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SoftwareLiteracy committed Apr 6, 2024
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</li>
<li class="toctree-l1"><a class="reference internal" href="/parameters/">Parameters</a>
</li>
<li class="toctree-l1"><a class="reference internal" href="/embedding_parameters/">Embedding Parameters</a>
<li class="toctree-l1"><a class="reference internal" href="/embedding_parameters/">Embedding</a>
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<li class="toctree-l1"><a class="reference internal" href="/simplex_/">Simplex</a>
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<ul>
<li class="toctree-l1"><a class="reference internal" href="/algorithms_in_depth/">EDM Algorithms in Depth</a>
</li>
<li class="toctree-l1"><a class="reference internal" href="/SMap_Demo/">S-map first principles</a>
<li class="toctree-l1"><a class="reference internal" href="/SMap_Demo/">S-map First Principles</a>
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<li class="toctree-l1"><a class="reference internal" href="/solvers_demo/">Custom Solvers</a>
<li class="toctree-l1"><a class="reference internal" href="/solvers_demo/">S-map Custom Solvers</a>
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</li>
<li class="toctree-l1"><a class="reference internal" href="../parameters/">Parameters</a>
</li>
<li class="toctree-l1"><a class="reference internal" href="../embedding_parameters/">Embedding Parameters</a>
<li class="toctree-l1"><a class="reference internal" href="../embedding_parameters/">Embedding</a>
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<li class="toctree-l1"><a class="reference internal" href="../simplex_/">Simplex</a>
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<li class="toctree-l1 current"><a class="reference internal current" href="./">EDM Algorithms in Depth</a>
<ul class="current">
<li class="toctree-l2"><a class="reference internal" href="#simplex-nearest-neighbor-projection">Simplex : Nearest Neighbor Projection</a>
<ul>
<li class="toctree-l3"><a class="reference internal" href="#video-of-simplex-prediction-at-varying-library-sizes">Video of Simplex prediction at varying library sizes</a>
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</ul>
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<li class="toctree-l2"><a class="reference internal" href="#s-map-sequential-locally-weighted-global-linear-maps">S-Map : Sequential Locally Weighted Global Linear Maps</a>
<ul>
<li class="toctree-l3"><a class="reference internal" href="#video-of-s-map-prediction-at-varying-theta">Video of S-Map prediction at varying theta</a>
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<li class="toctree-l3"><a class="reference internal" href="#s-map-interaction-coefficients">S-Map Interaction Coefficients</a>
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<li class="toctree-l2"><a class="reference internal" href="#ccm-convergent-cross-mapping">CCM : Convergent Cross Mapping</a>
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<li class="toctree-l1"><a class="reference internal" href="../SMap_Demo/">S-map First Principles</a>
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<li class="toctree-l1"><a class="reference internal" href="../solvers_demo/">Custom Solvers</a>
<li class="toctree-l1"><a class="reference internal" href="../solvers_demo/">S-map Custom Solvers</a>
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<h1 id="edm-algorithms">EDM Algorithms</h1>
<div><h1 id="edm-algorithms">EDM Algorithms</h1>
<h2 id="simplex-nearest-neighbor-projection">Simplex : Nearest Neighbor Projection</h2>
<p><code>Simplex</code> is a nearest neighbor projection. It locates the <code>knn</code>
nearest neighbors to the query point (location in the state-space from
Expand All @@ -147,6 +157,10 @@ <h2 id="simplex-nearest-neighbor-projection">Simplex : Nearest Neighbor Projecti
<a href="../edm_functions/#predictinterval"><code>PredictInterval()</code></a>,
convergent cross mapping <a href="../edm_functions/#ccm"><code>CCM()</code></a>,
and multiview embedding <a href="../edm_functions/#multiview"><code>Multiview()</code></a>.</p>
