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docs/pyerrors/correlators.html
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docs/pyerrors/covobs.html
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docs/pyerrors/dirac.html
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docs/pyerrors/fits.html
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@ -74,9 +74,6 @@
<li>
<a class="function" href="#residual_plot">residual_plot</a>
</li>
<li>
<a class="function" href="#covariance_matrix">covariance_matrix</a>
</li>
<li>
<a class="function" href="#error_band">error_band</a>
</li>
@ -343,7 +340,7 @@
<span class="k">if</span> <span class="n">kwargs</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s1">&#39;covariance&#39;</span><span class="p">)</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span><span class="p">:</span>
<span class="n">cov</span> <span class="o">=</span> <span class="n">kwargs</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s1">&#39;covariance&#39;</span><span class="p">)</span>
<span class="k">else</span><span class="p">:</span>
<span class="n">cov</span> <span class="o">=</span> <span class="n">covariance_matrix</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">concatenate</span><span class="p">((</span><span class="n">y</span><span class="p">,</span> <span class="n">x</span><span class="o">.</span><span class="n">ravel</span><span class="p">())))</span>
<span class="n">cov</span> <span class="o">=</span> <span class="n">covariance</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">concatenate</span><span class="p">((</span><span class="n">y</span><span class="p">,</span> <span class="n">x</span><span class="o">.</span><span class="n">ravel</span><span class="p">())))</span>
<span class="n">number_of_x_parameters</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">m</span> <span class="o">/</span> <span class="n">x_f</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">])</span>
@ -559,7 +556,7 @@
<span class="n">x0</span> <span class="o">=</span> <span class="p">[</span><span class="mf">0.1</span><span class="p">]</span> <span class="o">*</span> <span class="n">n_parms</span>
<span class="k">if</span> <span class="n">kwargs</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s1">&#39;correlated_fit&#39;</span><span class="p">)</span> <span class="ow">is</span> <span class="kc">True</span><span class="p">:</span>
<span class="n">cov</span> <span class="o">=</span> <span class="n">covariance_matrix</span><span class="p">(</span><span class="n">y</span><span class="p">)</span>
<span class="n">cov</span> <span class="o">=</span> <span class="n">covariance</span><span class="p">(</span><span class="n">y</span><span class="p">)</span>
<span class="n">covdiag</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">diag</span><span class="p">(</span><span class="mf">1.</span> <span class="o">/</span> <span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">diag</span><span class="p">(</span><span class="n">cov</span><span class="p">)))</span>
<span class="n">corr</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">copy</span><span class="p">(</span><span class="n">cov</span><span class="p">)</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">y</span><span class="p">)):</span>
@ -631,7 +628,7 @@
<span class="k">if</span> <span class="n">kwargs</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s1">&#39;expected_chisquare&#39;</span><span class="p">)</span> <span class="ow">is</span> <span class="kc">True</span><span class="p">:</span>
<span class="k">if</span> <span class="n">kwargs</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s1">&#39;correlated_fit&#39;</span><span class="p">)</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">True</span><span class="p">:</span>
<span class="n">W</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">diag</span><span class="p">(</span><span class="mi">1</span> <span class="o">/</span> <span class="n">np</span><span class="o">.</span><span class="n">asarray</span><span class="p">(</span><span class="n">dy_f</span><span class="p">))</span>
<span class="n">cov</span> <span class="o">=</span> <span class="n">covariance_matrix</span><span class="p">(</span><span class="n">y</span><span class="p">)</span>
<span class="n">cov</span> <span class="o">=</span> <span class="n">covariance</span><span class="p">(</span><span class="n">y</span><span class="p">)</span>
<span class="n">A</span> <span class="o">=</span> <span class="n">W</span> <span class="o">@</span> <span class="n">jacobian</span><span class="p">(</span><span class="n">func</span><span class="p">)(</span><span class="n">fit_result</span><span class="o">.</span><span class="n">x</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
<span class="n">P_phi</span> <span class="o">=</span> <span class="n">A</span> <span class="o">@</span> <span class="n">np</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">inv</span><span class="p">(</span><span class="n">A</span><span class="o">.</span><span class="n">T</span> <span class="o">@</span> <span class="n">A</span><span class="p">)</span> <span class="o">@</span> <span class="n">A</span><span class="o">.</span><span class="n">T</span>
<span class="n">expected_chisquare</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">trace</span><span class="p">((</span><span class="n">np</span><span class="o">.</span><span class="n">identity</span><span class="p">(</span><span class="n">x</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">])</span> <span class="o">-</span> <span class="n">P_phi</span><span class="p">)</span> <span class="o">@</span> <span class="n">W</span> <span class="o">@</span> <span class="n">cov</span> <span class="o">@</span> <span class="n">W</span><span class="p">)</span>
@ -755,33 +752,9 @@
<span class="n">plt</span><span class="o">.</span><span class="n">draw</span><span class="p">()</span>
<span class="k">def</span> <span class="nf">covariance_matrix</span><span class="p">(</span><span class="n">y</span><span class="p">):</span>
<span class="sd">&quot;&quot;&quot;Returns the covariance matrix of y.</span>
<span class="sd"> Parameters</span>
<span class="sd"> ----------</span>
<span class="sd"> y : list or numpy.ndarray</span>
<span class="sd"> List or one dimensional array of Obs</span>
<span class="sd"> &quot;&quot;&quot;</span>
<span class="n">length</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">y</span><span class="p">)</span>
<span class="n">cov</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">length</span><span class="p">,</span> <span class="n">length</span><span class="p">))</span>
<span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">item</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">y</span><span class="p">):</span>
<span class="k">for</span> <span class="n">j</span><span class="p">,</span> <span class="n">jtem</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">y</span><span class="p">[:</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]):</span>
<span class="k">if</span> <span class="n">i</span> <span class="o">==</span> <span class="n">j</span><span class="p">:</span>
<span class="n">cov</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">item</span><span class="o">.</span><span class="n">dvalue</span> <span class="o">**</span> <span class="mi">2</span>
<span class="k">else</span><span class="p">:</span>
<span class="n">cov</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">covariance</span><span class="p">(</span><span class="n">item</span><span class="p">,</span> <span class="n">jtem</span><span class="p">)</span>
<span class="n">cov</span> <span class="o">=</span> <span class="n">cov</span> <span class="o">+</span> <span class="n">cov</span><span class="o">.</span><span class="n">T</span> <span class="o">-</span> <span class="n">np</span><span class="o">.</span><span class="n">diag</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">diag</span><span class="p">(</span><span class="n">cov</span><span class="p">))</span>
<span class="n">eigenvalues</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">eigh</span><span class="p">(</span><span class="n">cov</span><span class="p">)[</span><span class="mi">0</span><span class="p">]</span>
<span class="k">if</span> <span class="ow">not</span> <span class="n">np</span><span class="o">.</span><span class="n">all</span><span class="p">(</span><span class="n">eigenvalues</span> <span class="o">&gt;=</span> <span class="mi">0</span><span class="p">):</span>
<span class="n">warnings</span><span class="o">.</span><span class="n">warn</span><span class="p">(</span><span class="s2">&quot;Covariance matrix is not positive semi-definite&quot;</span><span class="p">,</span> <span class="ne">RuntimeWarning</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;Eigenvalues of the covariance matrix:&quot;</span><span class="p">,</span> <span class="n">eigenvalues</span><span class="p">)</span>
<span class="k">return</span> <span class="n">cov</span>
<span class="k">def</span> <span class="nf">error_band</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="n">beta</span><span class="p">):</span>
<span class="sd">&quot;&quot;&quot;Returns the error band for an array of sample values x, for given fit function func with optimized parameters beta.&quot;&quot;&quot;</span>
