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<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>Home · QG3</title><link href="https://fonts.googleapis.com/css?family=Lato|Roboto+Mono" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.15.0/css/fontawesome.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.15.0/css/solid.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.15.0/css/brands.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.11.1/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="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></head><body><div id="documenter"><nav class="docs-sidebar"><div class="docs-package-name"><span class="docs-autofit">QG3</span></div><form class="docs-search" action="search/"><input class="docs-search-query" id="documenter-search-query" name="q" type="text" placeholder="Search docs"/></form><ul class="docs-menu"><li class="is-active"><a class="tocitem" href>Home</a></li><li><a class="tocitem" href="ref/">Reference</a></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"><nav class="breadcrumb"><ul class="is-hidden-mobile"><li class="is-active"><a href>Home</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Home</a></li></ul></nav><div class="docs-right"><a class="docs-edit-link" href="https://github.com//blob/master/docs/src/index.md" title="Edit on GitHub"><span class="docs-icon fab"></span><span class="docs-label is-hidden-touch">Edit on GitHub</span></a><a class="docs-settings-button fas fa-cog" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-sidebar-button fa fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a></div></header><article class="content" id="documenter-page"><h1 id="QG3-Model"><a class="docs-heading-anchor" href="#QG3-Model">QG3 Model</a><a id="QG3-Model-1"></a><a class="docs-heading-anchor-permalink" href="#QG3-Model" title="Permalink"></a></h1><p>This code provides a Julia implementation of the Marshall, Molteni QG3 Model. It runs on GPUs and is differentiable (tested only with Zygote).</p><p>The model is solved with a pseudo-spectral approach. Currently there is a naive implemention of Real Spherical Harmonics with transforms defined for a Gaussian Grid. There are bindings to FastTransforms.jl for a regular grid as well, however this is experimental and suffers from aliasing problems when integrating the equation.</p><p>So far this documentation is not yet fully fledged out. It just provides the docstrings and some very basic how-to.</p><p>Check out the <code>basic_test.jl</code> for a simple minimal example of the running the QG3 system. </p></article><nav class="docs-footer"><a class="docs-footer-nextpage" href="ref/">Reference »</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="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> on <span class="colophon-date" title="Thursday 11 November 2021 17:01">Thursday 11 November 2021</span>. Using Julia version 1.6.2.</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>Home · MCDTS</title><link href="https://fonts.googleapis.com/css?family=Lato|Roboto+Mono" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.15.0/css/fontawesome.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.15.0/css/solid.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.15.0/css/brands.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.11.1/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="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></head><body><div id="documenter"><nav class="docs-sidebar"><div class="docs-package-name"><span class="docs-autofit">MCDTS</span></div><form class="docs-search" action="search/"><input class="docs-search-query" id="documenter-search-query" name="q" type="text" placeholder="Search docs"/></form><ul class="docs-menu"><li class="is-active"><a class="tocitem" href>Home</a><ul class="internal"><li><a class="tocitem" href="#Usage"><span>Usage</span></a></li></ul></li><li><a class="tocitem" href="ref/">Reference</a></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"><nav class="breadcrumb"><ul class="is-hidden-mobile"><li class="is-active"><a href>Home</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Home</a></li></ul></nav><div class="docs-right"><a class="docs-edit-link" href="https://github.com//blob/master/docs/src/index.md" title="Edit on GitHub"><span class="docs-icon fab"></span><span class="docs-label is-hidden-touch">Edit on GitHub</span></a><a class="docs-settings-button fas fa-cog" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-sidebar-button fa fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a></div></header><article class="content" id="documenter-page"><h1 id="Monte-Carlo-Decision-Tree-Search-for-optimal-embedding"><a class="docs-heading-anchor" href="#Monte-Carlo-Decision-Tree-Search-for-optimal-embedding">Monte Carlo Decision Tree Search for optimal embedding</a><a id="Monte-Carlo-Decision-Tree-Search-for-optimal-embedding-1"></a><a class="docs-heading-anchor-permalink" href="#Monte-Carlo-Decision-Tree-Search-for-optimal-embedding" title="Permalink"></a></h1><p>This project implements the MCDTS algorithm outlined in the paper ____. It aims to provide an optimal time delay state space reconstruction from time series data with the help of decisions trees and suitable statistics that guide the decisions done during the rollout of these trees. For all details of the algorithm the reader is referred to the accompanying paper. Here we provide an implementation of all the variants described in the paper. All major functions have docstrings that describe their use. In what follows basic use examples are outlined.</p><h2 id="Usage"><a class="docs-heading-anchor" href="#Usage">Usage</a><a id="Usage-1"></a><a class="docs-heading-anchor-permalink" href="#Usage" title="Permalink"></a></h2><p>As a first example, we embedd a Lorenz63 system. This is meant as a toy example: we generate data from a Lorenz63 system and then try to reconstructe the full state space from only one of the three observables.</p><p>For this we also make use of DynamicalSystems.jl.</p><p>First we import the needed modules and generate a long trajectory of the Lorenz system (and discard any transient dynamics)</p><pre><code class="language-julia">using DynamicalSystems, MCDTS, Random, Test, DelayEmbeddings
# Check Lorenz System
Random.seed!(1234)
ds = Systems.lorenz()
data = trajectory(ds,200)