<h3 id="video-of-simplex-prediction-at-varying-library-sizes">Video of Simplex prediction at varying library sizes</h3>
<p>This video illustrates how denser state space representations with increasing
number of library vectors improves Simplex prediction.</p>
<p><div class="video-container"><video style="width:75%" muted controls alt="type:video"><source src="../imgs/TentMap_Library.mp4" type="video/mp4"></source></video></div><br></p>
<h2 id="s-map-sequential-locally-weighted-global-linear-maps">S-Map : Sequential Locally Weighted Global Linear Maps</h2>
<p>S-Map extends the state-space prediction in <code>Simplex</code> from an
average of the <code>E</code>+1 knn neighbors, to a linear regression fit to all
Expand All @@ -161,8 +175,12 @@ <h2 id="s-map-sequential-locally-weighted-global-linear-maps">S-Map : Sequential
over a spectrum of localization parameters θ allows one to infer
state-dependence, and thus nonlinearity of the dynamics. See
<a href="../edm_functions/#predictnonlinear"><code>PredictNonlinear()</code></a>.</p>
<p>For a demonstration of S-Map state-space localization, see the
<h3 id="video-of-s-map-prediction-at-varying-theta">Video of S-Map prediction at varying theta</h3>
<p>This video illustrates S-Map predictions with increasing values (higher localization) of <code>theta</code>.</p>
<p><div class="video-container"><video style="width:75%" muted controls alt="type:video"><source src="../imgs/Rossler_Theta.mp4" type="video/mp4"></source></video></div><br></p>
<p>For a first-principles exploration of S-Map state-space localization, see the
<a href="../SMap_Demo/">S-Map Notebook</a>.</p>
<h3 id="s-map-interaction-coefficients">S-Map Interaction Coefficients</h3>
<p>Another feature of S-Map is that for a properly fit model, the regression
coefficients between variables have been shown to approximate
the gradient (directional derivative) of variables along the manifold
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entire <em>My</em> is used, this provides evidence that <em>X</em> is casually
influencing <em>Y</em> as described in
<a href="https://science.sciencemag.org/content/338/6106/496">Sugihara et al. 2012</a>.</p>
<iframe width="100%" height="335" src="https://www.youtube.com/embed/NrFdIz-D2yM"
frameborder="0" allow="autoplay; gyroscope; picture-in-picture" allowfullscreen></iframe>
<iframe width="100%" height="335" src="https://www.youtube.com/embed/NrFdIz-D2yM" frameborder="0" allow="autoplay; gyroscope; picture-in-picture" allowfullscreen></iframe></div>

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<a href="../basic_examples/" class="btn btn-neutral float-left" title="Basic EDM Examples"><span class="icon icon-circle-arrow-left"></span> Previous</a>
<a href="../SMap_Demo/" class="btn btn-neutral float-right" title="S-map first principles">Next <span class="icon icon-circle-arrow-right"></span></a>
<a href="../SMap_Demo/" class="btn btn-neutral float-right" title="S-map First Principles">Next <span class="icon icon-circle-arrow-right"></span></a>
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</li>
<li class="toctree-l1"><a class="reference internal" href="../parameters/">Parameters</a>
</li>
<li class="toctree-l1"><a class="reference internal" href="../embedding_parameters/">Embedding Parameters</a>
<li class="toctree-l1"><a class="reference internal" href="../embedding_parameters/">Embedding</a>
</li>
<li class="toctree-l1"><a class="reference internal" href="../simplex_/">Simplex</a>
</li>
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<ul>
<li class="toctree-l1"><a class="reference internal" href="../algorithms_in_depth/">EDM Algorithms in Depth</a>
</li>
<li class="toctree-l1"><a class="reference internal" href="../SMap_Demo/">S-map first principles</a>
<li class="toctree-l1"><a class="reference internal" href="../SMap_Demo/">S-map First Principles</a>
</li>
<li class="toctree-l1"><a class="reference internal" href="../solvers_demo/">Custom Solvers</a>
<li class="toctree-l1"><a class="reference internal" href="../solvers_demo/">S-map Custom Solvers</a>
</li>
<li class="toctree-l1"><a class="reference internal" href="../cond_emb_demo/">Conditional Embedding</a>
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<h2 id="ccm"><function> CCM </function></h2>
