultiResponseVarianceComponentModels.lrt","text":"lrt(model1::MRVCModel, model0::MRVCModel)\n\nPerform a variation of the likelihood ratio test for univariate variance components models as in Molenberghs and Verbeke 2007 with model1 and model0 being the full and nested models, respectively.\n\n\n\n\n\n","category":"method"},{"location":"api/#MultiResponseVarianceComponentModels.permute-Union{Tuple{AbstractArray{Union{Missing, T}, 2}}, Tuple{T}} where T<:Union{Float32, Float64}","page":"API","title":"MultiResponseVarianceComponentModels.permute","text":"permute(Y::AbstractVecOrMat)\n\nReturn permutation P such that vec(Y)[P] rearranges vec(Y) with missing values spliced after non-missing values. Also return inverse permutation invP such that vec(Y)[P][invP] = vec(Y).\n\n\n\n\n\n","category":"method"},{"location":"api/#MultiResponseVarianceComponentModels.rg-Union{Tuple{MRVCModel{T}}, Tuple{T}} where T<:Union{Float32, Float64}","page":"API","title":"MultiResponseVarianceComponentModels.rg","text":"rg(model::MRVCModel)\n\nCalculate genetic/residual correlation estimates and their standard errors.\n\n\n\n\n\n","category":"method"},{"location":"api/#MultiResponseVarianceComponentModels.update_B!-Union{Tuple{MRVCModel{T}}, Tuple{T}} where T<:Union{Float32, Float64}","page":"API","title":"MultiResponseVarianceComponentModels.update_B!","text":"update_B!(model::MRVCModel)\n\nUpdate the regression coefficients model.B, assuming inverse of covariance matrix model.Ω is available at model.storage_nd_nd.\n\n\n\n\n\n","category":"method"},{"location":"api/#MultiResponseVarianceComponentModels.update_B_miss!-Union{Tuple{MRVCModel{T}}, Tuple{T}} where T<:Union{Float32, Float64}","page":"API","title":"MultiResponseVarianceComponentModels.update_B_miss!","text":"update_B_miss!(model::MRVCModel)\n\nUpdate the regression coefficients model.B, assuming inverse of covariance matrix model.Ω is available at model.storage_nd_nd and conditional covariance model.storage_n_miss_n_obs_1 precomputed.\n\n\n\n\n\n","category":"method"},{"location":"api/#MultiResponseVarianceComponentModels.update_Γk!-Union{Tuple{T}, Tuple{MRVCModel{T}, Integer}} where T<:Union{Float32, Float64}","page":"API","title":"MultiResponseVarianceComponentModels.update_Γk!","text":"update_Γk!(model, k)\n\nUpdate the parameters model.Γ[k] and model.Ψ[:,k] assuming a diagonal plus low rank structure for model.Σ[k]. Assumes covariance matrix model.Ω is available at model.storage_nd_nd and model.Ω⁻¹R precomputed.\n\n\n\n\n\n","category":"method"},{"location":"api/#MultiResponseVarianceComponentModels.update_Σ!-Union{Tuple{MRVCModel{T}}, Tuple{T}} where T<:Union{Float32, Float64}","page":"API","title":"MultiResponseVarianceComponentModels.update_Σ!","text":"update_Σ!(model::MRVCModel)\n\nUpdate the variance component parameters model.Σ, assuming inverse of covariance matrix model.Ω is available at model.storage_nd_nd. For missing response, assume conditional variance model.storage_n_miss_n_miss_1 is precomputed.\n\n\n\n\n\n","category":"method"},{"location":"api/#MultiResponseVarianceComponentModels.update_Σk!-Union{Tuple{T}, Tuple{MRVCModel{T}, Integer, Integer}} where T<:Union{Float32, Float64}","page":"API","title":"MultiResponseVarianceComponentModels.update_Σk!","text":"update_Σk!(model, k, rk)\n\nUpdate the model.Σ[k] assuming it has rank rk < d, assuming inverse of covariance matrix model.Ω is available at model.storage_nd_nd and model.Ω⁻¹R precomputed.\n\n\n\n\n\n","category":"method"},{"location":"api/#MultiResponseVarianceComponentModels.update_Σk!-Union{Tuple{T}, Tuple{MRVCModel{T}, Integer, Val{:EM}}} where T<:Union{Float32, Float64}","page":"API","title":"MultiResponseVarianceComponentModels.update_Σk!","text":"update_Σk!(model::MRVCModel, k, Val(:EM))\n\nEM update the model.Σ[k] assuming it has full rank d, inverse of covariance matrix model.Ω is available at model.storage_nd_nd, and model.Ω⁻¹R precomputed.\n\n\n\n\n\n","category":"method"},{"location":"api/#MultiResponseVarianceComponentModels.update_Σk!-Union{Tuple{T}, Tuple{MRVCModel{T}, Integer, Val{:MM}}} where T<:Union{Float32, Float64}","page":"API","title":"MultiResponseVarianceComponentModels.update_Σk!","text":"update_Σk!(model::MRVCModel, k, Val(:MM))\n\nMM update the model.Σ[k] assuming it has full rank d, inverse of covariance matrix model.Ω is available at model.storage_nd_nd, and model.Ω⁻¹R precomputed.\n\n\n\n\n\n","category":"method"},{"location":"api/#MultiResponseVarianceComponentModels.update_Σk_miss!-Union{Tuple{T}, Tuple{MRVCModel{T}, Integer, Val{:MM}}} where T<:Union{Float32, Float64}","page":"API","title":"MultiResponseVarianceComponentModels.update_Σk_miss!","text":"update_Σk_miss!(model::MRVCModel, k, Val(:MM))\n\nMM update the model.Σ[k] assuming it has full rank d, inverse of covariance matrix model.Ω is available at model.storage_nd_nd, and model.Ω⁻¹R, conditional variance model.storage_n_miss_n_miss_1, and Ω⁻¹P'CPΩ⁻¹ precomputed.\n\n\n\n\n\n","category":"method"},{"location":"examples/#Examples","page":"Examples","title":"Examples","text":"","category":"section"},{"location":"examples/#Simulate-data","page":"Examples","title":"Simulate data","text":"","category":"section"},{"location":"examples/","page":"Examples","title":"Examples","text":"using MultiResponseVarianceComponentModels, LinearAlgebra, Random\nRandom.seed!(6789)\nn = 1_000; # n of observations\nd = 4; # n of responses\np = 10; # n of covariates\nm = 5; # n of variance components\nX = rand(n, p);\nB = rand(p, d)\nV = [zeros(n, n) for _ in 1:m]; # kernel matrices\nΣ = [zeros(d, d) for _ in 1:m]; # variance components\nfor i in 1:m\n Vi = randn(n, n)\n copy!(V[i], Vi' * Vi)\n Σi = randn(d, d)\n copy!(Σ[i], Σi' * Σi)\nend\nΩ = zeros(n * d, n * d); # overall nd-by-nd covariance matrix Ω\nfor i = 1:m\n Ω += kron(Σ[i], V[i])\nend\nΩchol = cholesky(Ω);\nY = X * B + reshape(Ωchol.L * randn(n * d), n, d)","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"note: Note\nIn the case of heritability and genetic correlation analyses, one can use classic genetic relationship matrices (GRMs) for boldsymbolV_i's, which in turn can be constructed using SnpArrays.jl.","category":"page"},{"location":"examples/#Maximum-likelihood-estimation","page":"Examples","title":"Maximum likelihood estimation","text":"","category":"section"},{"location":"examples/","page":"Examples","title":"Examples","text":"model = MRVCModel(Y, X, V)\n@timev fit!(model)","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"Then variance components and mean effects estimates can be accessed through","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"model.Σ\nmodel.B\nhcat(vec(B), vec(model.B))\nreduce(hcat, [hcat(vec(Σ[i]), vec(model.Σ[i])) for i in 1:m])","category":"page"},{"location":"examples/#Standard-errors","page":"Examples","title":"Standard errors","text":"","category":"section"},{"location":"examples/","page":"Examples","title":"Examples","text":"Sampling variance and covariance of these estimates are","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"model.Σcov\nmodel.Bcov","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"Corresponding standard error of these estimates are","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"sqrt.(diag(model.Σcov))\nsqrt.(diag(model.Bcov))","category":"page"},{"location":"examples/#Residual-maximum-likelihood-estimation","page":"Examples","title":"Residual maximum likelihood estimation","text":"","category":"section"},{"location":"examples/","page":"Examples","title":"Examples","text":"For REML estimation, you can instead:","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"model = MRVCModel(Y, X, V; reml = true)\n@timev fit!(model)","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"Variance components and mean effects estimates and their standard errors can be accessed through:","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"model.Σ\nmodel.B_reml\nhcat(vec(B), vec(model.B_reml))\nreduce(hcat, [hcat(vec(Σ[i]), vec(model.Σ[i])) for i in 1:m])\nmodel.Σcov\nmodel.Bcov_reml\nsqrt.(diag(model.Σcov))\nsqrt.(diag(model.Bcov_reml))","category":"page"},{"location":"examples/#Estimation-only","page":"Examples","title":"Estimation only","text":"","category":"section"},{"location":"examples/","page":"Examples","title":"Examples","text":"Calculating standard errors can be memory-consuming, so you could instead forego such calculation via:","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"model = MRVCModel(Y, X, V; se = false) # or model = MRVCModel(Y, X, V; se = false, reml = true)\n@timev fit!(model)","category":"page"},{"location":"examples/#Missing-response","page":"Examples","title":"Missing response","text":"","category":"section"},{"location":"examples/","page":"Examples","title":"Examples","text":"You can also fit data with missing response. For example:","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"Y_miss = Matrix{Union{eltype(Y), Missing}}(missing, size(Y))\ncopy!(Y_miss, Y)\nY_miss[rand(1:length(Y_miss), n)] .= missing\n\nmodel = MRVCModel(Y_miss, X, V, se = false)\n@timev fit!(model)","category":"page"},{"location":"advanced/#Advanced-details","page":"Advanced details","title":"Advanced details","text":"","category":"section"},{"location":"advanced/#Estimation","page":"Advanced details","title":"Estimation","text":"","category":"section"},{"location":"advanced/","page":"Advanced details","title":"Advanced details","text":"For the MM algorithm, the updates in each iteration are","category":"page"},{"location":"advanced/","page":"Advanced details","title":"Advanced details","text":"beginaligned\ntextvec boldsymbolB^(t) = (boldsymbolI_d otimes boldsymbolX^T) boldsymbolOmega^-(t) (boldsymbolI_d otimes boldsymbolX)^-1 (boldsymbolI_d otimes boldsymbolX^T) boldsymbolOmega^-(t) textvec boldsymbolY \nboldsymbolGamma_i^(t + 1) = boldsymbolL_i^-(t)TboldsymbolL_i^(t)T(boldsymbolGamma_i^(t)boldsymbolR^(t)TboldsymbolV_iboldsymbolR^(t)boldsymbolGamma_i^(t))boldsymbolL_i^(t)^12 boldsymbolL_i^-(t)\nendaligned","category":"page"},{"location":"advanced/","page":"Advanced details","title":"Advanced details","text":"where boldsymbolOmega^(t) = sum_i=1^m boldsymbolGamma_i^(t) otimes boldsymbolV_i and boldsymbolL_i^(t) is the Cholesky factor of boldsymbolM_i^(t) = (boldsymbolI_d otimes boldsymbol1_n)^T (boldsymbol1_d boldsymbol1_d^T otimes boldsymbolV_i) odot boldsymbolOmega^-(t) (boldsymbolI_d otimes boldsymbol1_n), while boldsymbolR^(t) is the n times d matrix such that textvec boldsymbolR^(t) = boldsymbolOmega^-(t) textvec(boldsymbolY - boldsymbolX boldsymbolB^(t)).","category":"page"},{"location":"advanced/","page":"Advanced details","title":"Advanced details","text":"For the EM algorithm, the updates in each iteration are","category":"page"},{"location":"advanced/","page":"Advanced details","title":"Advanced details","text":"beginaligned\ntextvec boldsymbolB^(t) = (boldsymbolI_d otimes boldsymbolX^T) boldsymbolOmega^-(t) (boldsymbolI_d otimes boldsymbolX)^-1 (boldsymbolI_d otimes boldsymbolX^T) boldsymbolOmega^-(t) textvec boldsymbolY \nboldsymbolGamma_i^(t + 1) = frac1r_i boldsymbolGamma_i^(t) boldsymbolR^(t)T boldsymbolV_i boldsymbolR^(t) - boldsymbolM_i^(t) boldsymbolGamma_i^(t) + boldsymbolGamma_i^(t)\nendaligned","category":"page"},{"location":"advanced/","page":"Advanced details","title":"Advanced details","text":"where r_i = textrank(boldsymbolV_i). As seen, the updates for mean effects boldsymbolB are the same for these two algorithms.","category":"page"},{"location":"advanced/","page":"Advanced details","title":"Advanced details","text":"In the setting of missing response, the adjusted MM updates in each interation are","category":"page"},{"location":"advanced/","page":"Advanced details","title":"Advanced details","text":"beginaligned\ntextvec boldsymbolB^(t) = (boldsymbolI_d otimes boldsymbolX^T) boldsymbolOmega^-(t) (boldsymbolI_d otimes boldsymbolX)^-1 (boldsymbolI_d otimes boldsymbolX^T) boldsymbolOmega^-(t) textvec boldsymbolZ^(t) \nboldsymbolGamma_i^(t + 1) = boldsymbolL_i^-(t)TboldsymbolL_i^(t)TboldsymbolGamma_i^(t)(boldsymbolR^*(t)TboldsymbolV_iboldsymbolR^*(t) + boldsymbolM_i^*(t))boldsymbolGamma_i^(t)boldsymbolL_i^(t)^12 boldsymbolL_i^-(t)\nendaligned","category":"page"},{"location":"advanced/","page":"Advanced details","title":"Advanced details","text":"where boldsymbolZ^(t) is the completed response matrix from conditional mean and boldsymbolM_i^*(t) = (boldsymbolI_d otimes boldsymbol1_n)^T (boldsymbol1_d boldsymbol1_d^T otimes boldsymbolV_i) odot (boldsymbolOmega^-(t) boldsymbolP^T boldsymbolC^(t)boldsymbolPboldsymbolOmega^-(t)) (boldsymbolI_d otimes boldsymbol1_n), while boldsymbolR^*(t) is the n times d matrix such that textvec boldsymbolR^*(t) = boldsymbolOmega^-(t) textvec(boldsymbolZ^(t) - boldsymbolX boldsymbolB^(t)). Additionally, boldsymbolP is the permutation matrix such that boldsymbolP cdot textvec(boldsymbolY) = beginbmatrix boldsymboly_textobs boldsymboly_textmis endbmatrix, where boldsymboly_textobs and boldsymboly_textmis are vectors of observed and missing response values, respectively, in column-major order, and the block matrix boldsymbolC^(t) is boldsymbol0 except for a lower-right block consisting of conditional variance. As seen, the two MM updates for both mean effects and variance components are of similar form.","category":"page"},{"location":"advanced/#Inference","page":"Advanced details","title":"Inference","text":"","category":"section"},{"location":"advanced/","page":"Advanced details","title":"Advanced details","text":"Standard errors for our estimates are calculated using the Fisher information matrix, where","category":"page"},{"location":"advanced/","page":"Advanced details","title":"Advanced details","text":"beginaligned\ntextE left- fracpartial^2partial(textvec boldsymbolB)^T partial(textvec boldsymbolB) mathcalL right = (boldsymbolI_d otimes boldsymbolX^T) boldsymbolOmega^-1 (boldsymbolI_d otimes boldsymbolX) \ntextE left - fracpartial^2partial (textvech boldsymbolGamma_i)^T partial (textvec boldsymbolB) mathcalL right = boldsymbol0 \ntextE left - fracpartial^2partial (textvech boldsymbolGamma_j)^T partial (textvech boldsymbolGamma_i) mathcalL right = frac12 boldsymbolU_i^T (boldsymbolOmega^-1 otimes boldsymbolOmega^-1) boldsymbolU_j\nendaligned","category":"page"},{"location":"advanced/","page":"Advanced details","title":"Advanced details","text":"and boldsymbolU_i = (boldsymbolI_d otimes boldsymbolK_nd otimes boldsymbolI_n) (boldsymbolI_d^2 otimes textvec boldsymbolV_i) boldsymbolD_d. Here, boldsymbolK_nd is the nd times nd commutation matrix and boldsymbolD_d the d^2 times fracd(d+1)2 duplication matrix. textvech boldsymbolGamma_i creates an fracd(d+1)2 times 1 vector from boldsymbolGamma_i by stacking its lower triangular part.","category":"page"},{"location":"","page":"Home","title":"Home","text":"CurrentModule = MultiResponseVarianceComponentModels","category":"page"},{"location":"#MultiResponseVarianceComponentModels","page":"Home","title":"MultiResponseVarianceComponentModels","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"MRVCModels.jl is a package for fitting and testing multivariate response variance components linear mixed models of form","category":"page"},{"location":"","page":"Home","title":"Home","text":"textvec boldsymbolY sim mathcalN(textvec(boldsymbolX boldsymbolB) sum_i=1^m boldsymbolGamma_i otimes boldsymbolV_i)","category":"page"},{"location":"","page":"Home","title":"Home","text":"where boldsymbolY and boldsymbolX are n times d response and n times p predictor matrices, respectively, and boldsymbolV_1 ldots boldsymbolV_m are m known n times n positive semidefinite matrices. textvec boldsymbolY creates an nd times 1 vector from boldsymbolY by stacking its columns and otimes denotes the Kronecker product. The parameters of the model include p times d mean effects boldsymbolB and d times d variance components (boldsymbolGamma_1 dots boldsymbolGamma_m), which MRVCModels.jl estimates through either minorization-maximization (MM) or expectation–maximization (EM) algorithms.","category":"page"},{"location":"","page":"Home","title":"Home","text":"note: Note\nMRVCModels.jl is not suitable for biobank-scale data. We recommend using this package for datasets of size up to n cdot d approx 50000. This package can also handle data with missing response.","category":"page"},{"location":"#Installation","page":"Home","title":"Installation","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"To use MRVCModels.jl, type:","category":"page"},{"location":"","page":"Home","title":"Home","text":"using Pkg\nPkg.add(url = \"https://github.com/Hua-Zhou/MultiResponseVarianceComponentModels.jl.git\")","category":"page"},{"location":"","page":"Home","title":"Home","text":"This documentation was built using Documenter.jl.","category":"page"},{"location":"","page":"Home","title":"Home","text":"using Dates # hide\nprintln(\"Documentation built $(Dates.now()) with Julia $(VERSION)\") # hide","category":"page"}] +} diff --git a/v0.3.0/siteinfo.js b/v0.3.0/siteinfo.js new file mode 100644 index 0000000..bcac4fe --- /dev/null +++ b/v0.3.0/siteinfo.js @@ -0,0 +1 @@ +var DOCUMENTER_CURRENT_VERSION = "v0.3.0"; diff --git a/v0.3.1/.documenter-siteinfo.json b/v0.3.1/.documenter-siteinfo.json new file mode 100644 index 0000000..0e51385 --- /dev/null +++ b/v0.3.1/.documenter-siteinfo.json @@ -0,0 +1 @@ +{"documenter":{"julia_version":"1.10.4","generation_timestamp":"2024-06-20T18:59:14","documenter_version":"1.4.1"}} \ No newline at end of file diff --git a/v0.3.1/advanced/index.html b/v0.3.1/advanced/index.html new file mode 100644 index 0000000..b83addf --- /dev/null +++ b/v0.3.1/advanced/index.html @@ -0,0 +1,27 @@ + +Advanced details · MRVCModels
    GitHub

    Advanced details

    Estimation

    For the MM algorithm, the updates in each iteration are

    \[\begin{aligned} +\text{vec}\ \boldsymbol{B}^{(t)} &= [(\boldsymbol{I}_d \otimes \boldsymbol{X}^T) \boldsymbol{\Omega}^{-(t)} (\boldsymbol{I}_d \otimes \boldsymbol{X})]^{-1} (\boldsymbol{I}_d \otimes \boldsymbol{X}^T) \boldsymbol{\Omega}^{-(t)} \text{vec}\ \boldsymbol{Y} \\ +\boldsymbol{\Gamma}_i^{(t + 1)} &= \boldsymbol{L}_i^{-(t)T}[\boldsymbol{L}_i^{(t)T}\boldsymbol{\Gamma}_i^{(t)}(\boldsymbol{R}^{(t)T}\boldsymbol{V}_i\boldsymbol{R}^{(t)})\boldsymbol{\Gamma}_i^{(t)}\boldsymbol{L}_i^{(t)}]^{1/2} \boldsymbol{L}_i^{-(t)}, +\end{aligned}\]

    where $\boldsymbol{\Omega}^{(t)} = \sum_{i=1}^m \boldsymbol{\Gamma}_i^{(t)} \otimes \boldsymbol{V}_i$ and $\boldsymbol{L}_i^{(t)}$ is the Cholesky factor of $\boldsymbol{M}_i^{(t)} = (\boldsymbol{I}_d \otimes \boldsymbol{1}_n)^T [(\boldsymbol{1}_d \boldsymbol{1}_d^T \otimes \boldsymbol{V}_i) \odot \boldsymbol{\Omega}^{-(t)}] (\boldsymbol{I}_d \otimes \boldsymbol{1}_n)$, while $\boldsymbol{R}^{(t)}$ is the $n \times d$ matrix such that $\text{vec}\ \boldsymbol{R}^{(t)} = \boldsymbol{\Omega}^{-(t)} \text{vec}(\boldsymbol{Y} - \boldsymbol{X} \boldsymbol{B}^{(t)})$. $\odot$ denotes the Hadamard product.

