Aggregate documentation

chaostools/dev/.documenter-siteinfo.json

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chaostools/dev/dimreduction/index.html

@@ -5,7 +5,7 @@
 \vdots & \vdots & \vdots & \vdots \\
 x_{N-d+1} & x_{N-d+2} &\ldots & x_N
 \end{array}
-\right) = U\cdot S \cdot V^{tr}.\]</p><p>where <span>$x := s - \bar{s}$</span>. The columns of <span>$U$</span> can then be used as a new coordinate system, and by considering the values of the singular values <span>$S$</span> you can decide how many columns of <span>$U$</span> are "important".</p></div><a class="docs-sourcelink" href="https://github.com/JuliaDynamics/ChaosTools.jl/blob/a9b834cf10bf01d7ccc1ec1828214c53fa34e8a6/src/dimreduction/broomhead_king.jl#L7-L37" target="_blank">source</a></section></article><p>This alternative/improvement of the traditional delay coordinates can be a very powerful tool. An example where it shines is noisy data where there is the effect of superficial dimensions due to noise.</p><p>Take the following example where we produce noisy data from a system and then use Broomhead-King coordinates as an alternative to "vanilla" delay coordinates:</p><pre><code class="language-julia hljs">using ChaosTools, CairoMakie
+\right) = U\cdot S \cdot V^{tr}.\]</p><p>where <span>$x := s - \bar{s}$</span>. The columns of <span>$U$</span> can then be used as a new coordinate system, and by considering the values of the singular values <span>$S$</span> you can decide how many columns of <span>$U$</span> are "important".</p></div><a class="docs-sourcelink" href="https://github.com/JuliaDynamics/ChaosTools.jl/blob/a7634b7bd083978ea3bfe3de89a62068dd92a49a/src/dimreduction/broomhead_king.jl#L7-L37" target="_blank">source</a></section></article><p>This alternative/improvement of the traditional delay coordinates can be a very powerful tool. An example where it shines is noisy data where there is the effect of superficial dimensions due to noise.</p><p>Take the following example where we produce noisy data from a system and then use Broomhead-King coordinates as an alternative to "vanilla" delay coordinates:</p><pre><code class="language-julia hljs">using ChaosTools, CairoMakie
 
 function gissinger_rule(u, p, t)
     μ, ν, Γ = p
@@ -33,5 +33,5 @@
 R = embed(s, 3, estimate_delay(x, "mi_min"))
 lines!(axs[2], columns(R)...)
 axs[2].title = "2D embedding of s"
-fig</code></pre><img alt="Example block output" src="74378711.png"/><p>we have used the same system as in the <a href="@ref">Delay Coordinates Embedding</a> example, and picked the optimal delay time of <code>τ = 30</code> (for same <code>Δt = 0.05</code>). Regardless, the vanilla delay coordinates is much worse than the Broomhead-King coordinates.</p><h2 id="DyCA-Dynamical-Component-Analysis"><a class="docs-heading-anchor" href="#DyCA-Dynamical-Component-Analysis">DyCA - Dynamical Component Analysis</a><a id="DyCA-Dynamical-Component-Analysis-1"></a><a class="docs-heading-anchor-permalink" href="#DyCA-Dynamical-Component-Analysis" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" href="#ChaosTools.dyca" id="ChaosTools.dyca"><code>ChaosTools.dyca</code></a> — <span class="docstring-category">Function</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">dyca(data, eig_threshold) -&gt; eigenvalues, proj_mat, projected_data</code></pre><p>Compute the Dynamical Component analysis (DyCA) of the given <code>data</code><sup class="footnote-reference"><a href="#footnote-Uhl2018" id="citeref-Uhl2018">[Uhl2018]</a></sup> used for dimensionality reduction.</p><p>Return the eigenvalues, projection matrix, and reduced-dimension data (which are just <code>data*proj_mat</code>).