pyerrors/docs/pyerrors/fits.html
2022-01-06 10:58:19 +00:00

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<title>pyerrors.fits API documentation</title>
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<h2>API Documentation</h2>
<ul class="memberlist">
<li>
<a class="class" href="#Fit_result">Fit_result</a>
<ul class="memberlist">
<li>
<a class="function" href="#Fit_result.__init__">Fit_result</a>
</li>
<li>
<a class="function" href="#Fit_result.gamma_method">gamma_method</a>
</li>
</ul>
</li>
<li>
<a class="function" href="#least_squares">least_squares</a>
</li>
<li>
<a class="function" href="#total_least_squares">total_least_squares</a>
</li>
<li>
<a class="function" href="#prior_fit">prior_fit</a>
</li>
<li>
<a class="function" href="#standard_fit">standard_fit</a>
</li>
<li>
<a class="function" href="#odr_fit">odr_fit</a>
</li>
<li>
<a class="function" href="#fit_lin">fit_lin</a>
</li>
<li>
<a class="function" href="#qqplot">qqplot</a>
</li>
<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>
</ul>
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<section>
<h1 class="modulename">
<a href="./../pyerrors.html">pyerrors</a><wbr>.fits </h1>
<details>
<summary>View Source</summary>
<div class="codehilite"><pre><span></span><span class="kn">from</span> <span class="nn">collections.abc</span> <span class="kn">import</span> <span class="n">Sequence</span>
<span class="kn">import</span> <span class="nn">warnings</span>
<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="kn">import</span> <span class="nn">autograd.numpy</span> <span class="k">as</span> <span class="nn">anp</span>
<span class="kn">import</span> <span class="nn">scipy.optimize</span>
<span class="kn">import</span> <span class="nn">scipy.stats</span>
<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
<span class="kn">from</span> <span class="nn">matplotlib</span> <span class="kn">import</span> <span class="n">gridspec</span>
<span class="kn">from</span> <span class="nn">scipy.odr</span> <span class="kn">import</span> <span class="n">ODR</span><span class="p">,</span> <span class="n">Model</span><span class="p">,</span> <span class="n">RealData</span>
<span class="kn">import</span> <span class="nn">iminuit</span>
<span class="kn">from</span> <span class="nn">autograd</span> <span class="kn">import</span> <span class="n">jacobian</span>
<span class="kn">from</span> <span class="nn">autograd</span> <span class="kn">import</span> <span class="n">elementwise_grad</span> <span class="k">as</span> <span class="n">egrad</span>
<span class="kn">from</span> <span class="nn">.obs</span> <span class="kn">import</span> <span class="n">Obs</span><span class="p">,</span> <span class="n">derived_observable</span><span class="p">,</span> <span class="n">covariance</span><span class="p">,</span> <span class="n">cov_Obs</span>
<span class="k">class</span> <span class="nc">Fit_result</span><span class="p">(</span><span class="n">Sequence</span><span class="p">):</span>
<span class="sd">&quot;&quot;&quot;Represents fit results.</span>
<span class="sd"> Attributes</span>
<span class="sd"> ----------</span>
<span class="sd"> fit_parameters : list</span>
<span class="sd"> results for the individual fit parameters,</span>
<span class="sd"> also accessible via indices.</span>
<span class="sd"> &quot;&quot;&quot;</span>
<span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
<span class="bp">self</span><span class="o">.</span><span class="n">fit_parameters</span> <span class="o">=</span> <span class="kc">None</span>
<span class="k">def</span> <span class="fm">__getitem__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">idx</span><span class="p">):</span>
<span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">fit_parameters</span><span class="p">[</span><span class="n">idx</span><span class="p">]</span>
<span class="k">def</span> <span class="fm">__len__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
<span class="k">return</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">fit_parameters</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">gamma_method</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
<span class="sd">&quot;&quot;&quot;Apply the gamma method to all fit parameters&quot;&quot;&quot;</span>
<span class="p">[</span><span class="n">o</span><span class="o">.</span><span class="n">gamma_method</span><span class="p">()</span> <span class="k">for</span> <span class="n">o</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">fit_parameters</span><span class="p">]</span>
<span class="k">def</span> <span class="fm">__str__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
<span class="bp">self</span><span class="o">.</span><span class="n">gamma_method</span><span class="p">()</span>
<span class="n">my_str</span> <span class="o">=</span> <span class="s1">&#39;Goodness of fit:</span><span class="se">\n</span><span class="s1">&#39;</span>
<span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="s1">&#39;chisquare_by_dof&#39;</span><span class="p">):</span>
<span class="n">my_str</span> <span class="o">+=</span> <span class="s1">&#39;</span><span class="se">\u03C7\u00b2</span><span class="s1">/d.o.f. = &#39;</span> <span class="o">+</span> <span class="sa">f</span><span class="s1">&#39;</span><span class="si">{</span><span class="bp">self</span><span class="o">.</span><span class="n">chisquare_by_dof</span><span class="si">:</span><span class="s1">2.6f</span><span class="si">}</span><span class="s1">&#39;</span> <span class="o">+</span> <span class="s1">&#39;</span><span class="se">\n</span><span class="s1">&#39;</span>
<span class="k">elif</span> <span class="nb">hasattr</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="s1">&#39;residual_variance&#39;</span><span class="p">):</span>
<span class="n">my_str</span> <span class="o">+=</span> <span class="s1">&#39;residual variance = &#39;</span> <span class="o">+</span> <span class="sa">f</span><span class="s1">&#39;</span><span class="si">{</span><span class="bp">self</span><span class="o">.</span><span class="n">residual_variance</span><span class="si">:</span><span class="s1">2.6f</span><span class="si">}</span><span class="s1">&#39;</span> <span class="o">+</span> <span class="s1">&#39;</span><span class="se">\n</span><span class="s1">&#39;</span>
<span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="s1">&#39;chisquare_by_expected_chisquare&#39;</span><span class="p">):</span>
<span class="n">my_str</span> <span class="o">+=</span> <span class="s1">&#39;</span><span class="se">\u03C7\u00b2</span><span class="s1">/</span><span class="se">\u03C7\u00b2</span><span class="s1">exp = &#39;</span> <span class="o">+</span> <span class="sa">f</span><span class="s1">&#39;</span><span class="si">{</span><span class="bp">self</span><span class="o">.</span><span class="n">chisquare_by_expected_chisquare</span><span class="si">:</span><span class="s1">2.6f</span><span class="si">}</span><span class="s1">&#39;</span> <span class="o">+</span> <span class="s1">&#39;</span><span class="se">\n</span><span class="s1">&#39;</span>
<span class="n">my_str</span> <span class="o">+=</span> <span class="s1">&#39;Fit parameters:</span><span class="se">\n</span><span class="s1">&#39;</span>
<span class="k">for</span> <span class="n">i_par</span><span class="p">,</span> <span class="n">par</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">fit_parameters</span><span class="p">):</span>
<span class="n">my_str</span> <span class="o">+=</span> <span class="nb">str</span><span class="p">(</span><span class="n">i_par</span><span class="p">)</span> <span class="o">+</span> <span class="s1">&#39;</span><span class="se">\t</span><span class="s1">&#39;</span> <span class="o">+</span> <span class="s1">&#39; &#39;</span> <span class="o">*</span> <span class="nb">int</span><span class="p">(</span><span class="n">par</span> <span class="o">&gt;=</span> <span class="mi">0</span><span class="p">)</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">par</span><span class="p">)</span><span class="o">.</span><span class="n">rjust</span><span class="p">(</span><span class="nb">int</span><span class="p">(</span><span class="n">par</span> <span class="o">&lt;</span> <span class="mf">0.0</span><span class="p">))</span> <span class="o">+</span> <span class="s1">&#39;</span><span class="se">\n</span><span class="s1">&#39;</span>
<span class="k">return</span> <span class="n">my_str</span>
<span class="k">def</span> <span class="fm">__repr__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
<span class="n">m</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="nb">len</span><span class="p">,</span> <span class="nb">list</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="vm">__dict__</span><span class="o">.</span><span class="n">keys</span><span class="p">())))</span> <span class="o">+</span> <span class="mi">1</span>
<span class="k">return</span> <span class="s1">&#39;</span><span class="se">\n</span><span class="s1">&#39;</span><span class="o">.</span><span class="n">join</span><span class="p">([</span><span class="n">key</span><span class="o">.</span><span class="n">rjust</span><span class="p">(</span><span class="n">m</span><span class="p">)</span> <span class="o">+</span> <span class="s1">&#39;: &#39;</span> <span class="o">+</span> <span class="nb">repr</span><span class="p">(</span><span class="n">value</span><span class="p">)</span> <span class="k">for</span> <span class="n">key</span><span class="p">,</span> <span class="n">value</span> <span class="ow">in</span> <span class="nb">sorted</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="vm">__dict__</span><span class="o">.</span><span class="n">items</span><span class="p">())])</span>
<span class="k">def</span> <span class="nf">least_squares</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="n">priors</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">silent</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="sa">r</span><span class="sd">&#39;&#39;&#39;Performs a non-linear fit to y = func(x).</span>
<span class="sd"> Parameters</span>
<span class="sd"> ----------</span>
<span class="sd"> x : list</span>
<span class="sd"> list of floats.</span>
<span class="sd"> y : list</span>
<span class="sd"> list of Obs.</span>
<span class="sd"> func : object</span>
<span class="sd"> fit function, has to be of the form</span>
<span class="sd"> ```python</span>
<span class="sd"> def func(a, x):</span>
<span class="sd"> y = a[0] + a[1] * x + a[2] * anp.sinh(x)</span>
<span class="sd"> return y</span>
<span class="sd"> ```</span>
<span class="sd"> For multiple x values func can be of the form</span>
<span class="sd"> ```python</span>
<span class="sd"> def func(a, x):</span>
<span class="sd"> (x1, x2) = x</span>
<span class="sd"> return a[0] * x1 ** 2 + a[1] * x2</span>
<span class="sd"> ```</span>
<span class="sd"> It is important that all numpy functions refer to autograd.numpy, otherwise the differentiation</span>
<span class="sd"> will not work</span>
<span class="sd"> priors : list, optional</span>
<span class="sd"> priors has to be a list with an entry for every parameter in the fit. The entries can either be</span>
<span class="sd"> Obs (e.g. results from a previous fit) or strings containing a value and an error formatted like</span>
<span class="sd"> 0.548(23), 500(40) or 0.5(0.4)</span>
<span class="sd"> silent : bool, optional</span>
<span class="sd"> If true all output to the console is omitted (default False).</span>
<span class="sd"> initial_guess : list</span>
<span class="sd"> can provide an initial guess for the input parameters. Relevant for</span>
<span class="sd"> non-linear fits with many parameters.</span>
<span class="sd"> method : str</span>
<span class="sd"> can be used to choose an alternative method for the minimization of chisquare.</span>
<span class="sd"> The possible methods are the ones which can be used for scipy.optimize.minimize and</span>
<span class="sd"> migrad of iminuit. If no method is specified, Levenberg-Marquard is used.</span>
<span class="sd"> Reliable alternatives are migrad, Powell and Nelder-Mead.</span>
<span class="sd"> resplot : bool</span>
<span class="sd"> If true, a plot which displays fit, data and residuals is generated (default False).</span>
<span class="sd"> qqplot : bool</span>
<span class="sd"> If true, a quantile-quantile plot of the fit result is generated (default False).</span>
<span class="sd"> expected_chisquare : bool</span>
<span class="sd"> If true prints the expected chisquare which is</span>
<span class="sd"> corrected by effects caused by correlated input data.</span>
<span class="sd"> This can take a while as the full correlation matrix</span>
<span class="sd"> has to be calculated (default False).</span>
<span class="sd"> correlated_fit : bool</span>
<span class="sd"> If true, use the full correlation matrix in the definition of the chisquare</span>
<span class="sd"> (only works for prior==None and when no method is given, at the moment).</span>
<span class="sd"> const_par : list, optional</span>
<span class="sd"> List of N Obs that are used to constrain the last N fit parameters of func.</span>
<span class="sd"> &#39;&#39;&#39;</span>
<span class="k">if</span> <span class="n">priors</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span><span class="p">:</span>
<span class="k">return</span> <span class="n">_prior_fit</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="n">priors</span><span class="p">,</span> <span class="n">silent</span><span class="o">=</span><span class="n">silent</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
<span class="k">else</span><span class="p">:</span>
<span class="k">return</span> <span class="n">_standard_fit</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="n">silent</span><span class="o">=</span><span class="n">silent</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">total_least_squares</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="n">silent</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="sa">r</span><span class="sd">&#39;&#39;&#39;Performs a non-linear fit to y = func(x) and returns a list of Obs corresponding to the fit parameters.</span>
<span class="sd"> Parameters</span>
<span class="sd"> ----------</span>
<span class="sd"> x : list</span>
<span class="sd"> list of Obs, or a tuple of lists of Obs</span>
<span class="sd"> y : list</span>
<span class="sd"> list of Obs. The dvalues of the Obs are used as x- and yerror for the fit.</span>
<span class="sd"> func : object</span>
<span class="sd"> func has to be of the form</span>
<span class="sd"> ```python</span>
<span class="sd"> def func(a, x):</span>
<span class="sd"> y = a[0] + a[1] * x + a[2] * anp.sinh(x)</span>
<span class="sd"> return y</span>
<span class="sd"> ```</span>
<span class="sd"> For multiple x values func can be of the form</span>
<span class="sd"> ```python</span>
<span class="sd"> def func(a, x):</span>
<span class="sd"> (x1, x2) = x</span>
<span class="sd"> return a[0] * x1 ** 2 + a[1] * x2</span>
<span class="sd"> ```</span>
<span class="sd"> It is important that all numpy functions refer to autograd.numpy, otherwise the differentiation</span>
<span class="sd"> will not work.</span>
<span class="sd"> silent : bool, optional</span>
<span class="sd"> If true all output to the console is omitted (default False).</span>
<span class="sd"> initial_guess : list</span>
<span class="sd"> can provide an initial guess for the input parameters. Relevant for non-linear</span>
<span class="sd"> fits with many parameters.</span>
<span class="sd"> expected_chisquare : bool</span>
<span class="sd"> If true prints the expected chisquare which is</span>
<span class="sd"> corrected by effects caused by correlated input data.</span>
<span class="sd"> This can take a while as the full correlation matrix</span>
<span class="sd"> has to be calculated (default False).</span>
<span class="sd"> const_par : list, optional</span>
<span class="sd"> List of N Obs that are used to constrain the last N fit parameters of func.</span>
<span class="sd"> Based on the orthogonal distance regression module of scipy</span>
<span class="sd"> &#39;&#39;&#39;</span>
<span class="n">output</span> <span class="o">=</span> <span class="n">Fit_result</span><span class="p">()</span>
<span class="n">output</span><span class="o">.</span><span class="n">fit_function</span> <span class="o">=</span> <span class="n">func</span>
<span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
<span class="n">x_shape</span> <span class="o">=</span> <span class="n">x</span><span class="o">.</span><span class="n">shape</span>
<span class="k">if</span> <span class="ow">not</span> <span class="n">callable</span><span class="p">(</span><span class="n">func</span><span class="p">):</span>
<span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s1">&#39;func has to be a function.&#39;</span><span class="p">)</span>
<span class="n">func_aug</span> <span class="o">=</span> <span class="n">func</span>
<span class="k">if</span> <span class="s1">&#39;const_par&#39;</span> <span class="ow">in</span> <span class="n">kwargs</span><span class="p">:</span>
<span class="n">const_par</span> <span class="o">=</span> <span class="n">kwargs</span><span class="p">[</span><span class="s1">&#39;const_par&#39;</span><span class="p">]</span>
<span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">const_par</span><span class="p">,</span> <span class="n">Obs</span><span class="p">):</span>
<span class="n">const_par</span> <span class="o">=</span> <span class="p">[</span><span class="n">const_par</span><span class="p">]</span>
<span class="k">def</span> <span class="nf">func</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
<span class="k">return</span> <span class="n">func_aug</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">p</span><span class="p">,</span> <span class="p">[</span><span class="n">o</span><span class="o">.</span><span class="n">value</span> <span class="k">for</span> <span class="n">o</span> <span class="ow">in</span> <span class="n">const_par</span><span class="p">])),</span> <span class="n">x</span><span class="p">)</span>
<span class="k">else</span><span class="p">:</span>
<span class="n">const_par</span> <span class="o">=</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="mi">25</span><span class="p">):</span>
<span class="k">try</span><span class="p">:</span>
<span class="n">func</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">i</span><span class="p">),</span> <span class="n">x</span><span class="o">.</span><span class="n">T</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
<span class="k">except</span> <span class="ne">Exception</span><span class="p">:</span>
<span class="k">pass</span>
<span class="k">else</span><span class="p">:</span>
<span class="k">break</span>
<span class="n">n_parms</span> <span class="o">=</span> <span class="n">i</span>
<span class="k">if</span> <span class="ow">not</span> <span class="n">silent</span><span class="p">:</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;Fit with&#39;</span><span class="p">,</span> <span class="n">n_parms</span><span class="p">,</span> <span class="s1">&#39;parameters&#39;</span><span class="p">)</span>
<span class="k">if</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">const_par</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">):</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;</span><span class="se">\t</span><span class="s1"> and </span><span class="si">%d</span><span class="s1"> constrained parameter</span><span class="si">%s</span><span class="s1">&#39;</span> <span class="o">%</span> <span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">const_par</span><span class="p">),</span> <span class="s1">&#39;s&#39;</span> <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">const_par</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">1</span> <span class="k">else</span> <span class="s1">&#39;&#39;</span><span class="p">),</span> <span class="n">const_par</span><span class="p">)</span>
<span class="n">x_f</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">vectorize</span><span class="p">(</span><span class="k">lambda</span> <span class="n">o</span><span class="p">:</span> <span class="n">o</span><span class="o">.</span><span class="n">value</span><span class="p">)(</span><span class="n">x</span><span class="p">)</span>
<span class="n">dx_f</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">vectorize</span><span class="p">(</span><span class="k">lambda</span> <span class="n">o</span><span class="p">:</span> <span class="n">o</span><span class="o">.</span><span class="n">dvalue</span><span class="p">)(</span><span class="n">x</span><span class="p">)</span>
<span class="n">y_f</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="n">o</span><span class="o">.</span><span class="n">value</span> <span class="k">for</span> <span class="n">o</span> <span class="ow">in</span> <span class="n">y</span><span class="p">])</span>
<span class="n">dy_f</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</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">y</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">asarray</span><span class="p">(</span><span class="n">dx_f</span><span class="p">)</span> <span class="o">&lt;=</span> <span class="mf">0.0</span><span class="p">):</span>
<span class="k">raise</span> <span class="ne">Exception</span><span class="p">(</span><span class="s1">&#39;No x errors available, run the gamma method first.&#39;</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">asarray</span><span class="p">(</span><span class="n">dy_f</span><span class="p">)</span> <span class="o">&lt;=</span> <span class="mf">0.0</span><span class="p">):</span>
