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{"documenter":{"julia_version":"1.10.2","generation_timestamp":"2024-03-03T13:31:28","documenter_version":"1.3.0"}}
{"documenter":{"julia_version":"1.10.2","generation_timestamp":"2024-04-08T01:09:25","documenter_version":"1.3.0"}}
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<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>General State Space Examples · DifferenceEquations.jl</title><meta name="title" content="General State Space Examples · DifferenceEquations.jl"/><meta property="og:title" content="General State Space Examples · DifferenceEquations.jl"/><meta property="twitter:title" content="General State Space Examples · DifferenceEquations.jl"/><meta name="description" content="Documentation for DifferenceEquations.jl."/><meta property="og:description" content="Documentation for DifferenceEquations.jl."/><meta property="twitter:description" content="Documentation for DifferenceEquations.jl."/><meta property="og:url" content="https://DifferenceEquations.sciml.ai/stable/examples/general_state_space_examples/"/><meta property="twitter:url" content="https://DifferenceEquations.sciml.ai/stable/examples/general_state_space_examples/"/><link rel="canonical" href="https://DifferenceEquations.sciml.ai/stable/examples/general_state_space_examples/"/><script data-outdated-warner src="../../assets/warner.js"></script><link href="https://cdnjs.cloudflare.com/ajax/libs/lato-font/3.0.0/css/lato-font.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/juliamono/0.050/juliamono.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/6.4.2/css/fontawesome.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/6.4.2/css/solid.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/6.4.2/css/brands.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.8/katex.min.css" rel="stylesheet" type="text/css"/><script>documenterBaseURL="../.."</script><script 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class="docs-search-query input is-rounded is-small is-clickable my-2 mx-auto py-1 px-2" id="documenter-search-query">Search docs (Ctrl + /)</button><ul class="docs-menu"><li><a class="tocitem" href="../../">DifferenceEquations.jl: Discrete-Time State Space Solution Methods</a></li><li><span class="tocitem">Examples</span><ul><li><a class="tocitem" href="../linear_state_space_examples/">Linear State Space Examples</a></li><li><a class="tocitem" href="../quadratic_state_space_examples/">Quadratic State Space Examples</a></li><li class="is-active"><a class="tocitem" href>General State Space Examples</a></li></ul></li></ul><div class="docs-version-selector field has-addons"><div class="control"><span class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">Examples</a></li><li class="is-active"><a href>General State Space Examples</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>General State Space Examples</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/SciML/DifferenceEquations.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/SciML/DifferenceEquations.jl/blob/main/docs/src/examples/general_state_space_examples.md" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="General-State-Space-Examples"><a class="docs-heading-anchor" href="#General-State-Space-Examples">General State Space Examples</a><a id="General-State-Space-Examples-1"></a><a class="docs-heading-anchor-permalink" href="#General-State-Space-Examples" title="Permalink"></a></h1><div class="admonition is-info"><header class="admonition-header">Note</header><div class="admonition-body"><p>This is a placeholder for future support for general nonlinear state-space problems. The basic implementation is a relatively simple variation on the linear version, where you call back into AD for the <code>f,g,h</code> calls in the <code>rrule</code> definition. Because of the mixture of AD calls and rules, it may make sense to wait for <code>Enzyme.jl</code> to be ready.</p></div></div><p>A future feature, if anyone is interested in writing it, is full support for</p><p class="math-container">\[u_{n+1} = f(u_n,p,t_n) + g(u_n,p,t_n) w_{n+1}\]</p><p>for some functions <span>$f$</span> and <span>$g$</span>, where <span>$w_{n+1}$</span> are IID random shocks to the evolution equation. The <span>$p$</span> is a vector of potentially differentiable parameters.</p><p>In addition, there is an optional observation equation</p><p class="math-container">\[z_n = h(u_n, p, t_n) + v_n\]</p><p>This could involve both the simulation and the calculation of the joint likelihood conditional on the noise, as in the other examples.</p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../quadratic_state_space_examples/">« Quadratic State Space Examples</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.3.0 on <span class="colophon-date" title="Sunday 3 March 2024 13:31">Sunday 3 March 2024</span>. Using Julia version 1.10.2.