<span class="n">cov</span> <span class="o">=</span> <span class="n">covariance_matrix</span><span class="p">(</span><span class="n">beta</span><span class="p">)</span>
<span class="n">cov</span> <span class="o">=</span> <span class="n">covariance</span><span class="p">(</span><span class="n">beta</span><span class="p">)</span>
<span class="k">if</span> <span class="n">np</span><span class="o">.</span><span class="n">any</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">cov</span> <span class="o">-</span> <span class="n">cov</span><span class="o">.</span><span class="n">T</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">1000</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">finfo</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">float64</span><span class="p">)</span><span class="o">.</span><span class="n">eps</span><span class="p">):</span>
<span class="n">warnings</span><span class="o">.</span><span class="n">warn</span><span class="p">(</span><span class="s2">&quot;Covariance matrix is not symmetric within floating point precision&quot;</span><span class="p">,</span> <span class="ne">RuntimeWarning</span><span class="p">)</span>
@ -1220,7 +1193,7 @@ If true, use the full correlation matrix in the definition of the chisquare
<span class="k">if</span> <span class="n">kwargs</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s1">&#39;covariance&#39;</span><span class="p">)</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span><span class="p">:</span>
<span class="n">cov</span> <span class="o">=</span> <span class="n">kwargs</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s1">&#39;covariance&#39;</span><span class="p">)</span>
<span class="k">else</span><span class="p">:</span>
<span class="n">cov</span> <span class="o">=</span> <span class="n">covariance_matrix</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">concatenate</span><span class="p">((</span><span class="n">y</span><span class="p">,</span> <span class="n">x</span><span class="o">.</span><span class="n">ravel</span><span class="p">())))</span>
<span class="n">cov</span> <span class="o">=</span> <span class="n">covariance</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">concatenate</span><span class="p">((</span><span class="n">y</span><span class="p">,</span> <span class="n">x</span><span class="o">.</span><span class="n">ravel</span><span class="p">())))</span>
<span class="n">number_of_x_parameters</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="n">m</span> <span class="o">/</span> <span class="n">x_f</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">])</span>
@ -1451,54 +1424,6 @@ check if the residuals of the fit are gaussian distributed.</p>
</div>
</section>
<section id="covariance_matrix">
<div class="attr function"><a class="headerlink" href="#covariance_matrix">#&nbsp;&nbsp</a>
<span class="def">def</span>
<span class="name">covariance_matrix</span><span class="signature">(y)</span>:
</div>
<details>
<summary>View Source</summary>
<div class="pdoc-code codehilite"><pre><span></span><span class="k">def</span> <span class="nf">covariance_matrix</span><span class="p">(</span><span class="n">y</span><span class="p">):</span>
<span class="sd">&quot;&quot;&quot;Returns the covariance matrix of y.</span>
<span class="sd"> Parameters</span>
<span class="sd"> ----------</span>
<span class="sd"> y : list or numpy.ndarray</span>
<span class="sd"> List or one dimensional array of Obs</span>
<span class="sd"> &quot;&quot;&quot;</span>
<span class="n">length</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">y</span><span class="p">)</span>
<span class="n">cov</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">length</span><span class="p">,</span> <span class="n">length</span><span class="p">))</span>
<span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">item</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">y</span><span class="p">):</span>
<span class="k">for</span> <span class="n">j</span><span class="p">,</span> <span class="n">jtem</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">y</span><span class="p">[:</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]):</span>
<span class="k">if</span> <span class="n">i</span> <span class="o">==</span> <span class="n">j</span><span class="p">:</span>
<span class="n">cov</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">item</span><span class="o">.</span><span class="n">dvalue</span> <span class="o">**</span> <span class="mi">2</span>
<span class="k">else</span><span class="p">:</span>
<span class="n">cov</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">covariance</span><span class="p">(</span><span class="n">item</span><span class="p">,</span> <span class="n">jtem</span><span class="p">)</span>
<span class="n">cov</span> <span class="o">=</span> <span class="n">cov</span> <span class="o">+</span> <span class="n">cov</span><span class="o">.</span><span class="n">T</span> <span class="o">-</span> <span class="n">np</span><span class="o">.</span><span class="n">diag</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">diag</span><span class="p">(</span><span class="n">cov</span><span class="p">))</span>
<span class="n">eigenvalues</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">eigh</span><span class="p">(</span><span class="n">cov</span><span class="p">)[</span><span class="mi">0</span><span class="p">]</span>
<span class="k">if</span> <span class="ow">not</span> <span class="n">np</span><span class="o">.</span><span class="n">all</span><span class="p">(</span><span class="n">eigenvalues</span> <span class="o">&gt;=</span> <span class="mi">0</span><span class="p">):</span>
<span class="n">warnings</span><span class="o">.</span><span class="n">warn</span><span class="p">(</span><span class="s2">&quot;Covariance matrix is not positive semi-definite&quot;</span><span class="p">,</span> <span class="ne">RuntimeWarning</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;Eigenvalues of the covariance matrix:&quot;</span><span class="p">,</span> <span class="n">eigenvalues</span><span class="p">)</span>
<span class="k">return</span> <span class="n">cov</span>
</pre></div>
</details>
<div class="docstring"><p>Returns the covariance matrix of y.</p>
<h6 id="parameters">Parameters</h6>
<ul>
<li><strong>y</strong> (list or numpy.ndarray):
List or one dimensional array of Obs</li>
</ul>
</div>
</section>
<section id="error_band">
<div class="attr function"><a class="headerlink" href="#error_band">#&nbsp;&nbsp</a>
@ -1512,7 +1437,7 @@ List or one dimensional array of Obs</li>
<summary>View Source</summary>
<div class="pdoc-code codehilite"><pre><span></span><span class="k">def</span> <span class="nf">error_band</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="n">beta</span><span class="p">):</span>
<span class="sd">&quot;&quot;&quot;Returns the error band for an array of sample values x, for given fit function func with optimized parameters beta.&quot;&quot;&quot;</span>
<span class="n">cov</span> <span class="o">=</span> <span class="n">covariance_matrix</span><span class="p">(</span><span class="n">beta</span><span class="p">)</span>
<span class="n">cov</span> <span class="o">=</span> <span class="n">covariance</span><span class="p">(</span><span class="n">beta</span><span class="p">)</span>
<span class="k">if</span> <span class="n">np</span><span class="o">.</span><span class="n">any</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">cov</span> <span class="o">-</span> <span class="n">cov</span><span class="o">.</span><span class="n">T</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">1000</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">finfo</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">float64</span><span class="p">)</span><span class="o">.</span><span class="n">eps</span><span class="p">):</span>
<span class="n">warnings</span><span class="o">.</span><span class="n">warn</span><span class="p">(</span><span class="s2">&quot;Covariance matrix is not symmetric within floating point precision&quot;</span><span class="p">,</span> <span class="ne">RuntimeWarning</span><span class="p">)</span>

6
docs/pyerrors/input.html
View file

@ -3,7 +3,7 @@
<head>
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<meta name="generator" content="pdoc 10.0.1"/>
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<h1 class="modulename">
<a href="./../pyerrors.html">pyerrors</a><wbr>.input </h1>
<div class="docstring"><p><code><a href="../pyerrors.html">pyerrors</a></code> includes an <code>input</code> submodule in which input routines and parsers for the output of various numerical programs are contained.</p>
<div class="docstring"><p><code>pyerrors</code> includes an <code>input</code> submodule in which input routines and parsers for the output of various numerical programs are contained.</p>
<h1 id="jackknife-samples">Jackknife samples</h1>
<p>For comparison with other analysis workflows <code><a href="../pyerrors.html">pyerrors</a></code> can also generate jackknife samples from an <code>Obs</code> object or import jackknife samples into an <code>Obs</code> object.
<p>For comparison with other analysis workflows <code>pyerrors</code> can also generate jackknife samples from an <code>Obs</code> object or import jackknife samples into an <code>Obs</code> object.