</p><p><strong>Keyword Arguments</strong></p><ul><li>norm_eigenvectors=false : if true, normalize the eigenvectors</li></ul><p><strong>Description</strong></p><p>Dynamical Component Analysis (DyCA) is a method to detect projection vectors to reduce the dimensionality of multi-variate, high-dimensional deterministic datasets. Unlike methods like PCA or ICA that make a stochasticity assumption, DyCA relies on a determinacy assumption on the time-series and is based on the solution of a generalized eigenvalue problem. After choosing an appropriate eigenvalue threshold and solving the eigenvalue problem, the obtained eigenvectors are used to project the high-dimensional dataset onto a lower dimension. The obtained eigenvalues measure the quality of the assumption of linear determinism for the investigated data. Furthermore, the number of the generalized eigenvalues with a value of approximately 1.0 are a measure of the number of linear equations contained in the dataset. This property is useful in detecting regions with highly deterministic parts in the time-series and also as a preprocessing step for reservoir computing of high-dimensional spatio-temporal data.</p><p>The generalised eigenvalue problem we solve is:</p><p class="math-container">\[C_1 C_0^{-1} C_1^{\top} \bar{u} = \lambda C_2 \bar{u}
-\]</p><p>where <span>$C_0$</span> is the correlation matrix of the data with itself, <span>$C_1$</span> the correlation matrix of the data with its derivative, and <span>$C_2$</span> the correlation matrix of the derivative of the data with itself. The eigenvectors <span>$\bar{u}$</span> with eigenvalues approximately 1 and their <span>$C_1^{-1} C_2 u$</span> counterpart, form the space where the data is projected onto.</p></div><a class="docs-sourcelink" href="https://github.com/JuliaDynamics/ChaosTools.jl/blob/a9b834cf10bf01d7ccc1ec1828214c53fa34e8a6/src/dimreduction/dyca.jl#L4-L45" target="_blank">source</a></section></article><section class="footnotes is-size-7"><ul><li class="footnote" id="footnote-Broomhead1987"><a class="tag is-link" href="#citeref-Broomhead1987">Broomhead1987</a>Broomhead, Jones, King, J. Phys. A <strong>20</strong>, 9, pp L563 (1987)</li><li class="footnote" id="footnote-Uhl2018"><a class="tag is-link" href="#citeref-Uhl2018">Uhl2018</a>B Seifert, K Korn, S Hartmann, C Uhl, <em>Dynamical Component Analysis (DYCA): Dimensionality Reduction for High-Dimensional Deterministic Time-Series</em>, 10.1109/mlsp.2018.8517024, 2018 IEEE 28th International Workshop on Machine Learning for Signal Processing (MLSP)</li></ul></section></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../chaos_detection/">« Detecting &amp; Categorizing Chaos</a><a class="docs-footer-nextpage" href="../periodicity/">Fixed points &amp; Periodicity »</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></p><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><option value="catppuccin-latte">catppuccin-latte</option><option value="catppuccin-frappe">catppuccin-frappe</option><option value="catppuccin-macchiato">catppuccin-macchiato</option><option value="catppuccin-mocha">catppuccin-mocha</option></select></div><p></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.8.0 on <span class="colophon-date" title="Monday 18 November 2024 09:22">Monday 18 November 2024</span>. Using Julia version 1.11.1.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></HTML>
+fig</code></pre><img alt="Example block output" src="84db019b.png"/><p>we have used the same system as in the <a href="@ref">Delay Coordinates Embedding</a> example, and picked the optimal delay time of <code>τ = 30</code> (for same <code>Δt = 0.05</code>). Regardless, the vanilla delay coordinates is much worse than the Broomhead-King coordinates.</p><h2 id="DyCA-Dynamical-Component-Analysis"><a class="docs-heading-anchor" href="#DyCA-Dynamical-Component-Analysis">DyCA - Dynamical Component Analysis</a><a id="DyCA-Dynamical-Component-Analysis-1"></a><a class="docs-heading-anchor-permalink" href="#DyCA-Dynamical-Component-Analysis" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" href="#ChaosTools.dyca" id="ChaosTools.dyca"><code>ChaosTools.dyca</code></a> — <span class="docstring-category">Function</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">dyca(data, eig_threshold) -&gt; eigenvalues, proj_mat, projected_data</code></pre><p>Compute the Dynamical Component analysis (DyCA) of the given <code>data</code><sup class="footnote-reference"><a href="#footnote-Uhl2018" id="citeref-Uhl2018">[Uhl2018]</a></sup> used for dimensionality reduction.