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<title>Calculation of confidence intervals — Non-Linear Least-Squares Minimization and Curve-Fitting for Python</title>
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<li><a class="reference internal" href="#">Calculation of confidence intervals</a><ul>
<li><a class="reference internal" href="#method-used-for-calculating-confidence-intervals">Method used for calculating confidence intervals</a></li>
<li><a class="reference internal" href="#a-basic-example">A basic example</a></li>
<li><a class="reference internal" href="#working-without-standard-error-estimates">Working without standard error estimates</a></li>
<li><a class="reference internal" href="#calculating-and-visualizing-maps-of-chi-2">Calculating and visualizing maps of <span class="math notranslate nohighlight">\(\chi^2\)</span></a></li>
<li><a class="reference internal" href="#an-advanced-example-for-evaluating-confidence-intervals">An advanced example for evaluating confidence intervals</a></li>
<li><a class="reference internal" href="#confidence-interval-functions">Confidence Interval Functions</a><ul>
<li><a class="reference internal" href="#lmfit.conf_interval"><code class="docutils literal notranslate"><span class="pre">conf_interval()</span></code></a></li>
<li><a class="reference internal" href="#lmfit.conf_interval2d"><code class="docutils literal notranslate"><span class="pre">conf_interval2d()</span></code></a></li>
<li><a class="reference internal" href="#lmfit.ci_report"><code class="docutils literal notranslate"><span class="pre">ci_report()</span></code></a></li>
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<section id="module-lmfit.confidence">
<span id="calculation-of-confidence-intervals"></span><span id="confidence-chapter"></span><h1>Calculation of confidence intervals<a class="headerlink" href="#module-lmfit.confidence" title="Link to this heading">¶</a></h1>
<p>The lmfit <code class="xref py py-mod docutils literal notranslate"><span class="pre">confidence</span></code> module allows you to explicitly calculate
confidence intervals for variable parameters. For most models, it is not
necessary since the estimation of the standard error from the estimated
covariance matrix is normally quite good.</p>
<p>But for some models, the sum of two exponentials for example, the approximation
begins to fail. For this case, lmfit has the function <code class="xref py py-func docutils literal notranslate"><span class="pre">conf_interval()</span></code>
to calculate confidence intervals directly. This is substantially slower
than using the errors estimated from the covariance matrix, but the results
are more robust.</p>
<p>Please note that <code class="xref py py-func docutils literal notranslate"><span class="pre">conf_interval()</span></code> is not suited to work with
fit results obtained by <code class="docutils literal notranslate"><span class="pre">emcee</span></code>.</p>
<section id="method-used-for-calculating-confidence-intervals">
<h2>Method used for calculating confidence intervals<a class="headerlink" href="#method-used-for-calculating-confidence-intervals" title="Link to this heading">¶</a></h2>
<p>The F-test is used to compare our null model, which is the best fit we have
found, with an alternate model, where one of the parameters is fixed to a
specific value. The value is changed until the difference between <span class="math notranslate nohighlight">\(\chi^2_0\)</span>
and <span class="math notranslate nohighlight">\(\chi^2_{f}\)</span> can’t be explained by the loss of a degree of freedom
within a certain confidence.</p>
<div class="math notranslate nohighlight">
\[F(P_{fix},N-P) = \left(\frac{\chi^2_f}{\chi^2_{0}}-1\right)\frac{N-P}{P_{fix}}\]</div>
<p><code class="docutils literal notranslate"><span class="pre">N</span></code> is the number of data points and <code class="docutils literal notranslate"><span class="pre">P</span></code> the number of parameters of the null model.