See <code><a href="obs.html#Obs.export_jackknife">pyerrors.obs.Obs.export_jackknife</a></code> and <code><a href="obs.html#import_jackknife">pyerrors.obs.import_jackknife</a></code> for details.</p>
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@ -1643,21 +1643,60 @@
<span class="k">return</span> <span class="n">o</span>
<span class="k">def</span> <span class="nf">covariance</span><span class="p">(</span><span class="n">obs1</span><span class="p">,</span> <span class="n">obs2</span><span class="p">,</span> <span class="n">correlation</span><span class="o">=</span><span class="kc">False</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
<span class="sd">&quot;&quot;&quot;Calculates the covariance of two observables.</span>
<span class="k">def</span> <span class="nf">covariance</span><span class="p">(</span><span class="n">obs</span><span class="p">,</span> <span class="n">visualize</span><span class="o">=</span><span class="kc">False</span><span class="p">,</span> <span class="n">correlation</span><span class="o">=</span><span class="kc">False</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
<span class="sd">&quot;&quot;&quot;Calculates the covariance matrix of a set of observables.</span>
<span class="sd"> covariance(obs, obs) is equal to obs.dvalue ** 2</span>
<span class="sd"> The gamma method has to be applied first to both observables.</span>
<span class="sd"> If abs(covariance(obs1, obs2)) &gt; obs1.dvalue * obs2.dvalue, the covariance</span>
<span class="sd"> is constrained to the maximum value.</span>
<span class="sd"> The gamma method has to be applied first to all observables.</span>
<span class="sd"> Parameters</span>
<span class="sd"> ----------</span>
<span class="sd"> obs : list or numpy.ndarray</span>
<span class="sd"> List or one dimensional array of Obs</span>
<span class="sd"> visualize : bool</span>
<span class="sd"> If True plots the corresponding normalized correlation matrix (default False).</span>
<span class="sd"> correlation : bool</span>
<span class="sd"> if true the correlation instead of the covariance is returned (default False)</span>
<span class="sd"> If True the correlation instead of the covariance is returned (default False).</span>
<span class="sd"> Notes</span>
<span class="sd"> -----</span>
<span class="sd"> The covariance is estimated by calculating the correlation matrix assuming no autocorrelation and then rescaling the correlation matrix by the full errors including the previous gamma method estimate for the autocorrelation of the observables. For observables defined on a single ensemble this is equivalent to assuming that the integrated autocorrelation time of an off-diagonal element is equal to the geometric mean of the integrated autocorrelation times of the corresponding diagonal elements.</span>
<span class="sd"> $$</span>
<span class="sd"> \tau_{\mathrm{int}, ij}=\sqrt{\tau_{\mathrm{int}, i}\times \tau_{\mathrm{int}, j}}</span>
<span class="sd"> $$</span>
<span class="sd"> This construction ensures that the estimated covariance matrix is positive semi-definite (up to numerical rounding errors).</span>
<span class="sd"> &quot;&quot;&quot;</span>
<span class="n">length</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">obs</span><span class="p">)</span>
<span class="n">cov</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">length</span><span class="p">,</span> <span class="n">length</span><span class="p">))</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">length</span><span class="p">):</span>
<span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">length</span><span class="p">):</span>
<span class="n">cov</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">_covariance_element</span><span class="p">(</span><span class="n">obs</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">obs</span><span class="p">[</span><span class="n">j</span><span class="p">])</span>
<span class="n">cov</span> <span class="o">=</span> <span class="n">cov</span> <span class="o">+</span> <span class="n">cov</span><span class="o">.</span><span class="n">T</span> <span class="o">-</span> <span class="n">np</span><span class="o">.</span><span class="n">diag</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">diag</span><span class="p">(</span><span class="n">cov</span><span class="p">))</span>
<span class="n">corr</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">diag</span><span class="p">(</span><span class="mi">1</span> <span class="o">/</span> <span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">diag</span><span class="p">(</span><span class="n">cov</span><span class="p">)))</span> <span class="o">@</span> <span class="n">cov</span> <span class="o">@</span> <span class="n">np</span><span class="o">.</span><span class="n">diag</span><span class="p">(</span><span class="mi">1</span> <span class="o">/</span> <span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">diag</span><span class="p">(</span><span class="n">cov</span><span class="p">)))</span>
<span class="n">errors</span> <span class="o">=</span> <span class="p">[</span><span class="n">o</span><span class="o">.</span><span class="n">dvalue</span> <span class="k">for</span> <span class="n">o</span> <span class="ow">in</span> <span class="n">obs</span><span class="p">]</span>
<span class="n">cov</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">diag</span><span class="p">(</span><span class="n">errors</span><span class="p">)</span> <span class="o">@</span> <span class="n">corr</span> <span class="o">@</span> <span class="n">np</span><span class="o">.</span><span class="n">diag</span><span class="p">(</span><span class="n">errors</span><span class="p">)</span>
<span class="n">eigenvalues</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">eigh</span><span class="p">(</span><span class="n">cov</span><span class="p">)[</span><span class="mi">0</span><span class="p">]</span>
<span class="k">if</span> <span class="ow">not</span> <span class="n">np</span><span class="o">.</span><span class="n">all</span><span class="p">(</span><span class="n">eigenvalues</span> <span class="o">&gt;=</span> <span class="mi">0</span><span class="p">):</span>
<span class="n">warnings</span><span class="o">.</span><span class="n">warn</span><span class="p">(</span><span class="s2">&quot;Covariance matrix is not positive semi-definite (Eigenvalues: &quot;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">eigenvalues</span><span class="p">)</span> <span class="o">+</span> <span class="s2">&quot;)&quot;</span><span class="p">,</span> <span class="ne">RuntimeWarning</span><span class="p">)</span>
<span class="k">if</span> <span class="n">visualize</span><span class="p">:</span>
<span class="n">plt</span><span class="o">.</span><span class="n">matshow</span><span class="p">(</span><span class="n">corr</span><span class="p">,</span> <span class="n">vmin</span><span class="o">=-</span><span class="mi">1</span><span class="p">,</span> <span class="n">vmax</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">set_cmap</span><span class="p">(</span><span class="s1">&#39;RdBu&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">colorbar</span><span class="p">()</span>
<span class="n">plt</span><span class="o">.</span><span class="n">draw</span><span class="p">()</span>
<span class="k">if</span> <span class="n">correlation</span> <span class="ow">is</span> <span class="kc">True</span><span class="p">:</span>
<span class="k">return</span> <span class="n">corr</span>
<span class="k">else</span><span class="p">:</span>
<span class="k">return</span> <span class="n">cov</span>
<span class="k">def</span> <span class="nf">_covariance_element</span><span class="p">(</span><span class="n">obs1</span><span class="p">,</span> <span class="n">obs2</span><span class="p">):</span>
<span class="sd">&quot;&quot;&quot;Estimates the covariance of two Obs objects, neglecting autocorrelations.&quot;&quot;&quot;</span>
<span class="k">def</span> <span class="nf">expand_deltas</span><span class="p">(</span><span class="n">deltas</span><span class="p">,</span> <span class="n">idx</span><span class="p">,</span> <span class="n">shape</span><span class="p">,</span> <span class="n">new_idx</span><span class="p">):</span>
<span class="sd">&quot;&quot;&quot;Expand deltas defined on idx to a contiguous range [new_idx[0], new_idx[-1]].</span>
<span class="sd"> New, empy entries are filled by 0. If idx and new_idx are of type range, the smallest</span>
@ -1680,29 +1719,18 @@
<span class="n">ret</span><span class="p">[</span><span class="n">idx</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">-</span> <span class="n">new_idx</span><span class="p">[</span><span class="mi">0</span><span class="p">]]</span> <span class="o">=</span> <span class="n">deltas</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
<span class="k">return</span> <span class="n">ret</span>
<span class="k">def</span> <span class="nf">calc_gamma</span><span class="p">(</span><span class="n">deltas1</span><span class="p">,</span> <span class="n">deltas2</span><span class="p">,</span> <span class="n">idx1</span><span class="p">,</span> <span class="n">idx2</span><span class="p">,</span> <span class="n">new_idx</span><span class="p">,</span> <span class="n">w_max</span><span class="p">):</span>
<span class="n">gamma</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">w_max</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">calc_gamma</span><span class="p">(</span><span class="n">deltas1</span><span class="p">,</span> <span class="n">deltas2</span><span class="p">,</span> <span class="n">idx1</span><span class="p">,</span> <span class="n">idx2</span><span class="p">,</span> <span class="n">new_idx</span><span class="p">):</span>
<span class="n">deltas1</span> <span class="o">=</span> <span class="n">expand_deltas</span><span class="p">(</span><span class="n">deltas1</span><span class="p">,</span> <span class="n">idx1</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">idx1</span><span class="p">),</span> <span class="n">new_idx</span><span class="p">)</span>