<span class="k">raise</span> <span class="ne">Exception</span><span class="p">(</span><span class="s1">&#39;No y errors available, run the gamma method first.&#39;</span><span class="p">)</span>
<span class="k">if</span> <span class="s1">&#39;initial_guess&#39;</span> <span class="ow">in</span> <span class="n">kwargs</span><span class="p">:</span>
<span class="n">x0</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;initial_guess&#39;</span><span class="p">)</span>
<span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">x0</span><span class="p">)</span> <span class="o">!=</span> <span class="n">n_parms</span><span class="p">:</span>
<span class="k">raise</span> <span class="ne">Exception</span><span class="p">(</span><span class="s1">&#39;Initial guess does not have the correct length: </span><span class="si">%d</span><span class="s1"> vs. </span><span class="si">%d</span><span class="s1">&#39;</span> <span class="o">%</span> <span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">x0</span><span class="p">),</span> <span class="n">n_parms</span><span class="p">))</span>
<span class="k">else</span><span class="p">:</span>
<span class="n">x0</span> <span class="o">=</span> <span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">*</span> <span class="n">n_parms</span>
<span class="n">data</span> <span class="o">=</span> <span class="n">RealData</span><span class="p">(</span><span class="n">x_f</span><span class="p">,</span> <span class="n">y_f</span><span class="p">,</span> <span class="n">sx</span><span class="o">=</span><span class="n">dx_f</span><span class="p">,</span> <span class="n">sy</span><span class="o">=</span><span class="n">dy_f</span><span class="p">)</span>
<span class="n">model</span> <span class="o">=</span> <span class="n">Model</span><span class="p">(</span><span class="n">func</span><span class="p">)</span>
<span class="n">odr</span> <span class="o">=</span> <span class="n">ODR</span><span class="p">(</span><span class="n">data</span><span class="p">,</span> <span class="n">model</span><span class="p">,</span> <span class="n">x0</span><span class="p">,</span> <span class="n">partol</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">odr</span><span class="o">.</span><span class="n">set_job</span><span class="p">(</span><span class="n">fit_type</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="n">deriv</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
<span class="n">out</span> <span class="o">=</span> <span class="n">odr</span><span class="o">.</span><span class="n">run</span><span class="p">()</span>
<span class="n">output</span><span class="o">.</span><span class="n">residual_variance</span> <span class="o">=</span> <span class="n">out</span><span class="o">.</span><span class="n">res_var</span>
<span class="n">output</span><span class="o">.</span><span class="n">method</span> <span class="o">=</span> <span class="s1">&#39;ODR&#39;</span>
<span class="n">output</span><span class="o">.</span><span class="n">message</span> <span class="o">=</span> <span class="n">out</span><span class="o">.</span><span class="n">stopreason</span>
<span class="n">output</span><span class="o">.</span><span class="n">xplus</span> <span class="o">=</span> <span class="n">out</span><span class="o">.</span><span class="n">xplus</span>
<span class="k">if</span> <span class="ow">not</span> <span class="n">silent</span><span class="p">:</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;Method: ODR&#39;</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="o">*</span><span class="n">out</span><span class="o">.</span><span class="n">stopreason</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;Residual variance:&#39;</span><span class="p">,</span> <span class="n">output</span><span class="o">.</span><span class="n">residual_variance</span><span class="p">)</span>
<span class="k">if</span> <span class="n">out</span><span class="o">.</span><span class="n">info</span> <span class="o">&gt;</span> <span class="mi">3</span><span class="p">:</span>
<span class="k">raise</span> <span class="ne">Exception</span><span class="p">(</span><span class="s1">&#39;The minimization procedure did not converge.&#39;</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">size</span>
<span class="n">n_parms_aug</span> <span class="o">=</span> <span class="n">n_parms</span> <span class="o">+</span> <span class="nb">len</span><span class="p">(</span><span class="n">const_par</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">odr_chisquare</span><span class="p">(</span><span class="n">p</span><span class="p">):</span>
<span class="n">model</span> <span class="o">=</span> <span class="n">func</span><span class="p">(</span><span class="n">p</span><span class="p">[:</span><span class="n">n_parms</span><span class="p">],</span> <span class="n">p</span><span class="p">[</span><span class="n">n_parms</span><span class="p">:]</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">x_shape</span><span class="p">))</span>
<span class="n">chisq</span> <span class="o">=</span> <span class="n">anp</span><span class="o">.</span><span class="n">sum</span><span class="p">(((</span><span class="n">y_f</span> <span class="o">-</span> <span class="n">model</span><span class="p">)</span> <span class="o">/</span> <span class="n">dy_f</span><span class="p">)</span> <span class="o">**</span> <span class="mi">2</span><span class="p">)</span> <span class="o">+</span> <span class="n">anp</span><span class="o">.</span><span class="n">sum</span><span class="p">(((</span><span class="n">x_f</span> <span class="o">-</span> <span class="n">p</span><span class="p">[</span><span class="n">n_parms</span><span class="p">:]</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">x_shape</span><span class="p">))</span> <span class="o">/</span> <span class="n">dx_f</span><span class="p">)</span> <span class="o">**</span> <span class="mi">2</span><span class="p">)</span>
<span class="k">return</span> <span class="n">chisq</span>
<span class="k">def</span> <span class="nf">odr_chisquare_aug</span><span class="p">(</span><span class="n">p</span><span class="p">):</span>
<span class="n">model</span> <span class="o">=</span> <span class="n">func_aug</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">p</span><span class="p">[:</span><span class="n">n_parms_aug</span><span class="p">],</span> <span class="p">[</span><span class="n">o</span><span class="o">.</span><span class="n">value</span> <span class="k">for</span> <span class="n">o</span> <span class="ow">in</span> <span class="n">const_par</span><span class="p">])),</span> <span class="n">p</span><span class="p">[</span><span class="n">n_parms_aug</span><span class="p">:]</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">x_shape</span><span class="p">))</span>
<span class="n">chisq</span> <span class="o">=</span> <span class="n">anp</span><span class="o">.</span><span class="n">sum</span><span class="p">(((</span><span class="n">y_f</span> <span class="o">-</span> <span class="n">model</span><span class="p">)</span> <span class="o">/</span> <span class="n">dy_f</span><span class="p">)</span> <span class="o">**</span> <span class="mi">2</span><span class="p">)</span> <span class="o">+</span> <span class="n">anp</span><span class="o">.</span><span class="n">sum</span><span class="p">(((</span><span class="n">x_f</span> <span class="o">-</span> <span class="n">p</span><span class="p">[</span><span class="n">n_parms_aug</span><span class="p">:]</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">x_shape</span><span class="p">))</span> <span class="o">/</span> <span class="n">dx_f</span><span class="p">)</span> <span class="o">**</span> <span class="mi">2</span><span class="p">)</span>
<span class="k">return</span> <span class="n">chisq</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;expected_chisquare&#39;</span><span class="p">)</span> <span class="ow">is</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">np</span><span class="o">.</span><span class="n">concatenate</span><span class="p">((</span><span class="n">dy_f</span><span class="o">.</span><span class="n">ravel</span><span class="p">(),</span> <span class="n">dx_f</span><span class="o">.</span><span class="n">ravel</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;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">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>
<span class="n">old_jac</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">out</span><span class="o">.</span><span class="n">beta</span><span class="p">,</span> <span class="n">out</span><span class="o">.</span><span class="n">xplus</span><span class="p">)</span>
<span class="n">fused_row1</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">concatenate</span><span class="p">((</span><span class="n">old_jac</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">number_of_x_parameters</span> <span class="o">*</span> <span class="p">[</span><span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">old_jac</span><span class="o">.</span><span class="n">shape</span><span class="p">)]),</span> <span class="n">axis</span><span class="o">=</span><span class="mi">0</span><span class="p">)))</span>
<span class="n">fused_row2</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">concatenate</span><span class="p">((</span><span class="n">jacobian</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">:</span> <span class="n">func</span><span class="p">(</span><span class="n">y</span><span class="p">,</span> <span class="n">x</span><span class="p">))(</span><span class="n">out</span><span class="o">.</span><span class="n">xplus</span><span class="p">,</span> <span class="n">out</span><span class="o">.</span><span class="n">beta</span><span class="p">)</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</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> <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> <span class="o">*</span> <span class="n">number_of_x_parameters</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">number_of_x_parameters</span> <span class="o">*</span> <span class="n">old_jac</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">])))</span>
<span class="n">new_jac</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">concatenate</span><span class="p">((</span><span class="n">fused_row1</span><span class="p">,</span> <span class="n">fused_row2</span><span class="p">),</span> <span class="n">axis</span><span class="o">=</span><span class="mi">1</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">new_jac</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">P_phi</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</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>
<span class="k">if</span> <span class="n">expected_chisquare</span> <span class="o">&lt;=</span> <span class="mf">0.0</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;Negative expected_chisquare.&quot;</span><span class="p">,</span> <span class="ne">RuntimeWarning</span><span class="p">)</span>
<span class="n">expected_chisquare</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">expected_chisquare</span><span class="p">)</span>
<span class="n">output</span><span class="o">.</span><span class="n">chisquare_by_expected_chisquare</span> <span class="o">=</span> <span class="n">odr_chisquare</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">out</span><span class="o">.</span><span class="n">beta</span><span class="p">,</span> <span class="n">out</span><span class="o">.</span><span class="n">xplus</span><span class="o">.</span><span class="n">ravel</span><span class="p">())))</span> <span class="o">/</span> <span class="n">expected_chisquare</span>
<span class="k">if</span> <span class="ow">not</span> <span class="n">silent</span><span class="p">:</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;chisquare/expected_chisquare:&#39;</span><span class="p">,</span>
<span class="n">output</span><span class="o">.</span><span class="n">chisquare_by_expected_chisquare</span><span class="p">)</span>
<span class="n">fitp</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">concatenate</span><span class="p">((</span><span class="n">out</span><span class="o">.</span><span class="n">beta</span><span class="p">,</span> <span class="p">[</span><span class="n">o</span><span class="o">.</span><span class="n">value</span> <span class="k">for</span> <span class="n">o</span> <span class="ow">in</span> <span class="n">const_par</span><span class="p">]))</span>
<span class="n">hess_inv</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">pinv</span><span class="p">(</span><span class="n">jacobian</span><span class="p">(</span><span class="n">jacobian</span><span class="p">(</span><span class="n">odr_chisquare_aug</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">fitp</span><span class="p">,</span> <span class="n">out</span><span class="o">.</span><span class="n">xplus</span><span class="o">.</span><span class="n">ravel</span><span class="p">()))))</span>
<span class="k">def</span> <span class="nf">odr_chisquare_compact_x</span><span class="p">(</span><span class="n">d</span><span class="p">):</span>
<span class="n">model</span> <span class="o">=</span> <span class="n">func_aug</span><span class="p">(</span><span class="n">d</span><span class="p">[:</span><span class="n">n_parms_aug</span><span class="p">],</span> <span class="n">d</span><span class="p">[</span><span class="n">n_parms_aug</span><span class="p">:</span><span class="n">n_parms_aug</span> <span class="o">+</span> <span class="n">m</span><span class="p">]</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">x_shape</span><span class="p">))</span>
<span class="n">chisq</span> <span class="o">=</span> <span class="n">anp</span><span class="o">.</span><span class="n">sum</span><span class="p">(((</span><span class="n">y_f</span> <span class="o">-</span> <span class="n">model</span><span class="p">)</span> <span class="o">/</span> <span class="n">dy_f</span><span class="p">)</span> <span class="o">**</span> <span class="mi">2</span><span class="p">)</span> <span class="o">+</span> <span class="n">anp</span><span class="o">.</span><span class="n">sum</span><span class="p">(((</span><span class="n">d</span><span class="p">[</span><span class="n">n_parms_aug</span> <span class="o">+</span> <span class="n">m</span><span class="p">:]</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">x_shape</span><span class="p">)</span> <span class="o">-</span> <span class="n">d</span><span class="p">[</span><span class="n">n_parms_aug</span><span class="p">:</span><span class="n">n_parms_aug</span> <span class="o">+</span> <span class="n">m</span><span class="p">]</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">x_shape</span><span class="p">))</span> <span class="o">/</span> <span class="n">dx_f</span><span class="p">)</span> <span class="o">**</span> <span class="mi">2</span><span class="p">)</span>
<span class="k">return</span> <span class="n">chisq</span>
<span class="n">jac_jac_x</span> <span class="o">=</span> <span class="n">jacobian</span><span class="p">(</span><span class="n">jacobian</span><span class="p">(</span><span class="n">odr_chisquare_compact_x</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">fitp</span><span class="p">,</span> <span class="n">out</span><span class="o">.</span><span class="n">xplus</span><span class="o">.</span><span class="n">ravel</span><span class="p">(),</span> <span class="n">x_f</span><span class="o">.</span><span class="n">ravel</span><span class="p">())))</span>
<span class="n">deriv_x</span> <span class="o">=</span> <span class="o">-</span><span class="n">hess_inv</span> <span class="o">@</span> <span class="n">jac_jac_x</span><span class="p">[:</span><span class="n">n_parms_aug</span> <span class="o">+</span> <span class="n">m</span><span class="p">,</span> <span class="n">n_parms_aug</span> <span class="o">+</span> <span class="n">m</span><span class="p">:]</span>
<span class="k">def</span> <span class="nf">odr_chisquare_compact_y</span><span class="p">(</span><span class="n">d</span><span class="p">):</span>
<span class="n">model</span> <span class="o">=</span> <span class="n">func_aug</span><span class="p">(</span><span class="n">d</span><span class="p">[:</span><span class="n">n_parms_aug</span><span class="p">],</span> <span class="n">d</span><span class="p">[</span><span class="n">n_parms_aug</span><span class="p">:</span><span class="n">n_parms_aug</span> <span class="o">+</span> <span class="n">m</span><span class="p">]</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">x_shape</span><span class="p">))</span>
<span class="n">chisq</span> <span class="o">=</span> <span class="n">anp</span><span class="o">.</span><span class="n">sum</span><span class="p">(((</span><span class="n">d</span><span class="p">[</span><span class="n">n_parms_aug</span> <span class="o">+</span> <span class="n">m</span><span class="p">:]</span> <span class="o">-</span> <span class="n">model</span><span class="p">)</span> <span class="o">/</span> <span class="n">dy_f</span><span class="p">)</span> <span class="o">**</span> <span class="mi">2</span><span class="p">)</span> <span class="o">+</span> <span class="n">anp</span><span class="o">.</span><span class="n">sum</span><span class="p">(((</span><span class="n">x_f</span> <span class="o">-</span> <span class="n">d</span><span class="p">[</span><span class="n">n_parms_aug</span><span class="p">:</span><span class="n">n_parms_aug</span> <span class="o">+</span> <span class="n">m</span><span class="p">]</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">x_shape</span><span class="p">))</span> <span class="o">/</span> <span class="n">dx_f</span><span class="p">)</span> <span class="o">**</span> <span class="mi">2</span><span class="p">)</span>
<span class="k">return</span> <span class="n">chisq</span>
<span class="n">jac_jac_y</span> <span class="o">=</span> <span class="n">jacobian</span><span class="p">(</span><span class="n">jacobian</span><span class="p">(</span><span class="n">odr_chisquare_compact_y</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">fitp</span><span class="p">,</span> <span class="n">out</span><span class="o">.</span><span class="n">xplus</span><span class="o">.</span><span class="n">ravel</span><span class="p">(),</span> <span class="n">y_f</span><span class="p">)))</span>
<span class="n">deriv_y</span> <span class="o">=</span> <span class="o">-</span><span class="n">hess_inv</span> <span class="o">@</span> <span class="n">jac_jac_y</span><span class="p">[:</span><span class="n">n_parms_aug</span> <span class="o">+</span> <span class="n">m</span><span class="p">,</span> <span class="n">n_parms_aug</span> <span class="o">+</span> <span class="n">m</span><span class="p">:]</span>
<span class="n">result</span> <span class="o">=</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">n_parms</span><span class="p">):</span>
<span class="n">result</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">derived_observable</span><span class="p">(</span><span class="k">lambda</span> <span class="n">my_var</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">:</span> <span class="p">(</span><span class="n">my_var</span><span class="p">[</span><span class="mi">0</span><span class="p">]</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="o">/</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="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">value</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="o">*</span> <span class="n">out</span><span class="o">.</span><span class="n">beta</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="nb">list</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="o">+</span> <span class="nb">list</span><span class="p">(</span><span class="n">y</span><span class="p">),</span> <span class="n">man_grad</span><span class="o">=</span><span class="nb">list</span><span class="p">(</span><span class="n">deriv_x</span><span class="p">[</span><span class="n">i</span><span class="p">])</span> <span class="o">+</span> <span class="nb">list</span><span class="p">(</span><span class="n">deriv_y</span><span class="p">[</span><span class="n">i</span><span class="p">])))</span>
<span class="n">output</span><span class="o">.</span><span class="n">fit_parameters</span> <span class="o">=</span> <span class="n">result</span> <span class="o">+</span> <span class="n">const_par</span>
<span class="n">output</span><span class="o">.</span><span class="n">odr_chisquare</span> <span class="o">=</span> <span class="n">odr_chisquare</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">out</span><span class="o">.</span><span class="n">beta</span><span class="p">,</span> <span class="n">out</span><span class="o">.</span><span class="n">xplus</span><span class="o">.</span><span class="n">ravel</span><span class="p">())))</span>
<span class="n">output</span><span class="o">.</span><span class="n">dof</span> <span class="o">=</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">n_parms</span>
<span class="k">return</span> <span class="n">output</span>
<span class="k">def</span> <span class="nf">prior_fit</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="n">priors</span><span class="p">,</span> <span class="n">silent</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="n">warnings</span><span class="o">.</span><span class="n">warn</span><span class="p">(</span><span class="s2">&quot;prior_fit renamed to least_squares&quot;</span><span class="p">,</span> <span class="ne">DeprecationWarning</span><span class="p">)</span>