    For the EM algorithm, the updates in each iteration are

    \[\begin{aligned} +\text{vec}\ \boldsymbol{B}^{(t)} &= [(\boldsymbol{I}_d \otimes \boldsymbol{X}^T) \boldsymbol{\Omega}^{-(t)} (\boldsymbol{I}_d \otimes \boldsymbol{X})]^{-1} (\boldsymbol{I}_d \otimes \boldsymbol{X}^T) \boldsymbol{\Omega}^{-(t)} \text{vec}\ \boldsymbol{Y} \\ +\boldsymbol{\Gamma}_i^{(t + 1)} &= \frac{1}{r_i} \boldsymbol{\Gamma}_i^{(t)} ( \boldsymbol{R}^{(t)T} \boldsymbol{V}_i \boldsymbol{R}^{(t)} - \boldsymbol{M}_i^{(t)} ) \boldsymbol{\Gamma}_i^{(t)} + \boldsymbol{\Gamma}_i^{(t)}, +\end{aligned}\]

    where $r_i = \text{rank}(\boldsymbol{V}_i)$. As seen, the updates for mean effects $\boldsymbol{B}$ are the same for these two algorithms.

    Inference

    Standard errors for our estimates are calculated using the Fisher information matrix, where

    \[\begin{aligned} +\text{E} \left[- \frac{\partial^2}{\partial(\text{vec}\ \boldsymbol{B})^T \partial(\text{vec}\ \boldsymbol{B})} \mathcal{L} \right] &= (\boldsymbol{I}_d \otimes \boldsymbol{X}^T) \boldsymbol{\Omega}^{-1} (\boldsymbol{I}_d \otimes \boldsymbol{X}) \\ +\text{E} \left[ - \frac{\partial^2}{\partial (\text{vech}\ \boldsymbol{\Gamma}_i)^T \partial (\text{vec}\ \boldsymbol{B})} \mathcal{L} \right] &= \boldsymbol{0} \\ +\text{E} \left[ - \frac{\partial^2}{\partial (\text{vech}\ \boldsymbol{\Gamma}_j)^T \partial (\text{vech}\ \boldsymbol{\Gamma}_i)} \mathcal{L} \right] &= \frac{1}{2} \boldsymbol{U}_i^T (\boldsymbol{\Omega}^{-1} \otimes \boldsymbol{\Omega}^{-1}) \boldsymbol{U}_j +\end{aligned}\]

    and $\boldsymbol{U}_i = (\boldsymbol{I}_d \otimes \boldsymbol{K}_{nd} \otimes \boldsymbol{I}_n) (\boldsymbol{I}_{d^2} \otimes \text{vec}\ \boldsymbol{V}_i) \boldsymbol{D}_{d}$. Here, $\boldsymbol{K}_{nd}$ is the $nd \times nd$ commutation matrix and $\boldsymbol{D}_{d}$ the $d^2 \times \frac{d(d+1)}{2}$ duplication matrix. $\text{vech}\ \boldsymbol{\Gamma}_i$ creates an $\frac{d(d+1)}{2} \times 1$ vector from $\boldsymbol{\Gamma}_i$ by stacking its lower triangular part.

    Special case: missing response

    In the setting of missing response, the adjusted MM updates in each interation are

    \[\begin{aligned} +\text{vec}\ \boldsymbol{B}^{(t)} &= [(\boldsymbol{I}_d \otimes \boldsymbol{X}^T) \boldsymbol{\Omega}^{-(t)} (\boldsymbol{I}_d \otimes \boldsymbol{X})]^{-1} (\boldsymbol{I}_d \otimes \boldsymbol{X}^T) \boldsymbol{\Omega}^{-(t)} \text{vec}\ \boldsymbol{Z}^{(t)} \\ +\boldsymbol{\Gamma}_i^{(t + 1)} &= \boldsymbol{L}_i^{-(t)T}[\boldsymbol{L}_i^{(t)T}\boldsymbol{\Gamma}_i^{(t)}(\boldsymbol{R}^{*(t)T}\boldsymbol{V}_i\boldsymbol{R}^{*(t)} + \boldsymbol{M}_i^{*(t)})\boldsymbol{\Gamma}_i^{(t)}\boldsymbol{L}_i^{(t)}]^{1/2} \boldsymbol{L}_i^{-(t)}, +\end{aligned}\]

    where $\boldsymbol{Z}^{(t)}$ is the completed response matrix from conditional mean and $\boldsymbol{M}_i^{*(t)} = (\boldsymbol{I}_d \otimes \boldsymbol{1}_n)^T [(\boldsymbol{1}_d \boldsymbol{1}_d^T \otimes \boldsymbol{V}_i) \odot (\boldsymbol{\Omega}^{-(t)} \boldsymbol{P}^T \boldsymbol{C}^{(t)}\boldsymbol{P}\boldsymbol{\Omega}^{-(t)})] (\boldsymbol{I}_d \otimes \boldsymbol{1}_n)$, while $\boldsymbol{R}^{*(t)}$ is the $n \times d$ matrix such that $\text{vec}\ \boldsymbol{R}^{*(t)} = \boldsymbol{\Omega}^{-(t)} \text{vec}(\boldsymbol{Z}^{(t)} - \boldsymbol{X} \boldsymbol{B}^{(t)})$. Additionally, $\boldsymbol{P}$ is the $nd \times nd$ permutation matrix such that $\boldsymbol{P} \cdot \text{vec}\ \boldsymbol{Y} = \begin{bmatrix} \boldsymbol{y}_{\text{obs}} \\ \boldsymbol{y}_{\text{mis}} \end{bmatrix}$, where $\boldsymbol{y}_{\text{obs}}$ and $\boldsymbol{y}_{\text{mis}}$ are vectors of observed and missing response values, respectively, in column-major order, and the block matrix $\boldsymbol{C}^{(t)}$ is $\boldsymbol{0}$ except for a lower-right block consisting of conditional variance. As seen, the two MM updates are of similar form.

    Special case: $m = 2$

    When there are $m = 2$ variance components such that $\boldsymbol{\Omega} = \boldsymbol{\Gamma}_1 \otimes \boldsymbol{V}_1 + \boldsymbol{\Gamma}_2 \otimes \boldsymbol{V}_2$, repeated inversion of the $nd \times nd$ matrix $\boldsymbol{\Omega}$ can be avoided and reduced to one $d \times d$ generalized eigen-decomposition per iteration. Without loss of generality, if we assume $\boldsymbol{V}_2$ to be positive definite, the generalized eigen-decomposition of the matrix pair $(\boldsymbol{V}_1, \boldsymbol{V}_2)$ yields generalized eigenvalues $\boldsymbol{d} = (d_1, \dots, d_n)^T$ and generalized eigenvectors $\boldsymbol{U}$ such that $\boldsymbol{U}^T \boldsymbol{V}_1 \boldsymbol{U} = \boldsymbol{D} = \text{diag}(\boldsymbol{d})$ and $\boldsymbol{U}^T \boldsymbol{V}_2 \boldsymbol{U} = \boldsymbol{I}_n$. Similarly, if we let the generalized eigen-decomposition of $(\boldsymbol{\Gamma}_1^{(t)}, \boldsymbol{\Gamma}_2^{(t)})$ be $(\boldsymbol{\Lambda}^{(t)}, \boldsymbol{\Phi}^{(t)})$ such that $\boldsymbol{\Phi}^{(t)T} \boldsymbol{\Gamma}_1^{(t)} \boldsymbol{\Phi}^{(t)} = \boldsymbol{\Lambda}^{(t)} = \text{diag}(\boldsymbol{\lambda^{(t)}})$ and $\boldsymbol{\Phi}^{(t)T} \boldsymbol{\Gamma}_2^{(t)} \boldsymbol{\Phi}^{(t)} = \boldsymbol{I}_d$, then the MM updates in each iteration become

    \[\begin{aligned} +\text{vec}\ \boldsymbol{B}^{(t)} &= [(\boldsymbol{\Phi}^{(t)T}\otimes \tilde{\boldsymbol{X}})^T (\boldsymbol{\Lambda}^{(t)} \otimes \boldsymbol{D} + \boldsymbol{I}_d \otimes \boldsymbol{I}_n)^{-1} (\boldsymbol{\Phi}^{(t)T}\otimes \tilde{\boldsymbol{X}})]^{-1} \\ +&\quad \cdot (\boldsymbol{\Phi}^{(t)T}\otimes \tilde{\boldsymbol{X}})^T (\boldsymbol{\Lambda}^{(t)} \otimes \boldsymbol{D} + \boldsymbol{I}_d \otimes \boldsymbol{I}_n)^{-1} \text{vec}(\tilde{\boldsymbol{Y}} \boldsymbol{\Phi}^{(t)}) \\ +\boldsymbol{\Gamma}_i^{(t + 1)} &= \boldsymbol{L}_i^{-(t)T}[\boldsymbol{L}_i^{(t)T}\boldsymbol{N}_i^{(t)T}\boldsymbol{N}_i^{(t)}\boldsymbol{L}_i^{(t)}]^{1/2} \boldsymbol{L}_i^{-(t)}, +\end{aligned}\]

    where $\tilde{\boldsymbol{X}} = \boldsymbol{U}^T \boldsymbol{X}$, $\tilde{\boldsymbol{Y}} = \boldsymbol{U}^T \boldsymbol{Y}$, $\boldsymbol{L}_1^{(t)}$ is the Cholesky factor of $\boldsymbol{\Phi}^{(t)}\text{diag}(\text{tr}(\boldsymbol{D}(\lambda_k^{(t)}\boldsymbol{D} + \boldsymbol{I}_n)^{-1}), k = 1,\dots, d)\boldsymbol{\Phi}^{(t)T}$, $\boldsymbol{L}_2^{(t)}$ is the Cholesky factor of $\boldsymbol{\Phi}^{(t)}\text{diag}(\text{tr}((\lambda_k^{(t)}\boldsymbol{D} + \boldsymbol{I}_n)^{-1}), k = 1,\dots, d)\boldsymbol{\Phi}^{(t)T}$, $\boldsymbol{N}_1^{(t)} = \boldsymbol{D}^{1/2}\{[(\tilde{\boldsymbol{Y}} - \tilde{\boldsymbol{X}}\boldsymbol{B})\boldsymbol{\Phi}^{(t)}]\oslash(\boldsymbol{d}\boldsymbol{\lambda}^{(t)T} + \boldsymbol{1}_n\boldsymbol{1}_d^T) \} \boldsymbol{\Lambda}^{(t)}\boldsymbol{\Phi}^{-(t)}$, and $\boldsymbol{N}_2^{(t)} = \{[(\tilde{\boldsymbol{Y}} - \tilde{\boldsymbol{X}}\boldsymbol{B})\boldsymbol{\Phi}^{(t)}]\oslash(\boldsymbol{d}\boldsymbol{\lambda}^{(t)T} + \boldsymbol{1}_n\boldsymbol{1}_d^T) \} \boldsymbol{\Phi}^{-(t)}$. $\oslash$ denotes the Hadamard quotient.

    In this setting, the Fisher information matrix is equivalent to

    \[\begin{aligned} +\text{E} \left[- \frac{\partial^2}{\partial(\text{vec}\ \boldsymbol{B})^T \partial(\text{vec}\ \boldsymbol{B})} \mathcal{L} \right] &= (\boldsymbol{\Phi}^{T}\otimes \tilde{\boldsymbol{X}})^T (\boldsymbol{\Lambda} \otimes \boldsymbol{D} + \boldsymbol{I}_d \otimes \boldsymbol{I}_n)^{-1} (\boldsymbol{\Phi}^{T}\otimes \tilde{\boldsymbol{X}}) \\ +\text{E} \left[ - \frac{\partial^2}{\partial (\text{vech} \ \boldsymbol{\Gamma}_i)^T \partial (\text{vec}\ \boldsymbol{B})} \mathcal{L} \right] &= \boldsymbol{0} \\ +\text{E} \left[ - \frac{\partial^2}{\partial (\text{vech}\ \boldsymbol{\Gamma}_j)^T \partial (\text{vech}\ \boldsymbol{\Gamma}_i)} \mathcal{L} \right] &= \frac{1}{2} \boldsymbol{D}_d^T(\boldsymbol{\Phi}\otimes \boldsymbol{\Phi}) \text{diag}(\text{vec}\ \boldsymbol{W}_{ij}) (\boldsymbol{\Phi} \otimes \boldsymbol{\Phi})^T\boldsymbol{D}_d, +\end{aligned}\]

    where $\boldsymbol{W}_{ij}$ is the $d \times d$ matrix that has entries

    \[\begin{aligned} +(\boldsymbol{W}_{11})_{kl} &= \text{tr}(\boldsymbol{D}^2(\boldsymbol{\lambda}_k \boldsymbol{D} + \boldsymbol{I}_n)^{-1}(\boldsymbol{\lambda}_l \boldsymbol{D} + \boldsymbol{I}_n)^{-1}) \\ +(\boldsymbol{W}_{12})_{kl} &= \text{tr}(\boldsymbol{D}(\boldsymbol{\lambda}_k \boldsymbol{D} + \boldsymbol{I}_n)^{-1}(\boldsymbol{\lambda}_l \boldsymbol{D} + \boldsymbol{I}_n)^{-1}) \\ +(\boldsymbol{W}_{22})_{kl} &= \text{tr}((\boldsymbol{\lambda}_k \boldsymbol{D} + \boldsymbol{I}_n)^{-1}(\boldsymbol{\lambda}_l \boldsymbol{D} + \boldsymbol{I}_n)^{-1}). +\end{aligned}\]

    diff --git a/v0.3.1/api/index.html b/v0.3.1/api/index.html new file mode 100644 index 0000000..ed587f4 --- /dev/null +++ b/v0.3.1/api/index.html @@ -0,0 +1,19 @@ + +API · MRVCModels
    GitHub

    API

    MultiResponseVarianceComponentModels.MRVCModel — Method
    MRVCModel(
+    Y::AbstractVecOrMat,
+    X::Union{Nothing, AbstractVecOrMat},
+    V::Union{AbstractMatrix, Vector{<:AbstractMatrix}}
+    )
+MRTVCModel(
+    Y::AbstractVecOrMat,
+    X::Union{Nothing, AbstractVecOrMat},
+    V::Vector{<:AbstractMatrix}
+    )

    Create a new MRVCModel or MRTVCModel instance from response matrix Y, predictor matrix X, and kernel matrices V. For MRTVCModel instance, the number of variance components must be two.

    Keyword arguments

    se::Bool        calculate standard errors; default true
+reml::Bool      pursue REML estimation instead of ML estimation; default false
    source
    MultiResponseVarianceComponentModels.fisher_B! — Method
    fisher_B!(model::MRVCModel)

    Compute the sampling variance-covariance model.Bcov of regression coefficients model.B, assuming inverse of covariance matrix model.Ω is available at model.storage_nd_nd.

    source
    MultiResponseVarianceComponentModels.fisher_Σ! — Method
    fisher_Σ!(model::MRVCModel)

    Compute the sampling variance-covariance model.Σcov of variance component estimates model.Σ, assuming inverse of covariance matrix model.Ω is available at model.storage_nd_nd.

    source
    MultiResponseVarianceComponentModels.fit! — Method
    fit!(model::MRVCModel)
+fit!(model::MRTVCModel)

    Fit a multivariate response variance components model by MM or EM algorithm.

    Keyword arguments

    maxiter::Int        maximum number of iterations; default 1000
+reltol::Real        relative tolerance for convergence; default 1e-6
+verbose::Bool       display algorithmic information; default true
+init::Symbol        initialization strategy; :default initializes by least squares, while
+    :user uses user-supplied values at model.B and model.Σ
+algo::Symbol        optimization algorithm; :MM (default) or EM (for MRVCModel)
+log::Bool           record iterate history or not; default false
    source
    MultiResponseVarianceComponentModels.h2 — Method
    h2(model::MRVCModel)

    Calculate heritability estimates and their standard errors, assuming that all variance components capture genetic effects except the last term. Also return total heritability from sum of individual contributions and its standard error.

    source
    MultiResponseVarianceComponentModels.loglikelihood! — Method
    loglikelihood!(model::MRVCModel)

    Overwrite model.storage_nd_nd by inverse of the covariance matrix model.Ω, overwrite model.storage_nd by U' \ vec(model.R), and return the log-likelihood. Assume model.Ω and model.R are already updated according to model.Σ and model.B.

    source
    MultiResponseVarianceComponentModels.loglikelihood_miss! — Method
    loglikelihood_miss!(model::MRVCModel)

    Overwrite model.storage_nd_nd by inverse of the covariance matrix model.Ω, overwrite model.storage_nd by U' \ vec(model.R), overwrite model.storage_n_miss_n_miss_1 by conditional variance, precompute model.storage_n_miss_n_obs_1 for conditional mean, and return the value of surrogate Q-function of log-likelihood. Assume model.Ω and model.R are already updated according to model.Σ and model.B.

    source
    MultiResponseVarianceComponentModels.lrt — Method
    lrt(model1::MRVCModel, model0::MRVCModel)

    Perform a variation of the likelihood ratio test for univariate variance components models as in Molenberghs and Verbeke 2007 with model1 and model0 being the full and nested models, respectively.

    source
    MultiResponseVarianceComponentModels.permute — Method
    permute(Y::AbstractVecOrMat)

    Return permutation P such that vec(Y)[P] rearranges vec(Y) with missing values spliced after non-missing values. Also return inverse permutation invP such that vec(Y)[P][invP] = vec(Y).

    source
    MultiResponseVarianceComponentModels.update_B_miss! — Method
    update_B_miss!(model::MRVCModel)

    Update the regression coefficients model.B, assuming inverse of covariance matrix model.Ω is available at model.storage_nd_nd and model.storage_n_miss_n_obs_1 for conditional mean is precomputed.

    source
    MultiResponseVarianceComponentModels.update_Γk! — Method
    update_Γk!(model, k)

    Update the parameters model.Γ[k] and model.Ψ[:,k] assuming a diagonal plus low rank structure for model.Σ[k]. Assumes covariance matrix model.Ω is available at model.storage_nd_nd and model.Ω⁻¹R precomputed.

    source
    MultiResponseVarianceComponentModels.update_Σ! — Method
    update_Σ!(model::MRVCModel)

    Update the variance component parameters model.Σ, assuming inverse of covariance matrix model.Ω is available at model.storage_nd_nd. If missing response, assume conditional variance model.storage_n_miss_n_miss_1 is precomputed.

    source
    MultiResponseVarianceComponentModels.update_Σk! — Method
    update_Σk!(model::MRVCModel, k, Val(:EM))

    EM update the model.Σ[k] assuming it has full rank d, inverse of covariance matrix model.Ω is available at model.storage_nd_nd, and model.Ω⁻¹R is precomputed.

    source
    MultiResponseVarianceComponentModels.update_Σk! — Method
    update_Σk!(model::MRVCModel, k, Val(:MM))

    MM update the model.Σ[k] assuming it has full rank d, inverse of covariance matrix model.Ω is available at model.storage_nd_nd, and model.Ω⁻¹R is precomputed.

    source
    MultiResponseVarianceComponentModels.update_Σk_miss! — Method
    update_Σk_miss!(model::MRVCModel, k, Val(:MM))