<span class="math notranslate nohighlight">\(P_{fix}\)</span> is the number of fixed parameters (or to be more clear, the
difference of number of parameters between our null model and the alternate
model).</p>
<p>Adding a log-likelihood method is under consideration.</p>
</section>
<section id="a-basic-example">
<h2>A basic example<a class="headerlink" href="#a-basic-example" title="Link to this heading">¶</a></h2>
<p>First we create an example problem:</p>
<div class="jupyter_cell jupyter_container docutils container">
<div class="cell_input code_cell docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></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">lmfit</span>
<span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mf">0.3</span><span class="p">,</span> <span class="mi">10</span><span class="p">,</span> <span class="mi">100</span><span class="p">)</span>
<span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">seed</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
<span class="n">y</span> <span class="o">=</span> <span class="mi">1</span><span class="o">/</span><span class="p">(</span><span class="mf">0.1</span><span class="o">*</span><span class="n">x</span><span class="p">)</span> <span class="o">+</span> <span class="mi">2</span> <span class="o">+</span> <span class="mf">0.1</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="n">x</span><span class="o">.</span><span class="n">size</span><span class="p">)</span>
<span class="n">pars</span> <span class="o">=</span> <span class="n">lmfit</span><span class="o">.</span><span class="n">Parameters</span><span class="p">()</span>
<span class="n">pars</span><span class="o">.</span><span class="n">add_many</span><span class="p">((</span><span class="s1">'a'</span><span class="p">,</span> <span class="mf">0.1</span><span class="p">),</span> <span class="p">(</span><span class="s1">'b'</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span>
<span class="k">def</span> <span class="nf">residual</span><span class="p">(</span><span class="n">p</span><span class="p">):</span>
<span class="k">return</span> <span class="mi">1</span><span class="o">/</span><span class="p">(</span><span class="n">p</span><span class="p">[</span><span class="s1">'a'</span><span class="p">]</span><span class="o">*</span><span class="n">x</span><span class="p">)</span> <span class="o">+</span> <span class="n">p</span><span class="p">[</span><span class="s1">'b'</span><span class="p">]</span> <span class="o">-</span> <span class="n">y</span>
</pre></div>
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<p>before we can generate the confidence intervals, we have to run a fit, so
that the automated estimate of the standard errors can be used as a
starting point:</p>
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<div class="cell_input code_cell docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">mini</span> <span class="o">=</span> <span class="n">lmfit</span><span class="o">.</span><span class="n">Minimizer</span><span class="p">(</span><span class="n">residual</span><span class="p">,</span> <span class="n">pars</span><span class="p">)</span>
<span class="n">result</span> <span class="o">=</span> <span class="n">mini</span><span class="o">.</span><span class="n">minimize</span><span class="p">()</span>
<span class="nb">print</span><span class="p">(</span><span class="n">lmfit</span><span class="o">.</span><span class="n">fit_report</span><span class="p">(</span><span class="n">result</span><span class="o">.</span><span class="n">params</span><span class="p">))</span>
</pre></div>
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<div class="output stream highlight-none notranslate"><div class="highlight"><pre><span></span>[[Variables]]
a: 0.09943896 +/- 1.9322e-04 (0.19%) (init = 0.1)
b: 1.98476942 +/- 0.01222678 (0.62%) (init = 1)
[[Correlations]] (unreported correlations are < 0.100)
C(a, b) = +0.6008
</pre></div>
</div>
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</div>
<p>Now it is just a simple function call to calculate the confidence
intervals:</p>
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<div class="cell_input code_cell docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">ci</span> <span class="o">=</span> <span class="n">lmfit</span><span class="o">.</span><span class="n">conf_interval</span><span class="p">(</span><span class="n">mini</span><span class="p">,</span> <span class="n">result</span><span class="p">)</span>
<span class="n">lmfit</span><span class="o">.</span><span class="n">printfuncs</span><span class="o">.</span><span class="n">report_ci</span><span class="p">(</span><span class="n">ci</span><span class="p">)</span>