</p><p>Return the eigenvalues, projection matrix, and reduced-dimension data (which are just <code>data*proj_mat</code>).</p><p><strong>Keyword Arguments</strong></p><ul><li>norm_eigenvectors=false : if true, normalize the eigenvectors</li></ul><p><strong>Description</strong></p><p>Dynamical Component Analysis (DyCA) is a method to detect projection vectors to reduce the dimensionality of multi-variate, high-dimensional deterministic datasets. Unlike methods like PCA or ICA that make a stochasticity assumption, DyCA relies on a determinacy assumption on the time-series and is based on the solution of a generalized eigenvalue problem. After choosing an appropriate eigenvalue threshold and solving the eigenvalue problem, the obtained eigenvectors are used to project the high-dimensional dataset onto a lower dimension. The obtained eigenvalues measure the quality of the assumption of linear determinism for the investigated data. Furthermore, the number of the generalized eigenvalues with a value of approximately 1.0 are a measure of the number of linear equations contained in the dataset. This property is useful in detecting regions with highly deterministic parts in the time-series and also as a preprocessing step for reservoir computing of high-dimensional spatio-temporal data.</p><p>The generalised eigenvalue problem we solve is:</p><p class="math-container">\[C_1 C_0^{-1} C_1^{\top} \bar{u} = \lambda C_2 \bar{u}
+\]</p><p>where <span>$C_0$</span> is the correlation matrix of the data with itself, <span>$C_1$</span> the correlation matrix of the data with its derivative, and <span>$C_2$</span> the correlation matrix of the derivative of the data with itself. The eigenvectors <span>$\bar{u}$</span> with eigenvalues approximately 1 and their <span>$C_1^{-1} C_2 u$</span> counterpart, form the space where the data is projected onto.</p></div><a class="docs-sourcelink" href="https://github.com/JuliaDynamics/ChaosTools.jl/blob/a7634b7bd083978ea3bfe3de89a62068dd92a49a/src/dimreduction/dyca.jl#L4-L45" target="_blank">source</a></section></article><section class="footnotes is-size-7"><ul><li class="footnote" id="footnote-Broomhead1987"><a class="tag is-link" href="#citeref-Broomhead1987">Broomhead1987</a>Broomhead, Jones, King, J. Phys. A <strong>20</strong>, 9, pp L563 (1987)</li><li class="footnote" id="footnote-Uhl2018"><a class="tag is-link" href="#citeref-Uhl2018">Uhl2018</a>B Seifert, K Korn, S Hartmann, C Uhl, <em>Dynamical Component Analysis (DYCA): Dimensionality Reduction for High-Dimensional Deterministic Time-Series</em>, 10.1109/mlsp.2018.8517024, 2018 IEEE 28th International Workshop on Machine Learning for Signal Processing (MLSP)</li></ul></section></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../chaos_detection/">« Detecting &amp; Categorizing Chaos</a><a class="docs-footer-nextpage" href="../periodicity/">Fixed points &amp; Periodicity »</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></p><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><option value="catppuccin-latte">catppuccin-latte</option><option value="catppuccin-frappe">catppuccin-frappe</option><option value="catppuccin-macchiato">catppuccin-macchiato</option><option value="catppuccin-mocha">catppuccin-mocha</option></select></div><p></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.8.0 on <span class="colophon-date" title="Thursday 16 January 2025 12:02">Thursday 16 January 2025</span>. Using Julia version 1.11.2.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></HTML>