<span class="k">return</span> <span class="n">least_squares</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="n">priors</span><span class="o">=</span><span class="n">priors</span><span class="p">,</span> <span class="n">silent</span><span class="o">=</span><span class="n">silent</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">_prior_fit</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="n">priors</span><span class="p">,</span> <span class="n">silent</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="n">output</span> <span class="o">=</span> <span class="n">Fit_result</span><span class="p">()</span>
<span class="n">output</span><span class="o">.</span><span class="n">fit_function</span> <span class="o">=</span> <span class="n">func</span>
<span class="n">x</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">x</span><span class="p">)</span>
<span class="k">if</span> <span class="ow">not</span> <span class="n">callable</span><span class="p">(</span><span class="n">func</span><span class="p">):</span>
<span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s1">&#39;func has to be a function.&#39;</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="mi">100</span><span class="p">):</span>
<span class="k">try</span><span class="p">:</span>
<span class="n">func</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">i</span><span class="p">),</span> <span class="mi">0</span><span class="p">)</span>
<span class="k">except</span> <span class="ne">Exception</span><span class="p">:</span>
<span class="k">pass</span>
<span class="k">else</span><span class="p">:</span>
<span class="k">break</span>
<span class="n">n_parms</span> <span class="o">=</span> <span class="n">i</span>
<span class="k">if</span> <span class="n">n_parms</span> <span class="o">!=</span> <span class="nb">len</span><span class="p">(</span><span class="n">priors</span><span class="p">):</span>
<span class="k">raise</span> <span class="ne">Exception</span><span class="p">(</span><span class="s1">&#39;Priors does not have the correct length.&#39;</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">extract_val_and_dval</span><span class="p">(</span><span class="n">string</span><span class="p">):</span>
<span class="n">split_string</span> <span class="o">=</span> <span class="n">string</span><span class="o">.</span><span class="n">split</span><span class="p">(</span><span class="s1">&#39;(&#39;</span><span class="p">)</span>
<span class="k">if</span> <span class="s1">&#39;.&#39;</span> <span class="ow">in</span> <span class="n">split_string</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="ow">and</span> <span class="s1">&#39;.&#39;</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">split_string</span><span class="p">[</span><span class="mi">1</span><span class="p">][:</span><span class="o">-</span><span class="mi">1</span><span class="p">]:</span>
<span class="n">factor</span> <span class="o">=</span> <span class="mi">10</span> <span class="o">**</span> <span class="o">-</span><span class="nb">len</span><span class="p">(</span><span class="n">split_string</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">partition</span><span class="p">(</span><span class="s1">&#39;.&#39;</span><span class="p">)[</span><span class="mi">2</span><span class="p">])</span>
<span class="k">else</span><span class="p">:</span>
<span class="n">factor</span> <span class="o">=</span> <span class="mi">1</span>
<span class="k">return</span> <span class="nb">float</span><span class="p">(</span><span class="n">split_string</span><span class="p">[</span><span class="mi">0</span><span class="p">]),</span> <span class="nb">float</span><span class="p">(</span><span class="n">split_string</span><span class="p">[</span><span class="mi">1</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">factor</span>
<span class="n">loc_priors</span> <span class="o">=</span> <span class="p">[]</span>
<span class="k">for</span> <span class="n">i_n</span><span class="p">,</span> <span class="n">i_prior</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">priors</span><span class="p">):</span>
<span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">i_prior</span><span class="p">,</span> <span class="n">Obs</span><span class="p">):</span>
<span class="n">loc_priors</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">i_prior</span><span class="p">)</span>
<span class="k">else</span><span class="p">:</span>
<span class="n">loc_val</span><span class="p">,</span> <span class="n">loc_dval</span> <span class="o">=</span> <span class="n">extract_val_and_dval</span><span class="p">(</span><span class="n">i_prior</span><span class="p">)</span>
<span class="n">loc_priors</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">cov_Obs</span><span class="p">(</span><span class="n">loc_val</span><span class="p">,</span> <span class="n">loc_dval</span> <span class="o">**</span> <span class="mi">2</span><span class="p">,</span> <span class="s1">&#39;#prior&#39;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">i_n</span><span class="p">)</span> <span class="o">+</span> <span class="sa">f</span><span class="s2">&quot;_</span><span class="si">{</span><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">randint</span><span class="p">(</span><span class="mi">2147483647</span><span class="p">)</span><span class="si">:</span><span class="s2">010d</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">))</span>
<span class="n">output</span><span class="o">.</span><span class="n">priors</span> <span class="o">=</span> <span class="n">loc_priors</span>
<span class="k">if</span> <span class="ow">not</span> <span class="n">silent</span><span class="p">:</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;Fit with&#39;</span><span class="p">,</span> <span class="n">n_parms</span><span class="p">,</span> <span class="s1">&#39;parameters&#39;</span><span class="p">)</span>
<span class="n">y_f</span> <span class="o">=</span> <span class="p">[</span><span class="n">o</span><span class="o">.</span><span class="n">value</span> <span class="k">for</span> <span class="n">o</span> <span class="ow">in</span> <span class="n">y</span><span class="p">]</span>
<span class="n">dy_f</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">y</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">asarray</span><span class="p">(</span><span class="n">dy_f</span><span class="p">)</span> <span class="o">&lt;=</span> <span class="mf">0.0</span><span class="p">):</span>
<span class="k">raise</span> <span class="ne">Exception</span><span class="p">(</span><span class="s1">&#39;No y errors available, run the gamma method first.&#39;</span><span class="p">)</span>
<span class="n">p_f</span> <span class="o">=</span> <span class="p">[</span><span class="n">o</span><span class="o">.</span><span class="n">value</span> <span class="k">for</span> <span class="n">o</span> <span class="ow">in</span> <span class="n">loc_priors</span><span class="p">]</span>
<span class="n">dp_f</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">loc_priors</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">asarray</span><span class="p">(</span><span class="n">dp_f</span><span class="p">)</span> <span class="o">&lt;=</span> <span class="mf">0.0</span><span class="p">):</span>
<span class="k">raise</span> <span class="ne">Exception</span><span class="p">(</span><span class="s1">&#39;No prior errors available, run the gamma method first.&#39;</span><span class="p">)</span>
<span class="k">if</span> <span class="s1">&#39;initial_guess&#39;</span> <span class="ow">in</span> <span class="n">kwargs</span><span class="p">:</span>
<span class="n">x0</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;initial_guess&#39;</span><span class="p">)</span>
<span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">x0</span><span class="p">)</span> <span class="o">!=</span> <span class="n">n_parms</span><span class="p">:</span>
<span class="k">raise</span> <span class="ne">Exception</span><span class="p">(</span><span class="s1">&#39;Initial guess does not have the correct length.&#39;</span><span class="p">)</span>
<span class="k">else</span><span class="p">:</span>
<span class="n">x0</span> <span class="o">=</span> <span class="n">p_f</span>
<span class="k">def</span> <span class="nf">chisqfunc</span><span class="p">(</span><span class="n">p</span><span class="p">):</span>
<span class="n">model</span> <span class="o">=</span> <span class="n">func</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
<span class="n">chisq</span> <span class="o">=</span> <span class="n">anp</span><span class="o">.</span><span class="n">sum</span><span class="p">(((</span><span class="n">y_f</span> <span class="o">-</span> <span class="n">model</span><span class="p">)</span> <span class="o">/</span> <span class="n">dy_f</span><span class="p">)</span> <span class="o">**</span> <span class="mi">2</span><span class="p">)</span> <span class="o">+</span> <span class="n">anp</span><span class="o">.</span><span class="n">sum</span><span class="p">(((</span><span class="n">p_f</span> <span class="o">-</span> <span class="n">p</span><span class="p">)</span> <span class="o">/</span> <span class="n">dp_f</span><span class="p">)</span> <span class="o">**</span> <span class="mi">2</span><span class="p">)</span>
<span class="k">return</span> <span class="n">chisq</span>
<span class="k">if</span> <span class="ow">not</span> <span class="n">silent</span><span class="p">:</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;Method: migrad&#39;</span><span class="p">)</span>
<span class="n">m</span> <span class="o">=</span> <span class="n">iminuit</span><span class="o">.</span><span class="n">Minuit</span><span class="p">(</span><span class="n">chisqfunc</span><span class="p">,</span> <span class="n">x0</span><span class="p">)</span>
<span class="n">m</span><span class="o">.</span><span class="n">errordef</span> <span class="o">=</span> <span class="mi">1</span>
<span class="n">m</span><span class="o">.</span><span class="n">print_level</span> <span class="o">=</span> <span class="mi">0</span>
<span class="k">if</span> <span class="s1">&#39;tol&#39;</span> <span class="ow">in</span> <span class="n">kwargs</span><span class="p">:</span>
<span class="n">m</span><span class="o">.</span><span class="n">tol</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;tol&#39;</span><span class="p">)</span>
<span class="k">else</span><span class="p">:</span>
<span class="n">m</span><span class="o">.</span><span class="n">tol</span> <span class="o">=</span> <span class="mf">1e-4</span>
<span class="n">m</span><span class="o">.</span><span class="n">migrad</span><span class="p">()</span>
<span class="n">params</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">m</span><span class="o">.</span><span class="n">values</span><span class="p">)</span>
<span class="n">output</span><span class="o">.</span><span class="n">chisquare_by_dof</span> <span class="o">=</span> <span class="n">m</span><span class="o">.</span><span class="n">fval</span> <span class="o">/</span> <span class="nb">len</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
<span class="n">output</span><span class="o">.</span><span class="n">method</span> <span class="o">=</span> <span class="s1">&#39;migrad&#39;</span>
<span class="k">if</span> <span class="ow">not</span> <span class="n">silent</span><span class="p">:</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;chisquare/d.o.f.:&#39;</span><span class="p">,</span> <span class="n">output</span><span class="o">.</span><span class="n">chisquare_by_dof</span><span class="p">)</span>
<span class="k">if</span> <span class="ow">not</span> <span class="n">m</span><span class="o">.</span><span class="n">fmin</span><span class="o">.</span><span class="n">is_valid</span><span class="p">:</span>
<span class="k">raise</span> <span class="ne">Exception</span><span class="p">(</span><span class="s1">&#39;The minimization procedure did not converge.&#39;</span><span class="p">)</span>
<span class="n">hess_inv</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">pinv</span><span class="p">(</span><span class="n">jacobian</span><span class="p">(</span><span class="n">jacobian</span><span class="p">(</span><span class="n">chisqfunc</span><span class="p">))(</span><span class="n">params</span><span class="p">))</span>
<span class="k">def</span> <span class="nf">chisqfunc_compact</span><span class="p">(</span><span class="n">d</span><span class="p">):</span>
<span class="n">model</span> <span class="o">=</span> <span class="n">func</span><span class="p">(</span><span class="n">d</span><span class="p">[:</span><span class="n">n_parms</span><span class="p">],</span> <span class="n">x</span><span class="p">)</span>
<span class="n">chisq</span> <span class="o">=</span> <span class="n">anp</span><span class="o">.</span><span class="n">sum</span><span class="p">(((</span><span class="n">d</span><span class="p">[</span><span class="n">n_parms</span><span class="p">:</span> <span class="n">n_parms</span> <span class="o">+</span> <span class="nb">len</span><span class="p">(</span><span class="n">x</span><span class="p">)]</span> <span class="o">-</span> <span class="n">model</span><span class="p">)</span> <span class="o">/</span> <span class="n">dy_f</span><span class="p">)</span> <span class="o">**</span> <span class="mi">2</span><span class="p">)</span> <span class="o">+</span> <span class="n">anp</span><span class="o">.</span><span class="n">sum</span><span class="p">(((</span><span class="n">d</span><span class="p">[</span><span class="n">n_parms</span> <span class="o">+</span> <span class="nb">len</span><span class="p">(</span><span class="n">x</span><span class="p">):]</span> <span class="o">-</span> <span class="n">d</span><span class="p">[:</span><span class="n">n_parms</span><span class="p">])</span> <span class="o">/</span> <span class="n">dp_f</span><span class="p">)</span> <span class="o">**</span> <span class="mi">2</span><span class="p">)</span>
<span class="k">return</span> <span class="n">chisq</span>
<span class="n">jac_jac</span> <span class="o">=</span> <span class="n">jacobian</span><span class="p">(</span><span class="n">jacobian</span><span class="p">(</span><span class="n">chisqfunc_compact</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">params</span><span class="p">,</span> <span class="n">y_f</span><span class="p">,</span> <span class="n">p_f</span><span class="p">)))</span>
<span class="n">deriv</span> <span class="o">=</span> <span class="o">-</span><span class="n">hess_inv</span> <span class="o">@</span> <span class="n">jac_jac</span><span class="p">[:</span><span class="n">n_parms</span><span class="p">,</span> <span class="n">n_parms</span><span class="p">:]</span>
<span class="n">result</span> <span class="o">=</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">n_parms</span><span class="p">):</span>
<span class="n">result</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">derived_observable</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">:</span> <span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="mi">0</span><span class="p">]</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="o">/</span> <span class="p">(</span><span class="n">y</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">value</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="o">*</span> <span class="n">params</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="nb">list</span><span class="p">(</span><span class="n">y</span><span class="p">)</span> <span class="o">+</span> <span class="nb">list</span><span class="p">(</span><span class="n">loc_priors</span><span class="p">),</span> <span class="n">man_grad</span><span class="o">=</span><span class="nb">list</span><span class="p">(</span><span class="n">deriv</span><span class="p">[</span><span class="n">i</span><span class="p">])))</span>
<span class="n">output</span><span class="o">.</span><span class="n">fit_parameters</span> <span class="o">=</span> <span class="n">result</span>
<span class="n">output</span><span class="o">.</span><span class="n">chisquare</span> <span class="o">=</span> <span class="n">chisqfunc</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">asarray</span><span class="p">(</span><span class="n">params</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;resplot&#39;</span><span class="p">)</span> <span class="ow">is</span> <span class="kc">True</span><span class="p">:</span>
<span class="n">residual_plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="n">result</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;qqplot&#39;</span><span class="p">)</span> <span class="ow">is</span> <span class="kc">True</span><span class="p">:</span>
<span class="n">qqplot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="n">result</span><span class="p">)</span>
<span class="k">return</span> <span class="n">output</span>
<span class="k">def</span> <span class="nf">standard_fit</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="n">silent</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="n">warnings</span><span class="o">.</span><span class="n">warn</span><span class="p">(</span><span class="s2">&quot;standard_fit renamed to least_squares&quot;</span><span class="p">,</span> <span class="ne">DeprecationWarning</span><span class="p">)</span>
<span class="k">return</span> <span class="n">least_squares</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="n">silent</span><span class="o">=</span><span class="n">silent</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">_standard_fit</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="n">silent</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="n">output</span> <span class="o">=</span> <span class="n">Fit_result</span><span class="p">()</span>
<span class="n">output</span><span class="o">.</span><span class="n">fit_function</span> <span class="o">=</span> <span class="n">func</span>
<span class="n">x</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">x</span><span class="p">)</span>
<span class="k">if</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="nb">len</span><span class="p">(</span><span class="n">y</span><span class="p">):</span>
<span class="k">raise</span> <span class="ne">Exception</span><span class="p">(</span><span class="s1">&#39;x and y input have to have the same length&#39;</span><span class="p">)</span>
<span class="k">if</span> <span class="nb">len</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">&gt;</span> <span class="mi">2</span><span class="p">:</span>
<span class="k">raise</span> <span class="ne">Exception</span><span class="p">(</span><span class="s1">&#39;Unknown format for x values&#39;</span><span class="p">)</span>
<span class="k">if</span> <span class="ow">not</span> <span class="n">callable</span><span class="p">(</span><span class="n">func</span><span class="p">):</span>
<span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s1">&#39;func has to be a function.&#39;</span><span class="p">)</span>
<span class="n">func_aug</span> <span class="o">=</span> <span class="n">func</span>
<span class="k">if</span> <span class="s1">&#39;const_par&#39;</span> <span class="ow">in</span> <span class="n">kwargs</span><span class="p">:</span>
<span class="n">const_par</span> <span class="o">=</span> <span class="n">kwargs</span><span class="p">[</span><span class="s1">&#39;const_par&#39;</span><span class="p">]</span>
<span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">const_par</span><span class="p">,</span> <span class="n">Obs</span><span class="p">):</span>
<span class="n">const_par</span> <span class="o">=</span> <span class="p">[</span><span class="n">const_par</span><span class="p">]</span>
<span class="k">def</span> <span class="nf">func</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
<span class="k">return</span> <span class="n">func_aug</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">p</span><span class="p">,</span> <span class="p">[</span><span class="n">o</span><span class="o">.</span><span class="n">value</span> <span class="k">for</span> <span class="n">o</span> <span class="ow">in</span> <span class="n">const_par</span><span class="p">])),</span> <span class="n">x</span><span class="p">)</span>
<span class="k">else</span><span class="p">:</span>
<span class="n">const_par</span> <span class="o">=</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="mi">25</span><span class="p">):</span>
<span class="k">try</span><span class="p">:</span>
<span class="n">func</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">i</span><span class="p">),</span> <span class="n">x</span><span class="o">.</span><span class="n">T</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
<span class="k">except</span> <span class="ne">Exception</span><span class="p">:</span>
<span class="k">pass</span>
<span class="k">else</span><span class="p">:</span>
<span class="k">break</span>
<span class="n">n_parms</span> <span class="o">=</span> <span class="n">i</span>
<span class="k">if</span> <span class="ow">not</span> <span class="n">silent</span><span class="p">:</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;Fit with&#39;</span><span class="p">,</span> <span class="n">n_parms</span><span class="p">,</span> <span class="s1">&#39;parameters&#39;</span><span class="p">)</span>
<span class="k">if</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">const_par</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">):</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;</span><span class="se">\t</span><span class="s1"> and </span><span class="si">%d</span><span class="s1"> constrained parameter</span><span class="si">%s</span><span class="s1">&#39;</span> <span class="o">%</span> <span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">const_par</span><span class="p">),</span> <span class="s1">&#39;s&#39;</span> <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">const_par</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">1</span> <span class="k">else</span> <span class="s1">&#39;&#39;</span><span class="p">),</span> <span class="n">const_par</span><span class="p">)</span>