<div><h2 id="ccm"><function> CCM </function></h2>
<p>Convergent cross mapping between <code>columns</code> : <code>target</code>; and, reverse mapping between <code>target</code> : <code>columns</code>.</p>
<p><strong> Python </strong> : </p>
<pre><code class="language-python">CCM(pathIn='./', dataFile='', dataFrame=None, pathOut='./',
Expand All @@ -132,13 +132,13 @@ <h2 id="ccm"><function> CCM </function></h2>
parameterList=False, verbose=False, showPlot=False, noTime=False)
</code></pre>
<p><strong> R </strong> : </p>
<pre><code class="language-R">CCM(pathIn=&quot;./&quot;, dataFile=&quot;&quot;, dataFrame=NULL, E=0,
<pre><code class="language-R">CCM(pathIn="./", dataFile="", dataFrame=NULL, E=0,
Tp=0, knn=0, tau=-1, exclusionRadius=0,
columns=&quot;&quot;, target=&quot;&quot;, libSizes=&quot;&quot;, sample=0,
columns="", target="", libSizes="", sample=0,
random=TRUE, seed=0, embedded=FALSE, includeData=FALSE,
parameterList=FALSE, verbose=FALSE, showPlot=FALSE, noTime=FALSE)
</code></pre>
<hr />
<hr>
<table>
<thead>
<tr>
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</tr>
<tr>
<td>dataFrame</td>
<td>pyEDM: pandas DataFrame<br/>rEDM: data.frame</td>
<td>pyEDM: pandas DataFrame<br>rEDM: data.frame</td>
<td>None</td>
<td>Input DataFrame</td>
</tr>
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</tr>
</tbody>
</table>
<p><br/>
<p><br>
Refer to the <a href="../parameters/">parameters</a> table for general parameter definitions.</p>
<p><strong> Notes </strong> :<br />
<p><strong> Notes </strong> :<br>
Normally, one column and one target are specified. The column time series is time-delay embedded to dimension <code>E</code>, then cross mapped with the target time series. In a separate thread, the target time series is embedded to <code>E</code> and cross mapped against the column acting as the "target". </p>
<p>If there are multiple <code>columns</code> and <code>embedded</code> is false, each column is time-delay embedded to dimension <code>E</code> creating an N-columns * E dimensional "mixed" embedding. If <code>embedded</code> is true, no time-delay embedding is done, creating a multivariate embedding of the speficied columns. The same logic applies if multiple target are specified for the "reverse" mapping. If embedded is false, each target is time-delay embedded to dimension <code>E</code> creating an N-target * E dimensional "mixed" embedding cross mapped to <em>only the first column</em> as the cross map target. If <code>embedded</code> is true, no time-delay embedding is done, creating a multivariate embedding of the specified target(s).</p>
<p><code>libSizes</code> specifies a string of whitespace or comma separated library sizes. If the string has 3 values, and, if the third value is less than the second value, the three values are interpreted as a sequence generator specifying "start stop increment" row values, i.e. "10 80 10" will evaluate library sizes from 10 to 80 in increments of 10.</p>
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current library size are used. Note this is <em>not</em> convergent cross mapping. </p>
<p>If <code>seed=0</code> , then a random seed is generated for the random number generator.
Otherwise, seed is used to initialise the random number generator.</p>
<p><strong> Returns </strong> :<br />
If <code>includeData</code> is <code>False</code>: returns DataFrame with 3 columns. <br />
<p><strong> Returns </strong> :<br>
If <code>includeData</code> is <code>False</code>: returns DataFrame with 3 columns. <br>
The first column is <code>LibSize</code>, the second and third columns
are Pearson correlation coefficients for <code>column</code> : <code>target</code>
and <code>target</code> : <code>column</code> cross mapping.</p>
<p>If <code>includeData</code> is <code>True</code>: returns a list with the above DataFrame
and a DataFrame of all Simplex projection statistics. If <code>parameterList = True</code>, a dictionary of <code>parameters</code> is added.</p>
and a DataFrame of all Simplex projection statistics. If <code>parameterList = True</code>, a dictionary of <code>parameters</code> is added.</p></div>

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