    MM update the model.Σ[k] assuming it has full rank d, inverse of covariance matrix model.Ω is available at model.storage_nd_nd, model.Ω⁻¹R is precomputed, and Ω⁻¹P'CPΩ⁻¹ is available at model.storage_nd_nd_miss.

    source
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0 : 400; + + debounce(() => { + if (isExpanded) { + $(this).removeClass("fa-chevron-up").addClass("fa-chevron-down"); + $(".docstring-article-toggle-button") + .removeClass("fa-chevron-down") + .addClass("fa-chevron-right"); + + isExpanded = false; + + $(".docstring section").slideUp(animationSpeed); + } else { + $(this).removeClass("fa-chevron-down").addClass("fa-chevron-up"); + $(".docstring-article-toggle-button") + .removeClass("fa-chevron-right") + .addClass("fa-chevron-down"); + + isExpanded = true; + articleToggleTitle = "Collapse docstring"; + navArticleToggleTitle = "Collapse all docstrings"; + + $(".docstring section").slideDown(animationSpeed); + } + + $(this).prop("title", navArticleToggleTitle); + $(".docstring-article-toggle-button").prop("title", articleToggleTitle); + }); +}); + +function debounce(callback, timeout = 300) { + if (Date.now() - timer > timeout) { + callback(); + } + + clearTimeout(timer); + + timer = Date.now(); +} + +}) +//////////////////////////////////////////////////////////////////////////////// +require([], function() { +function addCopyButtonCallbacks() { + for (const el of document.getElementsByTagName("pre")) { + const button = document.createElement("button"); + button.classList.add("copy-button", "fa-solid", "fa-copy"); + button.setAttribute("aria-label", "Copy this code block"); + button.setAttribute("title", "Copy"); + + el.appendChild(button); + + const success = function () { + button.classList.add("success", "fa-check"); + button.classList.remove("fa-copy"); + }; + + const failure = function () { + button.classList.add("error", "fa-xmark"); + button.classList.remove("fa-copy"); + }; + + button.addEventListener("click", function () { + copyToClipboard(el.innerText).then(success, failure); + + setTimeout(function () { + button.classList.add("fa-copy"); + button.classList.remove("success", "fa-check", "fa-xmark"); + }, 5000); + }); + } +} + +function copyToClipboard(text) { + // clipboard API is only available in secure contexts + if (window.navigator && window.navigator.clipboard) { + return window.navigator.clipboard.writeText(text); + } else { + return new Promise(function (resolve, reject) { + try { + const el = document.createElement("textarea"); + el.textContent = text; + el.style.position = "fixed"; + el.style.opacity = 0; + document.body.appendChild(el); + el.select(); + document.execCommand("copy"); + + resolve(); + } catch (err) { + reject(err); + } finally { + document.body.removeChild(el); + } + }); + } +} + +if (document.readyState === "loading") { + document.addEventListener("DOMContentLoaded", addCopyButtonCallbacks); +} else { + addCopyButtonCallbacks(); +} + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery', 'headroom', 'headroom-jquery'], function($, Headroom) { + +// Manages the top navigation bar (hides it when the user starts scrolling down on the +// mobile). +window.Headroom = Headroom; // work around buggy module loading? +$(document).ready(function () { + $("#documenter .docs-navbar").headroom({ + tolerance: { up: 10, down: 10 }, + }); +}); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +$(document).ready(function () { + let meta = $("div[data-docstringscollapsed]").data(); + + if (meta?.docstringscollapsed) { + $("#documenter-article-toggle-button").trigger({ + type: "click", + noToggleAnimation: true, + }); + } +}); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +/* +To get an in-depth about the thought process you can refer: https://hetarth02.hashnode.dev/series/gsoc + +PSEUDOCODE: + +Searching happens automatically as the user types or adjusts the selected filters. +To preserve responsiveness, as much as possible of the slow parts of the search are done +in a web worker. Searching and result generation are done in the worker, and filtering and +DOM updates are done in the main thread. The filters are in the main thread as they should +be very quick to apply. This lets filters be changed without re-searching with minisearch +(which is possible even if filtering is on the worker thread) and also lets filters be +changed _while_ the worker is searching and without message passing (neither of which are +possible if filtering is on the worker thread) + +SEARCH WORKER: + +Import minisearch + +Build index + +On message from main thread + run search + find the first 200 unique results from each category, and compute their divs for display + note that this is necessary and sufficient information for the main thread to find the + first 200 unique results from any given filter set + post results to main thread + +MAIN: + +Launch worker + +Declare nonconstant globals (worker_is_running, last_search_text, unfiltered_results) + +On text update + if worker is not running, launch_search() + +launch_search + set worker_is_running to true, set last_search_text to the search text + post the search query to worker + +on message from worker + if last_search_text is not the same as the text in the search field, + the latest search result is not reflective of the latest search query, so update again + launch_search() + otherwise + set worker_is_running to false + + regardless, display the new search results to the user + save the unfiltered_results as a global + update_search() + +on filter click + adjust the filter selection + update_search() + +update_search + apply search filters by looping through the unfiltered_results and finding the first 200 + unique results that match the filters + + Update the DOM +*/ + +/////// SEARCH WORKER /////// + +function worker_function(documenterSearchIndex, documenterBaseURL, filters) { + importScripts( + "https://cdn.jsdelivr.net/npm/minisearch@6.1.0/dist/umd/index.min.js" + ); + + let data = documenterSearchIndex.map((x, key) => { + x["id"] = key; // minisearch requires a unique for each object + return x; + }); + + // list below is the lunr 2.1.3 list minus the intersect with names(Base) + // (all, any, get, in, is, only, which) and (do, else, for, let, where, while, with) + // ideally we'd just filter the original list but it's not available as a variable + const stopWords = new Set([ + "a", + "able", + "about", + "across", + "after", + "almost", + "also", + "am", + "among", + "an", + "and", + "are", + "as", + "at", + "be", + "because", + "been", + "but", + "by", + "can", + "cannot", + "could", + "dear", + "did", + "does", + "either", + "ever", + "every", + "from", + "got", + "had", + "has", + "have", + "he", + "her", + "hers", + "him", + "his", + "how", + "however", + "i", + "if", + "into", + "it", + "its", + "just", + "least", + "like", + "likely", + "may", + "me", + "might", + "most", + "must", + "my", + "neither", + "no", + "nor", + "not", + "of", + "off", + "often", + "on", + "or", + "other", + "our", + "own", + "rather", + "said", + "say", + "says", + "she", + "should", + "since", + "so", + "some", + "than", + "that", + "the", + "their", + "them", + "then", + "there", + "these", + "they", + "this", + "tis", + "to", + "too", + "twas", + "us", + "wants", + "was", + "we", + "were", + "what", + "when", + "who", + "whom", + "why", + "will", + "would", + "yet", + "you", + "your", + ]); + + let index = new MiniSearch({ + fields: ["title", "text"], // fields to index for full-text search + storeFields: ["location", "title", "text", "category", "page"], // fields to return with results + processTerm: (term) => { + let word = stopWords.has(term) ? null : term; + if (word) { + // custom trimmer that doesn't strip @ and !, which are used in julia macro and function names + word = word + .replace(/^[^a-zA-Z0-9@!]+/, "") + .replace(/[^a-zA-Z0-9@!]+$/, ""); + + word = word.toLowerCase(); + } + + return word ?? null; + }, + // add . as a separator, because otherwise "title": "Documenter.Anchors.add!", would not + // find anything if searching for "add!", only for the entire qualification + tokenize: (string) => string.split(/[\s\-\.]+/), + // options which will be applied during the search + searchOptions: { + prefix: true, + boost: { title: 100 }, + fuzzy: 2, + }, + }); + + index.addAll(data); + + /** + * Used to map characters to HTML entities. + * Refer: https://github.com/lodash/lodash/blob/main/src/escape.ts + */ + const htmlEscapes = { + "&": "&", + "<": "<", + ">": ">", + '"': """, + "'": "'", + }; + + /** + * Used to match HTML entities and HTML characters. + * Refer: https://github.com/lodash/lodash/blob/main/src/escape.ts + */ + const reUnescapedHtml = /[&<>"']/g; + const reHasUnescapedHtml = RegExp(reUnescapedHtml.source); + + /** + * Escape function from lodash + * Refer: https://github.com/lodash/lodash/blob/main/src/escape.ts + */ + function escape(string) { + return string && reHasUnescapedHtml.test(string) + ? string.replace(reUnescapedHtml, (chr) => htmlEscapes[chr]) + : string || ""; + } + + /** + * Make the result component given a minisearch result data object and the value + * of the search input as queryString. To view the result object structure, refer: + * https://lucaong.github.io/minisearch/modules/_minisearch_.html#searchresult + * + * @param {object} result + * @param {string} querystring + * @returns string + */ + function make_search_result(result, querystring) { + let search_divider = `
    `; + let display_link = + result.location.slice(Math.max(0), Math.min(50, result.location.length)) + + (result.location.length > 30 ? "..." : ""); // To cut-off the link because it messes with the overflow of the whole div + + if (result.page !== "") { + display_link += ` (${result.page})`; + } + + let textindex = new RegExp(`${querystring}`, "i").exec(result.text); + let text = + textindex !== null + ? result.text.slice( + Math.max(textindex.index - 100, 0), + Math.min( + textindex.index + querystring.length + 100, + result.text.length + ) + ) + : ""; // cut-off text before and after from the match + + text = text.length ? escape(text) : ""; + + let display_result = text.length + ? "..." + + text.replace( + new RegExp(`${escape(querystring)}`, "i"), // For first occurrence + '$&' + ) + + "..." + : ""; // highlights the match + + let in_code = false; + if (!["page", "section"].includes(result.category.toLowerCase())) { + in_code = true; + } + + // We encode the full url to escape some special characters which can lead to broken links + let result_div = ` + +
    +
    ${escape(result.title)}
    +
    ${result.category}
    +
    +

    + ${display_result} +

    +
    + ${display_link} +
    +     + ${search_divider} + `; + + return result_div; + } + + self.onmessage = function (e) { + let query = e.data; + let results = index.search(query, { + filter: (result) => { + // Only return relevant results + return result.score >= 1; + }, + }); + + // Pre-filter to deduplicate and limit to 200 per category to the extent + // possible without knowing what the filters are. + let filtered_results = []; + let counts = {}; + for (let filter of filters) { + counts[filter] = 0; + } + let present = {}; + + for (let result of results) { + cat = result.category; + cnt = counts[cat]; + if (cnt < 200) { + id = cat + "---" + result.location; + if (present[id]) { + continue; + } + present[id] = true; + filtered_results.push({ + location: result.location, + category: cat, + div: make_search_result(result, query), + }); + } + } + + postMessage(filtered_results); + }; +} + +// `worker = Threads.@spawn worker_function(documenterSearchIndex)`, but in JavaScript! +const filters = [ + ...new Set(documenterSearchIndex["docs"].map((x) => x.category)), +]; +const worker_str = + "(" + + worker_function.toString() + + ")(" + + JSON.stringify(documenterSearchIndex["docs"]) + + "," + + JSON.stringify(documenterBaseURL) + + "," + + JSON.stringify(filters) + + ")"; +const worker_blob = new Blob([worker_str], { type: "text/javascript" }); +const worker = new Worker(URL.createObjectURL(worker_blob)); + +/////// SEARCH MAIN /////// + +// Whether the worker is currently handling a search. This is a boolean +// as the worker only ever handles 1 or 0 searches at a time. +var worker_is_running = false; + +// The last search text that was sent to the worker. This is used to determine +// if the worker should be launched again when it reports back results. +var last_search_text = ""; + +// The results of the last search. This, in combination with the state of the filters +// in the DOM, is used compute the results to display on calls to update_search. +var unfiltered_results = []; + +// Which filter is currently selected +var selected_filter = ""; + +$(document).on("input", ".documenter-search-input", function (event) { + if (!worker_is_running) { + launch_search(); + } +}); + +function launch_search() { + worker_is_running = true; + last_search_text = $(".documenter-search-input").val(); + worker.postMessage(last_search_text); +} + +worker.onmessage = function (e) { + if (last_search_text !== $(".documenter-search-input").val()) { + launch_search(); + } else { + worker_is_running = false; + } + + unfiltered_results = e.data; + update_search(); +}; + +$(document).on("click", ".search-filter", function () { + if ($(this).hasClass("search-filter-selected")) { + selected_filter = ""; + } else { + selected_filter = $(this).text().toLowerCase(); + } + + // This updates search results and toggles classes for UI: + update_search(); +}); + +/** + * Make/Update the search component + */ +function update_search() { + let querystring = $(".documenter-search-input").val(); + + if (querystring.trim()) { + if (selected_filter == "") { + results = unfiltered_results; + } else { + results = unfiltered_results.filter((result) => { + return selected_filter == result.category.toLowerCase(); + }); + } + + let search_result_container = ``; + let modal_filters = make_modal_body_filters(); + let search_divider = `
    `; + + if (results.length) { + let links = []; + let count = 0; + let search_results = ""; + + for (var i = 0, n = results.length; i < n && count < 200; ++i) { + let result = results[i]; + if (result.location && !links.includes(result.location)) { + search_results += result.div; + count++; + links.push(result.location); + } + } + + if (count == 1) { + count_str = "1 result"; + } else if (count == 200) { + count_str = "200+ results"; + } else { + count_str = count + " results"; + } + let result_count = `
    ${count_str}
    `; + + search_result_container = ` +
    + ${modal_filters} + ${search_divider} + ${result_count} +
    + ${search_results} +
    +
    + `; + } else { + search_result_container = ` +
    + ${modal_filters} + ${search_divider} +
    0 result(s)
    +
    +
    No result found!
    + `; + } + + if ($(".search-modal-card-body").hasClass("is-justify-content-center")) { + $(".search-modal-card-body").removeClass("is-justify-content-center"); + } + + $(".search-modal-card-body").html(search_result_container); + } else { + if (!$(".search-modal-card-body").hasClass("is-justify-content-center")) { + $(".search-modal-card-body").addClass("is-justify-content-center"); + } + + $(".search-modal-card-body").html(` +
    Type something to get started!
    + `); + } +} + +/** + * Make the modal filter html + * + * @returns string + */ +function make_modal_body_filters() { + let str = filters + .map((val) => { + if (selected_filter == val.toLowerCase()) { + return `${val}`; + } else { + return `${val}`; + } + }) + .join(""); + + return ` +
    + Filters: + ${str} +
    `; +} + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +// Modal settings dialog +$(document).ready(function () { + var settings = $("#documenter-settings"); + $("#documenter-settings-button").click(function () { + settings.toggleClass("is-active"); + }); + // Close the dialog if X is clicked + $("#documenter-settings button.delete").click(function () { + settings.removeClass("is-active"); + }); + // Close dialog if ESC is pressed + $(document).keyup(function (e) { + if (e.keyCode == 27) settings.removeClass("is-active"); + }); +}); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +$(document).ready(function () { + let search_modal_header = ` +
    +
    +