data = data[10001:end,:]</code></pre><p>The easiest way to use MCDTS for embedding is to use it with its default options just as</p><pre><code class="language-julia">tree = mcdts_embedding(data, 100)</code></pre><p>here with <code>N=100</code> trials. This will perform the MCDTS algorithm and return the full decision tree. The best embedding can then just be printed as</p><pre><code class="language-julia">best_node = MCDTS.best_embedding(tree)
println(best_node)</code></pre><p>e.g.</p><pre><code class="language-none">Node with τ=12, i_t=1 ,L=-1.5795030971438209 - full embd. τ=[0, 61, 48, 12] ,i_ts=[1, 1, 1, 1]</code></pre><p>This version of <code>mcdts_embedding</code> uses default options to estimate all parameters of the algorithm. It is of course also possible to choose these individually and thus also use all different versions that are presented in the paper.</p><p>The most important paremters are the Theiler window, the minimum temporal distance for points that are considered neighours in phase space. We can get an estimate with the help of DelayEmbeddings.jl (part of DynamicalSystems.jl)</p><pre><code class="language-julia">w1 = DelayEmbeddings.estimate_delay(data[:,1],&quot;mi_min&quot;)
w2 = DelayEmbeddings.estimate_delay(data[:,2],&quot;mi_min&quot;)
w3 = DelayEmbeddings.estimate_delay(data[:,3],&quot;mi_min&quot;)
w = maximum(hcat(w1,w2,w3))
delays = 0:100
N_trials = 100</code></pre><p>Also, we set the range of delays that we want to consider, here <code>0:100</code>. Next, we speficy the wanted optimization. The default optimziation goal is the Pecuzal algorithm, as outlined in the paper, with</p><pre><code class="language-julia">pecuzal = MCDTS.PecuzalOptim()</code></pre><p>Then we can call the proper <code>mcdts_embedding</code> function with</p><pre><code class="language-julia">MCDTS.mcdts_embedding(Dataset(data[:,1]), pecuzal, w1, delays, runs)
best_node = MCDTS.best_embedding(tree)
println(best_node)</code></pre><p>Further optimisation goals can be set by combining an <a href="@ref"><code>AbstractDelayPreselection</code></a> with an <a href="@ref"><code>AbstractLoss</code></a>, e.g. via</p><pre><code class="language-none">optimgoal = MCDTS.MCDTSOptimGoal(MCDTS.FNN_statistic(0.05), MCDTS.Continuity_function())</code></pre><p>which uses the FNN statistic with a threshold of 0.05 as a loss function and the continuity as a delay preselection method. </p></article><nav class="docs-footer"><a class="docs-footer-nextpage" href="ref/">Reference »</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="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> on <span class="colophon-date" title="Monday 22 November 2021 17:53">Monday 22 November 2021</span>. Using Julia version 1.6.2.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>Search · QG3</title><link href="https://fonts.googleapis.com/css?family=Lato|Roboto+Mono" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.15.0/css/fontawesome.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.15.0/css/solid.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.15.0/css/brands.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.11.1/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="../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></head><body><div id="documenter"><nav class="docs-sidebar"><div class="docs-package-name"><span class="docs-autofit">QG3</span></div><form class="docs-search" action><input class="docs-search-query" id="documenter-search-query" name="q" type="text" placeholder="Search docs"/></form><ul class="docs-menu"><li><a class="tocitem" href="../">Home</a></li><li><a class="tocitem" href="../ref/">Reference</a></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"><nav class="breadcrumb"><ul class="is-hidden-mobile"><li class="is-active"><a href>Search</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Search</a></li></ul></nav><div class="docs-right"><a class="docs-settings-button fas fa-cog" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-sidebar-button fa fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a></div></header><article><p id="documenter-search-info">Loading search...</p><ul id="documenter-search-results"></ul></article><nav class="docs-footer"><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="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> on <span class="colophon-date" title="Thursday 11 November 2021 17:01">Thursday 11 November 2021</span>. Using Julia version 1.6.2.</p></section><footer class="modal-card-foot"></footer></div></div></div></body><script src="../search_index.js"></script><script src="../assets/search.js"></script></html> <html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>Search · MCDTS</title><link href="https://fonts.googleapis.com/css?family=Lato|Roboto+Mono" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.15.0/css/fontawesome.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.15.0/css/solid.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.15.0/css/brands.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.11.1/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="../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></head><body><div id="documenter"><nav class="docs-sidebar"><div class="docs-package-name"><span class="docs-autofit">MCDTS</span></div><form class="docs-search" action><input class="docs-search-query" id="documenter-search-query" name="q" type="text" placeholder="Search docs"/></form><ul class="docs-menu"><li><a class="tocitem" href="../">Home</a></li><li><a class="tocitem" href="../ref/">Reference</a></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"><nav class="breadcrumb"><ul class="is-hidden-mobile"><li class="is-active"><a href>Search</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Search</a></li></ul></nav><div class="docs-right"><a class="docs-settings-button fas fa-cog" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-sidebar-button fa fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a></div></header><article><p id="documenter-search-info">Loading search...</p><ul id="documenter-search-results"></ul></article><nav class="docs-footer"><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="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> on <span class="colophon-date" title="Monday 22 November 2021 17:53">Monday 22 November 2021</span>. 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