<span class="n">y_f</span> <span class="o">=</span> <span class="p">[</span><span class="n">o</span><span class="o">.</span><span class="n">value</span> <span class="k">for</span> <span class="n">o</span> <span class="ow">in</span> <span class="n">y</span><span class="p">]</span>
<span class="n">dy_f</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">y</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">asarray</span><span class="p">(</span><span class="n">dy_f</span><span class="p">)</span> <span class="o">&lt;=</span> <span class="mf">0.0</span><span class="p">):</span>
<span class="k">raise</span> <span class="ne">Exception</span><span class="p">(</span><span class="s1">&#39;No y errors available, run the gamma method first.&#39;</span><span class="p">)</span>
<span class="k">if</span> <span class="s1">&#39;initial_guess&#39;</span> <span class="ow">in</span> <span class="n">kwargs</span><span class="p">:</span>
<span class="n">x0</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;initial_guess&#39;</span><span class="p">)</span>
<span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">x0</span><span class="p">)</span> <span class="o">!=</span> <span class="n">n_parms</span><span class="p">:</span>
<span class="k">raise</span> <span class="ne">Exception</span><span class="p">(</span><span class="s1">&#39;Initial guess does not have the correct length: </span><span class="si">%d</span><span class="s1"> vs. </span><span class="si">%d</span><span class="s1">&#39;</span> <span class="o">%</span> <span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">x0</span><span class="p">),</span> <span class="n">n_parms</span><span class="p">))</span>
<span class="k">else</span><span class="p">:</span>
<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">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>
<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="nb">len</span><span class="p">(</span><span class="n">y</span><span class="p">)):</span>
<span class="n">corr</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">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">np</span><span class="o">.</span><span class="n">sqrt</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">i</span><span class="p">]</span> <span class="o">*</span> <span class="n">cov</span><span class="p">[</span><span class="n">j</span><span class="p">][</span><span class="n">j</span><span class="p">])</span>
<span class="n">condn</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">cond</span><span class="p">(</span><span class="n">corr</span><span class="p">)</span>
<span class="k">if</span> <span class="n">condn</span> <span class="o">&gt;</span> <span class="mf">1e4</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;Correlation matrix may be ill-conditioned! condition number: </span><span class="si">%1.2e</span><span class="s2">&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">condn</span><span class="p">),</span> <span class="ne">RuntimeWarning</span><span class="p">)</span>
<span class="n">chol</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">cholesky</span><span class="p">(</span><span class="n">corr</span><span class="p">)</span>
<span class="n">chol_inv</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">chol</span><span class="p">)</span>
<span class="n">chol_inv</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">chol_inv</span><span class="p">,</span> <span class="n">covdiag</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">chisqfunc</span><span class="p">(</span><span class="n">p</span><span class="p">):</span>
<span class="n">model</span> <span class="o">=</span> <span class="n">func</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
<span class="n">chisq</span> <span class="o">=</span> <span class="n">anp</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">anp</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">chol_inv</span><span class="p">,</span> <span class="p">(</span><span class="n">y_f</span> <span class="o">-</span> <span class="n">model</span><span class="p">))</span> <span class="o">**</span> <span class="mi">2</span><span class="p">)</span>
<span class="k">return</span> <span class="n">chisq</span>
<span class="k">def</span> <span class="nf">chisqfunc_aug</span><span class="p">(</span><span class="n">p</span><span class="p">):</span>
<span class="n">model</span> <span class="o">=</span> <span class="n">func_aug</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">p</span><span class="p">,</span> <span class="p">[</span><span class="n">o</span><span class="o">.</span><span class="n">value</span> <span class="k">for</span> <span class="n">o</span> <span class="ow">in</span> <span class="n">const_par</span><span class="p">])),</span> <span class="n">x</span><span class="p">)</span>
<span class="n">chisq</span> <span class="o">=</span> <span class="n">anp</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">anp</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">chol_inv</span><span class="p">,</span> <span class="p">(</span><span class="n">y_f</span> <span class="o">-</span> <span class="n">model</span><span class="p">))</span> <span class="o">**</span> <span class="mi">2</span><span class="p">)</span>
<span class="k">return</span> <span class="n">chisq</span>
<span class="k">else</span><span class="p">:</span>
<span class="k">def</span> <span class="nf">chisqfunc</span><span class="p">(</span><span class="n">p</span><span class="p">):</span>
<span class="n">model</span> <span class="o">=</span> <span class="n">func</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
<span class="n">chisq</span> <span class="o">=</span> <span class="n">anp</span><span class="o">.</span><span class="n">sum</span><span class="p">(((</span><span class="n">y_f</span> <span class="o">-</span> <span class="n">model</span><span class="p">)</span> <span class="o">/</span> <span class="n">dy_f</span><span class="p">)</span> <span class="o">**</span> <span class="mi">2</span><span class="p">)</span>
<span class="k">return</span> <span class="n">chisq</span>
<span class="k">def</span> <span class="nf">chisqfunc_aug</span><span class="p">(</span><span class="n">p</span><span class="p">):</span>
<span class="n">model</span> <span class="o">=</span> <span class="n">func_aug</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">p</span><span class="p">,</span> <span class="p">[</span><span class="n">o</span><span class="o">.</span><span class="n">value</span> <span class="k">for</span> <span class="n">o</span> <span class="ow">in</span> <span class="n">const_par</span><span class="p">])),</span> <span class="n">x</span><span class="p">)</span>
<span class="n">chisq</span> <span class="o">=</span> <span class="n">anp</span><span class="o">.</span><span class="n">sum</span><span class="p">(((</span><span class="n">y_f</span> <span class="o">-</span> <span class="n">model</span><span class="p">)</span> <span class="o">/</span> <span class="n">dy_f</span><span class="p">)</span> <span class="o">**</span> <span class="mi">2</span><span class="p">)</span>
<span class="k">return</span> <span class="n">chisq</span>
<span class="k">if</span> <span class="s1">&#39;method&#39;</span> <span class="ow">in</span> <span class="n">kwargs</span><span class="p">:</span>
<span class="n">output</span><span class="o">.</span><span class="n">method</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;method&#39;</span><span class="p">)</span>
<span class="k">if</span> <span class="ow">not</span> <span class="n">silent</span><span class="p">:</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;Method:&#39;</span><span class="p">,</span> <span class="n">kwargs</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="s1">&#39;method&#39;</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;method&#39;</span><span class="p">)</span> <span class="o">==</span> <span class="s1">&#39;migrad&#39;</span><span class="p">:</span>
<span class="n">fit_result</span> <span class="o">=</span> <span class="n">iminuit</span><span class="o">.</span><span class="n">minimize</span><span class="p">(</span><span class="n">chisqfunc</span><span class="p">,</span> <span class="n">x0</span><span class="p">)</span>
<span class="n">fit_result</span> <span class="o">=</span> <span class="n">iminuit</span><span class="o">.</span><span class="n">minimize</span><span class="p">(</span><span class="n">chisqfunc</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="k">else</span><span class="p">:</span>
<span class="n">fit_result</span> <span class="o">=</span> <span class="n">scipy</span><span class="o">.</span><span class="n">optimize</span><span class="o">.</span><span class="n">minimize</span><span class="p">(</span><span class="n">chisqfunc</span><span class="p">,</span> <span class="n">x0</span><span class="p">,</span> <span class="n">method</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;method&#39;</span><span class="p">))</span>
<span class="n">fit_result</span> <span class="o">=</span> <span class="n">scipy</span><span class="o">.</span><span class="n">optimize</span><span class="o">.</span><span class="n">minimize</span><span class="p">(</span><span class="n">chisqfunc</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">method</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;method&#39;</span><span class="p">),</span> <span class="n">tol</span><span class="o">=</span><span class="mf">1e-12</span><span class="p">)</span>
<span class="n">chisquare</span> <span class="o">=</span> <span class="n">fit_result</span><span class="o">.</span><span class="n">fun</span>
<span class="n">output</span><span class="o">.</span><span class="n">iterations</span> <span class="o">=</span> <span class="n">fit_result</span><span class="o">.</span><span class="n">nit</span>
<span class="k">else</span><span class="p">:</span>
<span class="n">output</span><span class="o">.</span><span class="n">method</span> <span class="o">=</span> <span class="s1">&#39;Levenberg-Marquardt&#39;</span>
<span class="k">if</span> <span class="ow">not</span> <span class="n">silent</span><span class="p">:</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;Method: Levenberg-Marquardt&#39;</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="kc">True</span><span class="p">:</span>
<span class="k">def</span> <span class="nf">chisqfunc_residuals</span><span class="p">(</span><span class="n">p</span><span class="p">):</span>
<span class="n">model</span> <span class="o">=</span> <span class="n">func</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
<span class="n">chisq</span> <span class="o">=</span> <span class="n">anp</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">chol_inv</span><span class="p">,</span> <span class="p">(</span><span class="n">y_f</span> <span class="o">-</span> <span class="n">model</span><span class="p">))</span>
<span class="k">return</span> <span class="n">chisq</span>
<span class="k">else</span><span class="p">:</span>
<span class="k">def</span> <span class="nf">chisqfunc_residuals</span><span class="p">(</span><span class="n">p</span><span class="p">):</span>
<span class="n">model</span> <span class="o">=</span> <span class="n">func</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
<span class="n">chisq</span> <span class="o">=</span> <span class="p">((</span><span class="n">y_f</span> <span class="o">-</span> <span class="n">model</span><span class="p">)</span> <span class="o">/</span> <span class="n">dy_f</span><span class="p">)</span>
<span class="k">return</span> <span class="n">chisq</span>
<span class="n">fit_result</span> <span class="o">=</span> <span class="n">scipy</span><span class="o">.</span><span class="n">optimize</span><span class="o">.</span><span class="n">least_squares</span><span class="p">(</span><span class="n">chisqfunc_residuals</span><span class="p">,</span> <span class="n">x0</span><span class="p">,</span> <span class="n">method</span><span class="o">=</span><span class="s1">&#39;lm&#39;</span><span class="p">,</span> <span class="n">ftol</span><span class="o">=</span><span class="mf">1e-15</span><span class="p">,</span> <span class="n">gtol</span><span class="o">=</span><span class="mf">1e-15</span><span class="p">,</span> <span class="n">xtol</span><span class="o">=</span><span class="mf">1e-15</span><span class="p">)</span>
<span class="n">chisquare</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">fit_result</span><span class="o">.</span><span class="n">fun</span> <span class="o">**</span> <span class="mi">2</span><span class="p">)</span>
<span class="n">output</span><span class="o">.</span><span class="n">iterations</span> <span class="o">=</span> <span class="n">fit_result</span><span class="o">.</span><span class="n">nfev</span>
<span class="k">if</span> <span class="ow">not</span> <span class="n">fit_result</span><span class="o">.</span><span class="n">success</span><span class="p">:</span>
<span class="k">raise</span> <span class="ne">Exception</span><span class="p">(</span><span class="s1">&#39;The minimization procedure did not converge.&#39;</span><span class="p">)</span>
<span class="k">if</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">n_parms</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">:</span>
<span class="n">output</span><span class="o">.</span><span class="n">chisquare_by_dof</span> <span class="o">=</span> <span class="n">chisquare</span> <span class="o">/</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">n_parms</span><span class="p">)</span>
<span class="k">else</span><span class="p">:</span>
<span class="n">output</span><span class="o">.</span><span class="n">chisquare_by_dof</span> <span class="o">=</span> <span class="nb">float</span><span class="p">(</span><span class="s1">&#39;nan&#39;</span><span class="p">)</span>
<span class="n">output</span><span class="o">.</span><span class="n">message</span> <span class="o">=</span> <span class="n">fit_result</span><span class="o">.</span><span class="n">message</span>
<span class="k">if</span> <span class="ow">not</span> <span class="n">silent</span><span class="p">:</span>
<span class="nb">print</span><span class="p">(</span><span class="n">fit_result</span><span class="o">.</span><span class="n">message</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;chisquare/d.o.f.:&#39;</span><span class="p">,</span> <span class="n">output</span><span class="o">.</span><span class="n">chisquare_by_dof</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;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">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>
<span class="n">output</span><span class="o">.</span><span class="n">chisquare_by_expected_chisquare</span> <span class="o">=</span> <span class="n">chisquare</span> <span class="o">/</span> <span class="n">expected_chisquare</span>
<span class="k">if</span> <span class="ow">not</span> <span class="n">silent</span><span class="p">:</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;chisquare/expected_chisquare:&#39;</span><span class="p">,</span>
<span class="n">output</span><span class="o">.</span><span class="n">chisquare_by_expected_chisquare</span><span class="p">)</span>
<span class="n">fitp</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">concatenate</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="p">[</span><span class="n">o</span><span class="o">.</span><span class="n">value</span> <span class="k">for</span> <span class="n">o</span> <span class="ow">in</span> <span class="n">const_par</span><span class="p">]))</span>
<span class="n">hess_inv</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">pinv</span><span class="p">(</span><span class="n">jacobian</span><span class="p">(</span><span class="n">jacobian</span><span class="p">(</span><span class="n">chisqfunc_aug</span><span class="p">))(</span><span class="n">fitp</span><span class="p">))</span>
<span class="n">n_parms_aug</span> <span class="o">=</span> <span class="n">n_parms</span> <span class="o">+</span> <span class="nb">len</span><span class="p">(</span><span class="n">const_par</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="kc">True</span><span class="p">:</span>
<span class="k">def</span> <span class="nf">chisqfunc_compact</span><span class="p">(</span><span class="n">d</span><span class="p">):</span>
<span class="n">model</span> <span class="o">=</span> <span class="n">func_aug</span><span class="p">(</span><span class="n">d</span><span class="p">[:</span><span class="n">n_parms_aug</span><span class="p">],</span> <span class="n">x</span><span class="p">)</span>
<span class="n">chisq</span> <span class="o">=</span> <span class="n">anp</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">anp</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">chol_inv</span><span class="p">,</span> <span class="p">(</span><span class="n">d</span><span class="p">[</span><span class="n">n_parms_aug</span><span class="p">:]</span> <span class="o">-</span> <span class="n">model</span><span class="p">))</span> <span class="o">**</span> <span class="mi">2</span><span class="p">)</span>
<span class="k">return</span> <span class="n">chisq</span>
<span class="k">else</span><span class="p">:</span>
<span class="k">def</span> <span class="nf">chisqfunc_compact</span><span class="p">(</span><span class="n">d</span><span class="p">):</span>
<span class="n">model</span> <span class="o">=</span> <span class="n">func_aug</span><span class="p">(</span><span class="n">d</span><span class="p">[:</span><span class="n">n_parms_aug</span><span class="p">],</span> <span class="n">x</span><span class="p">)</span>
<span class="n">chisq</span> <span class="o">=</span> <span class="n">anp</span><span class="o">.</span><span class="n">sum</span><span class="p">(((</span><span class="n">d</span><span class="p">[</span><span class="n">n_parms_aug</span><span class="p">:]</span> <span class="o">-</span> <span class="n">model</span><span class="p">)</span> <span class="o">/</span> <span class="n">dy_f</span><span class="p">)</span> <span class="o">**</span> <span class="mi">2</span><span class="p">)</span>
<span class="k">return</span> <span class="n">chisq</span>
<span class="n">jac_jac</span> <span class="o">=</span> <span class="n">jacobian</span><span class="p">(</span><span class="n">jacobian</span><span class="p">(</span><span class="n">chisqfunc_compact</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">fitp</span><span class="p">,</span> <span class="n">y_f</span><span class="p">)))</span>
<span class="n">deriv</span> <span class="o">=</span> <span class="o">-</span><span class="n">hess_inv</span> <span class="o">@</span> <span class="n">jac_jac</span><span class="p">[:</span><span class="n">n_parms_aug</span><span class="p">,</span> <span class="n">n_parms_aug</span><span class="p">:]</span>
<span class="n">result</span> <span class="o">=</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">n_parms</span><span class="p">):</span>
<span class="n">result</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">derived_observable</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">:</span> <span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="mi">0</span><span class="p">]</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="o">/</span> <span class="p">(</span><span class="n">y</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">value</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="o">*</span> <span class="n">fit_result</span><span class="o">.</span><span class="n">x</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="nb">list</span><span class="p">(</span><span class="n">y</span><span class="p">),</span> <span class="n">man_grad</span><span class="o">=</span><span class="nb">list</span><span class="p">(</span><span class="n">deriv</span><span class="p">[</span><span class="n">i</span><span class="p">])))</span>
<span class="n">output</span><span class="o">.</span><span class="n">fit_parameters</span> <span class="o">=</span> <span class="n">result</span> <span class="o">+</span> <span class="n">const_par</span>
<span class="n">output</span><span class="o">.</span><span class="n">chisquare</span> <span class="o">=</span> <span class="n">chisqfunc</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">output</span><span class="o">.</span><span class="n">dof</span> <span class="o">=</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">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;resplot&#39;</span><span class="p">)</span> <span class="ow">is</span> <span class="kc">True</span><span class="p">:</span>
<span class="n">residual_plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="n">result</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;qqplot&#39;</span><span class="p">)</span> <span class="ow">is</span> <span class="kc">True</span><span class="p">:</span>
<span class="n">qqplot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="n">result</span><span class="p">)</span>
<span class="k">return</span> <span class="n">output</span>
<span class="k">def</span> <span class="nf">odr_fit</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="n">silent</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="n">warnings</span><span class="o">.</span><span class="n">warn</span><span class="p">(</span><span class="s2">&quot;odr_fit renamed to total_least_squares&quot;</span><span class="p">,</span> <span class="ne">DeprecationWarning</span><span class="p">)</span>