    + + + + +

    +
    +
    + +
    +
    + `; + + let initial_search_body = ` +
    Type something to get started!
    + `; + + let search_modal_footer = ` +
    + + Ctrl + + / to search + + esc to close +
    + `; + + $(document.body).append( + ` + + ` + ); + + document.querySelector(".docs-search-query").addEventListener("click", () => { + openModal(); + }); + + document + .querySelector(".close-search-modal") + .addEventListener("click", () => { + closeModal(); + }); + + $(document).on("click", ".search-result-link", function () { + closeModal(); + }); + + document.addEventListener("keydown", (event) => { + if ((event.ctrlKey || event.metaKey) && event.key === "/") { + openModal(); + } else if (event.key === "Escape") { + closeModal(); + } + + return false; + }); + + // Functions to open and close a modal + function openModal() { + let searchModal = document.querySelector("#search-modal"); + + searchModal.classList.add("is-active"); + document.querySelector(".documenter-search-input").focus(); + } + + function closeModal() { + let searchModal = document.querySelector("#search-modal"); + let initial_search_body = ` +
    Type something to get started!
    + `; + + searchModal.classList.remove("is-active"); + document.querySelector(".documenter-search-input").blur(); + + if (!$(".search-modal-card-body").hasClass("is-justify-content-center")) { + $(".search-modal-card-body").addClass("is-justify-content-center"); + } + + $(".documenter-search-input").val(""); + $(".search-modal-card-body").html(initial_search_body); + } + + document + .querySelector("#search-modal .modal-background") + .addEventListener("click", () => { + closeModal(); + }); +}); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +// Manages the showing and hiding of the sidebar. +$(document).ready(function () { + var sidebar = $("#documenter > .docs-sidebar"); + var sidebar_button = $("#documenter-sidebar-button"); + sidebar_button.click(function (ev) { + ev.preventDefault(); + sidebar.toggleClass("visible"); + if (sidebar.hasClass("visible")) { + // Makes sure that the current menu item is visible in the sidebar. + $("#documenter .docs-menu a.is-active").focus(); + } + }); + $("#documenter > .docs-main").bind("click", function (ev) { + if ($(ev.target).is(sidebar_button)) { + return; + } + if (sidebar.hasClass("visible")) { + sidebar.removeClass("visible"); + } + }); +}); + +// Resizes the package name / sitename in the sidebar if it is too wide. +// Inspired by: https://github.com/davatron5000/FitText.js +$(document).ready(function () { + e = $("#documenter .docs-autofit"); + function resize() { + var L = parseInt(e.css("max-width"), 10); + var L0 = e.width(); + if (L0 > L) { + var h0 = parseInt(e.css("font-size"), 10); + e.css("font-size", (L * h0) / L0); + // TODO: make sure it survives resizes? + } + } + // call once and then register events + resize(); + $(window).resize(resize); + $(window).on("orientationchange", resize); +}); + +// Scroll the navigation bar to the currently selected menu item +$(document).ready(function () { + var sidebar = $("#documenter .docs-menu").get(0); + var active = $("#documenter .docs-menu .is-active").get(0); + if (typeof active !== "undefined") { + sidebar.scrollTop = active.offsetTop - sidebar.offsetTop - 15; + } +}); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +// Theme picker setup +$(document).ready(function () { + // onchange callback + $("#documenter-themepicker").change(function themepick_callback(ev) { + var themename = $("#documenter-themepicker option:selected").attr("value"); + if (themename === "auto") { + // set_theme(window.matchMedia('(prefers-color-scheme: dark)').matches ? 'dark' : 'light'); + window.localStorage.removeItem("documenter-theme"); + } else { + // set_theme(themename); + window.localStorage.setItem("documenter-theme", themename); + } + // We re-use the global function from themeswap.js to actually do the swapping. + set_theme_from_local_storage(); + }); + + // Make sure that the themepicker displays the correct theme when the theme is retrieved + // from localStorage + if (typeof window.localStorage !== "undefined") { + var theme = window.localStorage.getItem("documenter-theme"); + if (theme !== null) { + $("#documenter-themepicker option").each(function (i, e) { + e.selected = e.value === theme; + }); + } + } +}); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +// update the version selector with info from the siteinfo.js and ../versions.js files +$(document).ready(function () { + // If the version selector is disabled with DOCUMENTER_VERSION_SELECTOR_DISABLED in the + // siteinfo.js file, we just return immediately and not display the version selector. + if ( + typeof DOCUMENTER_VERSION_SELECTOR_DISABLED === "boolean" && + DOCUMENTER_VERSION_SELECTOR_DISABLED + ) { + return; + } + + var version_selector = $("#documenter .docs-version-selector"); + var version_selector_select = $("#documenter .docs-version-selector select"); + + version_selector_select.change(function (x) { + target_href = version_selector_select + .children("option:selected") + .get(0).value; + window.location.href = target_href; + }); + + // add the current version to the selector based on siteinfo.js, but only if the selector is empty + if ( + typeof DOCUMENTER_CURRENT_VERSION !== "undefined" && + $("#version-selector > option").length == 0 + ) { + var option = $( + "" + ); + version_selector_select.append(option); + } + + if (typeof DOC_VERSIONS !== "undefined") { + var existing_versions = version_selector_select.children("option"); + var existing_versions_texts = existing_versions.map(function (i, x) { + return x.text; + }); + DOC_VERSIONS.forEach(function (each) { + var version_url = documenterBaseURL + "/../" + each + "/"; + var existing_id = $.inArray(each, existing_versions_texts); + // if not already in the version selector, add it as a new option, + // otherwise update the old option with the URL and enable it + if (existing_id == -1) { + var option = $( + "" + ); + version_selector_select.append(option); + } else { + var option = existing_versions[existing_id]; + option.value = version_url; + option.disabled = false; + } + }); + } + + // only show the version selector if the selector has been populated + if (version_selector_select.children("option").length > 0) { + version_selector.toggleClass("visible"); + } +}); + +}) diff --git a/v0.3.1/assets/logo.svg b/v0.3.1/assets/logo.svg new file mode 100644 index 0000000..df89daf --- /dev/null +++ b/v0.3.1/assets/logo.svg @@ -0,0 +1,169 @@ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +≈ ++ ++…+ + diff --git a/v0.3.1/assets/themes/documenter-dark.css b/v0.3.1/assets/themes/documenter-dark.css new file mode 100644 index 0000000..1d71701 --- /dev/null +++ b/v0.3.1/assets/themes/documenter-dark.css @@ -0,0 +1,7 @@ +html.theme--documenter-dark .pagination-previous,html.theme--documenter-dark .pagination-next,html.theme--documenter-dark .pagination-link,html.theme--documenter-dark .pagination-ellipsis,html.theme--documenter-dark .file-cta,html.theme--documenter-dark .file-name,html.theme--documenter-dark .select select,html.theme--documenter-dark .textarea,html.theme--documenter-dark .input,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input,html.theme--documenter-dark .button{-moz-appearance:none;-webkit-appearance:none;align-items:center;border:1px solid transparent;border-radius:.4em;box-shadow:none;display:inline-flex;font-size:1rem;height:2.5em;justify-content:flex-start;line-height:1.5;padding-bottom:calc(0.5em - 1px);padding-left:calc(0.75em - 1px);padding-right:calc(0.75em - 1px);padding-top:calc(0.5em - 1px);position:relative;vertical-align:top}html.theme--documenter-dark .pagination-previous:focus,html.theme--documenter-dark .pagination-next:focus,html.theme--documenter-dark .pagination-link:focus,html.theme--documenter-dark .pagination-ellipsis:focus,html.theme--documenter-dark .file-cta:focus,html.theme--documenter-dark .file-name:focus,html.theme--documenter-dark .select select:focus,html.theme--documenter-dark .textarea:focus,html.theme--documenter-dark .input:focus,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input:focus,html.theme--documenter-dark .button:focus,html.theme--documenter-dark .is-focused.pagination-previous,html.theme--documenter-dark .is-focused.pagination-next,html.theme--documenter-dark .is-focused.pagination-link,html.theme--documenter-dark .is-focused.pagination-ellipsis,html.theme--documenter-dark .is-focused.file-cta,html.theme--documenter-dark .is-focused.file-name,html.theme--documenter-dark .select select.is-focused,html.theme--documenter-dark .is-focused.textarea,html.theme--documenter-dark .is-focused.input,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.is-focused,html.theme--documenter-dark .is-focused.button,html.theme--documenter-dark .pagination-previous:active,html.theme--documenter-dark .pagination-next:active,html.theme--documenter-dark .pagination-link:active,html.theme--documenter-dark .pagination-ellipsis:active,html.theme--documenter-dark .file-cta:active,html.theme--documenter-dark .file-name:active,html.theme--documenter-dark .select select:active,html.theme--documenter-dark .textarea:active,html.theme--documenter-dark .input:active,html.theme--documenter-dark 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.button.is-white{background-color:#fff;border-color:#fff;box-shadow:none}html.theme--documenter-dark .button.is-white.is-inverted{background-color:#0a0a0a;color:#fff}html.theme--documenter-dark .button.is-white.is-inverted:hover,html.theme--documenter-dark .button.is-white.is-inverted.is-hovered{background-color:#000}html.theme--documenter-dark .button.is-white.is-inverted[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-white.is-inverted{background-color:#0a0a0a;border-color:transparent;box-shadow:none;color:#fff}html.theme--documenter-dark .button.is-white.is-loading::after{border-color:transparent transparent #0a0a0a #0a0a0a !important}html.theme--documenter-dark .button.is-white.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--documenter-dark .button.is-white.is-outlined:hover,html.theme--documenter-dark .button.is-white.is-outlined.is-hovered,html.theme--documenter-dark .button.is-white.is-outlined:focus,html.theme--documenter-dark .button.is-white.is-outlined.is-focused{background-color:#fff;border-color:#fff;color:#0a0a0a}html.theme--documenter-dark .button.is-white.is-outlined.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--documenter-dark .button.is-white.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-white.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-white.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-white.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #0a0a0a #0a0a0a !important}html.theme--documenter-dark .button.is-white.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-white.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--documenter-dark 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html.theme--documenter-dark .button.is-white.is-inverted.is-outlined{background-color:transparent;border-color:#0a0a0a;box-shadow:none;color:#0a0a0a}html.theme--documenter-dark .button.is-black{background-color:#0a0a0a;border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-black:hover,html.theme--documenter-dark .button.is-black.is-hovered{background-color:#040404;border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-black:focus,html.theme--documenter-dark .button.is-black.is-focused{border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-black:focus:not(:active),html.theme--documenter-dark .button.is-black.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(10,10,10,0.25)}html.theme--documenter-dark .button.is-black:active,html.theme--documenter-dark .button.is-black.is-active{background-color:#000;border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-black[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-black{background-color:#0a0a0a;border-color:#0a0a0a;box-shadow:none}html.theme--documenter-dark .button.is-black.is-inverted{background-color:#fff;color:#0a0a0a}html.theme--documenter-dark .button.is-black.is-inverted:hover,html.theme--documenter-dark .button.is-black.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--documenter-dark .button.is-black.is-inverted[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-black.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#0a0a0a}html.theme--documenter-dark .button.is-black.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--documenter-dark .button.is-black.is-outlined{background-color:transparent;border-color:#0a0a0a;color:#0a0a0a}html.theme--documenter-dark .button.is-black.is-outlined:hover,html.theme--documenter-dark .button.is-black.is-outlined.is-hovered,html.theme--documenter-dark .button.is-black.is-outlined:focus,html.theme--documenter-dark .button.is-black.is-outlined.is-focused{background-color:#0a0a0a;border-color:#0a0a0a;color:#fff}html.theme--documenter-dark .button.is-black.is-outlined.is-loading::after{border-color:transparent transparent #0a0a0a #0a0a0a !important}html.theme--documenter-dark .button.is-black.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-black.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-black.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-black.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--documenter-dark .button.is-black.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-black.is-outlined{background-color:transparent;border-color:#0a0a0a;box-shadow:none;color:#0a0a0a}html.theme--documenter-dark .button.is-black.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--documenter-dark .button.is-black.is-inverted.is-outlined:hover,html.theme--documenter-dark .button.is-black.is-inverted.is-outlined.is-hovered,html.theme--documenter-dark .button.is-black.is-inverted.is-outlined:focus,html.theme--documenter-dark .button.is-black.is-inverted.is-outlined.is-focused{background-color:#fff;color:#0a0a0a}html.theme--documenter-dark .button.is-black.is-inverted.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-black.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-black.is-inverted.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-black.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #0a0a0a #0a0a0a !important}html.theme--documenter-dark .button.is-black.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-black.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--documenter-dark .button.is-light{background-color:#ecf0f1;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--documenter-dark .button.is-light:hover,html.theme--documenter-dark .button.is-light.is-hovered{background-color:#e5eaec;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--documenter-dark .button.is-light:focus,html.theme--documenter-dark .button.is-light.is-focused{border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--documenter-dark .button.is-light:focus:not(:active),html.theme--documenter-dark .button.is-light.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(236,240,241,0.25)}html.theme--documenter-dark .button.is-light:active,html.theme--documenter-dark .button.is-light.is-active{background-color:#dde4e6;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--documenter-dark .button.is-light[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-light{background-color:#ecf0f1;border-color:#ecf0f1;box-shadow:none}html.theme--documenter-dark .button.is-light.is-inverted{background-color:rgba(0,0,0,0.7);color:#ecf0f1}html.theme--documenter-dark .button.is-light.is-inverted:hover,html.theme--documenter-dark .button.is-light.is-inverted.is-hovered{background-color:rgba(0,0,0,0.7)}html.theme--documenter-dark .button.is-light.is-inverted[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-light.is-inverted{background-color:rgba(0,0,0,0.7);border-color:transparent;box-shadow:none;color:#ecf0f1}html.theme--documenter-dark .button.is-light.is-loading::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--documenter-dark .button.is-light.is-outlined{background-color:transparent;border-color:#ecf0f1;color:#ecf0f1}html.theme--documenter-dark .button.is-light.is-outlined:hover,html.theme--documenter-dark .button.is-light.is-outlined.is-hovered,html.theme--documenter-dark .button.is-light.is-outlined:focus,html.theme--documenter-dark .button.is-light.is-outlined.is-focused{background-color:#ecf0f1;border-color:#ecf0f1;color:rgba(0,0,0,0.7)}html.theme--documenter-dark .button.is-light.is-outlined.is-loading::after{border-color:transparent transparent #ecf0f1 #ecf0f1 !important}html.theme--documenter-dark .button.is-light.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-light.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-light.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-light.is-outlined.is-loading.is-focused::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--documenter-dark .button.is-light.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-light.is-outlined{background-color:transparent;border-color:#ecf0f1;box-shadow:none;color:#ecf0f1}html.theme--documenter-dark .button.is-light.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);color:rgba(0,0,0,0.7)}html.theme--documenter-dark .button.is-light.is-inverted.is-outlined:hover,html.theme--documenter-dark .button.is-light.is-inverted.is-outlined.is-hovered,html.theme--documenter-dark .button.is-light.is-inverted.is-outlined:focus,html.theme--documenter-dark .button.is-light.is-inverted.is-outlined.is-focused{background-color:rgba(0,0,0,0.7);color:#ecf0f1}html.theme--documenter-dark .button.is-light.is-inverted.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-light.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-light.is-inverted.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-light.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #ecf0f1 #ecf0f1 !important}html.theme--documenter-dark .button.is-light.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-light.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--documenter-dark .button.is-dark,html.theme--documenter-dark .content kbd.button{background-color:#282f2f;border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-dark:hover,html.theme--documenter-dark .content kbd.button:hover,html.theme--documenter-dark .button.is-dark.is-hovered,html.theme--documenter-dark .content kbd.button.is-hovered{background-color:#232829;border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-dark:focus,html.theme--documenter-dark .content kbd.button:focus,html.theme--documenter-dark .button.is-dark.is-focused,html.theme--documenter-dark .content kbd.button.is-focused{border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-dark:focus:not(:active),html.theme--documenter-dark .content kbd.button:focus:not(:active),html.theme--documenter-dark .button.is-dark.is-focused:not(:active),html.theme--documenter-dark .content kbd.button.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(40,47,47,0.25)}html.theme--documenter-dark .button.is-dark:active,html.theme--documenter-dark .content kbd.button:active,html.theme--documenter-dark .button.is-dark.is-active,html.theme--documenter-dark .content kbd.button.is-active{background-color:#1d2122;border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-dark[disabled],html.theme--documenter-dark .content kbd.button[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-dark,fieldset[disabled] html.theme--documenter-dark .content kbd.button{background-color:#282f2f;border-color:#282f2f;box-shadow:none}html.theme--documenter-dark .button.is-dark.is-inverted,html.theme--documenter-dark .content kbd.button.is-inverted{background-color:#fff;color:#282f2f}html.theme--documenter-dark .button.is-dark.is-inverted:hover,html.theme--documenter-dark .content kbd.button.is-inverted:hover,html.theme--documenter-dark .button.is-dark.is-inverted.is-hovered,html.theme--documenter-dark .content kbd.button.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--documenter-dark .button.is-dark.is-inverted[disabled],html.theme--documenter-dark .content kbd.button.is-inverted[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-dark.is-inverted,fieldset[disabled] html.theme--documenter-dark .content kbd.button.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#282f2f}html.theme--documenter-dark .button.is-dark.is-loading::after,html.theme--documenter-dark .content kbd.button.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--documenter-dark .button.is-dark.is-outlined,html.theme--documenter-dark .content kbd.button.is-outlined{background-color:transparent;border-color:#282f2f;color:#282f2f}html.theme--documenter-dark .button.is-dark.is-outlined:hover,html.theme--documenter-dark .content kbd.button.is-outlined:hover,html.theme--documenter-dark .button.is-dark.is-outlined.is-hovered,html.theme--documenter-dark .content kbd.button.is-outlined.is-hovered,html.theme--documenter-dark .button.is-dark.is-outlined:focus,html.theme--documenter-dark .content kbd.button.is-outlined:focus,html.theme--documenter-dark .button.is-dark.is-outlined.is-focused,html.theme--documenter-dark .content kbd.button.is-outlined.is-focused{background-color:#282f2f;border-color:#282f2f;color:#fff}html.theme--documenter-dark .button.is-dark.is-outlined.is-loading::after,html.theme--documenter-dark .content kbd.button.is-outlined.is-loading::after{border-color:transparent transparent #282f2f #282f2f !important}html.theme--documenter-dark .button.is-dark.is-outlined.is-loading:hover::after,html.theme--documenter-dark .content kbd.button.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-dark.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .content kbd.button.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-dark.is-outlined.is-loading:focus::after,html.theme--documenter-dark .content kbd.button.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-dark.is-outlined.is-loading.is-focused::after,html.theme--documenter-dark .content kbd.button.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--documenter-dark .button.is-dark.is-outlined[disabled],html.theme--documenter-dark .content kbd.button.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-dark.is-outlined,fieldset[disabled] html.theme--documenter-dark .content kbd.button.is-outlined{background-color:transparent;border-color:#282f2f;box-shadow:none;color:#282f2f}html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined:hover,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined:hover,html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-hovered,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-hovered,html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined:focus,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined:focus,html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-focused,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-focused{background-color:#fff;color:#282f2f}html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-loading:hover::after,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-loading:focus::after,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-loading.is-focused::after,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #282f2f #282f2f !important}html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined[disabled],html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined,fieldset[disabled] html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--documenter-dark .button.is-primary,html.theme--documenter-dark .docstring>section>a.button.docs-sourcelink{background-color:#375a7f;border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-primary:hover,html.theme--documenter-dark .docstring>section>a.button.docs-sourcelink:hover,html.theme--documenter-dark .button.is-primary.is-hovered,html.theme--documenter-dark .docstring>section>a.button.is-hovered.docs-sourcelink{background-color:#335476;border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-primary:focus,html.theme--documenter-dark 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.button.is-warning.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-warning.is-outlined{background-color:transparent;border-color:#ad8100;box-shadow:none;color:#ad8100}html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined:hover,html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined.is-hovered,html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined:focus,html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined.is-focused{background-color:#fff;color:#ad8100}html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #ad8100 #ad8100 !important}html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--documenter-dark .button.is-warning.is-light{background-color:#fffaeb;color:#d19c00}html.theme--documenter-dark .button.is-warning.is-light:hover,html.theme--documenter-dark .button.is-warning.is-light.is-hovered{background-color:#fff7de;border-color:transparent;color:#d19c00}html.theme--documenter-dark .button.is-warning.is-light:active,html.theme--documenter-dark .button.is-warning.is-light.is-active{background-color:#fff3d1;border-color:transparent;color:#d19c00}html.theme--documenter-dark .button.is-danger{background-color:#9e1b0d;border-color:transparent;color:#fff}html.theme--documenter-dark 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.button.is-danger.is-inverted:hover,html.theme--documenter-dark .button.is-danger.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--documenter-dark .button.is-danger.is-inverted[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-danger.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#9e1b0d}html.theme--documenter-dark .button.is-danger.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--documenter-dark .button.is-danger.is-outlined{background-color:transparent;border-color:#9e1b0d;color:#9e1b0d}html.theme--documenter-dark .button.is-danger.is-outlined:hover,html.theme--documenter-dark .button.is-danger.is-outlined.is-hovered,html.theme--documenter-dark .button.is-danger.is-outlined:focus,html.theme--documenter-dark .button.is-danger.is-outlined.is-focused{background-color:#9e1b0d;border-color:#9e1b0d;color:#fff}html.theme--documenter-dark .button.is-danger.is-outlined.is-loading::after{border-color:transparent transparent #9e1b0d #9e1b0d !important}html.theme--documenter-dark .button.is-danger.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-danger.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-danger.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-danger.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--documenter-dark .button.is-danger.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-danger.is-outlined{background-color:transparent;border-color:#9e1b0d;box-shadow:none;color:#9e1b0d}html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined:hover,html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined.is-hovered,html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined:focus,html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined.is-focused{background-color:#fff;color:#9e1b0d}html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #9e1b0d #9e1b0d !important}html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--documenter-dark .button.is-danger.is-light{background-color:#fdeeec;color:#ec311d}html.theme--documenter-dark .button.is-danger.is-light:hover,html.theme--documenter-dark .button.is-danger.is-light.is-hovered{background-color:#fce3e0;border-color:transparent;color:#ec311d}html.theme--documenter-dark .button.is-danger.is-light:active,html.theme--documenter-dark .button.is-danger.is-light.is-active{background-color:#fcd8d5;border-color:transparent;color:#ec311d}html.theme--documenter-dark .button.is-small,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.button{font-size:.75rem}html.theme--documenter-dark .button.is-small:not(.is-rounded),html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.button:not(.is-rounded){border-radius:3px}html.theme--documenter-dark .button.is-normal{font-size:1rem}html.theme--documenter-dark .button.is-medium{font-size:1.25rem}html.theme--documenter-dark .button.is-large{font-size:1.5rem}html.theme--documenter-dark .button[disabled],fieldset[disabled] html.theme--documenter-dark .button{background-color:#8c9b9d;border-color:#5e6d6f;box-shadow:none;opacity:.5}html.theme--documenter-dark .button.is-fullwidth{display:flex;width:100%}html.theme--documenter-dark .button.is-loading{color:transparent !important;pointer-events:none}html.theme--documenter-dark .button.is-loading::after{position:absolute;left:calc(50% - (1em * 0.5));top:calc(50% - (1em * 0.5));position:absolute !important}html.theme--documenter-dark .button.is-static{background-color:#282f2f;border-color:#5e6d6f;color:#dbdee0;box-shadow:none;pointer-events:none}html.theme--documenter-dark .button.is-rounded,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.button{border-radius:9999px;padding-left:calc(1em + 0.25em);padding-right:calc(1em + 0.25em)}html.theme--documenter-dark .buttons{align-items:center;display:flex;flex-wrap:wrap;justify-content:flex-start}html.theme--documenter-dark .buttons .button{margin-bottom:0.5rem}html.theme--documenter-dark .buttons .button:not(:last-child):not(.is-fullwidth){margin-right:.5rem}html.theme--documenter-dark .buttons:last-child{margin-bottom:-0.5rem}html.theme--documenter-dark .buttons:not(:last-child){margin-bottom:1rem}html.theme--documenter-dark .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large){font-size:.75rem}html.theme--documenter-dark .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large):not(.is-rounded){border-radius:3px}html.theme--documenter-dark .buttons.are-medium .button:not(.is-small):not(.is-normal):not(.is-large){font-size:1.25rem}html.theme--documenter-dark .buttons.are-large .button:not(.is-small):not(.is-normal):not(.is-medium){font-size:1.5rem}html.theme--documenter-dark .buttons.has-addons .button:not(:first-child){border-bottom-left-radius:0;border-top-left-radius:0}html.theme--documenter-dark .buttons.has-addons .button:not(:last-child){border-bottom-right-radius:0;border-top-right-radius:0;margin-right:-1px}html.theme--documenter-dark .buttons.has-addons .button:last-child{margin-right:0}html.theme--documenter-dark .buttons.has-addons .button:hover,html.theme--documenter-dark .buttons.has-addons .button.is-hovered{z-index:2}html.theme--documenter-dark .buttons.has-addons .button:focus,html.theme--documenter-dark .buttons.has-addons .button.is-focused,html.theme--documenter-dark .buttons.has-addons .button:active,html.theme--documenter-dark .buttons.has-addons .button.is-active,html.theme--documenter-dark .buttons.has-addons .button.is-selected{z-index:3}html.theme--documenter-dark .buttons.has-addons .button:focus:hover,html.theme--documenter-dark .buttons.has-addons .button.is-focused:hover,html.theme--documenter-dark .buttons.has-addons .button:active:hover,html.theme--documenter-dark .buttons.has-addons .button.is-active:hover,html.theme--documenter-dark .buttons.has-addons .button.is-selected:hover{z-index:4}html.theme--documenter-dark .buttons.has-addons .button.is-expanded{flex-grow:1;flex-shrink:1}html.theme--documenter-dark .buttons.is-centered{justify-content:center}html.theme--documenter-dark .buttons.is-centered:not(.has-addons) .button:not(.is-fullwidth){margin-left:0.25rem;margin-right:0.25rem}html.theme--documenter-dark .buttons.is-right{justify-content:flex-end}html.theme--documenter-dark .buttons.is-right:not(.has-addons) .button:not(.is-fullwidth){margin-left:0.25rem;margin-right:0.25rem}@media screen and (max-width: 768px){html.theme--documenter-dark .button.is-responsive.is-small,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.is-responsive{font-size:.5625rem}html.theme--documenter-dark .button.is-responsive,html.theme--documenter-dark .button.is-responsive.is-normal{font-size:.65625rem}html.theme--documenter-dark .button.is-responsive.is-medium{font-size:.75rem}html.theme--documenter-dark .button.is-responsive.is-large{font-size:1rem}}@media screen and (min-width: 769px) and (max-width: 1055px){html.theme--documenter-dark .button.is-responsive.is-small,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.is-responsive{font-size:.65625rem}html.theme--documenter-dark .button.is-responsive,html.theme--documenter-dark .button.is-responsive.is-normal{font-size:.75rem}html.theme--documenter-dark .button.is-responsive.is-medium{font-size:1rem}html.theme--documenter-dark .button.is-responsive.is-large{font-size:1.25rem}}html.theme--documenter-dark .container{flex-grow:1;margin:0 auto;position:relative;width:auto}html.theme--documenter-dark .container.is-fluid{max-width:none !important;padding-left:32px;padding-right:32px;width:100%}@media screen and (min-width: 1056px){html.theme--documenter-dark .container{max-width:992px}}@media screen and (max-width: 1215px){html.theme--documenter-dark .container.is-widescreen:not(.is-max-desktop){max-width:1152px}}@media screen and (max-width: 1407px){html.theme--documenter-dark .container.is-fullhd:not(.is-max-desktop):not(.is-max-widescreen){max-width:1344px}}@media screen and (min-width: 1216px){html.theme--documenter-dark .container:not(.is-max-desktop){max-width:1152px}}@media screen and (min-width: 1408px){html.theme--documenter-dark .container:not(.is-max-desktop):not(.is-max-widescreen){max-width:1344px}}html.theme--documenter-dark .content li+li{margin-top:0.25em}html.theme--documenter-dark .content p:not(:last-child),html.theme--documenter-dark .content dl:not(:last-child),html.theme--documenter-dark .content ol:not(:last-child),html.theme--documenter-dark .content ul:not(:last-child),html.theme--documenter-dark .content blockquote:not(:last-child),html.theme--documenter-dark .content pre:not(:last-child),html.theme--documenter-dark .content table:not(:last-child){margin-bottom:1em}html.theme--documenter-dark .content h1,html.theme--documenter-dark .content h2,html.theme--documenter-dark .content h3,html.theme--documenter-dark .content h4,html.theme--documenter-dark .content h5,html.theme--documenter-dark .content h6{color:#f2f2f2;font-weight:600;line-height:1.125}html.theme--documenter-dark .content h1{font-size:2em;margin-bottom:0.5em}html.theme--documenter-dark .content h1:not(:first-child){margin-top:1em}html.theme--documenter-dark .content h2{font-size:1.75em;margin-bottom:0.5714em}html.theme--documenter-dark .content h2:not(:first-child){margin-top:1.1428em}html.theme--documenter-dark .content h3{font-size:1.5em;margin-bottom:0.6666em}html.theme--documenter-dark .content h3:not(:first-child){margin-top:1.3333em}html.theme--documenter-dark .content h4{font-size:1.25em;margin-bottom:0.8em}html.theme--documenter-dark .content h5{font-size:1.125em;margin-bottom:0.8888em}html.theme--documenter-dark .content h6{font-size:1em;margin-bottom:1em}html.theme--documenter-dark .content blockquote{background-color:#282f2f;border-left:5px solid #5e6d6f;padding:1.25em 1.5em}html.theme--documenter-dark .content ol{list-style-position:outside;margin-left:2em;margin-top:1em}html.theme--documenter-dark .content ol:not([type]){list-style-type:decimal}html.theme--documenter-dark .content ol.is-lower-alpha:not([type]){list-style-type:lower-alpha}html.theme--documenter-dark .content ol.is-lower-roman:not([type]){list-style-type:lower-roman}html.theme--documenter-dark .content ol.is-upper-alpha:not([type]){list-style-type:upper-alpha}html.theme--documenter-dark .content ol.is-upper-roman:not([type]){list-style-type:upper-roman}html.theme--documenter-dark .content ul{list-style:disc outside;margin-left:2em;margin-top:1em}html.theme--documenter-dark .content ul ul{list-style-type:circle;margin-top:0.5em}html.theme--documenter-dark .content ul ul ul{list-style-type:square}html.theme--documenter-dark .content dd{margin-left:2em}html.theme--documenter-dark .content figure{margin-left:2em;margin-right:2em;text-align:center}html.theme--documenter-dark .content figure:not(:first-child){margin-top:2em}html.theme--documenter-dark .content figure:not(:last-child){margin-bottom:2em}html.theme--documenter-dark .content figure img{display:inline-block}html.theme--documenter-dark .content figure figcaption{font-style:italic}html.theme--documenter-dark .content pre{-webkit-overflow-scrolling:touch;overflow-x:auto;padding:0;white-space:pre;word-wrap:normal}html.theme--documenter-dark .content sup,html.theme--documenter-dark .content sub{font-size:75%}html.theme--documenter-dark .content table{width:100%}html.theme--documenter-dark .content table td,html.theme--documenter-dark .content table th{border:1px solid #5e6d6f;border-width:0 0 1px;padding:0.5em 0.75em;vertical-align:top}html.theme--documenter-dark .content table th{color:#f2f2f2}html.theme--documenter-dark .content table th:not([align]){text-align:inherit}html.theme--documenter-dark .content table thead td,html.theme--documenter-dark .content table thead th{border-width:0 0 2px;color:#f2f2f2}html.theme--documenter-dark .content table tfoot td,html.theme--documenter-dark .content table tfoot th{border-width:2px 0 0;color:#f2f2f2}html.theme--documenter-dark .content table tbody tr:last-child td,html.theme--documenter-dark .content table tbody tr:last-child th{border-bottom-width:0}html.theme--documenter-dark .content .tabs li+li{margin-top:0}html.theme--documenter-dark .content.is-small,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.content{font-size:.75rem}html.theme--documenter-dark .content.is-normal{font-size:1rem}html.theme--documenter-dark .content.is-medium{font-size:1.25rem}html.theme--documenter-dark .content.is-large{font-size:1.5rem}html.theme--documenter-dark .icon{align-items:center;display:inline-flex;justify-content:center;height:1.5rem;width:1.5rem}html.theme--documenter-dark .icon.is-small,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.icon{height:1rem;width:1rem}html.theme--documenter-dark .icon.is-medium{height:2rem;width:2rem}html.theme--documenter-dark .icon.is-large{height:3rem;width:3rem}html.theme--documenter-dark .icon-text{align-items:flex-start;color:inherit;display:inline-flex;flex-wrap:wrap;line-height:1.5rem;vertical-align:top}html.theme--documenter-dark .icon-text .icon{flex-grow:0;flex-shrink:0}html.theme--documenter-dark .icon-text .icon:not(:last-child){margin-right:.25em}html.theme--documenter-dark .icon-text .icon:not(:first-child){margin-left:.25em}html.theme--documenter-dark div.icon-text{display:flex}html.theme--documenter-dark .image,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img{display:block;position:relative}html.theme--documenter-dark .image img,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img img{display:block;height:auto;width:100%}html.theme--documenter-dark .image img.is-rounded,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img img.is-rounded{border-radius:9999px}html.theme--documenter-dark .image.is-fullwidth,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-fullwidth{width:100%}html.theme--documenter-dark .image.is-square img,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-square img,html.theme--documenter-dark .image.is-square .has-ratio,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-square .has-ratio,html.theme--documenter-dark .image.is-1by1 img,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-1by1 img,html.theme--documenter-dark .image.is-1by1 .has-ratio,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-1by1 .has-ratio,html.theme--documenter-dark .image.is-5by4 img,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-5by4 img,html.theme--documenter-dark .image.is-5by4 .has-ratio,html.theme--documenter-dark #documenter 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+ Maintainer: @highlightjs/core-team + Website: https://highlightjs.org/ + License: see project LICENSE + Touched: 2021 +*/pre code.hljs{display:block;overflow-x:auto;padding:1em}code.hljs{padding:3px 5px}.hljs{background:#F3F3F3;color:#444}.hljs-comment{color:#697070}.hljs-tag,.hljs-punctuation{color:#444a}.hljs-tag .hljs-name,.hljs-tag .hljs-attr{color:#444}.hljs-keyword,.hljs-attribute,.hljs-selector-tag,.hljs-meta .hljs-keyword,.hljs-doctag,.hljs-name{font-weight:bold}.hljs-type,.hljs-string,.hljs-number,.hljs-selector-id,.hljs-selector-class,.hljs-quote,.hljs-template-tag,.hljs-deletion{color:#880000}.hljs-title,.hljs-section{color:#880000;font-weight:bold}.hljs-regexp,.hljs-symbol,.hljs-variable,.hljs-template-variable,.hljs-link,.hljs-selector-attr,.hljs-operator,.hljs-selector-pseudo{color:#ab5656}.hljs-literal{color:#695}.hljs-built_in,.hljs-bullet,.hljs-code,.hljs-addition{color:#397300}.hljs-meta{color:#1f7199}.hljs-meta .hljs-string{color:#38a}.hljs-emphasis{font-style:italic}.hljs-strong{font-weight:bold}.gap-4{gap:1rem} diff --git a/v0.3.1/assets/themeswap.js b/v0.3.1/assets/themeswap.js new file mode 100644 index 0000000..9f5eebe --- /dev/null +++ b/v0.3.1/assets/themeswap.js @@ -0,0 +1,84 @@ +// Small function to quickly swap out themes. Gets put into the tag.. +function set_theme_from_local_storage() { + // Initialize the theme to null, which means default + var theme = null; + // If the browser supports the localstorage and is not disabled then try to get the + // documenter theme + if (window.localStorage != null) { + // Get the user-picked theme from localStorage. May be `null`, which means the default + // theme. + theme = window.localStorage.getItem("documenter-theme"); + } + // Check if the users preference is for dark color scheme + var darkPreference = + window.matchMedia("(prefers-color-scheme: dark)").matches === true; + // Initialize a few variables for the loop: + // + // - active: will contain the index of the theme that should be active. Note that there + // is no guarantee that localStorage contains sane values. If `active` stays `null` + // we either could not find the theme or it is the default (primary) theme anyway. + // Either way, we then need to stick to the primary theme. + // + // - disabled: style sheets that should be disabled (i.e. all the theme style sheets + // that are not the currently active theme) + var active = null; + var disabled = []; + var primaryLightTheme = null; + var primaryDarkTheme = null; + for (var i = 0; i < document.styleSheets.length; i++) { + var ss = document.styleSheets[i]; + // The tag of each style sheet is expected to have a data-theme-name attribute + // which must contain the name of the theme. The names in localStorage much match this. + var themename = ss.ownerNode.getAttribute("data-theme-name"); + // attribute not set => non-theme stylesheet => ignore + if (themename === null) continue; + // To distinguish the default (primary) theme, it needs to have the data-theme-primary + // attribute set. + if (ss.ownerNode.getAttribute("data-theme-primary") !== null) { + primaryLightTheme = themename; + } + // Check if the theme is primary dark theme so that we could store its name in darkTheme + if (ss.ownerNode.getAttribute("data-theme-primary-dark") !== null) { + primaryDarkTheme = themename; + } + // If we find a matching theme (and it's not the default), we'll set active to non-null + if (themename === theme) active = i; + // Store the style sheets of inactive themes so that we could disable them + if (themename !== theme) disabled.push(ss); + } + var activeTheme = null; + if (active !== null) { + // If we did find an active theme, we'll (1) add the theme--$(theme) class to + document.getElementsByTagName("html")[0].className = "theme--" + theme; + activeTheme = theme; + } else { + // If we did _not_ find an active theme, then we need to fall back to the primary theme + // which can either be dark or light, depending on the user's OS preference. + var activeTheme = darkPreference ? primaryDarkTheme : primaryLightTheme; + // In case it somehow happens that the relevant primary theme was not found in the + // preceding loop, we abort without doing anything. + if (activeTheme === null) { + console.error("Unable to determine primary theme."); + return; + } + // When switching to the primary light theme, then we must not have a class name + // for the tag. That's only for non-primary or the primary dark theme. + if (darkPreference) { + document.getElementsByTagName("html")[0].className = + "theme--" + activeTheme; + } else { + document.getElementsByTagName("html")[0].className = ""; + } + } + for (var i = 0; i < document.styleSheets.length; i++) { + var ss = document.styleSheets[i]; + // The tag of each style sheet is expected to have a data-theme-name attribute + // which must contain the name of the theme. The names in localStorage much match this. + var themename = ss.ownerNode.getAttribute("data-theme-name"); + // attribute not set => non-theme stylesheet => ignore + if (themename === null) continue; + // we'll disable all the stylesheets, except for the active one + ss.disabled = !(themename == activeTheme); + } +} +set_theme_from_local_storage(); diff --git a/v0.3.1/assets/warner.js b/v0.3.1/assets/warner.js new file mode 100644 index 0000000..3f6f5d0 --- /dev/null +++ b/v0.3.1/assets/warner.js @@ -0,0 +1,52 @@ +function maybeAddWarning() { + // DOCUMENTER_NEWEST is defined in versions.js, DOCUMENTER_CURRENT_VERSION and DOCUMENTER_STABLE + // in siteinfo.js. + // If either of these are undefined something went horribly wrong, so we abort. + if ( + window.DOCUMENTER_NEWEST === undefined || + window.DOCUMENTER_CURRENT_VERSION === undefined || + window.DOCUMENTER_STABLE === undefined + ) { + return; + } + + // Current version is not a version number, so we can't tell if it's the newest version. Abort. + if (!/v(\d+\.)*\d+/.test(window.DOCUMENTER_CURRENT_VERSION)) { + return; + } + + // Current version is newest version, so no need to add a warning. + if (window.DOCUMENTER_NEWEST === window.DOCUMENTER_CURRENT_VERSION) { + return; + } + + // Add a noindex meta tag (unless one exists) so that search engines don't index this version of the docs. + if (document.body.querySelector('meta[name="robots"]') === null) { + const meta = document.createElement("meta"); + meta.name = "robots"; + meta.content = "noindex"; + + document.getElementsByTagName("head")[0].appendChild(meta); + } + + const div = document.createElement("div"); + div.classList.add("outdated-warning-overlay"); + const closer = document.createElement("button"); + closer.classList.add("outdated-warning-closer", "delete"); + closer.addEventListener("click", function () { + document.body.removeChild(div); + }); + const href = window.documenterBaseURL + "/../" + window.DOCUMENTER_STABLE; + div.innerHTML = + 'This documentation is not for the latest stable release, but for either the development version or an older release.
    Click here to go to the documentation for the latest stable release.'; + div.appendChild(closer); + document.body.appendChild(div); +} + +if (document.readyState === "loading") { + document.addEventListener("DOMContentLoaded", maybeAddWarning); +} else { + maybeAddWarning(); +} diff --git a/v0.3.1/examples/index.html b/v0.3.1/examples/index.html new file mode 100644 index 0000000..a12730a --- /dev/null +++ b/v0.3.1/examples/index.html @@ -0,0 +1,481 @@ + +Examples · MRVCModels
    GitHub