<span class="n">deltas2</span> <span class="o">=</span> <span class="n">expand_deltas</span><span class="p">(</span><span class="n">deltas2</span><span class="p">,</span> <span class="n">idx2</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">idx2</span><span class="p">),</span> <span class="n">new_idx</span><span class="p">)</span>
<span class="n">new_shape</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">deltas1</span><span class="p">)</span>
<span class="n">max_gamma</span> <span class="o">=</span> <span class="nb">min</span><span class="p">(</span><span class="n">new_shape</span><span class="p">,</span> <span class="n">w_max</span><span class="p">)</span>
<span class="c1"># The padding for the fft has to be even</span>
<span class="n">padding</span> <span class="o">=</span> <span class="n">new_shape</span> <span class="o">+</span> <span class="n">max_gamma</span> <span class="o">+</span> <span class="p">(</span><span class="n">new_shape</span> <span class="o">+</span> <span class="n">max_gamma</span><span class="p">)</span> <span class="o">%</span> <span class="mi">2</span>
<span class="n">gamma</span><span class="p">[:</span><span class="n">max_gamma</span><span class="p">]</span> <span class="o">+=</span> <span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">fft</span><span class="o">.</span><span class="n">irfft</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">fft</span><span class="o">.</span><span class="n">rfft</span><span class="p">(</span><span class="n">deltas1</span><span class="p">,</span> <span class="n">padding</span><span class="p">)</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">conjugate</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">fft</span><span class="o">.</span><span class="n">rfft</span><span class="p">(</span><span class="n">deltas2</span><span class="p">,</span> <span class="n">padding</span><span class="p">)))[:</span><span class="n">max_gamma</span><span class="p">]</span> <span class="o">+</span> <span class="n">np</span><span class="o">.</span><span class="n">fft</span><span class="o">.</span><span class="n">irfft</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">fft</span><span class="o">.</span><span class="n">rfft</span><span class="p">(</span><span class="n">deltas2</span><span class="p">,</span> <span class="n">padding</span><span class="p">)</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">conjugate</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">fft</span><span class="o">.</span><span class="n">rfft</span><span class="p">(</span><span class="n">deltas1</span><span class="p">,</span> <span class="n">padding</span><span class="p">)))[:</span><span class="n">max_gamma</span><span class="p">])</span> <span class="o">/</span> <span class="mf">2.0</span>
<span class="k">return</span> <span class="n">gamma</span>
<span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">deltas1</span> <span class="o">*</span> <span class="n">deltas2</span><span class="p">)</span>
<span class="k">if</span> <span class="nb">set</span><span class="p">(</span><span class="n">obs1</span><span class="o">.</span><span class="n">names</span><span class="p">)</span><span class="o">.</span><span class="n">isdisjoint</span><span class="p">(</span><span class="nb">set</span><span class="p">(</span><span class="n">obs2</span><span class="o">.</span><span class="n">names</span><span class="p">)):</span>
<span class="k">return</span> <span class="mf">0.</span>
<span class="k">return</span> <span class="mf">0.0</span>
<span class="k">if</span> <span class="ow">not</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">obs1</span><span class="p">,</span> <span class="s1">&#39;e_dvalue&#39;</span><span class="p">)</span> <span class="ow">or</span> <span class="ow">not</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">obs2</span><span class="p">,</span> <span class="s1">&#39;e_dvalue&#39;</span><span class="p">):</span>
<span class="k">raise</span> <span class="ne">Exception</span><span class="p">(</span><span class="s1">&#39;The gamma method has to be applied to both Obs first.&#39;</span><span class="p">)</span>
<span class="n">dvalue</span> <span class="o">=</span> <span class="mi">0</span>
<span class="n">e_gamma</span> <span class="o">=</span> <span class="p">{}</span>
<span class="n">e_dvalue</span> <span class="o">=</span> <span class="p">{}</span>
<span class="n">e_n_tauint</span> <span class="o">=</span> <span class="p">{}</span>
<span class="n">e_rho</span> <span class="o">=</span> <span class="p">{}</span>
<span class="n">dvalue</span> <span class="o">=</span> <span class="mf">0.0</span>
<span class="k">for</span> <span class="n">e_name</span> <span class="ow">in</span> <span class="n">obs1</span><span class="o">.</span><span class="n">mc_names</span><span class="p">:</span>
@ -1710,52 +1738,32 @@
<span class="k">continue</span>
<span class="n">idl_d</span> <span class="o">=</span> <span class="p">{}</span>
<span class="n">r_length</span> <span class="o">=</span> <span class="p">[]</span>
<span class="k">for</span> <span class="n">r_name</span> <span class="ow">in</span> <span class="n">obs1</span><span class="o">.</span><span class="n">e_content</span><span class="p">[</span><span class="n">e_name</span><span class="p">]:</span>
<span class="k">if</span> <span class="n">r_name</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">obs2</span><span class="o">.</span><span class="n">e_content</span><span class="p">[</span><span class="n">e_name</span><span class="p">]:</span>
<span class="k">continue</span>
<span class="n">idl_d</span><span class="p">[</span><span class="n">r_name</span><span class="p">]</span> <span class="o">=</span> <span class="n">_merge_idx</span><span class="p">([</span><span class="n">obs1</span><span class="o">.</span><span class="n">idl</span><span class="p">[</span><span class="n">r_name</span><span class="p">],</span> <span class="n">obs2</span><span class="o">.</span><span class="n">idl</span><span class="p">[</span><span class="n">r_name</span><span class="p">]])</span>
<span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">idl_d</span><span class="p">[</span><span class="n">r_name</span><span class="p">],</span> <span class="nb">range</span><span class="p">):</span>
<span class="n">r_length</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">idl_d</span><span class="p">[</span><span class="n">r_name</span><span class="p">]))</span>
<span class="k">else</span><span class="p">:</span>
<span class="n">r_length</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">idl_d</span><span class="p">[</span><span class="n">r_name</span><span class="p">][</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">-</span> <span class="n">idl_d</span><span class="p">[</span><span class="n">r_name</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span> <span class="o">+</span> <span class="mi">1</span><span class="p">))</span>
<span class="k">if</span> <span class="ow">not</span> <span class="n">r_length</span><span class="p">:</span>
<span class="k">return</span> <span class="mf">0.</span>
<span class="n">w_max</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="n">r_length</span><span class="p">)</span> <span class="o">//</span> <span class="mi">2</span>
<span class="n">e_gamma</span><span class="p">[</span><span class="n">e_name</span><span class="p">]</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">w_max</span><span class="p">)</span>
<span class="n">gamma</span> <span class="o">=</span> <span class="mf">0.0</span>
<span class="k">for</span> <span class="n">r_name</span> <span class="ow">in</span> <span class="n">obs1</span><span class="o">.</span><span class="n">e_content</span><span class="p">[</span><span class="n">e_name</span><span class="p">]:</span>
<span class="k">if</span> <span class="n">r_name</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">obs2</span><span class="o">.</span><span class="n">e_content</span><span class="p">[</span><span class="n">e_name</span><span class="p">]:</span>
<span class="k">continue</span>
<span class="n">e_gamma</span><span class="p">[</span><span class="n">e_name</span><span class="p">]</span> <span class="o">+=</span> <span class="n">calc_gamma</span><span class="p">(</span><span class="n">obs1</span><span class="o">.</span><span class="n">deltas</span><span class="p">[</span><span class="n">r_name</span><span class="p">],</span> <span class="n">obs2</span><span class="o">.</span><span class="n">deltas</span><span class="p">[</span><span class="n">r_name</span><span class="p">],</span> <span class="n">obs1</span><span class="o">.</span><span class="n">idl</span><span class="p">[</span><span class="n">r_name</span><span class="p">],</span> <span class="n">obs2</span><span class="o">.</span><span class="n">idl</span><span class="p">[</span><span class="n">r_name</span><span class="p">],</span> <span class="n">idl_d</span><span class="p">[</span><span class="n">r_name</span><span class="p">],</span> <span class="n">w_max</span><span class="p">)</span>
<span class="n">gamma</span> <span class="o">+=</span> <span class="n">calc_gamma</span><span class="p">(</span><span class="n">obs1</span><span class="o">.</span><span class="n">deltas</span><span class="p">[</span><span class="n">r_name</span><span class="p">],</span> <span class="n">obs2</span><span class="o">.</span><span class="n">deltas</span><span class="p">[</span><span class="n">r_name</span><span class="p">],</span> <span class="n">obs1</span><span class="o">.</span><span class="n">idl</span><span class="p">[</span><span class="n">r_name</span><span class="p">],</span> <span class="n">obs2</span><span class="o">.</span><span class="n">idl</span><span class="p">[</span><span class="n">r_name</span><span class="p">],</span> <span class="n">idl_d</span><span class="p">[</span><span class="n">r_name</span><span class="p">])</span>
<span class="k">if</span> <span class="n">np</span><span class="o">.</span><span class="n">all</span><span class="p">(</span><span class="n">e_gamma</span><span class="p">[</span><span class="n">e_name</span><span class="p">]</span> <span class="o">==</span> <span class="mf">0.0</span><span class="p">):</span>