<span class="k">return</span> <span class="n">total_least_squares</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="n">silent</span><span class="o">=</span><span class="n">silent</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">fit_lin</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
<span class="sd">&quot;&quot;&quot;Performs a linear fit to y = n + m * x and returns two Obs n, m.</span>
<span class="sd"> y has to be a list of Obs, the dvalues of the Obs are used as yerror for the fit.</span>
<span class="sd"> x can either be a list of floats in which case no xerror is assumed, or</span>
<span class="sd"> a list of Obs, where the dvalues of the Obs are used as xerror for the fit.</span>
<span class="sd"> &quot;&quot;&quot;</span>
<span class="k">def</span> <span class="nf">f</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
<span class="n">y</span> <span class="o">=</span> <span class="n">a</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">+</span> <span class="n">a</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">*</span> <span class="n">x</span>
<span class="k">return</span> <span class="n">y</span>
<span class="k">if</span> <span class="nb">all</span><span class="p">(</span><span class="nb">isinstance</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">Obs</span><span class="p">)</span> <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="n">x</span><span class="p">):</span>
<span class="n">out</span> <span class="o">=</span> <span class="n">total_least_squares</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
<span class="k">return</span> <span class="n">out</span><span class="o">.</span><span class="n">fit_parameters</span>
<span class="k">elif</span> <span class="nb">all</span><span class="p">(</span><span class="nb">isinstance</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="nb">float</span><span class="p">)</span> <span class="ow">or</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="nb">int</span><span class="p">)</span> <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="n">x</span><span class="p">)</span> <span class="ow">or</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">ndarray</span><span class="p">):</span>
<span class="n">out</span> <span class="o">=</span> <span class="n">least_squares</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
<span class="k">return</span> <span class="n">out</span><span class="o">.</span><span class="n">fit_parameters</span>
<span class="k">else</span><span class="p">:</span>
<span class="k">raise</span> <span class="ne">Exception</span><span class="p">(</span><span class="s1">&#39;Unsupported types for x&#39;</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">qqplot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">o_y</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="n">p</span><span class="p">):</span>
<span class="sd">&quot;&quot;&quot; Generates a quantile-quantile plot of the fit result which can be used to</span>
<span class="sd"> check if the residuals of the fit are gaussian distributed.</span>
<span class="sd"> &quot;&quot;&quot;</span>
<span class="n">residuals</span> <span class="o">=</span> <span class="p">[]</span>
<span class="k">for</span> <span class="n">i_x</span><span class="p">,</span> <span class="n">i_y</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">o_y</span><span class="p">):</span>
<span class="n">residuals</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">i_y</span> <span class="o">-</span> <span class="n">func</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">i_x</span><span class="p">))</span> <span class="o">/</span> <span class="n">i_y</span><span class="o">.</span><span class="n">dvalue</span><span class="p">)</span>
<span class="n">residuals</span> <span class="o">=</span> <span class="nb">sorted</span><span class="p">(</span><span class="n">residuals</span><span class="p">)</span>
<span class="n">my_y</span> <span class="o">=</span> <span class="p">[</span><span class="n">o</span><span class="o">.</span><span class="n">value</span> <span class="k">for</span> <span class="n">o</span> <span class="ow">in</span> <span class="n">residuals</span><span class="p">]</span>
<span class="n">probplot</span> <span class="o">=</span> <span class="n">scipy</span><span class="o">.</span><span class="n">stats</span><span class="o">.</span><span class="n">probplot</span><span class="p">(</span><span class="n">my_y</span><span class="p">)</span>
<span class="n">my_x</span> <span class="o">=</span> <span class="n">probplot</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span>
<span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">8</span><span class="p">,</span> <span class="mi">8</span> <span class="o">/</span> <span class="mf">1.618</span><span class="p">))</span>
<span class="n">plt</span><span class="o">.</span><span class="n">errorbar</span><span class="p">(</span><span class="n">my_x</span><span class="p">,</span> <span class="n">my_y</span><span class="p">,</span> <span class="n">fmt</span><span class="o">=</span><span class="s1">&#39;o&#39;</span><span class="p">)</span>
<span class="n">fit_start</span> <span class="o">=</span> <span class="n">my_x</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
<span class="n">fit_stop</span> <span class="o">=</span> <span class="n">my_x</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
<span class="n">samples</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">fit_start</span><span class="p">,</span> <span class="n">fit_stop</span><span class="p">,</span> <span class="mf">0.01</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">samples</span><span class="p">,</span> <span class="n">samples</span><span class="p">,</span> <span class="s1">&#39;k--&#39;</span><span class="p">,</span> <span class="n">zorder</span><span class="o">=</span><span class="mi">11</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s1">&#39;Standard normal distribution&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">samples</span><span class="p">,</span> <span class="n">probplot</span><span class="p">[</span><span class="mi">1</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="n">samples</span> <span class="o">+</span> <span class="n">probplot</span><span class="p">[</span><span class="mi">1</span><span class="p">][</span><span class="mi">1</span><span class="p">],</span> <span class="n">zorder</span><span class="o">=</span><span class="mi">10</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s1">&#39;Least squares fit, r=&#39;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">around</span><span class="p">(</span><span class="n">probplot</span><span class="p">[</span><span class="mi">1</span><span class="p">][</span><span class="mi">2</span><span class="p">],</span> <span class="mi">3</span><span class="p">)))</span>
<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s1">&#39;Theoretical quantiles&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s1">&#39;Ordered Values&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">legend</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">def</span> <span class="nf">residual_plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="n">fit_res</span><span class="p">):</span>
<span class="sd">&quot;&quot;&quot; Generates a plot which compares the fit to the data and displays the corresponding residuals&quot;&quot;&quot;</span>
<span class="n">xstart</span> <span class="o">=</span> <span class="n">x</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">-</span> <span class="mf">0.5</span>
<span class="n">xstop</span> <span class="o">=</span> <span class="n">x</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">+</span> <span class="mf">0.5</span>
<span class="n">x_samples</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">xstart</span><span class="p">,</span> <span class="n">xstop</span><span class="p">,</span> <span class="mf">0.01</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">8</span><span class="p">,</span> <span class="mi">8</span> <span class="o">/</span> <span class="mf">1.618</span><span class="p">))</span>
<span class="n">gs</span> <span class="o">=</span> <span class="n">gridspec</span><span class="o">.</span><span class="n">GridSpec</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">height_ratios</span><span class="o">=</span><span class="p">[</span><span class="mi">3</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="n">wspace</span><span class="o">=</span><span class="mf">0.0</span><span class="p">,</span> <span class="n">hspace</span><span class="o">=</span><span class="mf">0.0</span><span class="p">)</span>
<span class="n">ax0</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="n">gs</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
<span class="n">ax0</span><span class="o">.</span><span class="n">errorbar</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="p">[</span><span class="n">o</span><span class="o">.</span><span class="n">value</span> <span class="k">for</span> <span class="n">o</span> <span class="ow">in</span> <span class="n">y</span><span class="p">],</span> <span class="n">yerr</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">y</span><span class="p">],</span> <span class="n">ls</span><span class="o">=</span><span class="s1">&#39;none&#39;</span><span class="p">,</span> <span class="n">fmt</span><span class="o">=</span><span class="s1">&#39;o&#39;</span><span class="p">,</span> <span class="n">capsize</span><span class="o">=</span><span class="mi">3</span><span class="p">,</span> <span class="n">markersize</span><span class="o">=</span><span class="mi">5</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s1">&#39;Data&#39;</span><span class="p">)</span>
<span class="n">ax0</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x_samples</span><span class="p">,</span> <span class="n">func</span><span class="p">([</span><span class="n">o</span><span class="o">.</span><span class="n">value</span> <span class="k">for</span> <span class="n">o</span> <span class="ow">in</span> <span class="n">fit_res</span><span class="p">],</span> <span class="n">x_samples</span><span class="p">),</span> <span class="n">label</span><span class="o">=</span><span class="s1">&#39;Fit&#39;</span><span class="p">,</span> <span class="n">zorder</span><span class="o">=</span><span class="mi">10</span><span class="p">,</span> <span class="n">ls</span><span class="o">=</span><span class="s1">&#39;-&#39;</span><span class="p">,</span> <span class="n">ms</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
<span class="n">ax0</span><span class="o">.</span><span class="n">set_xticklabels</span><span class="p">([])</span>
<span class="n">ax0</span><span class="o">.</span><span class="n">set_xlim</span><span class="p">([</span><span class="n">xstart</span><span class="p">,</span> <span class="n">xstop</span><span class="p">])</span>
<span class="n">ax0</span><span class="o">.</span><span class="n">set_xticklabels</span><span class="p">([])</span>
<span class="n">ax0</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
<span class="n">residuals</span> <span class="o">=</span> <span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">asarray</span><span class="p">([</span><span class="n">o</span><span class="o">.</span><span class="n">value</span> <span class="k">for</span> <span class="n">o</span> <span class="ow">in</span> <span class="n">y</span><span class="p">])</span> <span class="o">-</span> <span class="n">func</span><span class="p">([</span><span class="n">o</span><span class="o">.</span><span class="n">value</span> <span class="k">for</span> <span class="n">o</span> <span class="ow">in</span> <span class="n">fit_res</span><span class="p">],</span> <span class="n">x</span><span class="p">))</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">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">y</span><span class="p">])</span>
<span class="n">ax1</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="n">gs</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
<span class="n">ax1</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">residuals</span><span class="p">,</span> <span class="s1">&#39;ko&#39;</span><span class="p">,</span> <span class="n">ls</span><span class="o">=</span><span class="s1">&#39;none&#39;</span><span class="p">,</span> <span class="n">markersize</span><span class="o">=</span><span class="mi">5</span><span class="p">)</span>
<span class="n">ax1</span><span class="o">.</span><span class="n">tick_params</span><span class="p">(</span><span class="n">direction</span><span class="o">=</span><span class="s1">&#39;out&#39;</span><span class="p">)</span>
<span class="n">ax1</span><span class="o">.</span><span class="n">tick_params</span><span class="p">(</span><span class="n">axis</span><span class="o">=</span><span class="s2">&quot;x&quot;</span><span class="p">,</span> <span class="n">bottom</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span> <span class="n">top</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span> <span class="n">labelbottom</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
<span class="n">ax1</span><span class="o">.</span><span class="n">axhline</span><span class="p">(</span><span class="n">y</span><span class="o">=</span><span class="mf">0.0</span><span class="p">,</span> <span class="n">ls</span><span class="o">=</span><span class="s1">&#39;--&#39;</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;k&#39;</span><span class="p">,</span> <span class="n">marker</span><span class="o">=</span><span class="s2">&quot; &quot;</span><span class="p">)</span>
<span class="n">ax1</span><span class="o">.</span><span class="n">fill_between</span><span class="p">(</span><span class="n">x_samples</span><span class="p">,</span> <span class="o">-</span><span class="mf">1.0</span><span class="p">,</span> <span class="mf">1.0</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.1</span><span class="p">,</span> <span class="n">facecolor</span><span class="o">=</span><span class="s1">&#39;k&#39;</span><span class="p">)</span>
<span class="n">ax1</span><span class="o">.</span><span class="n">set_xlim</span><span class="p">([</span><span class="n">xstart</span><span class="p">,</span> <span class="n">xstop</span><span class="p">])</span>
<span class="n">ax1</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">&#39;Residuals&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">subplots_adjust</span><span class="p">(</span><span class="n">wspace</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">hspace</span><span class="o">=</span><span class="kc">None</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">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.&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="k">return</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="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="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>
<span class="n">deriv</span> <span class="o">=</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">x</span><span class="p">):</span>
<span class="n">deriv</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">egrad</span><span class="p">(</span><span class="n">func</span><span class="p">)([</span><span class="n">o</span><span class="o">.</span><span class="n">value</span> <span class="k">for</span> <span class="n">o</span> <span class="ow">in</span> <span class="n">beta</span><span class="p">],</span> <span class="n">item</span><span class="p">)))</span>
<span class="n">err</span> <span class="o">=</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">x</span><span class="p">):</span>
<span class="n">err</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="n">deriv</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">@</span> <span class="n">cov</span> <span class="o">@</span> <span class="n">deriv</span><span class="p">[</span><span class="n">i</span><span class="p">]))</span>
<span class="n">err</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">err</span><span class="p">)</span>
<span class="k">return</span> <span class="n">err</span>
</pre></div>
</details>
</section>
<section id="Fit_result">
<div class="attr class">
<a class="headerlink" href="#Fit_result">#&nbsp;&nbsp</a>
<span class="def">class</span>
<span class="name">Fit_result</span><wbr>(<span class="base">collections.abc.Sequence</span>):
</div>
<details>
<summary>View Source</summary>
<div class="codehilite"><pre><span></span><span class="k">class</span> <span class="nc">Fit_result</span><span class="p">(</span><span class="n">Sequence</span><span class="p">):</span>
<span class="sd">&quot;&quot;&quot;Represents fit results.</span>
<span class="sd"> Attributes</span>
<span class="sd"> ----------</span>
<span class="sd"> fit_parameters : list</span>
<span class="sd"> results for the individual fit parameters,</span>
<span class="sd"> also accessible via indices.</span>
<span class="sd"> &quot;&quot;&quot;</span>
<span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
<span class="bp">self</span><span class="o">.</span><span class="n">fit_parameters</span> <span class="o">=</span> <span class="kc">None</span>
<span class="k">def</span> <span class="fm">__getitem__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">idx</span><span class="p">):</span>
<span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">fit_parameters</span><span class="p">[</span><span class="n">idx</span><span class="p">]</span>
<span class="k">def</span> <span class="fm">__len__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
<span class="k">return</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">fit_parameters</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">gamma_method</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
<span class="sd">&quot;&quot;&quot;Apply the gamma method to all fit parameters&quot;&quot;&quot;</span>
<span class="p">[</span><span class="n">o</span><span class="o">.</span><span class="n">gamma_method</span><span class="p">()</span> <span class="k">for</span> <span class="n">o</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">fit_parameters</span><span class="p">]</span>
<span class="k">def</span> <span class="fm">__str__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
<span class="bp">self</span><span class="o">.</span><span class="n">gamma_method</span><span class="p">()</span>
<span class="n">my_str</span> <span class="o">=</span> <span class="s1">&#39;Goodness of fit:</span><span class="se">\n</span><span class="s1">&#39;</span>
<span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="s1">&#39;chisquare_by_dof&#39;</span><span class="p">):</span>
<span class="n">my_str</span> <span class="o">+=</span> <span class="s1">&#39;</span><span class="se">\u03C7\u00b2</span><span class="s1">/d.o.f. = &#39;</span> <span class="o">+</span> <span class="sa">f</span><span class="s1">&#39;</span><span class="si">{</span><span class="bp">self</span><span class="o">.</span><span class="n">chisquare_by_dof</span><span class="si">:</span><span class="s1">2.6f</span><span class="si">}</span><span class="s1">&#39;</span> <span class="o">+</span> <span class="s1">&#39;</span><span class="se">\n</span><span class="s1">&#39;</span>
<span class="k">elif</span> <span class="nb">hasattr</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="s1">&#39;residual_variance&#39;</span><span class="p">):</span>
<span class="n">my_str</span> <span class="o">+=</span> <span class="s1">&#39;residual variance = &#39;</span> <span class="o">+</span> <span class="sa">f</span><span class="s1">&#39;</span><span class="si">{</span><span class="bp">self</span><span class="o">.</span><span class="n">residual_variance</span><span class="si">:</span><span class="s1">2.6f</span><span class="si">}</span><span class="s1">&#39;</span> <span class="o">+</span> <span class="s1">&#39;</span><span class="se">\n</span><span class="s1">&#39;</span>
<span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="s1">&#39;chisquare_by_expected_chisquare&#39;</span><span class="p">):</span>
<span class="n">my_str</span> <span class="o">+=</span> <span class="s1">&#39;</span><span class="se">\u03C7\u00b2</span><span class="s1">/</span><span class="se">\u03C7\u00b2</span><span class="s1">exp = &#39;</span> <span class="o">+</span> <span class="sa">f</span><span class="s1">&#39;</span><span class="si">{</span><span class="bp">self</span><span class="o">.</span><span class="n">chisquare_by_expected_chisquare</span><span class="si">:</span><span class="s1">2.6f</span><span class="si">}</span><span class="s1">&#39;</span> <span class="o">+</span> <span class="s1">&#39;</span><span class="se">\n</span><span class="s1">&#39;</span>
<span class="n">my_str</span> <span class="o">+=</span> <span class="s1">&#39;Fit parameters:</span><span class="se">\n</span><span class="s1">&#39;</span>
<span class="k">for</span> <span class="n">i_par</span><span class="p">,</span> <span class="n">par</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">fit_parameters</span><span class="p">):</span>
<span class="n">my_str</span> <span class="o">+=</span> <span class="nb">str</span><span class="p">(</span><span class="n">i_par</span><span class="p">)</span> <span class="o">+</span> <span class="s1">&#39;</span><span class="se">\t</span><span class="s1">&#39;</span> <span class="o">+</span> <span class="s1">&#39; &#39;</span> <span class="o">*</span> <span class="nb">int</span><span class="p">(</span><span class="n">par</span> <span class="o">&gt;=</span> <span class="mi">0</span><span class="p">)</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">par</span><span class="p">)</span><span class="o">.</span><span class="n">rjust</span><span class="p">(</span><span class="nb">int</span><span class="p">(</span><span class="n">par</span> <span class="o">&lt;</span> <span class="mf">0.0</span><span class="p">))</span> <span class="o">+</span> <span class="s1">&#39;</span><span class="se">\n</span><span class="s1">&#39;</span>