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>General State Space Examples · DifferenceEquations.jl</title><meta name="title" content="General State Space Examples · DifferenceEquations.jl"/><meta property="og:title" content="General State Space Examples · DifferenceEquations.jl"/><meta property="twitter:title" content="General State Space Examples · DifferenceEquations.jl"/><meta name="description" content="Documentation for DifferenceEquations.jl."/><meta property="og:description" content="Documentation for DifferenceEquations.jl."/><meta property="twitter:description" content="Documentation for DifferenceEquations.jl."/><meta property="og:url" content="https://DifferenceEquations.sciml.ai/stable/examples/general_state_space_examples/"/><meta property="twitter:url" content="https://DifferenceEquations.sciml.ai/stable/examples/general_state_space_examples/"/><link rel="canonical" href="https://DifferenceEquations.sciml.ai/stable/examples/general_state_space_examples/"/><script data-outdated-warner src="../../assets/warner.js"></script><link href="https://cdnjs.cloudflare.com/ajax/libs/lato-font/3.0.0/css/lato-font.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/juliamono/0.050/juliamono.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/6.4.2/css/fontawesome.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/6.4.2/css/solid.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/6.4.2/css/brands.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.8/katex.min.css" rel="stylesheet" type="text/css"/><script>documenterBaseURL="../.."</script><script src="https://cdnjs.cloudflare.com/ajax/libs/require.js/2.3.6/require.min.js" data-main="../../assets/documenter.js"></script><script src="../../search_index.js"></script><script src="../../siteinfo.js"></script><script src="../../../versions.js"></script><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../../assets/themes/documenter-dark.css" data-theme-name="documenter-dark" data-theme-primary-dark/><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../../assets/themes/documenter-light.css" data-theme-name="documenter-light" data-theme-primary/><script src="../../assets/themeswap.js"></script><link href="../../assets/favicon.ico" rel="icon" type="image/x-icon"/></head><body><div id="documenter"><nav class="docs-sidebar"><a class="docs-logo" href="../../"><img src="../../assets/logo.png" alt="DifferenceEquations.jl logo"/></a><div class="docs-package-name"><span class="docs-autofit"><a href="../../">DifferenceEquations.jl</a></span></div><button class="docs-search-query input is-rounded is-small is-clickable my-2 mx-auto py-1 px-2" id="documenter-search-query">Search docs (Ctrl + /)</button><ul class="docs-menu"><li><a class="tocitem" href="../../">DifferenceEquations.jl: Discrete-Time State Space Solution Methods</a></li><li><span class="tocitem">Examples</span><ul><li><a class="tocitem" href="../linear_state_space_examples/">Linear State Space Examples</a></li><li><a class="tocitem" href="../quadratic_state_space_examples/">Quadratic State Space Examples</a></li><li class="is-active"><a class="tocitem" href>General State Space Examples</a></li></ul></li></ul><div class="docs-version-selector field has-addons"><div class="control"><span class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><a 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class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="General-State-Space-Examples"><a class="docs-heading-anchor" href="#General-State-Space-Examples">General State Space Examples</a><a id="General-State-Space-Examples-1"></a><a class="docs-heading-anchor-permalink" href="#General-State-Space-Examples" title="Permalink"></a></h1><div class="admonition is-info"><header class="admonition-header">Note</header><div class="admonition-body"><p>This is a placeholder for future support for general nonlinear state-space problems. The basic implementation is a relatively simple variation on the linear version, where you call back into AD for the <code>f,g,h</code> calls in the <code>rrule</code> definition. Because of the mixture of AD calls and rules, it may make sense to wait for <code>Enzyme.jl</code> to be ready.</p></div></div><p>A future feature, if anyone is interested in writing it, is full support for</p><p class="math-container">\[u_{n+1} = f(u_n,p,t_n) + g(u_n,p,t_n) w_{n+1}\]</p><p>for some functions <span>$f$</span> and <span>$g$</span>, where <span>$w_{n+1}$</span> are IID random shocks to the evolution equation. The <span>$p$</span> is a vector of potentially differentiable parameters.</p><p>In addition, there is an optional observation equation</p><p class="math-container">\[z_n = h(u_n, p, t_n) + v_n\]</p><p>This could involve both the simulation and the calculation of the joint likelihood conditional on the noise, as in the other examples.</p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../quadratic_state_space_examples/">« Quadratic State Space Examples</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.3.0 on <span class="colophon-date" title="Monday 8 April 2024 01:09">Monday 8 April 2024</span>. Using Julia version 1.10.2.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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