<span class="k">if</span> <span class="n">gamma</span> <span class="o">==</span> <span class="mf">0.0</span><span class="p">:</span>
<span class="k">continue</span>
<span class="n">e_shapes</span> <span class="o">=</span> <span class="p">[]</span>
<span class="k">for</span> <span class="n">r_name</span> <span class="ow">in</span> <span class="n">obs1</span><span class="o">.</span><span class="n">e_content</span><span class="p">[</span><span class="n">e_name</span><span class="p">]:</span>
<span class="n">e_shapes</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">obs1</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="n">r_name</span><span class="p">])</span>
<span class="n">gamma_div</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">w_max</span><span class="p">)</span>
<span class="n">gamma_div</span> <span class="o">=</span> <span class="mf">0.0</span>
<span class="n">e_N</span> <span class="o">=</span> <span class="mi">0</span>
<span class="k">for</span> <span class="n">r_name</span> <span class="ow">in</span> <span class="n">obs1</span><span class="o">.</span><span class="n">e_content</span><span class="p">[</span><span class="n">e_name</span><span class="p">]:</span>
<span class="k">if</span> <span class="n">r_name</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">obs2</span><span class="o">.</span><span class="n">e_content</span><span class="p">[</span><span class="n">e_name</span><span class="p">]:</span>
<span class="k">continue</span>
<span class="n">gamma_div</span> <span class="o">+=</span> <span class="n">calc_gamma</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">ones</span><span class="p">(</span><span class="n">obs1</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="n">r_name</span><span class="p">]),</span> <span class="n">np</span><span class="o">.</span><span class="n">ones</span><span class="p">(</span><span class="n">obs2</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="n">r_name</span><span class="p">]),</span> <span class="n">obs1</span><span class="o">.</span><span class="n">idl</span><span class="p">[</span><span class="n">r_name</span><span class="p">],</span> <span class="n">obs2</span><span class="o">.</span><span class="n">idl</span><span class="p">[</span><span class="n">r_name</span><span class="p">],</span> <span class="n">idl_d</span><span class="p">[</span><span class="n">r_name</span><span class="p">],</span> <span class="n">w_max</span><span class="p">)</span>
<span class="n">gamma_div</span> <span class="o">+=</span> <span class="n">calc_gamma</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">ones</span><span class="p">(</span><span class="n">obs1</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="n">r_name</span><span class="p">]),</span> <span class="n">np</span><span class="o">.</span><span class="n">ones</span><span class="p">(</span><span class="n">obs2</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="n">r_name</span><span class="p">]),</span> <span class="n">obs1</span><span class="o">.</span><span class="n">idl</span><span class="p">[</span><span class="n">r_name</span><span class="p">],</span> <span class="n">obs2</span><span class="o">.</span><span class="n">idl</span><span class="p">[</span><span class="n">r_name</span><span class="p">],</span> <span class="n">idl_d</span><span class="p">[</span><span class="n">r_name</span><span class="p">])</span>
<span class="n">e_N</span> <span class="o">+=</span> <span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">ones_like</span><span class="p">(</span><span class="n">idl_d</span><span class="p">[</span><span class="n">r_name</span><span class="p">]))</span>
<span class="n">e_gamma</span><span class="p">[</span><span class="n">e_name</span><span class="p">]</span> <span class="o">/=</span> <span class="n">gamma_div</span><span class="p">[:</span><span class="n">w_max</span><span class="p">]</span>
<span class="n">gamma</span> <span class="o">/=</span> <span class="n">gamma_div</span>
<span class="n">e_rho</span><span class="p">[</span><span class="n">e_name</span><span class="p">]</span> <span class="o">=</span> <span class="n">e_gamma</span><span class="p">[</span><span class="n">e_name</span><span class="p">][:</span><span class="n">w_max</span><span class="p">]</span> <span class="o">/</span> <span class="n">e_gamma</span><span class="p">[</span><span class="n">e_name</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span>
<span class="n">e_n_tauint</span><span class="p">[</span><span class="n">e_name</span><span class="p">]</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">cumsum</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">concatenate</span><span class="p">(([</span><span class="mf">0.5</span><span class="p">],</span> <span class="n">e_rho</span><span class="p">[</span><span class="n">e_name</span><span class="p">][</span><span class="mi">1</span><span class="p">:])))</span>
<span class="c1"># Make sure no entry of tauint is smaller than 0.5</span>
<span class="n">e_n_tauint</span><span class="p">[</span><span class="n">e_name</span><span class="p">][</span><span class="n">e_n_tauint</span><span class="p">[</span><span class="n">e_name</span><span class="p">]</span> <span class="o">&lt;</span> <span class="mf">0.5</span><span class="p">]</span> <span class="o">=</span> <span class="mf">0.500000000001</span>
<span class="n">window</span> <span class="o">=</span> <span class="nb">min</span><span class="p">(</span><span class="n">obs1</span><span class="o">.</span><span class="n">e_windowsize</span><span class="p">[</span><span class="n">e_name</span><span class="p">],</span> <span class="n">obs2</span><span class="o">.</span><span class="n">e_windowsize</span><span class="p">[</span><span class="n">e_name</span><span class="p">])</span>
<span class="c1"># Bias correction hep-lat/0306017 eq. (49)</span>
<span class="n">e_dvalue</span><span class="p">[</span><span class="n">e_name</span><span class="p">]</span> <span class="o">=</span> <span class="mi">2</span> <span class="o">*</span> <span class="p">(</span><span class="n">e_n_tauint</span><span class="p">[</span><span class="n">e_name</span><span class="p">][</span><span class="n">window</span><span class="p">]</span> <span class="o">+</span> <span class="n">obs1</span><span class="o">.</span><span class="n">tau_exp</span><span class="p">[</span><span class="n">e_name</span><span class="p">]</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">e_rho</span><span class="p">[</span><span class="n">e_name</span><span class="p">][</span><span class="n">window</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]))</span> <span class="o">*</span> <span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="p">(</span><span class="mi">2</span> <span class="o">*</span> <span class="n">window</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="o">/</span> <span class="n">e_N</span><span class="p">)</span> <span class="o">*</span> <span class="n">e_gamma</span><span class="p">[</span><span class="n">e_name</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span> <span class="o">/</span> <span class="n">e_N</span>
<span class="n">dvalue</span> <span class="o">+=</span> <span class="n">e_dvalue</span><span class="p">[</span><span class="n">e_name</span><span class="p">]</span>
<span class="n">dvalue</span> <span class="o">+=</span> <span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="mi">1</span> <span class="o">/</span> <span class="n">e_N</span><span class="p">)</span> <span class="o">*</span> <span class="n">gamma</span> <span class="o">/</span> <span class="n">e_N</span>
<span class="k">for</span> <span class="n">e_name</span> <span class="ow">in</span> <span class="n">obs1</span><span class="o">.</span><span class="n">cov_names</span><span class="p">:</span>
@ -1764,12 +1772,6 @@
<span class="n">dvalue</span> <span class="o">+=</span> <span class="nb">float</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">transpose</span><span class="p">(</span><span class="n">obs1</span><span class="o">.</span><span class="n">covobs</span><span class="p">[</span><span class="n">e_name</span><span class="p">]</span><span class="o">.</span><span class="n">grad</span><span class="p">),</span> <span class="n">np</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">obs1</span><span class="o">.</span><span class="n">covobs</span><span class="p">[</span><span class="n">e_name</span><span class="p">]</span><span class="o">.</span><span class="n">cov</span><span class="p">,</span> <span class="n">obs2</span><span class="o">.</span><span class="n">covobs</span><span class="p">[</span><span class="n">e_name</span><span class="p">]</span><span class="o">.</span><span class="n">grad</span><span class="p">)))</span>
<span class="k">if</span> <span class="n">np</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">dvalue</span> <span class="o">/</span> <span class="n">obs1</span><span class="o">.</span><span class="n">dvalue</span> <span class="o">/</span> <span class="n">obs2</span><span class="o">.</span><span class="n">dvalue</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mf">1.0</span><span class="p">:</span>
<span class="n">dvalue</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">sign</span><span class="p">(</span><span class="n">dvalue</span><span class="p">)</span> <span class="o">*</span> <span class="n">obs1</span><span class="o">.</span><span class="n">dvalue</span> <span class="o">*</span> <span class="n">obs2</span><span class="o">.</span><span class="n">dvalue</span>
<span class="k">if</span> <span class="n">correlation</span><span class="p">:</span>
<span class="n">dvalue</span> <span class="o">=</span> <span class="n">dvalue</span> <span class="o">/</span> <span class="n">obs1</span><span class="o">.</span><span class="n">dvalue</span> <span class="o">/</span> <span class="n">obs2</span><span class="o">.</span><span class="n">dvalue</span>
<span class="k">return</span> <span class="n">dvalue</span>
@ -4827,157 +4829,86 @@ Second observable</li>
<span class="def">def</span>