<span class="k">return</span> <span class="n">my_str</span>
<span class="k">def</span> <span class="fm">__repr__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
<span class="n">m</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="nb">len</span><span class="p">,</span> <span class="nb">list</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="vm">__dict__</span><span class="o">.</span><span class="n">keys</span><span class="p">())))</span> <span class="o">+</span> <span class="mi">1</span>
<span class="k">return</span> <span class="s1">&#39;</span><span class="se">\n</span><span class="s1">&#39;</span><span class="o">.</span><span class="n">join</span><span class="p">([</span><span class="n">key</span><span class="o">.</span><span class="n">rjust</span><span class="p">(</span><span class="n">m</span><span class="p">)</span> <span class="o">+</span> <span class="s1">&#39;: &#39;</span> <span class="o">+</span> <span class="nb">repr</span><span class="p">(</span><span class="n">value</span><span class="p">)</span> <span class="k">for</span> <span class="n">key</span><span class="p">,</span> <span class="n">value</span> <span class="ow">in</span> <span class="nb">sorted</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="vm">__dict__</span><span class="o">.</span><span class="n">items</span><span class="p">())])</span>
</pre></div>
</details>
<div class="docstring"><p>Represents fit results.</p>
<h6 id="attributes">Attributes</h6>
<ul>
<li><strong>fit_parameters</strong> (list):
results for the individual fit parameters,
also accessible via indices.</li>
</ul>
</div>
<div id="Fit_result.__init__" class="classattr">
<div class="attr function"><a class="headerlink" href="#Fit_result.__init__">#&nbsp;&nbsp</a>
<span class="name">Fit_result</span><span class="signature">()</span>
</div>
<details>
<summary>View Source</summary>
<div class="codehilite"><pre><span></span> <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
<span class="bp">self</span><span class="o">.</span><span class="n">fit_parameters</span> <span class="o">=</span> <span class="kc">None</span>
</pre></div>
</details>
</div>
<div id="Fit_result.gamma_method" class="classattr">
<div class="attr function"><a class="headerlink" href="#Fit_result.gamma_method">#&nbsp;&nbsp</a>
<span class="def">def</span>
<span class="name">gamma_method</span><span class="signature">(self)</span>:
</div>
<details>
<summary>View Source</summary>
<div class="codehilite"><pre><span></span> <span class="k">def</span> <span class="nf">gamma_method</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
<span class="sd">&quot;&quot;&quot;Apply the gamma method to all fit parameters&quot;&quot;&quot;</span>
<span class="p">[</span><span class="n">o</span><span class="o">.</span><span class="n">gamma_method</span><span class="p">()</span> <span class="k">for</span> <span class="n">o</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">fit_parameters</span><span class="p">]</span>
</pre></div>
</details>
<div class="docstring"><p>Apply the gamma method to all fit parameters</p>
</div>
</div>
<div class="inherited">
<h5>Inherited Members</h5>
<dl>
<div><dt>collections.abc.Sequence</dt>
<dd id="Fit_result.index" class="function">index</dd>
<dd id="Fit_result.count" class="function">count</dd>
</div>
</dl>
</div>
</section>
<section id="least_squares">
<div class="attr function"><a class="headerlink" href="#least_squares">#&nbsp;&nbsp</a>
<span class="def">def</span>
<span class="name">least_squares</span><span class="signature">(x, y, func, priors=None, silent=False, **kwargs)</span>:
</div>
<details>
<summary>View Source</summary>
<div class="codehilite"><pre><span></span><span class="k">def</span> <span class="nf">least_squares</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="n">priors</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">silent</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="sa">r</span><span class="sd">&#39;&#39;&#39;Performs a non-linear fit to y = func(x).</span>
<span class="sd"> Parameters</span>
<span class="sd"> ----------</span>
<span class="sd"> x : list</span>
<span class="sd"> list of floats.</span>
<span class="sd"> y : list</span>
<span class="sd"> list of Obs.</span>
<span class="sd"> func : object</span>
<span class="sd"> fit function, has to be of the form</span>
<span class="sd"> ```python</span>
<span class="sd"> def func(a, x):</span>
<span class="sd"> y = a[0] + a[1] * x + a[2] * anp.sinh(x)</span>
<span class="sd"> return y</span>
<span class="sd"> ```</span>
<span class="sd"> For multiple x values func can be of the form</span>
<span class="sd"> ```python</span>
<span class="sd"> def func(a, x):</span>
<span class="sd"> (x1, x2) = x</span>
<span class="sd"> return a[0] * x1 ** 2 + a[1] * x2</span>
<span class="sd"> ```</span>
<span class="sd"> It is important that all numpy functions refer to autograd.numpy, otherwise the differentiation</span>
<span class="sd"> will not work</span>
<span class="sd"> priors : list, optional</span>
<span class="sd"> priors has to be a list with an entry for every parameter in the fit. The entries can either be</span>
<span class="sd"> Obs (e.g. results from a previous fit) or strings containing a value and an error formatted like</span>
<span class="sd"> 0.548(23), 500(40) or 0.5(0.4)</span>
<span class="sd"> silent : bool, optional</span>
<span class="sd"> If true all output to the console is omitted (default False).</span>
<span class="sd"> initial_guess : list</span>
<span class="sd"> can provide an initial guess for the input parameters. Relevant for</span>
<span class="sd"> non-linear fits with many parameters.</span>
<span class="sd"> method : str</span>
<span class="sd"> can be used to choose an alternative method for the minimization of chisquare.</span>
<span class="sd"> The possible methods are the ones which can be used for scipy.optimize.minimize and</span>
<span class="sd"> migrad of iminuit. If no method is specified, Levenberg-Marquard is used.</span>
<span class="sd"> Reliable alternatives are migrad, Powell and Nelder-Mead.</span>
<span class="sd"> resplot : bool</span>
<span class="sd"> If true, a plot which displays fit, data and residuals is generated (default False).</span>
<span class="sd"> qqplot : bool</span>
<span class="sd"> If true, a quantile-quantile plot of the fit result is generated (default False).</span>
<span class="sd"> expected_chisquare : bool</span>
<span class="sd"> If true prints the expected chisquare which is</span>
<span class="sd"> corrected by effects caused by correlated input data.</span>
<span class="sd"> This can take a while as the full correlation matrix</span>
<span class="sd"> has to be calculated (default False).</span>
<span class="sd"> correlated_fit : bool</span>
<span class="sd"> If true, use the full correlation matrix in the definition of the chisquare</span>
<span class="sd"> (only works for prior==None and when no method is given, at the moment).</span>
<span class="sd"> const_par : list, optional</span>
<span class="sd"> List of N Obs that are used to constrain the last N fit parameters of func.</span>
<span class="sd"> &#39;&#39;&#39;</span>
<span class="k">if</span> <span class="n">priors</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span><span class="p">:</span>
<span class="k">return</span> <span class="n">_prior_fit</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="n">priors</span><span class="p">,</span> <span class="n">silent</span><span class="o">=</span><span class="n">silent</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
<span class="k">else</span><span class="p">:</span>
<span class="k">return</span> <span class="n">_standard_fit</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="n">silent</span><span class="o">=</span><span class="n">silent</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
</pre></div>
</details>
<div class="docstring"><p>Performs a non-linear fit to y = func(x).</p>
<h6 id="parameters">Parameters</h6>
<ul>
<li><strong>x</strong> (list):
list of floats.</li>
<li><strong>y</strong> (list):
list of Obs.</li>
<li><p><strong>func</strong> (object):
fit function, has to be of the form</p>
<div class="codehilite"><pre><span></span><code><span class="k">def</span> <span class="nf">func</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
<span class="n">y</span> <span class="o">=</span> <span class="n">a</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">+</span> <span class="n">a</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">*</span> <span class="n">x</span> <span class="o">+</span> <span class="n">a</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span> <span class="o">*</span> <span class="n">anp</span><span class="o">.</span><span class="n">sinh</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
<span class="k">return</span> <span class="n">y</span>
</code></pre></div>
<p>For multiple x values func can be of the form</p>
<div class="codehilite"><pre><span></span><code><span class="k">def</span> <span class="nf">func</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
<span class="p">(</span><span class="n">x1</span><span class="p">,</span> <span class="n">x2</span><span class="p">)</span> <span class="o">=</span> <span class="n">x</span>
<span class="k">return</span> <span class="n">a</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="n">x1</span> <span class="o">**</span> <span class="mi">2</span> <span class="o">+</span> <span class="n">a</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">*</span> <span class="n">x2</span>
</code></pre></div>
<p>It is important that all numpy functions refer to autograd.numpy, otherwise the differentiation
will not work</p></li>
<li><strong>priors</strong> (list, optional):
priors has to be a list with an entry for every parameter in the fit. The entries can either be
Obs (e.g. results from a previous fit) or strings containing a value and an error formatted like
0.548(23), 500(40) or 0.5(0.4)</li>
<li><strong>silent</strong> (bool, optional):
If true all output to the console is omitted (default False).</li>
<li><strong>initial_guess</strong> (list):
can provide an initial guess for the input parameters. Relevant for
non-linear fits with many parameters.</li>
<li><strong>method</strong> (str):
can be used to choose an alternative method for the minimization of chisquare.
The possible methods are the ones which can be used for scipy.optimize.minimize and
migrad of iminuit. If no method is specified, Levenberg-Marquard is used.
Reliable alternatives are migrad, Powell and Nelder-Mead.</li>
<li><strong>resplot</strong> (bool):
If true, a plot which displays fit, data and residuals is generated (default False).</li>
<li><strong>qqplot</strong> (bool):
If true, a quantile-quantile plot of the fit result is generated (default False).</li>
<li><strong>expected_chisquare</strong> (bool):
If true prints the expected chisquare which is
corrected by effects caused by correlated input data.
This can take a while as the full correlation matrix
has to be calculated (default False).</li>
<li><strong>correlated_fit</strong> (bool):
If true, use the full correlation matrix in the definition of the chisquare
(only works for prior==None and when no method is given, at the moment).</li>
<li><strong>const_par</strong> (list, optional):
List of N Obs that are used to constrain the last N fit parameters of func.</li>
</ul>
</div>
</section>
<section id="total_least_squares">
<div class="attr function"><a class="headerlink" href="#total_least_squares">#&nbsp;&nbsp</a>
<span class="def">def</span>
<span class="name">total_least_squares</span><span class="signature">(x, y, func, silent=False, **kwargs)</span>:
</div>
<details>
<summary>View Source</summary>
<div class="codehilite"><pre><span></span><span class="k">def</span> <span class="nf">total_least_squares</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="n">silent</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="sa">r</span><span class="sd">&#39;&#39;&#39;Performs a non-linear fit to y = func(x) and returns a list of Obs corresponding to the fit parameters.</span>
<span class="sd"> Parameters</span>
<span class="sd"> ----------</span>
<span class="sd"> x : list</span>
<span class="sd"> list of Obs, or a tuple of lists of Obs</span>
<span class="sd"> y : list</span>
<span class="sd"> list of Obs. The dvalues of the Obs are used as x- and yerror for the fit.</span>
<span class="sd"> func : object</span>
<span class="sd"> func has to be of the form</span>
<span class="sd"> ```python</span>
<span class="sd"> def func(a, x):</span>
<span class="sd"> y = a[0] + a[1] * x + a[2] * anp.sinh(x)</span>
<span class="sd"> return y</span>
<span class="sd"> ```</span>
<span class="sd"> For multiple x values func can be of the form</span>
<span class="sd"> ```python</span>
<span class="sd"> def func(a, x):</span>
<span class="sd"> (x1, x2) = x</span>
<span class="sd"> return a[0] * x1 ** 2 + a[1] * x2</span>
<span class="sd"> ```</span>
<span class="sd"> It is important that all numpy functions refer to autograd.numpy, otherwise the differentiation</span>
<span class="sd"> will not work.</span>
<span class="sd"> silent : bool, optional</span>
<span class="sd"> If true all output to the console is omitted (default False).</span>
<span class="sd"> initial_guess : list</span>
<span class="sd"> can provide an initial guess for the input parameters. Relevant for non-linear</span>
<span class="sd"> fits with many parameters.</span>
<span class="sd"> expected_chisquare : bool</span>
<span class="sd"> If true prints the expected chisquare which is</span>
<span class="sd"> corrected by effects caused by correlated input data.</span>
<span class="sd"> This can take a while as the full correlation matrix</span>
<span class="sd"> has to be calculated (default False).</span>
<span class="sd"> const_par : list, optional</span>
<span class="sd"> List of N Obs that are used to constrain the last N fit parameters of func.</span>
<span class="sd"> Based on the orthogonal distance regression module of scipy</span>
<span class="sd"> &#39;&#39;&#39;</span>
<span class="n">output</span> <span class="o">=</span> <span class="n">Fit_result</span><span class="p">()</span>
<span class="n">output</span><span class="o">.</span><span class="n">fit_function</span> <span class="o">=</span> <span class="n">func</span>
<span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
<span class="n">x_shape</span> <span class="o">=</span> <span class="n">x</span><span class="o">.</span><span class="n">shape</span>
<span class="k">if</span> <span class="ow">not</span> <span class="n">callable</span><span class="p">(</span><span class="n">func</span><span class="p">):</span>
<span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s1">&#39;func has to be a function.&#39;</span><span class="p">)</span>
<span class="n">func_aug</span> <span class="o">=</span> <span class="n">func</span>
<span class="k">if</span> <span class="s1">&#39;const_par&#39;</span> <span class="ow">in</span> <span class="n">kwargs</span><span class="p">:</span>
<span class="n">const_par</span> <span class="o">=</span> <span class="n">kwargs</span><span class="p">[</span><span class="s1">&#39;const_par&#39;</span><span class="p">]</span>
<span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">const_par</span><span class="p">,</span> <span class="n">Obs</span><span class="p">):</span>
<span class="n">const_par</span> <span class="o">=</span> <span class="p">[</span><span class="n">const_par</span><span class="p">]</span>
<span class="k">def</span> <span class="nf">func</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
<span class="k">return</span> <span class="n">func_aug</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">p</span><span class="p">,</span> <span class="p">[</span><span class="n">o</span><span class="o">.</span><span class="n">value</span> <span class="k">for</span> <span class="n">o</span> <span class="ow">in</span> <span class="n">const_par</span><span class="p">])),</span> <span class="n">x</span><span class="p">)</span>
<span class="k">else</span><span class="p">:</span>
<span class="n">const_par</span> <span class="o">=</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="mi">25</span><span class="p">):</span>
<span class="k">try</span><span class="p">:</span>
<span class="n">func</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">i</span><span class="p">),</span> <span class="n">x</span><span class="o">.</span><span class="n">T</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
<span class="k">except</span> <span class="ne">Exception</span><span class="p">:</span>
<span class="k">pass</span>
<span class="k">else</span><span class="p">:</span>
<span class="k">break</span>
<span class="n">n_parms</span> <span class="o">=</span> <span class="n">i</span>
<span class="k">if</span> <span class="ow">not</span> <span class="n">silent</span><span class="p">:</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;Fit with&#39;</span><span class="p">,</span> <span class="n">n_parms</span><span class="p">,</span> <span class="s1">&#39;parameters&#39;</span><span class="p">)</span>
<span class="k">if</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">const_par</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">):</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;</span><span class="se">\t</span><span class="s1"> and </span><span class="si">%d</span><span class="s1"> constrained parameter</span><span class="si">%s</span><span class="s1">&#39;</span> <span class="o">%</span> <span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">const_par</span><span class="p">),</span> <span class="s1">&#39;s&#39;</span> <span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">const_par</span><span class="p">)</span> <span class="o">&gt;</span> <span class="mi">1</span> <span class="k">else</span> <span class="s1">&#39;&#39;</span><span class="p">),</span> <span class="n">const_par</span><span class="p">)</span>
<span class="n">x_f</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">vectorize</span><span class="p">(</span><span class="k">lambda</span> <span class="n">o</span><span class="p">:</span> <span class="n">o</span><span class="o">.</span><span class="n">value</span><span class="p">)(</span><span class="n">x</span><span class="p">)</span>
<span class="n">dx_f</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">vectorize</span><span class="p">(</span><span class="k">lambda</span> <span class="n">o</span><span class="p">:</span> <span class="n">o</span><span class="o">.</span><span class="n">dvalue</span><span class="p">)(</span><span class="n">x</span><span class="p">)</span>
<span class="n">y_f</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="n">o</span><span class="o">.</span><span class="n">value</span> <span class="k">for</span> <span class="n">o</span> <span class="ow">in</span> <span class="n">y</span><span class="p">])</span>
<span class="n">dy_f</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</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">y</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">asarray</span><span class="p">(</span><span class="n">dx_f</span><span class="p">)</span> <span class="o">&lt;=</span> <span class="mf">0.0</span><span class="p">):</span>
<span class="k">raise</span> <span class="ne">Exception</span><span class="p">(</span><span class="s1">&#39;No x errors available, run the gamma method first.&#39;</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">asarray</span><span class="p">(</span><span class="n">dy_f</span><span class="p">)</span> <span class="o">&lt;=</span> <span class="mf">0.0</span><span class="p">):</span>
<span class="k">raise</span> <span class="ne">Exception</span><span class="p">(</span><span class="s1">&#39;No y errors available, run the gamma method first.&#39;</span><span class="p">)</span>
<span class="k">if</span> <span class="s1">&#39;initial_guess&#39;</span> <span class="ow">in</span> <span class="n">kwargs</span><span class="p">:</span>
<span class="n">x0</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;initial_guess&#39;</span><span class="p">)</span>
<span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">x0</span><span class="p">)</span> <span class="o">!=</span> <span class="n">n_parms</span><span class="p">:</span>
<span class="k">raise</span> <span class="ne">Exception</span><span class="p">(</span><span class="s1">&#39;Initial guess does not have the correct length: </span><span class="si">%d</span><span class="s1"> vs. </span><span class="si">%d</span><span class="s1">&#39;</span> <span class="o">%</span> <span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">x0</span><span class="p">),</span> <span class="n">n_parms</span><span class="p">))</span>
<span class="k">else</span><span class="p">:</span>
<span class="n">x0</span> <span class="o">=</span> <span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">*</span> <span class="n">n_parms</span>
<span class="n">data</span> <span class="o">=</span> <span class="n">RealData</span><span class="p">(</span><span class="n">x_f</span><span class="p">,</span> <span class="n">y_f</span><span class="p">,</span> <span class="n">sx</span><span class="o">=</span><span class="n">dx_f</span><span class="p">,</span> <span class="n">sy</span><span class="o">=</span><span class="n">dy_f</span><span class="p">)</span>
<span class="n">model</span> <span class="o">=</span> <span class="n">Model</span><span class="p">(</span><span class="n">func</span><span class="p">)</span>
