11_equations.Rmd

---
title: "Equations"
author: "randy"
date: "`r Sys.Date()`"
output:
  pdf_document: default
  html_document: default
---

```{r setup, include=FALSE}
knitr::opts_chunk$set(echo = TRUE)
```


$$
\begin{split}
Y_{i} | X_{i}, b_{i} & \sim N \Big(\mu_i = X^T \beta,\  V_i = X_i^TGX_i + R_i\Big)\\
Y_i & = X_i^T \beta + X_i^T b_i + \epsilon = X_i^T(\beta + b_i) + \epsilon_i = X_i^T \gamma_i + \epsilon_i \\
\end{split}
$$


$$
\begin{split}
Y_i & = X_i^T\beta + X_i^Tb + \epsilon_i = X_i^T (\beta + b_i) + \epsilon_i = X_i^T \gamma_i + \epsilon_i \\
\dot y_i & = X_i^T \gamma_i\\
\ddot y_i & = Z_i^T \alpha + e_i
\end{split}
$$

we can assume $\epsilon$ is independent, but $e$ is dependent across different time with in individuals 

based on linear mixed model, the BLUP:
$\dot y = x_*^T(X^TV^{-1}X)^{-1}X^TV^{-1}Y$

based on linear regression BLE: 
$\ddot y = Z^T(Z^TZ)^{-1}Z^T \dot y = H^T \dot y$, $H^T = Z^T(Z^TZ)^{-1}Z^T$

then $\ddot y = Z^T(Z^TZ)^{-1}Z^T x_*^T(X^T V^{-1}X)^{-1}X^TV^{-1}Y = W^T Y$, $W^T = Z^T(Z^TZ)^{-1}Z^T x_*^T(X^T V^{-1}X)^{-1}X^TV^{-1}$ is a linear transformation from $Y$ to $\ddot y$.

Assuming that $Y - X^T \beta \sim N (0, V = X^TGX + R)$, however there is violation on the independence, 
that $\ddot y = W^T Y \sim N (W^TX^T\beta, \Sigma \neq W^TVW)$.



Also we have estimations from brokenstick model: 

$$
\begin{split}
\hat \beta & = (X^TV^{-1}X)^{-1}X^TV^{-1}Y\\
V[\hat \beta] & = (X^TV^{-1}X)^{-1}\\
V[\hat {\dot y} ] & = x_*^T (X^TV^{-1}X)^{-1} x_*\\
\hat b & = \hat GX \hat V^{-1}(Y - X^T \hat \beta)\\
V[\hat b_i - b_i] & = G - \hat G X V^{-1}X^T G + GX V^{-1}X^T(XV^{-1}X)^{-1}X^TV^{-1}XG
\end{split}
$$
Hence 

$$
\begin{split}
\hat Y_i & =  X_i^T \beta + X_i^T \hat b_i\\
& = X_i^T\beta + X_i^T\hat G_i X_i \hat V_i^{-1} (Y_i - X_i^T \beta) \\
& = (I - X_i^T\hat G_i X_i \hat V_i^{-1}) X_i^T \beta + X_i^T \hat G_i X_i \hat V_i^{-1} Y_i\\
& = (\hat R_i \hat V_i ^{-1}) X_i^T \beta + (1 - \hat R_i V_i^{-1})Y_i
\end{split}
$$


and linear model $E[\dot y] = Z^T \alpha$ with plug-in:

$$
\begin{split}
\hat \alpha & = (Z^TZ)^{-1} Z^T \dot y\\
V[\hat \alpha] & = (Z^TZ)^{-1}\\
\end{split}
$$


Then we have the second estimations for $V[\alpha]$ and $V[\ddot y]$, both :

$$
\begin{split}
V[\tilde \alpha] & = (Z^{-1})^T(\dot y - \ddot y)^T(\dot y - \ddot y)Z^{-1}\\
& = (Z^{-1})^T\Big(x_*^T (X^TV^{-1}X)^{-1}X^TV^{-1} Y - W^T Y\Big)^T\Big(x_*^T (X^TV^{-1}X)^{-1}X^TV^{-1} Y - W^T Y\Big) Z^{-1}
\end{split}
$$