<span class="name">covariance</span><span class="signature">(obs1, obs2, correlation=False, **kwargs)</span>:
<span class="name">covariance</span><span class="signature">(obs, visualize=False, correlation=False, **kwargs)</span>:
</div>
<details>
<summary>View Source</summary>
<div class="pdoc-code codehilite"><pre><span></span><span class="k">def</span> <span class="nf">covariance</span><span class="p">(</span><span class="n">obs1</span><span class="p">,</span> <span class="n">obs2</span><span class="p">,</span> <span class="n">correlation</span><span class="o">=</span><span class="kc">False</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
<span class="sd">&quot;&quot;&quot;Calculates the covariance of two observables.</span>
<div class="pdoc-code codehilite"><pre><span></span><span class="k">def</span> <span class="nf">covariance</span><span class="p">(</span><span class="n">obs</span><span class="p">,</span> <span class="n">visualize</span><span class="o">=</span><span class="kc">False</span><span class="p">,</span> <span class="n">correlation</span><span class="o">=</span><span class="kc">False</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
<span class="sd">&quot;&quot;&quot;Calculates the covariance matrix of a set of observables.</span>
<span class="sd"> covariance(obs, obs) is equal to obs.dvalue ** 2</span>
<span class="sd"> The gamma method has to be applied first to both observables.</span>
<span class="sd"> If abs(covariance(obs1, obs2)) &gt; obs1.dvalue * obs2.dvalue, the covariance</span>
<span class="sd"> is constrained to the maximum value.</span>
<span class="sd"> The gamma method has to be applied first to all observables.</span>
<span class="sd"> Parameters</span>
<span class="sd"> ----------</span>
<span class="sd"> obs : list or numpy.ndarray</span>
<span class="sd"> List or one dimensional array of Obs</span>
<span class="sd"> visualize : bool</span>
<span class="sd"> If True plots the corresponding normalized correlation matrix (default False).</span>
<span class="sd"> correlation : bool</span>
<span class="sd"> if true the correlation instead of the covariance is returned (default False)</span>
<span class="sd"> If True the correlation instead of the covariance is returned (default False).</span>
<span class="sd"> Notes</span>
<span class="sd"> -----</span>
<span class="sd"> The covariance is estimated by calculating the correlation matrix assuming no autocorrelation and then rescaling the correlation matrix by the full errors including the previous gamma method estimate for the autocorrelation of the observables. For observables defined on a single ensemble this is equivalent to assuming that the integrated autocorrelation time of an off-diagonal element is equal to the geometric mean of the integrated autocorrelation times of the corresponding diagonal elements.</span>
<span class="sd"> $$</span>
<span class="sd"> \tau_{\mathrm{int}, ij}=\sqrt{\tau_{\mathrm{int}, i}\times \tau_{\mathrm{int}, j}}</span>
<span class="sd"> $$</span>
<span class="sd"> This construction ensures that the estimated covariance matrix is positive semi-definite (up to numerical rounding errors).</span>
<span class="sd"> &quot;&quot;&quot;</span>
<span class="k">def</span> <span class="nf">expand_deltas</span><span class="p">(</span><span class="n">deltas</span><span class="p">,</span> <span class="n">idx</span><span class="p">,</span> <span class="n">shape</span><span class="p">,</span> <span class="n">new_idx</span><span class="p">):</span>
<span class="sd">&quot;&quot;&quot;Expand deltas defined on idx to a contiguous range [new_idx[0], new_idx[-1]].</span>
<span class="sd"> New, empy entries are filled by 0. If idx and new_idx are of type range, the smallest</span>
<span class="sd"> common divisor of the step sizes is used as new step size.</span>
<span class="n">length</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">obs</span><span class="p">)</span>
<span class="n">cov</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">length</span><span class="p">,</span> <span class="n">length</span><span class="p">))</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">length</span><span class="p">):</span>
<span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">length</span><span class="p">):</span>
<span class="n">cov</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">_covariance_element</span><span class="p">(</span><span class="n">obs</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">obs</span><span class="p">[</span><span class="n">j</span><span class="p">])</span>
<span class="n">cov</span> <span class="o">=</span> <span class="n">cov</span> <span class="o">+</span> <span class="n">cov</span><span class="o">.</span><span class="n">T</span> <span class="o">-</span> <span class="n">np</span><span class="o">.</span><span class="n">diag</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">diag</span><span class="p">(</span><span class="n">cov</span><span class="p">))</span>
<span class="sd"> Parameters</span>
<span class="sd"> ----------</span>
<span class="sd"> deltas -- List of fluctuations</span>
<span class="sd"> idx -- List or range of configs on which the deltas are defined.</span>
<span class="sd"> Has to be a subset of new_idx.</span>
<span class="sd"> shape -- Number of configs in idx.</span>
<span class="sd"> new_idx -- List of configs that defines the new range.</span>
<span class="sd"> &quot;&quot;&quot;</span>
<span class="n">corr</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">diag</span><span class="p">(</span><span class="mi">1</span> <span class="o">/</span> <span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">diag</span><span class="p">(</span><span class="n">cov</span><span class="p">)))</span> <span class="o">@</span> <span class="n">cov</span> <span class="o">@</span> <span class="n">np</span><span class="o">.</span><span class="n">diag</span><span class="p">(</span><span class="mi">1</span> <span class="o">/</span> <span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">diag</span><span class="p">(</span><span class="n">cov</span><span class="p">)))</span>
<span class="k">if</span> <span class="nb">type</span><span class="p">(</span><span class="n">idx</span><span class="p">)</span> <span class="ow">is</span> <span class="nb">range</span> <span class="ow">and</span> <span class="nb">type</span><span class="p">(</span><span class="n">new_idx</span><span class="p">)</span> <span class="ow">is</span> <span class="nb">range</span><span class="p">:</span>
<span class="k">if</span> <span class="n">idx</span> <span class="o">==</span> <span class="n">new_idx</span><span class="p">:</span>
<span class="k">return</span> <span class="n">deltas</span>
<span class="n">ret</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">new_idx</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">-</span> <span class="n">new_idx</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">shape</span><span class="p">):</span>
<span class="n">ret</span><span class="p">[</span><span class="n">idx</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">-</span> <span class="n">new_idx</span><span class="p">[</span><span class="mi">0</span><span class="p">]]</span> <span class="o">=</span> <span class="n">deltas</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
<span class="k">return</span> <span class="n">ret</span>
<span class="n">errors</span> <span class="o">=</span> <span class="p">[</span><span class="n">o</span><span class="o">.</span><span class="n">dvalue</span> <span class="k">for</span> <span class="n">o</span> <span class="ow">in</span> <span class="n">obs</span><span class="p">]</span>
<span class="n">cov</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">diag</span><span class="p">(</span><span class="n">errors</span><span class="p">)</span> <span class="o">@</span> <span class="n">corr</span> <span class="o">@</span> <span class="n">np</span><span class="o">.</span><span class="n">diag</span><span class="p">(</span><span class="n">errors</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">calc_gamma</span><span class="p">(</span><span class="n">deltas1</span><span class="p">,</span> <span class="n">deltas2</span><span class="p">,</span> <span class="n">idx1</span><span class="p">,</span> <span class="n">idx2</span><span class="p">,</span> <span class="n">new_idx</span><span class="p">,</span> <span class="n">w_max</span><span class="p">):</span>
<span class="n">gamma</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">w_max</span><span class="p">)</span>
<span class="n">deltas1</span> <span class="o">=</span> <span class="n">expand_deltas</span><span class="p">(</span><span class="n">deltas1</span><span class="p">,</span> <span class="n">idx1</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">idx1</span><span class="p">),</span> <span class="n">new_idx</span><span class="p">)</span>
<span class="n">deltas2</span> <span class="o">=</span> <span class="n">expand_deltas</span><span class="p">(</span><span class="n">deltas2</span><span class="p">,</span> <span class="n">idx2</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">idx2</span><span class="p">),</span> <span class="n">new_idx</span><span class="p">)</span>
<span class="n">new_shape</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">deltas1</span><span class="p">)</span>
<span class="n">max_gamma</span> <span class="o">=</span> <span class="nb">min</span><span class="p">(</span><span class="n">new_shape</span><span class="p">,</span> <span class="n">w_max</span><span class="p">)</span>
<span class="c1"># The padding for the fft has to be even</span>
<span class="n">padding</span> <span class="o">=</span> <span class="n">new_shape</span> <span class="o">+</span> <span class="n">max_gamma</span> <span class="o">+</span> <span class="p">(</span><span class="n">new_shape</span> <span class="o">+</span> <span class="n">max_gamma</span><span class="p">)</span> <span class="o">%</span> <span class="mi">2</span>