    Examples

    Simulate data

    julia> using MultiResponseVarianceComponentModels, LinearAlgebra, Random
    julia> Random.seed!(6789)Random.TaskLocalRNG()
    julia> n = 1_000;  # n of observations
    julia> d = 4;      # n of responses
    julia> p = 10;     # n of covariates
    julia> m = 5;      # n of variance components
    julia> X = rand(n, p);
    julia> B = rand(p, d)10×4 Matrix{Float64}:
+ 0.866192  0.739958   0.951693   0.716391
+ 0.157873  0.633772   0.916356   0.068644
+ 0.818581  0.333021   0.0713489  0.451072
+ 0.707496  0.924568   0.0636189  0.618679
+ 0.357756  0.173016   0.982384   0.54638
+ 0.722163  0.0683381  0.343516   0.616161
+ 0.441486  0.64316    0.862809   0.400984
+ 0.812486  0.275565   0.477624   0.482087
+ 0.889787  0.669932   0.103051   0.582507
+ 0.331484  0.989292   0.785077   0.456195
    julia> V = [zeros(n, n) for _ in 1:m]; # kernel matrices
    julia> Σ = [zeros(d, d) for _ in 1:m]; # variance components
    julia> for i in 1:m
+           Vi = randn(n, n)
+           copy!(V[i], Vi' * Vi)
+           Σi = randn(d, d)
+           copy!(Σ[i], Σi' * Σi)
+       end
    julia> Ω = zeros(n * d, n * d); # overall nd-by-nd covariance matrix Ω
    julia> for i = 1:m
+           Ω += kron(Σ[i], V[i])
+       end
    julia> Ωchol = cholesky(Ω);
    julia> Y = X * B + reshape(Ωchol.L * randn(n * d), n, d)1000×4 Matrix{Float64}:
+ -270.382    -66.0256   233.653     158.94
+  168.935    354.423     14.8585    -54.193
+   85.8007   117.902   -217.707      65.1394
+ -136.574    -42.6014  -234.087      75.6404
+ -190.312   -210.051    105.117     -10.0791
+  153.414    187.179     20.691     -62.3589
+  164.97     -22.4103   -13.6301    161.682
+   21.9438  -184.643    231.526    -136.723
+ -284.549     79.7798   -37.1967   -136.943
+  -59.2492  -124.37    -160.587     112.235
+    ⋮
+  -82.4606   200.557     -4.16006   -47.2201
+  170.0       95.8103   222.034    -179.011
+   98.1247   -33.0932   -67.4875    -86.9752
+ -167.871    110.395   -184.76      -71.029
+  111.644    -23.6517   113.328    -219.276
+  -77.2094   168.859     65.7808     22.939
+  129.34    -155.123   -251.557     120.656
+ -359.121    -82.2576   -82.0063     10.8301
+  -77.7126   -90.7158   -79.5722     92.9695
    Note

    In the case of heritability and genetic correlation analyses, one can use classic genetic relationship matrices (GRMs) for $\boldsymbol{V}_i$'s, which in turn can be constructed using SnpArrays.jl.