<span class="n">odr</span> <span class="o">=</span> <span class="n">ODR</span><span class="p">(</span><span class="n">data</span><span class="p">,</span> <span class="n">model</span><span class="p">,</span> <span class="n">x0</span><span class="p">,</span> <span class="n">partol</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">odr</span><span class="o">.</span><span class="n">set_job</span><span class="p">(</span><span class="n">fit_type</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="n">deriv</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
<span class="n">out</span> <span class="o">=</span> <span class="n">odr</span><span class="o">.</span><span class="n">run</span><span class="p">()</span>
<span class="n">output</span><span class="o">.</span><span class="n">residual_variance</span> <span class="o">=</span> <span class="n">out</span><span class="o">.</span><span class="n">res_var</span>
<span class="n">output</span><span class="o">.</span><span class="n">method</span> <span class="o">=</span> <span class="s1">&#39;ODR&#39;</span>
<span class="n">output</span><span class="o">.</span><span class="n">message</span> <span class="o">=</span> <span class="n">out</span><span class="o">.</span><span class="n">stopreason</span>
<span class="n">output</span><span class="o">.</span><span class="n">xplus</span> <span class="o">=</span> <span class="n">out</span><span class="o">.</span><span class="n">xplus</span>
<span class="k">if</span> <span class="ow">not</span> <span class="n">silent</span><span class="p">:</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;Method: ODR&#39;</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="o">*</span><span class="n">out</span><span class="o">.</span><span class="n">stopreason</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;Residual variance:&#39;</span><span class="p">,</span> <span class="n">output</span><span class="o">.</span><span class="n">residual_variance</span><span class="p">)</span>
<span class="k">if</span> <span class="n">out</span><span class="o">.</span><span class="n">info</span> <span class="o">&gt;</span> <span class="mi">3</span><span class="p">:</span>
<span class="k">raise</span> <span class="ne">Exception</span><span class="p">(</span><span class="s1">&#39;The minimization procedure did not converge.&#39;</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">size</span>
<span class="n">n_parms_aug</span> <span class="o">=</span> <span class="n">n_parms</span> <span class="o">+</span> <span class="nb">len</span><span class="p">(</span><span class="n">const_par</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">odr_chisquare</span><span class="p">(</span><span class="n">p</span><span class="p">):</span>
<span class="n">model</span> <span class="o">=</span> <span class="n">func</span><span class="p">(</span><span class="n">p</span><span class="p">[:</span><span class="n">n_parms</span><span class="p">],</span> <span class="n">p</span><span class="p">[</span><span class="n">n_parms</span><span class="p">:]</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">x_shape</span><span class="p">))</span>
<span class="n">chisq</span> <span class="o">=</span> <span class="n">anp</span><span class="o">.</span><span class="n">sum</span><span class="p">(((</span><span class="n">y_f</span> <span class="o">-</span> <span class="n">model</span><span class="p">)</span> <span class="o">/</span> <span class="n">dy_f</span><span class="p">)</span> <span class="o">**</span> <span class="mi">2</span><span class="p">)</span> <span class="o">+</span> <span class="n">anp</span><span class="o">.</span><span class="n">sum</span><span class="p">(((</span><span class="n">x_f</span> <span class="o">-</span> <span class="n">p</span><span class="p">[</span><span class="n">n_parms</span><span class="p">:]</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">x_shape</span><span class="p">))</span> <span class="o">/</span> <span class="n">dx_f</span><span class="p">)</span> <span class="o">**</span> <span class="mi">2</span><span class="p">)</span>
<span class="k">return</span> <span class="n">chisq</span>
<span class="k">def</span> <span class="nf">odr_chisquare_aug</span><span class="p">(</span><span class="n">p</span><span class="p">):</span>
<span class="n">model</span> <span class="o">=</span> <span class="n">func_aug</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">p</span><span class="p">[:</span><span class="n">n_parms_aug</span><span class="p">],</span> <span class="p">[</span><span class="n">o</span><span class="o">.</span><span class="n">value</span> <span class="k">for</span> <span class="n">o</span> <span class="ow">in</span> <span class="n">const_par</span><span class="p">])),</span> <span class="n">p</span><span class="p">[</span><span class="n">n_parms_aug</span><span class="p">:]</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">x_shape</span><span class="p">))</span>
<span class="n">chisq</span> <span class="o">=</span> <span class="n">anp</span><span class="o">.</span><span class="n">sum</span><span class="p">(((</span><span class="n">y_f</span> <span class="o">-</span> <span class="n">model</span><span class="p">)</span> <span class="o">/</span> <span class="n">dy_f</span><span class="p">)</span> <span class="o">**</span> <span class="mi">2</span><span class="p">)</span> <span class="o">+</span> <span class="n">anp</span><span class="o">.</span><span class="n">sum</span><span class="p">(((</span><span class="n">x_f</span> <span class="o">-</span> <span class="n">p</span><span class="p">[</span><span class="n">n_parms_aug</span><span class="p">:]</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">x_shape</span><span class="p">))</span> <span class="o">/</span> <span class="n">dx_f</span><span class="p">)</span> <span class="o">**</span> <span class="mi">2</span><span class="p">)</span>
<span class="k">return</span> <span class="n">chisq</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;expected_chisquare&#39;</span><span class="p">)</span> <span class="ow">is</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">np</span><span class="o">.</span><span class="n">concatenate</span><span class="p">((</span><span class="n">dy_f</span><span class="o">.</span><span class="n">ravel</span><span class="p">(),</span> <span class="n">dx_f</span><span class="o">.</span><span class="n">ravel</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;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">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>
<span class="n">old_jac</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">out</span><span class="o">.</span><span class="n">beta</span><span class="p">,</span> <span class="n">out</span><span class="o">.</span><span class="n">xplus</span><span class="p">)</span>
<span class="n">fused_row1</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">concatenate</span><span class="p">((</span><span class="n">old_jac</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">number_of_x_parameters</span> <span class="o">*</span> <span class="p">[</span><span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">old_jac</span><span class="o">.</span><span class="n">shape</span><span class="p">)]),</span> <span class="n">axis</span><span class="o">=</span><span class="mi">0</span><span class="p">)))</span>
<span class="n">fused_row2</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">concatenate</span><span class="p">((</span><span class="n">jacobian</span><span class="p">(</span><span class="k">lambda</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">:</span> <span class="n">func</span><span class="p">(</span><span class="n">y</span><span class="p">,</span> <span class="n">x</span><span class="p">))(</span><span class="n">out</span><span class="o">.</span><span class="n">xplus</span><span class="p">,</span> <span class="n">out</span><span class="o">.</span><span class="n">beta</span><span class="p">)</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</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> <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> <span class="o">*</span> <span class="n">number_of_x_parameters</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">number_of_x_parameters</span> <span class="o">*</span> <span class="n">old_jac</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">])))</span>
<span class="n">new_jac</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">concatenate</span><span class="p">((</span><span class="n">fused_row1</span><span class="p">,</span> <span class="n">fused_row2</span><span class="p">),</span> <span class="n">axis</span><span class="o">=</span><span class="mi">1</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">new_jac</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">P_phi</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</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>
<span class="k">if</span> <span class="n">expected_chisquare</span> <span class="o">&lt;=</span> <span class="mf">0.0</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;Negative expected_chisquare.&quot;</span><span class="p">,</span> <span class="ne">RuntimeWarning</span><span class="p">)</span>
<span class="n">expected_chisquare</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">expected_chisquare</span><span class="p">)</span>
<span class="n">output</span><span class="o">.</span><span class="n">chisquare_by_expected_chisquare</span> <span class="o">=</span> <span class="n">odr_chisquare</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">out</span><span class="o">.</span><span class="n">beta</span><span class="p">,</span> <span class="n">out</span><span class="o">.</span><span class="n">xplus</span><span class="o">.</span><span class="n">ravel</span><span class="p">())))</span> <span class="o">/</span> <span class="n">expected_chisquare</span>
<span class="k">if</span> <span class="ow">not</span> <span class="n">silent</span><span class="p">:</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;chisquare/expected_chisquare:&#39;</span><span class="p">,</span>
<span class="n">output</span><span class="o">.</span><span class="n">chisquare_by_expected_chisquare</span><span class="p">)</span>
<span class="n">fitp</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">concatenate</span><span class="p">((</span><span class="n">out</span><span class="o">.</span><span class="n">beta</span><span class="p">,</span> <span class="p">[</span><span class="n">o</span><span class="o">.</span><span class="n">value</span> <span class="k">for</span> <span class="n">o</span> <span class="ow">in</span> <span class="n">const_par</span><span class="p">]))</span>
<span class="n">hess_inv</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">pinv</span><span class="p">(</span><span class="n">jacobian</span><span class="p">(</span><span class="n">jacobian</span><span class="p">(</span><span class="n">odr_chisquare_aug</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">fitp</span><span class="p">,</span> <span class="n">out</span><span class="o">.</span><span class="n">xplus</span><span class="o">.</span><span class="n">ravel</span><span class="p">()))))</span>
<span class="k">def</span> <span class="nf">odr_chisquare_compact_x</span><span class="p">(</span><span class="n">d</span><span class="p">):</span>
<span class="n">model</span> <span class="o">=</span> <span class="n">func_aug</span><span class="p">(</span><span class="n">d</span><span class="p">[:</span><span class="n">n_parms_aug</span><span class="p">],</span> <span class="n">d</span><span class="p">[</span><span class="n">n_parms_aug</span><span class="p">:</span><span class="n">n_parms_aug</span> <span class="o">+</span> <span class="n">m</span><span class="p">]</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">x_shape</span><span class="p">))</span>
<span class="n">chisq</span> <span class="o">=</span> <span class="n">anp</span><span class="o">.</span><span class="n">sum</span><span class="p">(((</span><span class="n">y_f</span> <span class="o">-</span> <span class="n">model</span><span class="p">)</span> <span class="o">/</span> <span class="n">dy_f</span><span class="p">)</span> <span class="o">**</span> <span class="mi">2</span><span class="p">)</span> <span class="o">+</span> <span class="n">anp</span><span class="o">.</span><span class="n">sum</span><span class="p">(((</span><span class="n">d</span><span class="p">[</span><span class="n">n_parms_aug</span> <span class="o">+</span> <span class="n">m</span><span class="p">:]</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">x_shape</span><span class="p">)</span> <span class="o">-</span> <span class="n">d</span><span class="p">[</span><span class="n">n_parms_aug</span><span class="p">:</span><span class="n">n_parms_aug</span> <span class="o">+</span> <span class="n">m</span><span class="p">]</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">x_shape</span><span class="p">))</span> <span class="o">/</span> <span class="n">dx_f</span><span class="p">)</span> <span class="o">**</span> <span class="mi">2</span><span class="p">)</span>
<span class="k">return</span> <span class="n">chisq</span>
<span class="n">jac_jac_x</span> <span class="o">=</span> <span class="n">jacobian</span><span class="p">(</span><span class="n">jacobian</span><span class="p">(</span><span class="n">odr_chisquare_compact_x</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">fitp</span><span class="p">,</span> <span class="n">out</span><span class="o">.</span><span class="n">xplus</span><span class="o">.</span><span class="n">ravel</span><span class="p">(),</span> <span class="n">x_f</span><span class="o">.</span><span class="n">ravel</span><span class="p">())))</span>
<span class="n">deriv_x</span> <span class="o">=</span> <span class="o">-</span><span class="n">hess_inv</span> <span class="o">@</span> <span class="n">jac_jac_x</span><span class="p">[:</span><span class="n">n_parms_aug</span> <span class="o">+</span> <span class="n">m</span><span class="p">,</span> <span class="n">n_parms_aug</span> <span class="o">+</span> <span class="n">m</span><span class="p">:]</span>
<span class="k">def</span> <span class="nf">odr_chisquare_compact_y</span><span class="p">(</span><span class="n">d</span><span class="p">):</span>
<span class="n">model</span> <span class="o">=</span> <span class="n">func_aug</span><span class="p">(</span><span class="n">d</span><span class="p">[:</span><span class="n">n_parms_aug</span><span class="p">],</span> <span class="n">d</span><span class="p">[</span><span class="n">n_parms_aug</span><span class="p">:</span><span class="n">n_parms_aug</span> <span class="o">+</span> <span class="n">m</span><span class="p">]</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">x_shape</span><span class="p">))</span>
<span class="n">chisq</span> <span class="o">=</span> <span class="n">anp</span><span class="o">.</span><span class="n">sum</span><span class="p">(((</span><span class="n">d</span><span class="p">[</span><span class="n">n_parms_aug</span> <span class="o">+</span> <span class="n">m</span><span class="p">:]</span> <span class="o">-</span> <span class="n">model</span><span class="p">)</span> <span class="o">/</span> <span class="n">dy_f</span><span class="p">)</span> <span class="o">**</span> <span class="mi">2</span><span class="p">)</span> <span class="o">+</span> <span class="n">anp</span><span class="o">.</span><span class="n">sum</span><span class="p">(((</span><span class="n">x_f</span> <span class="o">-</span> <span class="n">d</span><span class="p">[</span><span class="n">n_parms_aug</span><span class="p">:</span><span class="n">n_parms_aug</span> <span class="o">+</span> <span class="n">m</span><span class="p">]</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">x_shape</span><span class="p">))</span> <span class="o">/</span> <span class="n">dx_f</span><span class="p">)</span> <span class="o">**</span> <span class="mi">2</span><span class="p">)</span>
<span class="k">return</span> <span class="n">chisq</span>
<span class="n">jac_jac_y</span> <span class="o">=</span> <span class="n">jacobian</span><span class="p">(</span><span class="n">jacobian</span><span class="p">(</span><span class="n">odr_chisquare_compact_y</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">fitp</span><span class="p">,</span> <span class="n">out</span><span class="o">.</span><span class="n">xplus</span><span class="o">.</span><span class="n">ravel</span><span class="p">(),</span> <span class="n">y_f</span><span class="p">)))</span>
<span class="n">deriv_y</span> <span class="o">=</span> <span class="o">-</span><span class="n">hess_inv</span> <span class="o">@</span> <span class="n">jac_jac_y</span><span class="p">[:</span><span class="n">n_parms_aug</span> <span class="o">+</span> <span class="n">m</span><span class="p">,</span> <span class="n">n_parms_aug</span> <span class="o">+</span> <span class="n">m</span><span class="p">:]</span>
<span class="n">result</span> <span class="o">=</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">n_parms</span><span class="p">):</span>
<span class="n">result</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">derived_observable</span><span class="p">(</span><span class="k">lambda</span> <span class="n">my_var</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">:</span> <span class="p">(</span><span class="n">my_var</span><span class="p">[</span><span class="mi">0</span><span class="p">]</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="o">/</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="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">value</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="o">*</span> <span class="n">out</span><span class="o">.</span><span class="n">beta</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="nb">list</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="o">+</span> <span class="nb">list</span><span class="p">(</span><span class="n">y</span><span class="p">),</span> <span class="n">man_grad</span><span class="o">=</span><span class="nb">list</span><span class="p">(</span><span class="n">deriv_x</span><span class="p">[</span><span class="n">i</span><span class="p">])</span> <span class="o">+</span> <span class="nb">list</span><span class="p">(</span><span class="n">deriv_y</span><span class="p">[</span><span class="n">i</span><span class="p">])))</span>
<span class="n">output</span><span class="o">.</span><span class="n">fit_parameters</span> <span class="o">=</span> <span class="n">result</span> <span class="o">+</span> <span class="n">const_par</span>
<span class="n">output</span><span class="o">.</span><span class="n">odr_chisquare</span> <span class="o">=</span> <span class="n">odr_chisquare</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">out</span><span class="o">.</span><span class="n">beta</span><span class="p">,</span> <span class="n">out</span><span class="o">.</span><span class="n">xplus</span><span class="o">.</span><span class="n">ravel</span><span class="p">())))</span>
<span class="n">output</span><span class="o">.</span><span class="n">dof</span> <span class="o">=</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">n_parms</span>
<span class="k">return</span> <span class="n">output</span>
</pre></div>
</details>
<div class="docstring"><p>Performs a non-linear fit to y = func(x) and returns a list of Obs corresponding to the fit parameters.</p>
<h6 id="parameters">Parameters</h6>
<ul>
<li><strong>x</strong> (list):
list of Obs, or a tuple of lists of Obs</li>
<li><strong>y</strong> (list):
list of Obs. The dvalues of the Obs are used as x- and yerror for the fit.</li>
<li><p><strong>func</strong> (object):
func has to be of the form</p>
<div class="codehilite"><pre><span></span><code><span class="k">def</span> <span class="nf">func</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
<span class="n">y</span> <span class="o">=</span> <span class="n">a</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">+</span> <span class="n">a</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">*</span> <span class="n">x</span> <span class="o">+</span> <span class="n">a</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span> <span class="o">*</span> <span class="n">anp</span><span class="o">.</span><span class="n">sinh</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
<span class="k">return</span> <span class="n">y</span>
</code></pre></div>
<p>For multiple x values func can be of the form</p>
<div class="codehilite"><pre><span></span><code><span class="k">def</span> <span class="nf">func</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
<span class="p">(</span><span class="n">x1</span><span class="p">,</span> <span class="n">x2</span><span class="p">)</span> <span class="o">=</span> <span class="n">x</span>
<span class="k">return</span> <span class="n">a</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="n">x1</span> <span class="o">**</span> <span class="mi">2</span> <span class="o">+</span> <span class="n">a</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">*</span> <span class="n">x2</span>
</code></pre></div>
<p>It is important that all numpy functions refer to autograd.numpy, otherwise the differentiation
will not work.</p></li>
<li><strong>silent</strong> (bool, optional):
If true all output to the console is omitted (default False).</li>
<li><strong>initial_guess</strong> (list):
can provide an initial guess for the input parameters. Relevant for non-linear
fits with many parameters.</li>
<li><strong>expected_chisquare</strong> (bool):
If true prints the expected chisquare which is
corrected by effects caused by correlated input data.