<span class="n">gamma</span><span class="p">[:</span><span class="n">max_gamma</span><span class="p">]</span> <span class="o">+=</span> <span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">fft</span><span class="o">.</span><span class="n">irfft</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">fft</span><span class="o">.</span><span class="n">rfft</span><span class="p">(</span><span class="n">deltas1</span><span class="p">,</span> <span class="n">padding</span><span class="p">)</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">conjugate</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">fft</span><span class="o">.</span><span class="n">rfft</span><span class="p">(</span><span class="n">deltas2</span><span class="p">,</span> <span class="n">padding</span><span class="p">)))[:</span><span class="n">max_gamma</span><span class="p">]</span> <span class="o">+</span> <span class="n">np</span><span class="o">.</span><span class="n">fft</span><span class="o">.</span><span class="n">irfft</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">fft</span><span class="o">.</span><span class="n">rfft</span><span class="p">(</span><span class="n">deltas2</span><span class="p">,</span> <span class="n">padding</span><span class="p">)</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">conjugate</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">fft</span><span class="o">.</span><span class="n">rfft</span><span class="p">(</span><span class="n">deltas1</span><span class="p">,</span> <span class="n">padding</span><span class="p">)))[:</span><span class="n">max_gamma</span><span class="p">])</span> <span class="o">/</span> <span class="mf">2.0</span>
<span class="n">eigenvalues</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">eigh</span><span class="p">(</span><span class="n">cov</span><span class="p">)[</span><span class="mi">0</span><span class="p">]</span>
<span class="k">if</span> <span class="ow">not</span> <span class="n">np</span><span class="o">.</span><span class="n">all</span><span class="p">(</span><span class="n">eigenvalues</span> <span class="o">&gt;=</span> <span class="mi">0</span><span class="p">):</span>
<span class="n">warnings</span><span class="o">.</span><span class="n">warn</span><span class="p">(</span><span class="s2">&quot;Covariance matrix is not positive semi-definite (Eigenvalues: &quot;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">eigenvalues</span><span class="p">)</span> <span class="o">+</span> <span class="s2">&quot;)&quot;</span><span class="p">,</span> <span class="ne">RuntimeWarning</span><span class="p">)</span>
<span class="k">return</span> <span class="n">gamma</span>
<span class="k">if</span> <span class="n">visualize</span><span class="p">:</span>
<span class="n">plt</span><span class="o">.</span><span class="n">matshow</span><span class="p">(</span><span class="n">corr</span><span class="p">,</span> <span class="n">vmin</span><span class="o">=-</span><span class="mi">1</span><span class="p">,</span> <span class="n">vmax</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">set_cmap</span><span class="p">(</span><span class="s1">&#39;RdBu&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">colorbar</span><span class="p">()</span>
<span class="n">plt</span><span class="o">.</span><span class="n">draw</span><span class="p">()</span>
<span class="k">if</span> <span class="nb">set</span><span class="p">(</span><span class="n">obs1</span><span class="o">.</span><span class="n">names</span><span class="p">)</span><span class="o">.</span><span class="n">isdisjoint</span><span class="p">(</span><span class="nb">set</span><span class="p">(</span><span class="n">obs2</span><span class="o">.</span><span class="n">names</span><span class="p">)):</span>
<span class="k">return</span> <span class="mf">0.</span>
<span class="k">if</span> <span class="ow">not</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">obs1</span><span class="p">,</span> <span class="s1">&#39;e_dvalue&#39;</span><span class="p">)</span> <span class="ow">or</span> <span class="ow">not</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">obs2</span><span class="p">,</span> <span class="s1">&#39;e_dvalue&#39;</span><span class="p">):</span>
<span class="k">raise</span> <span class="ne">Exception</span><span class="p">(</span><span class="s1">&#39;The gamma method has to be applied to both Obs first.&#39;</span><span class="p">)</span>
<span class="n">dvalue</span> <span class="o">=</span> <span class="mi">0</span>
<span class="n">e_gamma</span> <span class="o">=</span> <span class="p">{}</span>
<span class="n">e_dvalue</span> <span class="o">=</span> <span class="p">{}</span>
<span class="n">e_n_tauint</span> <span class="o">=</span> <span class="p">{}</span>
<span class="n">e_rho</span> <span class="o">=</span> <span class="p">{}</span>
<span class="k">for</span> <span class="n">e_name</span> <span class="ow">in</span> <span class="n">obs1</span><span class="o">.</span><span class="n">mc_names</span><span class="p">:</span>
<span class="k">if</span> <span class="n">e_name</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">obs2</span><span class="o">.</span><span class="n">mc_names</span><span class="p">:</span>
<span class="k">continue</span>
<span class="n">idl_d</span> <span class="o">=</span> <span class="p">{}</span>
<span class="n">r_length</span> <span class="o">=</span> <span class="p">[]</span>
<span class="k">for</span> <span class="n">r_name</span> <span class="ow">in</span> <span class="n">obs1</span><span class="o">.</span><span class="n">e_content</span><span class="p">[</span><span class="n">e_name</span><span class="p">]:</span>
<span class="k">if</span> <span class="n">r_name</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">obs2</span><span class="o">.</span><span class="n">e_content</span><span class="p">[</span><span class="n">e_name</span><span class="p">]:</span>
<span class="k">continue</span>
<span class="n">idl_d</span><span class="p">[</span><span class="n">r_name</span><span class="p">]</span> <span class="o">=</span> <span class="n">_merge_idx</span><span class="p">([</span><span class="n">obs1</span><span class="o">.</span><span class="n">idl</span><span class="p">[</span><span class="n">r_name</span><span class="p">],</span> <span class="n">obs2</span><span class="o">.</span><span class="n">idl</span><span class="p">[</span><span class="n">r_name</span><span class="p">]])</span>
<span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">idl_d</span><span class="p">[</span><span class="n">r_name</span><span class="p">],</span> <span class="nb">range</span><span class="p">):</span>
<span class="n">r_length</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">idl_d</span><span class="p">[</span><span class="n">r_name</span><span class="p">]))</span>
<span class="k">else</span><span class="p">:</span>
<span class="n">r_length</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">idl_d</span><span class="p">[</span><span class="n">r_name</span><span class="p">][</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">-</span> <span class="n">idl_d</span><span class="p">[</span><span class="n">r_name</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span> <span class="o">+</span> <span class="mi">1</span><span class="p">))</span>
<span class="k">if</span> <span class="ow">not</span> <span class="n">r_length</span><span class="p">:</span>
<span class="k">return</span> <span class="mf">0.</span>
<span class="n">w_max</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="n">r_length</span><span class="p">)</span> <span class="o">//</span> <span class="mi">2</span>
<span class="n">e_gamma</span><span class="p">[</span><span class="n">e_name</span><span class="p">]</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">w_max</span><span class="p">)</span>
<span class="k">for</span> <span class="n">r_name</span> <span class="ow">in</span> <span class="n">obs1</span><span class="o">.</span><span class="n">e_content</span><span class="p">[</span><span class="n">e_name</span><span class="p">]:</span>
<span class="k">if</span> <span class="n">r_name</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">obs2</span><span class="o">.</span><span class="n">e_content</span><span class="p">[</span><span class="n">e_name</span><span class="p">]:</span>
<span class="k">continue</span>
<span class="n">e_gamma</span><span class="p">[</span><span class="n">e_name</span><span class="p">]</span> <span class="o">+=</span> <span class="n">calc_gamma</span><span class="p">(</span><span class="n">obs1</span><span class="o">.</span><span class="n">deltas</span><span class="p">[</span><span class="n">r_name</span><span class="p">],</span> <span class="n">obs2</span><span class="o">.</span><span class="n">deltas</span><span class="p">[</span><span class="n">r_name</span><span class="p">],</span> <span class="n">obs1</span><span class="o">.</span><span class="n">idl</span><span class="p">[</span><span class="n">r_name</span><span class="p">],</span> <span class="n">obs2</span><span class="o">.</span><span class="n">idl</span><span class="p">[</span><span class="n">r_name</span><span class="p">],</span> <span class="n">idl_d</span><span class="p">[</span><span class="n">r_name</span><span class="p">],</span> <span class="n">w_max</span><span class="p">)</span>
<span class="k">if</span> <span class="n">np</span><span class="o">.</span><span class="n">all</span><span class="p">(</span><span class="n">e_gamma</span><span class="p">[</span><span class="n">e_name</span><span class="p">]</span> <span class="o">==</span> <span class="mf">0.0</span><span class="p">):</span>
<span class="k">continue</span>
<span class="n">e_shapes</span> <span class="o">=</span> <span class="p">[]</span>
<span class="k">for</span> <span class="n">r_name</span> <span class="ow">in</span> <span class="n">obs1</span><span class="o">.</span><span class="n">e_content</span><span class="p">[</span><span class="n">e_name</span><span class="p">]:</span>
<span class="n">e_shapes</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">obs1</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="n">r_name</span><span class="p">])</span>
<span class="n">gamma_div</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">w_max</span><span class="p">)</span>