    Maximum likelihood estimation

    julia> model = MRVCModel(Y, X, V)A multivariate response variance component model
+   * number of responses: 4
+   * number of observations: 1000
+   * number of fixed effects: 10
+   * number of variance components: 5
    julia> @timev fit!(model)[ Info: Running MM algorithm for ML estimation
+iter = 0, logl = -26845.09612499673
+iter = 1, logl = -25695.041826875506
+iter = 2, logl = -25293.37779878975
+iter = 3, logl = -25159.24906847003
+iter = 4, logl = -25107.52993220223
+iter = 5, logl = -25081.38428962063
+iter = 6, logl = -25064.352419795112
+iter = 7, logl = -25051.625744592016
+iter = 8, logl = -25041.61006910993
+iter = 9, logl = -25033.604422863456
+iter = 10, logl = -25027.18042662134
+iter = 11, logl = -25022.017854787307
+iter = 12, logl = -25017.85980986688
+iter = 13, logl = -25014.498019681763
+iter = 14, logl = -25011.76503965001
+iter = 15, logl = -25009.527835987446
+iter = 16, logl = -25007.68180780364
+iter = 17, logl = -25006.145330198324
+iter = 18, logl = -25004.854995838356
+iter = 19, logl = -25003.761633006412
+iter = 20, logl = -25002.827079011357
+iter = 21, logl = -25002.02162797271
+iter = 22, logl = -25001.322047241047
+iter = 23, logl = -25000.710054799467
+iter = 24, logl = -25000.171160151207
+iter = 25, logl = -24999.693786243588
+iter = 26, logl = -24999.26860570048
+iter = 27, logl = -24998.888038955105
+iter = 28, logl = -24998.545873977288
+iter = 29, logl = -24998.236977041215
+iter = 30, logl = -24997.95707160835
+iter = 31, logl = -24997.702568235134
+iter = 32, logl = -24997.470432811737
+iter = 33, logl = -24997.258083718323
+iter = 34, logl = -24997.063310912818
+iter = 35, logl = -24996.884211761542
+iter = 36, logl = -24996.719139735058
+iter = 37, logl = -24996.566663066376
+iter = 38, logl = -24996.425531184308
+iter = 39, logl = -24996.294647261133
+iter = 40, logl = -24996.173045607276
+iter = 41, logl = -24996.05987293569
+iter = 42, logl = -24995.95437274284
+iter = 43, logl = -24995.855872212444
+iter = 44, logl = -24995.7637711789
+iter = 45, logl = -24995.67753277965
+iter = 46, logl = -24995.596675504443
+iter = 47, logl = -24995.520766401234
+iter = 48, logl = -24995.449415250514
+iter = 49, logl = -24995.38226954833
+iter = 50, logl = -24995.31901017304
+iter = 51, logl = -24995.259347628595
+iter = 52, logl = -24995.203018777862
+iter = 53, logl = -24995.14978399378
+iter = 54, logl = -24995.09942466663
+iter = 55, logl = -24995.051741018295
+iter = 56, logl = -24995.006550180144
+iter = 57, logl = -24994.963684497296
+iter = 58, logl = -24994.922990031344
+iter = 59, logl = -24994.88432523246
+iter = 60, logl = -24994.84755976209
+iter = 61, logl = -24994.812573442905
+iter = 62, logl = -24994.779255325782
+iter = 63, logl = -24994.74750285241
+iter = 64, logl = -24994.717221108924
+iter = 65, logl = -24994.688322154096
+iter = 66, logl = -24994.660724417583
+iter = 67, logl = -24994.63435215651
+iter = 68, logl = -24994.609134967097
+iter = 69, logl = -24994.585007342
+[ Info: Updates converged!
+[ Info: Calculating standard errors
+ 93.434873 seconds (15.63 M allocations: 992.454 MiB, 0.31% gc time, 11.80% compilation time)
+elapsed time (ns):  93434873125
+gc time (ns):       291148958
+bytes allocated:    1040663100
+pool allocs:        15602304
+non-pool GC allocs: 26823
+malloc() calls:     929
+realloc() calls:    107
+free() calls:       335
+minor collections:  1
+full collections:   0
+Converged after 70 iterations.

    Then variance components and mean effects estimates can be accessed through

    julia> model.Σ5-element Vector{Matrix{Float64}}:
+ [2.8964235833864374 -0.9547701186106777 1.6856925389781934 0.8185656173741515; -0.9547701186106778 7.474693619948036 -4.744935846247633 -0.601253052759175; 1.6856925389781932 -4.744935846247633 8.639624062398605 0.4049276157435974; 0.8185656173741512 -0.601253052759175 0.4049276157435973 1.0145168589537352]
+ [3.776260289140289 2.3919496497788035 -0.3666324933910243 -1.365148782876448; 2.391949649778804 1.9914356452751707 -0.2630505419731652 -1.5892702567044454; -0.36663249339102433 -0.26305054197316524 4.0912210911875855 -2.287177667967102; -1.365148782876448 -1.5892702567044454 -2.287177667967102 4.526163931034507]
+ [3.1157472905367367 -0.08262940589305556 -0.27318718635822953 0.4179078660737166; -0.08262940589305548 3.4556143634179275 -0.20341922639890309 0.6648870938984194; -0.27318718635822953 -0.20341922639890314 1.9155083377403541 0.49426216391703903; 0.41790786607371666 0.6648870938984194 0.49426216391703903 0.5435108874298807]
+ [3.6027030953820782 1.2366936673894326 0.3637181414535267 0.06851794976775202; 1.2366936673894326 0.9715515927327851 1.6405295499121633 0.5971896668720472; 0.3637181414535266 1.6405295499121637 5.828012183031251 0.9752007240509573; 0.06851794976775206 0.5971896668720472 0.9752007240509573 0.9502953338899155]
+ [5.669808082536949 -2.01083534935516 1.1222606709505223 1.4405860817195557; -2.0108353493551605 6.032248061565413 9.439926228569771 -2.150730923690082; 1.1222606709505227 9.439926228569773 19.179102992380216 -3.054827357292586; 1.4405860817195555 -2.150730923690082 -3.054827357292586 3.445429926626015]
    julia> model.B10×4 Matrix{Float64}:
+   7.76641     0.0581691   30.0176     10.8424
+  -1.70871    17.0653      26.3436      5.24079
+  -4.99575    28.325      -20.9966    -13.1088
+ -10.2939     -6.00867     12.3106    -16.3959
+  -3.07093     0.733837    15.8845      1.17996
+  14.1045     -3.92946     11.5408     11.2497
+  -1.47949    -8.30862      5.49722    -0.213233
+   6.03975     5.27422      0.189634    3.90586
+  -0.632607  -14.288      -29.194       4.9749
+   6.17054   -13.2489     -23.6247     -3.52297
    julia> hcat(vec(B), vec(model.B))40×2 Matrix{Float64}:
+ 0.866192    7.76641
+ 0.157873   -1.70871
+ 0.818581   -4.99575
+ 0.707496  -10.2939
+ 0.357756   -3.07093
+ 0.722163   14.1045
+ 0.441486   -1.47949
+ 0.812486    6.03975
+ 0.889787   -0.632607
+ 0.331484    6.17054
+ ⋮
+ 0.068644    5.24079
+ 0.451072  -13.1088
+ 0.618679  -16.3959
+ 0.54638     1.17996
+ 0.616161   11.2497
+ 0.400984   -0.213233
+ 0.482087    3.90586
+ 0.582507    4.9749
+ 0.456195   -3.52297
    julia> reduce(hcat, [hcat(vec(Σ[i]), vec(model.Σ[i])) for i in 1:m])16×10 Matrix{Float64}:
+  2.59353    2.89642    3.32588    …  3.6027      5.37835    5.66981
+ -0.908817  -0.95477    2.21659       1.23669    -1.43756   -2.01084
+  1.58682    1.68569   -0.0345923     0.363718    1.65263    1.12226
+  0.365451   0.818566  -1.36959       0.0685179   0.983537   1.44059
+ -0.908817  -0.95477    2.21659       1.23669    -1.43756   -2.01084
+  7.12022    7.47469    1.75764    …  0.971552    6.84955    6.03225
+ -4.72969   -4.74494   -0.0950186     1.64053     9.98583    9.43993
+ -0.430129  -0.601253  -1.55052       0.59719    -1.98954   -2.15073
+  1.58682    1.68569   -0.0345923     0.363718    1.65263    1.12226
+ -4.72969   -4.74494   -0.0950186     1.64053     9.98583    9.43993
+  6.22778    8.63962    4.06617    …  5.82801    19.2083    19.1791
+  0.969228   0.404928  -2.51336       0.975201   -3.5885    -3.05483
+  0.365451   0.818566  -1.36959       0.0685179   0.983537   1.44059
+ -0.430129  -0.601253  -1.55052       0.59719    -1.98954   -2.15073
+  0.969228   0.404928  -2.51336       0.975201   -3.5885    -3.05483
+  1.2446     1.01452    5.02933    …  0.950295    2.85342    3.44543

    Standard errors

    Sampling variance and covariance of these estimates are

    julia> model.Σcov50×50 Matrix{Float64}:
+  0.47412      -0.00272341    0.108306     …  -0.00114518   -0.000725468
+ -0.00272341    0.280366     -0.0353382       -0.000510353   0.000627446
+  0.108306     -0.0353382     0.483991        -0.00663509    0.000814462
+  0.0597015    -0.0241462    -0.00476179      -0.00287831   -0.00410306
+ -0.000531375  -0.000740352  -0.00227336       0.00175773   -0.000550985
+ -5.46266e-6    0.0615992    -0.0084444    …   0.00754037   -0.000713865
+ -7.80744e-5    0.0347709    -0.00356761      -0.00609519    0.00360991
+  0.0240561    -0.0132044     0.213683         0.0166641    -0.000929684
+  0.0133385    -0.00919492    0.0562126       -0.0423021     0.00471451
+  0.00739022   -0.00597224   -0.0030313        0.00471978   -0.0236748
+  ⋮                                        ⋱
+ -0.00769785   -0.0406075    -0.012298         0.0234008    -0.0161668
+ -0.0133522    -0.0122384    -0.0861289        0.094258     -0.0224369
+ -0.00838137    0.00335528    0.00422692       0.0168207     0.0502388
+ -0.000645754  -0.00632483   -0.00175497      -0.0798787     0.0198685
+ -0.00102918   -0.00637199   -0.00825901   …  -0.187875      0.0275525
+ -0.000640514  -0.00315513   -0.000651312      0.140072     -0.0624593
+ -0.00183032   -0.00321512   -0.0232591       -0.36789       0.0382077
+ -0.00114518   -0.000510353  -0.00663509       0.426481     -0.0867558
+ -0.000725468   0.000627446   0.000814462     -0.0867558     0.189954
    julia> model.Bcov40×40 Matrix{Float64}:
+ 157.174     -19.3776    -19.4223    …   -0.257223   -1.24371    -1.90444
+ -19.3776    161.611     -22.5065        -1.3117     -1.42903    -1.33411
+ -19.4223    -22.5065    150.341         -1.38577    -2.56388    -1.25275
+ -14.6304    -12.8425    -19.2089        -2.29579    -2.03511    -1.00321
+ -11.6747    -13.9337    -12.4649        -0.198984   -1.44227    -2.31342
+ -24.2206    -18.4552     -7.96731   …   -3.00702    -1.14626    -0.923721
+ -13.3431    -17.4951    -15.8672        -1.57057    -0.945035   -1.77244
+ -12.7248    -19.4206    -11.7579        13.2375     -1.25172    -1.08592
+ -17.2277    -19.1384    -19.5393        -1.22189    12.8571     -0.396249
+ -15.0949    -12.6969    -15.9363        -1.23333    -0.422978   12.5215
+   ⋮                                 ⋱
+  -2.12779    13.4419     -1.25447       -9.26816    -9.00194    -6.01701
+  -0.752161   -1.49914    13.5685        -5.62018    -9.24482    -8.16962
+  -0.737933   -0.925351   -1.71971      -11.7708     -8.59325   -10.0889
+  -1.49387    -1.2578     -1.04615       -5.98367   -12.911     -13.1438
+  -1.72094    -2.33376    -0.495492  …   -9.81655    -5.37389    -8.16594
+  -1.53932    -0.763366   -2.22368       -9.36946    -7.53599   -10.4206
+  -0.257223   -1.3117     -1.38577       79.0721     -7.3782     -9.48116
+  -1.24371    -1.42903    -2.56388       -7.3782     75.6156     -0.921828
+  -1.90444    -1.33411    -1.25275       -9.48116    -0.921828   78.0356

    Corresponding standard error of these estimates are

    julia> sqrt.(diag(model.Σcov))50-element Vector{Float64}:
+ 0.6885637727726415
+ 0.529495666135442
+ 0.6956949396229591
+ 0.33645040355906397
+ 0.8084217056989712
+ 0.7442584972630495
+ 0.3639694672208399
+ 1.3688056422005268
+ 0.4677057749056628
+ 0.3179680333750869
+ ⋮
+ 0.5524130983848243
+ 0.8823995615262973
+ 0.43111265513125097
+ 0.7402837878083752
+ 0.9231964895406697
+ 0.41384022887389565
+ 1.8662336914254827
+ 0.6530553700433331
+ 0.43583743574349515
    julia> sqrt.(diag(model.Bcov))40-element Vector{Float64}:
+ 12.536904651555659
+ 12.712633608569577
+ 12.26134556777767
+ 12.169443903108293
+ 12.663455285629615
+ 11.870615133041053
+ 12.528437624570504
+ 12.51912728097892
+ 12.39394332241073
+ 12.510940364508672
+  ⋮
+  8.869558232699935
+  8.568572559479021
+  8.594848108237025
+  8.973590296072445
+  8.287660643177881
+  8.80370479520269
+  8.892247260593642
+  8.695724537824539
+  8.833773378129575

    Residual maximum likelihood estimation

    For REML estimation, you can instead:

    julia> model = MRVCModel(Y, X, V; reml = true)A multivariate response variance component model
+   * number of responses: 4
+   * number of observations: 1000
+   * number of fixed effects: 10
+   * number of variance components: 5
    julia> @timev fit!(model)[ Info: Running MM algorithm for REML estimation
+iter = 0, logl = -26597.537258425444
+iter = 1, logl = -25455.46070096719
+iter = 2, logl = -25062.15815081145
+iter = 3, logl = -24929.900271068564
+iter = 4, logl = -24878.96910611469
+iter = 5, logl = -24853.289345291265
+iter = 6, logl = -24836.614727024702
+iter = 7, logl = -24824.192330128142
+iter = 8, logl = -24814.4419685779
+iter = 9, logl = -24806.666946417736
+iter = 10, logl = -24800.441634611918
+iter = 11, logl = -24795.448814457188
+iter = 12, logl = -24791.434959065882
+iter = 13, logl = -24788.19527844777
+iter = 14, logl = -24785.565671601264
+iter = 15, logl = -24783.416144877603
+iter = 16, logl = -24781.644754384706
+iter = 17, logl = -24780.17213209585
+iter = 18, logl = -24778.936749268694
+iter = 19, logl = -24777.890976339873
+iter = 20, logl = -24776.997905413984
+iter = 21, logl = -24776.228846668713
+iter = 22, logl = -24775.561388912047
+iter = 23, logl = -24774.977914973028
+iter = 24, logl = -24774.46447416872
+iter = 25, logl = -24774.009929862073
+iter = 26, logl = -24773.605316168836
+iter = 27, logl = -24773.243352268248
+iter = 28, logl = -24772.918074830915
+iter = 29, logl = -24772.62455872271
+iter = 30, logl = -24772.358703657163
+iter = 31, logl = -24772.117070189604
+iter = 32, logl = -24771.896752736797
+iter = 33, logl = -24771.695280510114
+iter = 34, logl = -24771.51053960551
+iter = 35, logl = -24771.340711232333
+iter = 36, logl = -24771.18422234522
+iter = 37, logl = -24771.03970587292
+iter = 38, logl = -24770.90596843845
+iter = 39, logl = -24770.781963965892
+iter = 40, logl = -24770.66677195723
+iter = 41, logl = -24770.55957949347
+iter = 42, logl = -24770.459666236435
+iter = 43, logl = -24770.36639185783
+iter = 44, logl = -24770.27918545024
+iter = 45, logl = -24770.19753656323
+iter = 46, logl = -24770.120987582057
+iter = 47, logl = -24770.049127218048
+iter = 48, logl = -24769.981584929763
+iter = 49, logl = -24769.918026121268
+iter = 50, logl = -24769.85814799603
+iter = 51, logl = -24769.80167596465
+iter = 52, logl = -24769.74836052275
+iter = 53, logl = -24769.69797452873
+iter = 54, logl = -24769.65031082431
+iter = 55, logl = -24769.60518014765
+iter = 56, logl = -24769.56240929961
+iter = 57, logl = -24769.521839527268
+iter = 58, logl = -24769.48332509584
+iter = 59, logl = -24769.446732024946
+iter = 60, logl = -24769.41193696552
+iter = 61, logl = -24769.378826202505
+iter = 62, logl = -24769.34729476516
+iter = 63, logl = -24769.317245631824
+iter = 64, logl = -24769.288589020118
+iter = 65, logl = -24769.261241748998
+iter = 66, logl = -24769.235126666277
+iter = 67, logl = -24769.210172133422
+iter = 68, logl = -24769.186311562113
+[ Info: Updates converged!
+[ Info: Calculating standard errors
+ 81.354343 seconds (5.65 k allocations: 15.296 MiB, 0.01% compilation time)
+elapsed time (ns):  81354342709
+gc time (ns):       0
+bytes allocated:    16039272
+pool allocs:        5451
+non-pool GC allocs: 196
+malloc() calls:     7
+free() calls:       0
+minor collections:  0
+full collections:   0
+Converged after 69 iterations.

    Variance components and mean effects estimates and their standard errors can be accessed through:

    julia> model.Σ5-element Vector{Matrix{Float64}}:
+ [2.9338500446816433 -0.9556577622339366 1.681944315565907 0.82097923904986; -0.9556577622339366 7.523441348688346 -4.724626084973893 -0.6045832212502729; 1.6819443155659073 -4.724626084973893 8.695787597891757 0.3975584198892441; 0.8209792390498599 -0.6045832212502729 0.397558419889244 1.0347822362989731]
+ [3.8129402322306207 2.3985158343212087 -0.3501640293021632 -1.361452145970671; 2.3985158343212087 2.0221274445059243 -0.26055606213368854 -1.595835001991699; -0.3501640293021632 -0.26055606213368826 4.164419136795955 -2.293103534511798; -1.3614521459706712 -1.595835001991699 -2.293103534511798 4.545249416292042]
+ [3.1531618876945027 -0.08093870739430353 -0.27534179496140915 0.4229588473703198; -0.08093870739430353 3.4956494601351706 -0.2073584361891652 0.6627085784490545; -0.27534179496140915 -0.20735843618916527 1.976397882530889 0.5036124449893044; 0.4229588473703198 0.6627085784490545 0.5036124449893044 0.5585662911372018]
+ [3.635801209038145 1.250204732685735 0.3660670881565596 0.07071405261549359; 1.2502047326857344 0.983771630227471 1.65237696435348 0.6004734046277869; 0.3660670881565597 1.6523769643534798 5.891632852330085 0.9668222589182407; 0.07071405261549359 0.6004734046277869 0.9668222589182407 0.9633632310978463]
+ [5.704238993549306 -2.02059232700069 1.131104825718486 1.4423754007372802; -2.0205923270006902 6.0631121455622194 9.453887154496428 -2.1594622671919748; 1.131104825718486 9.453887154496428 19.22805021076556 -3.065314862756238; 1.4423754007372802 -2.1594622671919748 -3.065314862756238 3.4673781742053422]
    julia> model.B_reml10×4 Matrix{Float64}:
+   7.73978     0.0626092   29.9317     10.8401
+  -1.72827    17.0507      26.4331      5.24119
+  -4.99965    28.2624     -20.984     -13.0466
+ -10.2505     -5.98131     12.3119    -16.3662
+  -3.06092     0.788225    15.9509      1.1041
+  14.1047     -3.98098     11.5646     11.254
+  -1.45427    -8.264        5.3133     -0.233005
+   6.05113     5.20727      0.224061    3.95755
+  -0.679987  -14.213      -29.2147      4.91466
+   6.17329   -13.234      -23.6101     -3.48756
    julia> hcat(vec(B), vec(model.B_reml))40×2 Matrix{Float64}:
+ 0.866192    7.73978
+ 0.157873   -1.72827
+ 0.818581   -4.99965
+ 0.707496  -10.2505
+ 0.357756   -3.06092
+ 0.722163   14.1047
+ 0.441486   -1.45427
+ 0.812486    6.05113
+ 0.889787   -0.679987
+ 0.331484    6.17329
+ ⋮
+ 0.068644    5.24119
+ 0.451072  -13.0466
+ 0.618679  -16.3662
+ 0.54638     1.1041
+ 0.616161   11.254
+ 0.400984   -0.233005
+ 0.482087    3.95755
+ 0.582507    4.91466
+ 0.456195   -3.48756
    julia> reduce(hcat, [hcat(vec(Σ[i]), vec(model.Σ[i])) for i in 1:m])16×10 Matrix{Float64}:
+  2.59353    2.93385    3.32588    …  3.6358      5.37835    5.70424
+ -0.908817  -0.955658   2.21659       1.2502     -1.43756   -2.02059
+  1.58682    1.68194   -0.0345923     0.366067    1.65263    1.1311
+  0.365451   0.820979  -1.36959       0.0707141   0.983537   1.44238
+ -0.908817  -0.955658   2.21659       1.2502     -1.43756   -2.02059
+  7.12022    7.52344    1.75764    …  0.983772    6.84955    6.06311
+ -4.72969   -4.72463   -0.0950186     1.65238     9.98583    9.45389
+ -0.430129  -0.604583  -1.55052       0.600473   -1.98954   -2.15946
+  1.58682    1.68194   -0.0345923     0.366067    1.65263    1.1311
+ -4.72969   -4.72463   -0.0950186     1.65238     9.98583    9.45389
+  6.22778    8.69579    4.06617    …  5.89163    19.2083    19.2281
+  0.969228   0.397558  -2.51336       0.966822   -3.5885    -3.06531
+  0.365451   0.820979  -1.36959       0.0707141   0.983537   1.44238
+ -0.430129  -0.604583  -1.55052       0.600473   -1.98954   -2.15946
+  0.969228   0.397558  -2.51336       0.966822   -3.5885    -3.06531
+  1.2446     1.03478    5.02933    …  0.963363    2.85342    3.46738
    julia> model.Σcov50×50 Matrix{Float64}:
+  0.49577      -0.00244778    0.111934     …  -0.00120099   -0.000758647
+ -0.00244778    0.29227      -0.0343723       -0.000533875   0.00067098
+  0.111934     -0.0343723     0.505528        -0.00697209    0.000865612
+  0.061864     -0.0253519    -0.00568503      -0.00302124   -0.00430738
+ -0.000533068  -0.000402843  -0.00233763       0.00189715   -0.000601019
+  5.60305e-5    0.0635175    -0.00787928   …   0.00809095   -0.000772694
+ -4.0096e-5     0.0359266    -0.00341432      -0.00646355    0.00386484
+  0.0245991    -0.0126117     0.220813         0.0177754    -0.000996935
+  0.0136707    -0.00925492    0.0581185       -0.0446667     0.0050127
+  0.00759217   -0.00622591   -0.00328015       0.00502603   -0.0249292
+  ⋮                                        ⋱
+ -0.00795163   -0.0428182    -0.0129844        0.0236794    -0.0165915
+ -0.0140199    -0.012921     -0.0907524        0.0966537    -0.0229699
+ -0.00877942    0.00359818    0.00449712       0.0176113     0.0517196
+ -0.000667567  -0.00656994   -0.0018067       -0.0818833     0.0205321
+ -0.00106014   -0.00669857   -0.00860587   …  -0.193476      0.0283973
+ -0.000657675  -0.0033219    -0.000694922      0.144019     -0.0647023
+ -0.0019256    -0.00341227   -0.024527        -0.379053      0.0392726
+ -0.00120099   -0.000533875  -0.00697209       0.441745     -0.089645
+ -0.000758647   0.00067098    0.000865612     -0.089645      0.197463
    julia> model.Bcov_reml40×40 Matrix{Float64}:
+ 158.891     -19.5994    -19.6371   …   -0.270249   -1.25569    -1.91414
+ -19.5994    163.384     -22.755        -1.32842    -1.44256    -1.34338
+ -19.6371    -22.755     151.987        -1.39442    -2.57948    -1.26777
+ -14.7913    -12.9925    -19.412        -2.31154    -2.05524    -1.01497
+ -11.8093    -14.0871    -12.5893       -0.204628   -1.46038    -2.33716
+ -24.4932    -18.6219     -8.06361  …   -3.02449    -1.15581    -0.939094
+ -13.4737    -17.7017    -16.0374       -1.58474    -0.955878   -1.787
+ -12.8751    -19.6214    -11.8827       13.362      -1.26469    -1.10389
+ -17.4026    -19.3504    -19.7707       -1.23472    12.9793     -0.399421
+ -15.2551    -12.8481    -16.103        -1.24913    -0.426619   12.6487
+   ⋮                                ⋱
+  -2.14636    13.5663     -1.27382      -9.38304    -9.11398    -6.09505
+  -0.765165   -1.51854    13.6867       -5.69139    -9.3694     -8.2553
+  -0.7517     -0.930899   -1.73573     -11.9126     -8.7008    -10.1908
+  -1.50467    -1.26692    -1.05451      -6.03856   -13.0461    -13.3113
+  -1.74002    -2.34396    -0.50063  …   -9.94374    -5.4518     -8.24799
+  -1.54677    -0.78006    -2.23457      -9.47418    -7.6325    -10.5424
+  -0.270249   -1.32842    -1.39442      80.0047     -7.46137    -9.59586
+  -1.25569    -1.44256    -2.57948      -7.46137    76.5166     -0.942239
+  -1.91414    -1.34338    -1.26777      -9.59586    -0.942239   78.9404
    julia> sqrt.(diag(model.Σcov))50-element Vector{Float64}:
+ 0.7041093807634108
+ 0.5406201064474182
+ 0.7110051601946071
+ 0.3444911257855926
+ 0.8243533172952108
+ 0.7594961262777526
+ 0.37223105634402
+ 1.399043387180137
+ 0.47895976215151104
+ 0.32621217092743393
+ ⋮
+ 0.5632649585154991
+ 0.8978995129849152
+ 0.4392731438996924
+ 0.7554591568000877
+ 0.9384463145136606
+ 0.4220581137074655
+ 1.8966703307591692
+ 0.6646389147784445
+ 0.44436833540465226
    julia> sqrt.(diag(model.Bcov_reml))40-element Vector{Float64}:
+ 12.60518354143135
+ 12.782193317524683
+ 12.328308752465436
+ 12.23646139990698
+ 12.732636509969742
+ 11.934986834081903
+ 12.597350995825984
+ 12.586984559735523
+ 12.461643282682688
+ 12.578545874909784
+  ⋮
+  8.923342209927311
+  8.620236102433564
+  8.645876453902996
+  9.026071287093929
+  8.337457783234893
+  8.856449014354963
+  8.944535606024061
+  8.747376305559317
+  8.884843293186195

    Estimation only

    Calculating standard errors can be memory-consuming, so you could instead forego such calculation via:

    model = MRVCModel(Y, X, V; se = false) # or model = MRVCModel(Y, X, V; se = false, reml = true)
+@timev fit!(model)

    Missing response

    You can also fit data with missing response. For example:

    Y_miss = Matrix{Union{eltype(Y), Missing}}(missing, size(Y))
+copy!(Y_miss, Y)
+Y_miss[rand(1:length(Y_miss), n)] .= missing
+
+model = MRVCModel(Y_miss, X, V; se = false)
+@timev fit!(model)
    diff --git a/v0.3.1/index.html b/v0.3.1/index.html new file mode 100644 index 0000000..ab68691 --- /dev/null +++ b/v0.3.1/index.html @@ -0,0 +1,3 @@ + +Home · MRVCModels
    GitHub

    MRVCModels

    MRVCModels.jl is a package for fitting and testing multivariate response variance components linear mixed models of form

    \[\text{vec}\ \boldsymbol{Y} \sim \mathcal{N}(\text{vec}(\boldsymbol{X} \boldsymbol{B}), \sum_{i=1}^m \boldsymbol{\Gamma}_i \otimes \boldsymbol{V}_i),\]

    where $\boldsymbol{Y}$ and $\boldsymbol{X}$ are $n \times d$ response and $n \times p$ predictor matrices, respectively, and $\boldsymbol{V}_1, \ldots, \boldsymbol{V}_m$ are $m$ known $n \times n$ positive semidefinite matrices. $\text{vec}\ \boldsymbol{Y}$ creates an $nd \times 1$ vector from $\boldsymbol{Y}$ by stacking its columns and $\otimes$ denotes the Kronecker product. The parameters of the model include $p \times d$ mean effects $\boldsymbol{B}$ and $d \times d$ variance components ($\boldsymbol{\Gamma}_1, \dots, \boldsymbol{\Gamma}_m$), which MRVCModels.jl estimates through either minorization-maximization (MM) or expectation–maximization (EM) algorithms.

    Info

    MRVCModels.jl can also handle data with missing response, which destroys the symmetry of the log-likelihood and complicates maximum likelihood estimation. MM algorithm easily adapts to this challenge.

    Warning

    MRVCModels.jl is not suitable for biobank-scale data, except in the setting of $m = 2$. For $m > 2$, we recommend using this package for datasets of size up to $n \cdot d \approx 60000$. Further note that large $m$ can affect memory needed when calculating standard errors, since it will require $m(nd)^2$ storage space.

    Installation

    To use MRVCModels.jl, type:

    using Pkg
+Pkg.add(url = "https://github.com/Hua-Zhou/MultiResponseVarianceComponentModels.jl.git")

    This documentation was built using Documenter.jl.