This can take a while as the full correlation matrix
has to be calculated (default False).</li>
<li><strong>const_par</strong> (list, optional):
List of N Obs that are used to constrain the last N fit parameters of func.</li>
<li><strong>Based on the orthogonal distance regression module of scipy</strong></li>
</ul>
</div>
</section>
<section id="prior_fit">
<div class="attr function"><a class="headerlink" href="#prior_fit">#&nbsp;&nbsp</a>
<span class="def">def</span>
<span class="name">prior_fit</span><span class="signature">(x, y, func, priors, silent=False, **kwargs)</span>:
</div>
<details>
<summary>View Source</summary>
<div class="codehilite"><pre><span></span><span class="k">def</span> <span class="nf">prior_fit</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="n">priors</span><span class="p">,</span> <span class="n">silent</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="n">warnings</span><span class="o">.</span><span class="n">warn</span><span class="p">(</span><span class="s2">&quot;prior_fit renamed to least_squares&quot;</span><span class="p">,</span> <span class="ne">DeprecationWarning</span><span class="p">)</span>
<span class="k">return</span> <span class="n">least_squares</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="n">priors</span><span class="o">=</span><span class="n">priors</span><span class="p">,</span> <span class="n">silent</span><span class="o">=</span><span class="n">silent</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
</pre></div>
</details>
</section>
<section id="standard_fit">
<div class="attr function"><a class="headerlink" href="#standard_fit">#&nbsp;&nbsp</a>
<span class="def">def</span>
<span class="name">standard_fit</span><span class="signature">(x, y, func, silent=False, **kwargs)</span>:
</div>
<details>
<summary>View Source</summary>
<div class="codehilite"><pre><span></span><span class="k">def</span> <span class="nf">standard_fit</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="n">silent</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="n">warnings</span><span class="o">.</span><span class="n">warn</span><span class="p">(</span><span class="s2">&quot;standard_fit renamed to least_squares&quot;</span><span class="p">,</span> <span class="ne">DeprecationWarning</span><span class="p">)</span>
<span class="k">return</span> <span class="n">least_squares</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="n">silent</span><span class="o">=</span><span class="n">silent</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
</pre></div>
</details>
</section>
<section id="odr_fit">
<div class="attr function"><a class="headerlink" href="#odr_fit">#&nbsp;&nbsp</a>
<span class="def">def</span>
<span class="name">odr_fit</span><span class="signature">(x, y, func, silent=False, **kwargs)</span>:
</div>
<details>
<summary>View Source</summary>
<div class="codehilite"><pre><span></span><span class="k">def</span> <span class="nf">odr_fit</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="n">silent</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="n">warnings</span><span class="o">.</span><span class="n">warn</span><span class="p">(</span><span class="s2">&quot;odr_fit renamed to total_least_squares&quot;</span><span class="p">,</span> <span class="ne">DeprecationWarning</span><span class="p">)</span>
<span class="k">return</span> <span class="n">total_least_squares</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="n">silent</span><span class="o">=</span><span class="n">silent</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
</pre></div>
</details>
</section>
<section id="fit_lin">
<div class="attr function"><a class="headerlink" href="#fit_lin">#&nbsp;&nbsp</a>
<span class="def">def</span>
<span class="name">fit_lin</span><span class="signature">(x, y, **kwargs)</span>:
</div>
<details>
<summary>View Source</summary>
<div class="codehilite"><pre><span></span><span class="k">def</span> <span class="nf">fit_lin</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
<span class="sd">&quot;&quot;&quot;Performs a linear fit to y = n + m * x and returns two Obs n, m.</span>
<span class="sd"> y has to be a list of Obs, the dvalues of the Obs are used as yerror for the fit.</span>
<span class="sd"> x can either be a list of floats in which case no xerror is assumed, or</span>
<span class="sd"> a list of Obs, where the dvalues of the Obs are used as xerror for the fit.</span>
<span class="sd"> &quot;&quot;&quot;</span>
<span class="k">def</span> <span class="nf">f</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
<span class="n">y</span> <span class="o">=</span> <span class="n">a</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">+</span> <span class="n">a</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">*</span> <span class="n">x</span>
<span class="k">return</span> <span class="n">y</span>
<span class="k">if</span> <span class="nb">all</span><span class="p">(</span><span class="nb">isinstance</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">Obs</span><span class="p">)</span> <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="n">x</span><span class="p">):</span>
<span class="n">out</span> <span class="o">=</span> <span class="n">total_least_squares</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
<span class="k">return</span> <span class="n">out</span><span class="o">.</span><span class="n">fit_parameters</span>
<span class="k">elif</span> <span class="nb">all</span><span class="p">(</span><span class="nb">isinstance</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="nb">float</span><span class="p">)</span> <span class="ow">or</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="nb">int</span><span class="p">)</span> <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="n">x</span><span class="p">)</span> <span class="ow">or</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">ndarray</span><span class="p">):</span>
<span class="n">out</span> <span class="o">=</span> <span class="n">least_squares</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
<span class="k">return</span> <span class="n">out</span><span class="o">.</span><span class="n">fit_parameters</span>
<span class="k">else</span><span class="p">:</span>
<span class="k">raise</span> <span class="ne">Exception</span><span class="p">(</span><span class="s1">&#39;Unsupported types for x&#39;</span><span class="p">)</span>
</pre></div>
</details>
<div class="docstring"><p>Performs a linear fit to y = n + m * x and returns two Obs n, m.</p>
<p>y has to be a list of Obs, the dvalues of the Obs are used as yerror for the fit.
x can either be a list of floats in which case no xerror is assumed, or
a list of Obs, where the dvalues of the Obs are used as xerror for the fit.</p>
</div>
</section>
<section id="qqplot">
<div class="attr function"><a class="headerlink" href="#qqplot">#&nbsp;&nbsp</a>
<span class="def">def</span>
<span class="name">qqplot</span><span class="signature">(x, o_y, func, p)</span>:
</div>
<details>
<summary>View Source</summary>
<div class="codehilite"><pre><span></span><span class="k">def</span> <span class="nf">qqplot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">o_y</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="n">p</span><span class="p">):</span>
<span class="sd">&quot;&quot;&quot; Generates a quantile-quantile plot of the fit result which can be used to</span>
<span class="sd"> check if the residuals of the fit are gaussian distributed.</span>
<span class="sd"> &quot;&quot;&quot;</span>
<span class="n">residuals</span> <span class="o">=</span> <span class="p">[]</span>
<span class="k">for</span> <span class="n">i_x</span><span class="p">,</span> <span class="n">i_y</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">o_y</span><span class="p">):</span>
<span class="n">residuals</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">i_y</span> <span class="o">-</span> <span class="n">func</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">i_x</span><span class="p">))</span> <span class="o">/</span> <span class="n">i_y</span><span class="o">.</span><span class="n">dvalue</span><span class="p">)</span>
<span class="n">residuals</span> <span class="o">=</span> <span class="nb">sorted</span><span class="p">(</span><span class="n">residuals</span><span class="p">)</span>
<span class="n">my_y</span> <span class="o">=</span> <span class="p">[</span><span class="n">o</span><span class="o">.</span><span class="n">value</span> <span class="k">for</span> <span class="n">o</span> <span class="ow">in</span> <span class="n">residuals</span><span class="p">]</span>
<span class="n">probplot</span> <span class="o">=</span> <span class="n">scipy</span><span class="o">.</span><span class="n">stats</span><span class="o">.</span><span class="n">probplot</span><span class="p">(</span><span class="n">my_y</span><span class="p">)</span>
<span class="n">my_x</span> <span class="o">=</span> <span class="n">probplot</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span>
<span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">8</span><span class="p">,</span> <span class="mi">8</span> <span class="o">/</span> <span class="mf">1.618</span><span class="p">))</span>
<span class="n">plt</span><span class="o">.</span><span class="n">errorbar</span><span class="p">(</span><span class="n">my_x</span><span class="p">,</span> <span class="n">my_y</span><span class="p">,</span> <span class="n">fmt</span><span class="o">=</span><span class="s1">&#39;o&#39;</span><span class="p">)</span>
<span class="n">fit_start</span> <span class="o">=</span> <span class="n">my_x</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
<span class="n">fit_stop</span> <span class="o">=</span> <span class="n">my_x</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
<span class="n">samples</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">fit_start</span><span class="p">,</span> <span class="n">fit_stop</span><span class="p">,</span> <span class="mf">0.01</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">samples</span><span class="p">,</span> <span class="n">samples</span><span class="p">,</span> <span class="s1">&#39;k--&#39;</span><span class="p">,</span> <span class="n">zorder</span><span class="o">=</span><span class="mi">11</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s1">&#39;Standard normal distribution&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">samples</span><span class="p">,</span> <span class="n">probplot</span><span class="p">[</span><span class="mi">1</span><span class="p">][</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="n">samples</span> <span class="o">+</span> <span class="n">probplot</span><span class="p">[</span><span class="mi">1</span><span class="p">][</span><span class="mi">1</span><span class="p">],</span> <span class="n">zorder</span><span class="o">=</span><span class="mi">10</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s1">&#39;Least squares fit, r=&#39;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">around</span><span class="p">(</span><span class="n">probplot</span><span class="p">[</span><span class="mi">1</span><span class="p">][</span><span class="mi">2</span><span class="p">],</span> <span class="mi">3</span><span class="p">)))</span>
<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s1">&#39;Theoretical quantiles&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s1">&#39;Ordered Values&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
<span class="n">plt</span><span class="o">.</span><span class="n">draw</span><span class="p">()</span>
</pre></div>
</details>
<div class="docstring"><p>Generates a quantile-quantile plot of the fit result which can be used to
check if the residuals of the fit are gaussian distributed.</p>
</div>
</section>
<section id="residual_plot">
<div class="attr function"><a class="headerlink" href="#residual_plot">#&nbsp;&nbsp</a>
<span class="def">def</span>
<span class="name">residual_plot</span><span class="signature">(x, y, func, fit_res)</span>:
</div>
<details>
<summary>View Source</summary>
<div class="codehilite"><pre><span></span><span class="k">def</span> <span class="nf">residual_plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">func</span><span class="p">,</span> <span class="n">fit_res</span><span class="p">):</span>
<span class="sd">&quot;&quot;&quot; Generates a plot which compares the fit to the data and displays the corresponding residuals&quot;&quot;&quot;</span>
<span class="n">xstart</span> <span class="o">=</span> <span class="n">x</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">-</span> <span class="mf">0.5</span>
<span class="n">xstop</span> <span class="o">=</span> <span class="n">x</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">+</span> <span class="mf">0.5</span>
<span class="n">x_samples</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">xstart</span><span class="p">,</span> <span class="n">xstop</span><span class="p">,</span> <span class="mf">0.01</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">8</span><span class="p">,</span> <span class="mi">8</span> <span class="o">/</span> <span class="mf">1.618</span><span class="p">))</span>
<span class="n">gs</span> <span class="o">=</span> <span class="n">gridspec</span><span class="o">.</span><span class="n">GridSpec</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">height_ratios</span><span class="o">=</span><span class="p">[</span><span class="mi">3</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="n">wspace</span><span class="o">=</span><span class="mf">0.0</span><span class="p">,</span> <span class="n">hspace</span><span class="o">=</span><span class="mf">0.0</span><span class="p">)</span>
<span class="n">ax0</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="n">gs</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
<span class="n">ax0</span><span class="o">.</span><span class="n">errorbar</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="p">[</span><span class="n">o</span><span class="o">.</span><span class="n">value</span> <span class="k">for</span> <span class="n">o</span> <span class="ow">in</span> <span class="n">y</span><span class="p">],</span> <span class="n">yerr</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">y</span><span class="p">],</span> <span class="n">ls</span><span class="o">=</span><span class="s1">&#39;none&#39;</span><span class="p">,</span> <span class="n">fmt</span><span class="o">=</span><span class="s1">&#39;o&#39;</span><span class="p">,</span> <span class="n">capsize</span><span class="o">=</span><span class="mi">3</span><span class="p">,</span> <span class="n">markersize</span><span class="o">=</span><span class="mi">5</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s1">&#39;Data&#39;</span><span class="p">)</span>
<span class="n">ax0</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x_samples</span><span class="p">,</span> <span class="n">func</span><span class="p">([</span><span class="n">o</span><span class="o">.</span><span class="n">value</span> <span class="k">for</span> <span class="n">o</span> <span class="ow">in</span> <span class="n">fit_res</span><span class="p">],</span> <span class="n">x_samples</span><span class="p">),</span> <span class="n">label</span><span class="o">=</span><span class="s1">&#39;Fit&#39;</span><span class="p">,</span> <span class="n">zorder</span><span class="o">=</span><span class="mi">10</span><span class="p">,</span> <span class="n">ls</span><span class="o">=</span><span class="s1">&#39;-&#39;</span><span class="p">,</span> <span class="n">ms</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
<span class="n">ax0</span><span class="o">.</span><span class="n">set_xticklabels</span><span class="p">([])</span>
<span class="n">ax0</span><span class="o">.</span><span class="n">set_xlim</span><span class="p">([</span><span class="n">xstart</span><span class="p">,</span> <span class="n">xstop</span><span class="p">])</span>
<span class="n">ax0</span><span class="o">.</span><span class="n">set_xticklabels</span><span class="p">([])</span>
<span class="n">ax0</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
<span class="n">residuals</span> <span class="o">=</span> <span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">asarray</span><span class="p">([</span><span class="n">o</span><span class="o">.</span><span class="n">value</span> <span class="k">for</span> <span class="n">o</span> <span class="ow">in</span> <span class="n">y</span><span class="p">])</span> <span class="o">-</span> <span class="n">func</span><span class="p">([</span><span class="n">o</span><span class="o">.</span><span class="n">value</span> <span class="k">for</span> <span class="n">o</span> <span class="ow">in</span> <span class="n">fit_res</span><span class="p">],</span> <span class="n">x</span><span class="p">))</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">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">y</span><span class="p">])</span>
<span class="n">ax1</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="n">gs</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
<span class="n">ax1</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">residuals</span><span class="p">,</span> <span class="s1">&#39;ko&#39;</span><span class="p">,</span> <span class="n">ls</span><span class="o">=</span><span class="s1">&#39;none&#39;</span><span class="p">,</span> <span class="n">markersize</span><span class="o">=</span><span class="mi">5</span><span class="p">)</span>
<span class="n">ax1</span><span class="o">.</span><span class="n">tick_params</span><span class="p">(</span><span class="n">direction</span><span class="o">=</span><span class="s1">&#39;out&#39;</span><span class="p">)</span>
<span class="n">ax1</span><span class="o">.</span><span class="n">tick_params</span><span class="p">(</span><span class="n">axis</span><span class="o">=</span><span class="s2">&quot;x&quot;</span><span class="p">,</span> <span class="n">bottom</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span> <span class="n">top</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span> <span class="n">labelbottom</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
<span class="n">ax1</span><span class="o">.</span><span class="n">axhline</span><span class="p">(</span><span class="n">y</span><span class="o">=</span><span class="mf">0.0</span><span class="p">,</span> <span class="n">ls</span><span class="o">=</span><span class="s1">&#39;--&#39;</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;k&#39;</span><span class="p">,</span> <span class="n">marker</span><span class="o">=</span><span class="s2">&quot; &quot;</span><span class="p">)</span>
<span class="n">ax1</span><span class="o">.</span><span class="n">fill_between</span><span class="p">(</span><span class="n">x_samples</span><span class="p">,</span> <span class="o">-</span><span class="mf">1.0</span><span class="p">,</span> <span class="mf">1.0</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.1</span><span class="p">,</span> <span class="n">facecolor</span><span class="o">=</span><span class="s1">&#39;k&#39;</span><span class="p">)</span>
<span class="n">ax1</span><span class="o">.</span><span class="n">set_xlim</span><span class="p">([</span><span class="n">xstart</span><span class="p">,</span> <span class="n">xstop</span><span class="p">])</span>
<span class="n">ax1</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">&#39;Residuals&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">subplots_adjust</span><span class="p">(</span><span class="n">wspace</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">hspace</span><span class="o">=</span><span class="kc">None</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">draw</span><span class="p">()</span>
</pre></div>
</details>
<div class="docstring"><p>Generates a plot which compares the fit to the data and displays the corresponding residuals</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="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.&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="k">return</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>
</pre></div>
</details>
<div class="docstring"><p>Returns the covariance matrix of y.</p>
</div>
</section>
<section id="error_band">
<div class="attr function"><a class="headerlink" href="#error_band">#&nbsp;&nbsp</a>
<span class="def">def</span>
<span class="name">error_band</span><span class="signature">(x, func, beta)</span>:
</div>
<details>
<summary>View Source</summary>
<div class="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="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>
<span class="n">deriv</span> <span class="o">=</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">x</span><span class="p">):</span>
<span class="n">deriv</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">egrad</span><span class="p">(</span><span class="n">func</span><span class="p">)([</span><span class="n">o</span><span class="o">.</span><span class="n">value</span> <span class="k">for</span> <span class="n">o</span> <span class="ow">in</span> <span class="n">beta</span><span class="p">],</span> <span class="n">item</span><span class="p">)))</span>
<span class="n">err</span> <span class="o">=</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">x</span><span class="p">):</span>
<span class="n">err</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="n">deriv</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">@</span> <span class="n">cov</span> <span class="o">@</span> <span class="n">deriv</span><span class="p">[</span><span class="n">i</span><span class="p">]))</span>
<span class="n">err</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">err</span><span class="p">)</span>
<span class="k">return</span> <span class="n">err</span>
</pre></div>
</details>
<div class="docstring"><p>Returns the error band for an array of sample values x, for given fit function func with optimized parameters beta.</p>
</div>
</section>
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