<span class="n">e_N</span> <span class="o">=</span> <span class="mi">0</span>
<span class="k">for</span> <span class="n">r_name</span> <span class="ow">in</span> <span class="n">obs1</span><span class="o">.</span><span class="n">e_content</span><span class="p">[</span><span class="n">e_name</span><span class="p">]:</span>
<span class="k">if</span> <span class="n">r_name</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">obs2</span><span class="o">.</span><span class="n">e_content</span><span class="p">[</span><span class="n">e_name</span><span class="p">]:</span>
<span class="k">continue</span>
<span class="n">gamma_div</span> <span class="o">+=</span> <span class="n">calc_gamma</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">ones</span><span class="p">(</span><span class="n">obs1</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="n">r_name</span><span class="p">]),</span> <span class="n">np</span><span class="o">.</span><span class="n">ones</span><span class="p">(</span><span class="n">obs2</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="n">r_name</span><span class="p">]),</span> <span class="n">obs1</span><span class="o">.</span><span class="n">idl</span><span class="p">[</span><span class="n">r_name</span><span class="p">],</span> <span class="n">obs2</span><span class="o">.</span><span class="n">idl</span><span class="p">[</span><span class="n">r_name</span><span class="p">],</span> <span class="n">idl_d</span><span class="p">[</span><span class="n">r_name</span><span class="p">],</span> <span class="n">w_max</span><span class="p">)</span>
<span class="n">e_N</span> <span class="o">+=</span> <span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">ones_like</span><span class="p">(</span><span class="n">idl_d</span><span class="p">[</span><span class="n">r_name</span><span class="p">]))</span>
<span class="n">e_gamma</span><span class="p">[</span><span class="n">e_name</span><span class="p">]</span> <span class="o">/=</span> <span class="n">gamma_div</span><span class="p">[:</span><span class="n">w_max</span><span class="p">]</span>
<span class="n">e_rho</span><span class="p">[</span><span class="n">e_name</span><span class="p">]</span> <span class="o">=</span> <span class="n">e_gamma</span><span class="p">[</span><span class="n">e_name</span><span class="p">][:</span><span class="n">w_max</span><span class="p">]</span> <span class="o">/</span> <span class="n">e_gamma</span><span class="p">[</span><span class="n">e_name</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span>
<span class="n">e_n_tauint</span><span class="p">[</span><span class="n">e_name</span><span class="p">]</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">cumsum</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">concatenate</span><span class="p">(([</span><span class="mf">0.5</span><span class="p">],</span> <span class="n">e_rho</span><span class="p">[</span><span class="n">e_name</span><span class="p">][</span><span class="mi">1</span><span class="p">:])))</span>
<span class="c1"># Make sure no entry of tauint is smaller than 0.5</span>
<span class="n">e_n_tauint</span><span class="p">[</span><span class="n">e_name</span><span class="p">][</span><span class="n">e_n_tauint</span><span class="p">[</span><span class="n">e_name</span><span class="p">]</span> <span class="o">&lt;</span> <span class="mf">0.5</span><span class="p">]</span> <span class="o">=</span> <span class="mf">0.500000000001</span>
<span class="n">window</span> <span class="o">=</span> <span class="nb">min</span><span class="p">(</span><span class="n">obs1</span><span class="o">.</span><span class="n">e_windowsize</span><span class="p">[</span><span class="n">e_name</span><span class="p">],</span> <span class="n">obs2</span><span class="o">.</span><span class="n">e_windowsize</span><span class="p">[</span><span class="n">e_name</span><span class="p">])</span>
<span class="c1"># Bias correction hep-lat/0306017 eq. (49)</span>
<span class="n">e_dvalue</span><span class="p">[</span><span class="n">e_name</span><span class="p">]</span> <span class="o">=</span> <span class="mi">2</span> <span class="o">*</span> <span class="p">(</span><span class="n">e_n_tauint</span><span class="p">[</span><span class="n">e_name</span><span class="p">][</span><span class="n">window</span><span class="p">]</span> <span class="o">+</span> <span class="n">obs1</span><span class="o">.</span><span class="n">tau_exp</span><span class="p">[</span><span class="n">e_name</span><span class="p">]</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">e_rho</span><span class="p">[</span><span class="n">e_name</span><span class="p">][</span><span class="n">window</span> <span class="o">+</span> <span class="mi">1</span><span class="p">]))</span> <span class="o">*</span> <span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="p">(</span><span class="mi">2</span> <span class="o">*</span> <span class="n">window</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="o">/</span> <span class="n">e_N</span><span class="p">)</span> <span class="o">*</span> <span class="n">e_gamma</span><span class="p">[</span><span class="n">e_name</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span> <span class="o">/</span> <span class="n">e_N</span>
<span class="n">dvalue</span> <span class="o">+=</span> <span class="n">e_dvalue</span><span class="p">[</span><span class="n">e_name</span><span class="p">]</span>
<span class="k">for</span> <span class="n">e_name</span> <span class="ow">in</span> <span class="n">obs1</span><span class="o">.</span><span class="n">cov_names</span><span class="p">:</span>
<span class="k">if</span> <span class="n">e_name</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">obs2</span><span class="o">.</span><span class="n">cov_names</span><span class="p">:</span>
<span class="k">continue</span>
<span class="n">dvalue</span> <span class="o">+=</span> <span class="nb">float</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">transpose</span><span class="p">(</span><span class="n">obs1</span><span class="o">.</span><span class="n">covobs</span><span class="p">[</span><span class="n">e_name</span><span class="p">]</span><span class="o">.</span><span class="n">grad</span><span class="p">),</span> <span class="n">np</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">obs1</span><span class="o">.</span><span class="n">covobs</span><span class="p">[</span><span class="n">e_name</span><span class="p">]</span><span class="o">.</span><span class="n">cov</span><span class="p">,</span> <span class="n">obs2</span><span class="o">.</span><span class="n">covobs</span><span class="p">[</span><span class="n">e_name</span><span class="p">]</span><span class="o">.</span><span class="n">grad</span><span class="p">)))</span>
<span class="k">if</span> <span class="n">np</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">dvalue</span> <span class="o">/</span> <span class="n">obs1</span><span class="o">.</span><span class="n">dvalue</span> <span class="o">/</span> <span class="n">obs2</span><span class="o">.</span><span class="n">dvalue</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mf">1.0</span><span class="p">:</span>
<span class="n">dvalue</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">sign</span><span class="p">(</span><span class="n">dvalue</span><span class="p">)</span> <span class="o">*</span> <span class="n">obs1</span><span class="o">.</span><span class="n">dvalue</span> <span class="o">*</span> <span class="n">obs2</span><span class="o">.</span><span class="n">dvalue</span>
<span class="k">if</span> <span class="n">correlation</span><span class="p">:</span>
<span class="n">dvalue</span> <span class="o">=</span> <span class="n">dvalue</span> <span class="o">/</span> <span class="n">obs1</span><span class="o">.</span><span class="n">dvalue</span> <span class="o">/</span> <span class="n">obs2</span><span class="o">.</span><span class="n">dvalue</span>
<span class="k">return</span> <span class="n">dvalue</span>
<span class="k">if</span> <span class="n">correlation</span> <span class="ow">is</span> <span class="kc">True</span><span class="p">:</span>
<span class="k">return</span> <span class="n">corr</span>
<span class="k">else</span><span class="p">:</span>
<span class="k">return</span> <span class="n">cov</span>
</pre></div>
</details>
<div class="docstring"><p>Calculates the covariance of two observables.</p>
<div class="docstring"><p>Calculates the covariance matrix of a set of observables.</p>
<p>covariance(obs, obs) is equal to obs.dvalue ** 2
The gamma method has to be applied first to both observables.</p>
<p>If abs(covariance(obs1, obs2)) &gt; obs1.dvalue * obs2.dvalue, the covariance
is constrained to the maximum value.</p>
<p>The gamma method has to be applied first to all observables.</p>
<h6 id="parameters">Parameters</h6>
<ul>
<li><strong>obs</strong> (list or numpy.ndarray):
List or one dimensional array of Obs</li>
<li><strong>visualize</strong> (bool):
If True plots the corresponding normalized correlation matrix (default False).</li>
<li><strong>correlation</strong> (bool):
if true the correlation instead of the covariance is returned (default False)</li>
If True the correlation instead of the covariance is returned (default False).</li>
</ul>
<h6 id="notes">Notes</h6>
<p>The covariance is estimated by calculating the correlation matrix assuming no autocorrelation and then rescaling the correlation matrix by the full errors including the previous gamma method estimate for the autocorrelation of the observables. For observables defined on a single ensemble this is equivalent to assuming that the integrated autocorrelation time of an off-diagonal element is equal to the geometric mean of the integrated autocorrelation times of the corresponding diagonal elements.
$$
au_{\mathrm{int}, ij}=\sqrt{ au_{\mathrm{int}, i} imes au_{\mathrm{int}, j}}
$$
This construction ensures that the estimated covariance matrix is positive semi-definite (up to numerical rounding errors).</p>
</div>

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