    Documentation built 2024-06-20T18:59:09.277 with Julia 1.10.4
    diff --git a/v0.3.1/objects.inv b/v0.3.1/objects.inv new file mode 100644 index 0000000..9c3f379 Binary files /dev/null and b/v0.3.1/objects.inv differ diff --git a/v0.3.1/search_index.js b/v0.3.1/search_index.js new file mode 100644 index 0000000..8e68e86 --- /dev/null +++ b/v0.3.1/search_index.js @@ -0,0 +1,3 @@ +var documenterSearchIndex = {"docs": +[{"location":"api/#API","page":"API","title":"API","text":"","category":"section"},{"location":"api/","page":"API","title":"API","text":"","category":"page"},{"location":"api/","page":"API","title":"API","text":"Modules = [MultiResponseVarianceComponentModels]","category":"page"},{"location":"api/#MultiResponseVarianceComponentModels.MultiResponseVarianceComponentModels","page":"API","title":"MultiResponseVarianceComponentModels.MultiResponseVarianceComponentModels","text":"MRVCModels.jl permits maximum likelihood (ML) and residual maximum likelihood (REML) estimation as well as inference for multivariate response variance components linear mixed models.\n\n\n\n\n\n","category":"module"},{"location":"api/#MultiResponseVarianceComponentModels.MRVCModel-Union{Tuple{T}, Tuple{Union{AbstractMatrix{T}, AbstractArray{Union{Missing, T}, 2}}, Union{Nothing, AbstractMatrix{T}}, Vector{<:AbstractMatrix{T}}}} where T<:Union{Float32, Float64}","page":"API","title":"MultiResponseVarianceComponentModels.MRVCModel","text":"MRVCModel(\n Y::AbstractVecOrMat,\n X::Union{Nothing, AbstractVecOrMat},\n V::Union{AbstractMatrix, Vector{<:AbstractMatrix}}\n )\nMRTVCModel(\n Y::AbstractVecOrMat,\n X::Union{Nothing, AbstractVecOrMat},\n V::Vector{<:AbstractMatrix}\n )\n\nCreate a new MRVCModel or MRTVCModel instance from response matrix Y, predictor matrix X, and kernel matrices V. For MRTVCModel instance, the number of variance components must be two.\n\nKeyword arguments\n\nse::Bool calculate standard errors; default true\nreml::Bool pursue REML estimation instead of ML estimation; default false\n\n\n\n\n\n","category":"method"},{"location":"api/#MultiResponseVarianceComponentModels.fisher_B!-Union{Tuple{MRVCModel{T}}, Tuple{T}} where T<:Union{Float32, Float64}","page":"API","title":"MultiResponseVarianceComponentModels.fisher_B!","text":"fisher_B!(model::MRVCModel)\n\nCompute the sampling variance-covariance model.Bcov of regression coefficients model.B, assuming inverse of covariance matrix model.Ω is available at model.storage_nd_nd.\n\n\n\n\n\n","category":"method"},{"location":"api/#MultiResponseVarianceComponentModels.fisher_Σ!-Union{Tuple{MRVCModel{T}}, Tuple{T}} where T<:Union{Float32, Float64}","page":"API","title":"MultiResponseVarianceComponentModels.fisher_Σ!","text":"fisher_Σ!(model::MRVCModel)\n\nCompute the sampling variance-covariance model.Σcov of variance component estimates model.Σ, assuming inverse of covariance matrix model.Ω is available at model.storage_nd_nd.\n\n\n\n\n\n","category":"method"},{"location":"api/#MultiResponseVarianceComponentModels.fit!-Union{Tuple{MRVCModel{T}}, Tuple{T}} where T<:Union{Float32, Float64}","page":"API","title":"MultiResponseVarianceComponentModels.fit!","text":"fit!(model::MRVCModel)\nfit!(model::MRTVCModel)\n\nFit a multivariate response variance components model by MM or EM algorithm.\n\nKeyword arguments\n\nmaxiter::Int maximum number of iterations; default 1000\nreltol::Real relative tolerance for convergence; default 1e-6\nverbose::Bool display algorithmic information; default true\ninit::Symbol initialization strategy; :default initializes by least squares, while\n :user uses user-supplied values at model.B and model.Σ\nalgo::Symbol optimization algorithm; :MM (default) or EM (for MRVCModel)\nlog::Bool record iterate history or not; default false\n\n\n\n\n\n","category":"method"},{"location":"api/#MultiResponseVarianceComponentModels.h2-Union{Tuple{MRVCModel{T}}, Tuple{T}} where T<:Union{Float32, Float64}","page":"API","title":"MultiResponseVarianceComponentModels.h2","text":"h2(model::MRVCModel)\n\nCalculate heritability estimates and their standard errors, assuming that all variance components capture genetic effects except the last term. Also return total heritability from sum of individual contributions and its standard error.\n\n\n\n\n\n","category":"method"},{"location":"api/#MultiResponseVarianceComponentModels.kron_axpy!-Union{Tuple{T}, Tuple{AbstractVecOrMat{T}, AbstractVecOrMat{T}, AbstractVecOrMat{T}}} where T<:Real","page":"API","title":"MultiResponseVarianceComponentModels.kron_axpy!","text":"kron_axpy!(A, X, Y)\n\nOverwrite Y with A ⊗ X + Y. Same as Y += kron(A, X), but more memory efficient.\n\n\n\n\n\n","category":"method"},{"location":"api/#MultiResponseVarianceComponentModels.kron_reduction!-Union{Tuple{T}, Tuple{AbstractMatrix{T}, AbstractMatrix{T}, AbstractMatrix{T}}, Tuple{AbstractMatrix{T}, AbstractMatrix{T}, AbstractMatrix{T}, Bool}} where T<:Real","page":"API","title":"MultiResponseVarianceComponentModels.kron_reduction!","text":"kron_reduction!(A, B, C; sym = false)\n\nOverwrite C with the derivative of tr(A' (X ⊗ B)) wrt X. C[i, j] = dot(Aij, B), where Aij is the (i , j) block of A. sym = true indicates A and B are symmetric.\n\n\n\n\n\n","category":"method"},{"location":"api/#MultiResponseVarianceComponentModels.loglikelihood!-Union{Tuple{MRVCModel{T}}, Tuple{T}} where T<:Union{Float32, Float64}","page":"API","title":"MultiResponseVarianceComponentModels.loglikelihood!","text":"loglikelihood!(model::MRVCModel)\n\nOverwrite model.storage_nd_nd by inverse of the covariance matrix model.Ω, overwrite model.storage_nd by U' \\ vec(model.R), and return the log-likelihood. Assume model.Ω and model.R are already updated according to model.Σ and model.B.\n\n\n\n\n\n","category":"method"},{"location":"api/#MultiResponseVarianceComponentModels.loglikelihood_miss!-Union{Tuple{MRVCModel{T}}, Tuple{T}} where T<:Union{Float32, Float64}","page":"API","title":"MultiResponseVarianceComponentModels.loglikelihood_miss!","text":"loglikelihood_miss!(model::MRVCModel)\n\nOverwrite model.storage_nd_nd by inverse of the covariance matrix model.Ω, overwrite model.storage_nd by U' \\ vec(model.R), overwrite model.storage_n_miss_n_miss_1 by conditional variance, precompute model.storage_n_miss_n_obs_1 for conditional mean, and return the value of surrogate Q-function of log-likelihood. Assume model.Ω and model.R are already updated according to model.Σ and model.B.\n\n\n\n\n\n","category":"method"},{"location":"api/#MultiResponseVarianceComponentModels.lrt-Tuple{MRVCModel, MRVCModel}","page":"API","title":"MultiResponseVarianceComponentModels.lrt","text":"lrt(model1::MRVCModel, model0::MRVCModel)\n\nPerform a variation of the likelihood ratio test for univariate variance components models as in Molenberghs and Verbeke 2007 with model1 and model0 being the full and nested models, respectively.\n\n\n\n\n\n","category":"method"},{"location":"api/#MultiResponseVarianceComponentModels.permute-Union{Tuple{AbstractArray{Union{Missing, T}, 2}}, Tuple{T}} where T<:Union{Float32, Float64}","page":"API","title":"MultiResponseVarianceComponentModels.permute","text":"permute(Y::AbstractVecOrMat)\n\nReturn permutation P such that vec(Y)[P] rearranges vec(Y) with missing values spliced after non-missing values. Also return inverse permutation invP such that vec(Y)[P][invP] = vec(Y).\n\n\n\n\n\n","category":"method"},{"location":"api/#MultiResponseVarianceComponentModels.rg-Union{Tuple{MRVCModel{T}}, Tuple{T}} where T<:Union{Float32, Float64}","page":"API","title":"MultiResponseVarianceComponentModels.rg","text":"rg(model::MRVCModel)\n\nCalculate genetic/residual correlation estimates and their standard errors.\n\n\n\n\n\n","category":"method"},{"location":"api/#MultiResponseVarianceComponentModels.update_B!-Union{Tuple{MRVCModel{T}}, Tuple{T}} where T<:Union{Float32, Float64}","page":"API","title":"MultiResponseVarianceComponentModels.update_B!","text":"update_B!(model::MRVCModel)\n\nUpdate the regression coefficients model.B, assuming inverse of covariance matrix model.Ω is available at model.storage_nd_nd.\n\n\n\n\n\n","category":"method"},{"location":"api/#MultiResponseVarianceComponentModels.update_B_miss!-Union{Tuple{MRVCModel{T}}, Tuple{T}} where T<:Union{Float32, Float64}","page":"API","title":"MultiResponseVarianceComponentModels.update_B_miss!","text":"update_B_miss!(model::MRVCModel)\n\nUpdate the regression coefficients model.B, assuming inverse of covariance matrix model.Ω is available at model.storage_nd_nd and model.storage_n_miss_n_obs_1 for conditional mean is precomputed.\n\n\n\n\n\n","category":"method"},{"location":"api/#MultiResponseVarianceComponentModels.update_Γk!-Union{Tuple{T}, Tuple{MRVCModel{T}, Integer}} where T<:Union{Float32, Float64}","page":"API","title":"MultiResponseVarianceComponentModels.update_Γk!","text":"update_Γk!(model, k)\n\nUpdate the parameters model.Γ[k] and model.Ψ[:,k] assuming a diagonal plus low rank structure for model.Σ[k]. Assumes covariance matrix model.Ω is available at model.storage_nd_nd and model.Ω⁻¹R precomputed.\n\n\n\n\n\n","category":"method"},{"location":"api/#MultiResponseVarianceComponentModels.update_Σ!-Union{Tuple{MRVCModel{T}}, Tuple{T}} where T<:Union{Float32, Float64}","page":"API","title":"MultiResponseVarianceComponentModels.update_Σ!","text":"update_Σ!(model::MRVCModel)\n\nUpdate the variance component parameters model.Σ, assuming inverse of covariance matrix model.Ω is available at model.storage_nd_nd. If missing response, assume conditional variance model.storage_n_miss_n_miss_1 is precomputed.\n\n\n\n\n\n","category":"method"},{"location":"api/#MultiResponseVarianceComponentModels.update_Σk!-Union{Tuple{T}, Tuple{MRVCModel{T}, Integer, Integer}} where T<:Union{Float32, Float64}","page":"API","title":"MultiResponseVarianceComponentModels.update_Σk!","text":"update_Σk!(model, k, rk)\n\nUpdate the model.Σ[k] assuming it has rank rk < d, assuming inverse of covariance matrix model.Ω is available at model.storage_nd_nd and model.Ω⁻¹R precomputed.\n\n\n\n\n\n","category":"method"},{"location":"api/#MultiResponseVarianceComponentModels.update_Σk!-Union{Tuple{T}, Tuple{MRVCModel{T}, Integer, Val{:EM}}} where T<:Union{Float32, Float64}","page":"API","title":"MultiResponseVarianceComponentModels.update_Σk!","text":"update_Σk!(model::MRVCModel, k, Val(:EM))\n\nEM update the model.Σ[k] assuming it has full rank d, inverse of covariance matrix model.Ω is available at model.storage_nd_nd, and model.Ω⁻¹R is precomputed.\n\n\n\n\n\n","category":"method"},{"location":"api/#MultiResponseVarianceComponentModels.update_Σk!-Union{Tuple{T}, Tuple{MRVCModel{T}, Integer, Val{:MM}}} where T<:Union{Float32, Float64}","page":"API","title":"MultiResponseVarianceComponentModels.update_Σk!","text":"update_Σk!(model::MRVCModel, k, Val(:MM))\n\nMM update the model.Σ[k] assuming it has full rank d, inverse of covariance matrix model.Ω is available at model.storage_nd_nd, and model.Ω⁻¹R is precomputed.\n\n\n\n\n\n","category":"method"},{"location":"api/#MultiResponseVarianceComponentModels.update_Σk_miss!-Union{Tuple{T}, Tuple{MRVCModel{T}, Integer, Val{:MM}}} where T<:Union{Float32, Float64}","page":"API","title":"MultiResponseVarianceComponentModels.update_Σk_miss!","text":"update_Σk_miss!(model::MRVCModel, k, Val(:MM))\n\nMM update the model.Σ[k] assuming it has full rank d, inverse of covariance matrix model.Ω is available at model.storage_nd_nd, model.Ω⁻¹R is precomputed, and Ω⁻¹P'CPΩ⁻¹ is available at model.storage_nd_nd_miss.\n\n\n\n\n\n","category":"method"},{"location":"examples/#Examples","page":"Examples","title":"Examples","text":"","category":"section"},{"location":"examples/#Simulate-data","page":"Examples","title":"Simulate data","text":"","category":"section"},{"location":"examples/","page":"Examples","title":"Examples","text":"using MultiResponseVarianceComponentModels, LinearAlgebra, Random\nRandom.seed!(6789)\nn = 1_000; # n of observations\nd = 4; # n of responses\np = 10; # n of covariates\nm = 5; # n of variance components\nX = rand(n, p);\nB = rand(p, d)\nV = [zeros(n, n) for _ in 1:m]; # kernel matrices\nΣ = [zeros(d, d) for _ in 1:m]; # variance components\nfor i in 1:m\n Vi = randn(n, n)\n copy!(V[i], Vi' * Vi)\n Σi = randn(d, d)\n copy!(Σ[i], Σi' * Σi)\nend\nΩ = zeros(n * d, n * d); # overall nd-by-nd covariance matrix Ω\nfor i = 1:m\n Ω += kron(Σ[i], V[i])\nend\nΩchol = cholesky(Ω);\nY = X * B + reshape(Ωchol.L * randn(n * d), n, d)","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"note: Note\nIn the case of heritability and genetic correlation analyses, one can use classic genetic relationship matrices (GRMs) for boldsymbolV_i's, which in turn can be constructed using SnpArrays.jl.","category":"page"},{"location":"examples/#Maximum-likelihood-estimation","page":"Examples","title":"Maximum likelihood estimation","text":"","category":"section"},{"location":"examples/","page":"Examples","title":"Examples","text":"model = MRVCModel(Y, X, V)\n@timev fit!(model)","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"Then variance components and mean effects estimates can be accessed through","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"model.Σ\nmodel.B\nhcat(vec(B), vec(model.B))\nreduce(hcat, [hcat(vec(Σ[i]), vec(model.Σ[i])) for i in 1:m])","category":"page"},{"location":"examples/#Standard-errors","page":"Examples","title":"Standard errors","text":"","category":"section"},{"location":"examples/","page":"Examples","title":"Examples","text":"Sampling variance and covariance of these estimates are","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"model.Σcov\nmodel.Bcov","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"Corresponding standard error of these estimates are","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"sqrt.(diag(model.Σcov))\nsqrt.(diag(model.Bcov))","category":"page"},{"location":"examples/#Residual-maximum-likelihood-estimation","page":"Examples","title":"Residual maximum likelihood estimation","text":"","category":"section"},{"location":"examples/","page":"Examples","title":"Examples","text":"For REML estimation, you can instead:","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"model = MRVCModel(Y, X, V; reml = true)\n@timev fit!(model)","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"Variance components and mean effects estimates and their standard errors can be accessed through:","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"model.Σ\nmodel.B_reml\nhcat(vec(B), vec(model.B_reml))\nreduce(hcat, [hcat(vec(Σ[i]), vec(model.Σ[i])) for i in 1:m])\nmodel.Σcov\nmodel.Bcov_reml\nsqrt.(diag(model.Σcov))\nsqrt.(diag(model.Bcov_reml))","category":"page"},{"location":"examples/#Estimation-only","page":"Examples","title":"Estimation only","text":"","category":"section"},{"location":"examples/","page":"Examples","title":"Examples","text":"Calculating standard errors can be memory-consuming, so you could instead forego such calculation via:","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"model = MRVCModel(Y, X, V; se = false) # or model = MRVCModel(Y, X, V; se = false, reml = true)\n@timev fit!(model)","category":"page"},{"location":"examples/#Missing-response","page":"Examples","title":"Missing response","text":"","category":"section"},{"location":"examples/","page":"Examples","title":"Examples","text":"You can also fit data with missing response. For example:","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"Y_miss = Matrix{Union{eltype(Y), Missing}}(missing, size(Y))\ncopy!(Y_miss, Y)\nY_miss[rand(1:length(Y_miss), n)] .= missing\n\nmodel = MRVCModel(Y_miss, X, V; se = false)\n@timev fit!(model)","category":"page"},{"location":"advanced/#Advanced-details","page":"Advanced details","title":"Advanced details","text":"","category":"section"},{"location":"advanced/#Estimation","page":"Advanced details","title":"Estimation","text":"","category":"section"},{"location":"advanced/","page":"Advanced details","title":"Advanced details","text":"For the MM algorithm, the updates in each iteration are","category":"page"},{"location":"advanced/","page":"Advanced details","title":"Advanced details","text":"beginaligned\ntextvec boldsymbolB^(t) = (boldsymbolI_d otimes boldsymbolX^T) boldsymbolOmega^-(t) (boldsymbolI_d otimes boldsymbolX)^-1 (boldsymbolI_d otimes boldsymbolX^T) boldsymbolOmega^-(t) textvec boldsymbolY \nboldsymbolGamma_i^(t + 1) = boldsymbolL_i^-(t)TboldsymbolL_i^(t)TboldsymbolGamma_i^(t)(boldsymbolR^(t)TboldsymbolV_iboldsymbolR^(t))boldsymbolGamma_i^(t)boldsymbolL_i^(t)^12 boldsymbolL_i^-(t)\nendaligned","category":"page"},{"location":"advanced/","page":"Advanced details","title":"Advanced details","text":"where boldsymbolOmega^(t) = sum_i=1^m boldsymbolGamma_i^(t) otimes boldsymbolV_i and boldsymbolL_i^(t) is the Cholesky factor of boldsymbolM_i^(t) = (boldsymbolI_d otimes boldsymbol1_n)^T (boldsymbol1_d boldsymbol1_d^T otimes boldsymbolV_i) odot boldsymbolOmega^-(t) (boldsymbolI_d otimes boldsymbol1_n), while boldsymbolR^(t) is the n times d matrix such that textvec boldsymbolR^(t) = boldsymbolOmega^-(t) textvec(boldsymbolY - boldsymbolX boldsymbolB^(t)). odot denotes the Hadamard product.","category":"page"},{"location":"advanced/","page":"Advanced details","title":"Advanced details","text":"For the EM algorithm, the updates in each iteration are","category":"page"},{"location":"advanced/","page":"Advanced details","title":"Advanced details","text":"beginaligned\ntextvec boldsymbolB^(t) = (boldsymbolI_d otimes boldsymbolX^T) boldsymbolOmega^-(t) (boldsymbolI_d otimes boldsymbolX)^-1 (boldsymbolI_d otimes boldsymbolX^T) boldsymbolOmega^-(t) textvec boldsymbolY \nboldsymbolGamma_i^(t + 1) = frac1r_i boldsymbolGamma_i^(t) ( boldsymbolR^(t)T boldsymbolV_i boldsymbolR^(t) - boldsymbolM_i^(t) ) boldsymbolGamma_i^(t) + boldsymbolGamma_i^(t)\nendaligned","category":"page"},{"location":"advanced/","page":"Advanced details","title":"Advanced details","text":"where r_i = textrank(boldsymbolV_i). As seen, the updates for mean effects boldsymbolB are the same for these two algorithms.","category":"page"},{"location":"advanced/#Inference","page":"Advanced details","title":"Inference","text":"","category":"section"},{"location":"advanced/","page":"Advanced details","title":"Advanced details","text":"Standard errors for our estimates are calculated using the Fisher information matrix, where","category":"page"},{"location":"advanced/","page":"Advanced details","title":"Advanced details","text":"beginaligned\ntextE left- fracpartial^2partial(textvec boldsymbolB)^T partial(textvec boldsymbolB) mathcalL right = (boldsymbolI_d otimes boldsymbolX^T) boldsymbolOmega^-1 (boldsymbolI_d otimes boldsymbolX) \ntextE left - fracpartial^2partial (textvech boldsymbolGamma_i)^T partial (textvec boldsymbolB) mathcalL right = boldsymbol0 \ntextE left - fracpartial^2partial (textvech boldsymbolGamma_j)^T partial (textvech boldsymbolGamma_i) mathcalL right = frac12 boldsymbolU_i^T (boldsymbolOmega^-1 otimes boldsymbolOmega^-1) boldsymbolU_j\nendaligned","category":"page"},{"location":"advanced/","page":"Advanced details","title":"Advanced details","text":"and boldsymbolU_i = (boldsymbolI_d otimes boldsymbolK_nd otimes boldsymbolI_n) (boldsymbolI_d^2 otimes textvec boldsymbolV_i) boldsymbolD_d. Here, boldsymbolK_nd is the nd times nd commutation matrix and boldsymbolD_d the d^2 times fracd(d+1)2 duplication matrix. textvech boldsymbolGamma_i creates an fracd(d+1)2 times 1 vector from boldsymbolGamma_i by stacking its lower triangular part.","category":"page"},{"location":"advanced/#Special-case:-missing-response","page":"Advanced details","title":"Special case: missing response","text":"","category":"section"},{"location":"advanced/","page":"Advanced details","title":"Advanced details","text":"In the setting of missing response, the adjusted MM updates in each interation are","category":"page"},{"location":"advanced/","page":"Advanced details","title":"Advanced details","text":"beginaligned\ntextvec boldsymbolB^(t) = (boldsymbolI_d otimes boldsymbolX^T) boldsymbolOmega^-(t) (boldsymbolI_d otimes boldsymbolX)^-1 (boldsymbolI_d otimes boldsymbolX^T) boldsymbolOmega^-(t) textvec boldsymbolZ^(t) \nboldsymbolGamma_i^(t + 1) = boldsymbolL_i^-(t)TboldsymbolL_i^(t)TboldsymbolGamma_i^(t)(boldsymbolR^*(t)TboldsymbolV_iboldsymbolR^*(t) + boldsymbolM_i^*(t))boldsymbolGamma_i^(t)boldsymbolL_i^(t)^12 boldsymbolL_i^-(t)\nendaligned","category":"page"},{"location":"advanced/","page":"Advanced details","title":"Advanced details","text":"where boldsymbolZ^(t) is the completed response matrix from conditional mean and boldsymbolM_i^*(t) = (boldsymbolI_d otimes boldsymbol1_n)^T (boldsymbol1_d boldsymbol1_d^T otimes boldsymbolV_i) odot (boldsymbolOmega^-(t) boldsymbolP^T boldsymbolC^(t)boldsymbolPboldsymbolOmega^-(t)) (boldsymbolI_d otimes boldsymbol1_n), while boldsymbolR^*(t) is the n times d matrix such that textvec boldsymbolR^*(t) = boldsymbolOmega^-(t) textvec(boldsymbolZ^(t) - boldsymbolX boldsymbolB^(t)). Additionally, boldsymbolP is the nd times nd permutation matrix such that boldsymbolP cdot textvec boldsymbolY = beginbmatrix boldsymboly_textobs boldsymboly_textmis endbmatrix, where boldsymboly_textobs and boldsymboly_textmis are vectors of observed and missing response values, respectively, in column-major order, and the block matrix boldsymbolC^(t) is boldsymbol0 except for a lower-right block consisting of conditional variance. As seen, the two MM updates are of similar form.","category":"page"},{"location":"advanced/#Special-case:-m-2","page":"Advanced details","title":"Special case: m = 2","text":"","category":"section"},{"location":"advanced/","page":"Advanced details","title":"Advanced details","text":"When there are m = 2 variance components such that boldsymbolOmega = boldsymbolGamma_1 otimes boldsymbolV_1 + boldsymbolGamma_2 otimes boldsymbolV_2, repeated inversion of the nd times nd matrix boldsymbolOmega can be avoided and reduced to one d times d generalized eigen-decomposition per iteration. Without loss of generality, if we assume boldsymbolV_2 to be positive definite, the generalized eigen-decomposition of the matrix pair (boldsymbolV_1 boldsymbolV_2) yields generalized eigenvalues boldsymbold = (d_1 dots d_n)^T and generalized eigenvectors boldsymbolU such that boldsymbolU^T boldsymbolV_1 boldsymbolU = boldsymbolD = textdiag(boldsymbold) and boldsymbolU^T boldsymbolV_2 boldsymbolU = boldsymbolI_n. Similarly, if we let the generalized eigen-decomposition of (boldsymbolGamma_1^(t) boldsymbolGamma_2^(t)) be (boldsymbolLambda^(t) boldsymbolPhi^(t)) such that boldsymbolPhi^(t)T boldsymbolGamma_1^(t) boldsymbolPhi^(t) = boldsymbolLambda^(t) = textdiag(boldsymbollambda^(t)) and boldsymbolPhi^(t)T boldsymbolGamma_2^(t) boldsymbolPhi^(t) = boldsymbolI_d, then the MM updates in each iteration become","category":"page"},{"location":"advanced/","page":"Advanced details","title":"Advanced details","text":"beginaligned\ntextvec boldsymbolB^(t) = (boldsymbolPhi^(t)Totimes tildeboldsymbolX)^T (boldsymbolLambda^(t) otimes boldsymbolD + boldsymbolI_d otimes boldsymbolI_n)^-1 (boldsymbolPhi^(t)Totimes tildeboldsymbolX)^-1 \nquad cdot (boldsymbolPhi^(t)Totimes tildeboldsymbolX)^T (boldsymbolLambda^(t) otimes boldsymbolD + boldsymbolI_d otimes boldsymbolI_n)^-1 textvec(tildeboldsymbolY boldsymbolPhi^(t)) \nboldsymbolGamma_i^(t + 1) = boldsymbolL_i^-(t)TboldsymbolL_i^(t)TboldsymbolN_i^(t)TboldsymbolN_i^(t)boldsymbolL_i^(t)^12 boldsymbolL_i^-(t)\nendaligned","category":"page"},{"location":"advanced/","page":"Advanced details","title":"Advanced details","text":"where tildeboldsymbolX = boldsymbolU^T boldsymbolX, tildeboldsymbolY = boldsymbolU^T boldsymbolY, boldsymbolL_1^(t) is the Cholesky factor of boldsymbolPhi^(t)textdiag(texttr(boldsymbolD(lambda_k^(t)boldsymbolD + boldsymbolI_n)^-1) k = 1dots d)boldsymbolPhi^(t)T, boldsymbolL_2^(t) is the Cholesky factor of boldsymbolPhi^(t)textdiag(texttr((lambda_k^(t)boldsymbolD + boldsymbolI_n)^-1) k = 1dots d)boldsymbolPhi^(t)T, boldsymbolN_1^(t) = boldsymbolD^12(tildeboldsymbolY - tildeboldsymbolXboldsymbolB)boldsymbolPhi^(t)oslash(boldsymboldboldsymbollambda^(t)T + boldsymbol1_nboldsymbol1_d^T) boldsymbolLambda^(t)boldsymbolPhi^-(t), and boldsymbolN_2^(t) = (tildeboldsymbolY - tildeboldsymbolXboldsymbolB)boldsymbolPhi^(t)oslash(boldsymboldboldsymbollambda^(t)T + boldsymbol1_nboldsymbol1_d^T) boldsymbolPhi^-(t). oslash denotes the Hadamard quotient.","category":"page"},{"location":"advanced/","page":"Advanced details","title":"Advanced details","text":"In this setting, the Fisher information matrix is equivalent to","category":"page"},{"location":"advanced/","page":"Advanced details","title":"Advanced details","text":"beginaligned\ntextE left- fracpartial^2partial(textvec boldsymbolB)^T partial(textvec boldsymbolB) mathcalL right = (boldsymbolPhi^Totimes tildeboldsymbolX)^T (boldsymbolLambda otimes boldsymbolD + boldsymbolI_d otimes boldsymbolI_n)^-1 (boldsymbolPhi^Totimes tildeboldsymbolX) \ntextE left - fracpartial^2partial (textvech boldsymbolGamma_i)^T partial (textvec boldsymbolB) mathcalL right = boldsymbol0 \ntextE left - fracpartial^2partial (textvech boldsymbolGamma_j)^T partial (textvech boldsymbolGamma_i) mathcalL right = frac12 boldsymbolD_d^T(boldsymbolPhiotimes boldsymbolPhi) textdiag(textvec boldsymbolW_ij) (boldsymbolPhi otimes boldsymbolPhi)^TboldsymbolD_d\nendaligned","category":"page"},{"location":"advanced/","page":"Advanced details","title":"Advanced details","text":"where boldsymbolW_ij is the d times d matrix that has entries","category":"page"},{"location":"advanced/","page":"Advanced details","title":"Advanced details","text":"beginaligned\n(boldsymbolW_11)_kl = texttr(boldsymbolD^2(boldsymbollambda_k boldsymbolD + boldsymbolI_n)^-1(boldsymbollambda_l boldsymbolD + boldsymbolI_n)^-1) \n(boldsymbolW_12)_kl = texttr(boldsymbolD(boldsymbollambda_k boldsymbolD + boldsymbolI_n)^-1(boldsymbollambda_l boldsymbolD + boldsymbolI_n)^-1) \n(boldsymbolW_22)_kl = texttr((boldsymbollambda_k boldsymbolD + boldsymbolI_n)^-1(boldsymbollambda_l boldsymbolD + boldsymbolI_n)^-1)\nendaligned","category":"page"},{"location":"","page":"Home","title":"Home","text":"CurrentModule = MultiResponseVarianceComponentModels","category":"page"},{"location":"#MRVCModels","page":"Home","title":"MRVCModels","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"MRVCModels.jl is a package for fitting and testing multivariate response variance components linear mixed models of form","category":"page"},{"location":"","page":"Home","title":"Home","text":"textvec boldsymbolY sim mathcalN(textvec(boldsymbolX boldsymbolB) sum_i=1^m boldsymbolGamma_i otimes boldsymbolV_i)","category":"page"},{"location":"","page":"Home","title":"Home","text":"where boldsymbolY and boldsymbolX are n times d response and n times p predictor matrices, respectively, and boldsymbolV_1 ldots boldsymbolV_m are m known n times n positive semidefinite matrices. textvec boldsymbolY creates an nd times 1 vector from boldsymbolY by stacking its columns and otimes denotes the Kronecker product. The parameters of the model include p times d mean effects boldsymbolB and d times d variance components (boldsymbolGamma_1 dots boldsymbolGamma_m), which MRVCModels.jl estimates through either minorization-maximization (MM) or expectation–maximization (EM) algorithms.","category":"page"},{"location":"","page":"Home","title":"Home","text":"info: Info\nMRVCModels.jl can also handle data with missing response, which destroys the symmetry of the log-likelihood and complicates maximum likelihood estimation. MM algorithm easily adapts to this challenge.","category":"page"},{"location":"","page":"Home","title":"Home","text":"warning: Warning\nMRVCModels.jl is not suitable for biobank-scale data, except in the setting of m = 2. For m 2, we recommend using this package for datasets of size up to n cdot d approx 60000. Further note that large m can affect memory needed when calculating standard errors, since it will require m(nd)^2 storage space.","category":"page"},{"location":"#Installation","page":"Home","title":"Installation","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"To use MRVCModels.jl, type:","category":"page"},{"location":"","page":"Home","title":"Home","text":"using Pkg\nPkg.add(url = \"https://github.com/Hua-Zhou/MultiResponseVarianceComponentModels.jl.git\")","category":"page"},{"location":"","page":"Home","title":"Home","text":"This documentation was built using Documenter.jl.","category":"page"},{"location":"","page":"Home","title":"Home","text":"using Dates # hide\nprintln(\"Documentation built $(Dates.now()) with Julia $(VERSION)\") # hide","category":"page"}] +} diff --git a/v0.3.1/siteinfo.js b/v0.3.1/siteinfo.js new file mode 100644 index 0000000..5057e96 --- /dev/null +++ b/v0.3.1/siteinfo.js @@ -0,0 +1 @@ +var DOCUMENTER_CURRENT_VERSION = "v0.3.1"; diff --git a/versions.js b/versions.js new file mode 100644 index 0000000..dbebcbc --- /dev/null +++ b/versions.js @@ -0,0 +1,8 @@ +var DOC_VERSIONS = [ + "stable", + "v0.3", + "v0.2", + "dev", +]; +var DOCUMENTER_NEWEST = "v0.3.1"; +var DOCUMENTER_STABLE = "stable";