</pre></div>
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<div class="cell_output docutils container">
<div class="output stream highlight-none notranslate"><div class="highlight"><pre><span></span> 99.73% 95.45% 68.27% _BEST_ 68.27% 95.45% 99.73%
a: -0.00059 -0.00039 -0.00019 0.09944 +0.00019 +0.00039 +0.00060
b: -0.03764 -0.02477 -0.01229 1.98477 +0.01229 +0.02477 +0.03764
</pre></div>
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</div>
</div>
<p>This shows the best-fit values for the parameters in the <code class="docutils literal notranslate"><span class="pre">_BEST_</span></code> column,
and parameter values that are at the varying confidence levels given by
steps in <span class="math notranslate nohighlight">\(\sigma\)</span>. As we can see, the estimated error is almost the
same, and the uncertainties are well behaved: Going from 1-<span class="math notranslate nohighlight">\(\sigma\)</span>
(68% confidence) to 3-<span class="math notranslate nohighlight">\(\sigma\)</span> (99.7% confidence) uncertainties is
fairly linear. It can also be seen that the errors are fairly symmetric
around the best fit value. For this problem, it is not necessary to
calculate confidence intervals, and the estimates of the uncertainties from
the covariance matrix are sufficient.</p>
</section>
<section id="working-without-standard-error-estimates">
<h2>Working without standard error estimates<a class="headerlink" href="#working-without-standard-error-estimates" title="Link to this heading">¶</a></h2>
<p>Sometimes the estimation of the standard errors from the covariance
matrix fails, especially if values are near given bounds. Hence, to
find the confidence intervals in these cases, it is necessary to set
the errors by hand. Note that the standard error is only used to find an
upper limit for each value, hence the exact value is not important.</p>
<p>To set the step-size to 10% of the initial value we loop through all
parameters and set it manually:</p>
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<div class="cell_input code_cell docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="k">for</span> <span class="n">p</span> <span class="ow">in</span> <span class="n">result</span><span class="o">.</span><span class="n">params</span><span class="p">:</span>
<span class="n">result</span><span class="o">.</span><span class="n">params</span><span class="p">[</span><span class="n">p</span><span class="p">]</span><span class="o">.</span><span class="n">stderr</span> <span class="o">=</span> <span class="nb">abs</span><span class="p">(</span><span class="n">result</span><span class="o">.</span><span class="n">params</span><span class="p">[</span><span class="n">p</span><span class="p">]</span><span class="o">.</span><span class="n">value</span> <span class="o">*</span> <span class="mf">0.1</span><span class="p">)</span>
</pre></div>
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</section>
<section id="calculating-and-visualizing-maps-of-chi-2">
<span id="label-confidence-chi2-maps"></span><h2>Calculating and visualizing maps of <span class="math notranslate nohighlight">\(\chi^2\)</span><a class="headerlink" href="#calculating-and-visualizing-maps-of-chi-2" title="Link to this heading">¶</a></h2>
<p>The estimated values for the <span class="math notranslate nohighlight">\(1-\sigma\)</span> standard error calculated by
default for each fit include the effects of correlation between pairs of
variables, but assumes the uncertainties are symmetric. While it doesn’t
exactly say what the values of the <span class="math notranslate nohighlight">\(n-\sigma\)</span> uncertainties would be, the
implication is that the <span class="math notranslate nohighlight">\(n-\sigma\)</span> error is simply <span class="math notranslate nohighlight">\(n^2\sigma\)</span>.</p>
<p>The <code class="xref py py-func docutils literal notranslate"><span class="pre">conf_interval()</span></code> function described above improves on these
automatically (and quickly) calculated uncertainies by explicitly finding
<span class="math notranslate nohighlight">\(n-\sigma\)</span> confidence levels in both directions – it does not assume
that the uncertainties are symmetric. This function also takes into account the
correlations between pairs of variables, but it does not convey this
information very well.</p>
<p>For even further exploration of the confidence levels of parameter values, it
can be useful to calculate maps of <span class="math notranslate nohighlight">\(\chi^2\)</span> values for pairs of
variables around their best fit values and visualize these as contour plots.
Typically, pairs of variables will have elliptical contours of constant
<span class="math notranslate nohighlight">\(n-\sigma\)</span> level, with highly-correlated pairs of variables having high
ratios of major and minor axes.</p>
<p>The <code class="xref py py-func docutils literal notranslate"><span class="pre">conf_interval2d()</span></code> can calculate 2-d arrays or maps of either
probability or <span class="math notranslate nohighlight">\(\delta \chi^2 = \chi^2 - \chi_{\mathrm{best}}^2\)</span> for any
pair of variables. Visualizing these can help better understand the nature of
the uncertainties and correlations between parameters. To illustrate this,
we’ll start with an example fit to data that we deliberately add components not
accounted for in the model, and with slightly non-Gaussian noise – a
constructed but “real-world” example:</p>
<div class="jupyter_cell jupyter_container docutils container">
<div class="cell_input code_cell docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># <examples/doc_confidence_chi2_maps.py></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">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="kn">from</span> <span class="nn">lmfit</span> <span class="kn">import</span> <span class="n">conf_interval</span><span class="p">,</span> <span class="n">conf_interval2d</span><span class="p">,</span> <span class="n">report_ci</span>
<span class="kn">from</span> <span class="nn">lmfit.lineshapes</span> <span class="kn">import</span> <span class="n">gaussian</span>
<span class="kn">from</span> <span class="nn">lmfit.models</span> <span class="kn">import</span> <span class="n">GaussianModel</span><span class="p">,</span> <span class="n">LinearModel</span>
<span class="n">sigma_levels</span> <span class="o">=</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">rng</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">default_rng</span><span class="p">(</span><span class="n">seed</span><span class="o">=</span><span class="mi">102</span><span class="p">)</span>
<span class="c1">#########################</span>
<span class="c1"># set up data -- deliberately adding imperfections and</span>
<span class="c1"># a small amount of non-Gaussian noise</span>
<span class="n">npts</span> <span class="o">=</span> <span class="mi">501</span>
<span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">100</span><span class="p">,</span> <span class="n">num</span><span class="o">=</span><span class="n">npts</span><span class="p">)</span>
<span class="n">noise</span> <span class="o">=</span> <span class="n">rng</span><span class="o">.</span><span class="n">normal</span><span class="p">(</span><span class="n">scale</span><span class="o">=</span><span class="mf">0.3</span><span class="p">,</span> <span class="n">size</span><span class="o">=</span><span class="n">npts</span><span class="p">)</span> <span class="o">+</span> <span class="mf">0.2</span><span class="o">*</span><span class="n">rng</span><span class="o">.</span><span class="n">f</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">9</span><span class="p">,</span> <span class="n">size</span><span class="o">=</span><span class="n">npts</span><span class="p">)</span>
<span class="n">y</span> <span class="o">=</span> <span class="p">(</span><span class="n">gaussian</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">amplitude</span><span class="o">=</span><span class="mi">83</span><span class="p">,</span> <span class="n">center</span><span class="o">=</span><span class="mf">47.</span><span class="p">,</span> <span class="n">sigma</span><span class="o">=</span><span class="mf">5.</span><span class="p">)</span>
<span class="o">+</span> <span class="mf">0.02</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mi">4</span> <span class="o">+</span> <span class="mf">0.25</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">cos</span><span class="p">((</span><span class="n">x</span><span class="o">-</span><span class="mi">20</span><span class="p">)</span><span class="o">/</span><span class="mf">8.0</span><span class="p">)</span> <span class="o">+</span> <span class="n">noise</span><span class="p">)</span>
<span class="n">mod</span> <span class="o">=</span> <span class="n">GaussianModel</span><span class="p">()</span> <span class="o">+</span> <span class="n">LinearModel</span><span class="p">()</span>
<span class="n">params</span> <span class="o">=</span> <span class="n">mod</span><span class="o">.</span><span class="n">make_params</span><span class="p">(</span><span class="n">amplitude</span><span class="o">=</span><span class="mi">100</span><span class="p">,</span> <span class="n">center</span><span class="o">=</span><span class="mi">50</span><span class="p">,</span> <span class="n">sigma</span><span class="o">=</span><span class="mi">5</span><span class="p">,</span>
<span class="n">slope</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="n">intecept</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
<span class="n">out</span> <span class="o">=</span> <span class="n">mod</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">y</span><span class="p">,</span> <span class="n">params</span><span class="p">,</span> <span class="n">x</span><span class="o">=</span><span class="n">x</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">out</span><span class="o">.</span><span class="n">fit_report</span><span class="p">())</span>
<span class="c1">#########################</span>
<span class="c1"># run conf_intervale, print report</span>
<span class="n">sigma_levels</span> <span class="o">=</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">ci</span> <span class="o">=</span> <span class="n">conf_interval</span><span class="p">(</span><span class="n">out</span><span class="p">,</span> <span class="n">out</span><span class="p">,</span> <span class="n">sigmas</span><span class="o">=</span><span class="n">sigma_levels</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">"## Confidence Report:"</span><span class="p">)</span>
<span class="n">report_ci</span><span class="p">(</span><span class="n">ci</span><span class="p">)</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<div class="output stream highlight-none notranslate"><div class="highlight"><pre><span></span>[[Model]]
(Model(gaussian) + Model(linear))
[[Fit Statistics]]
# fitting method = leastsq
# function evals = 31
# data points = 501
# variables = 5
chi-square = 103.861381
reduced chi-square = 0.20939794
Akaike info crit = -778.348033
Bayesian info crit = -757.265003
R-squared = 0.93782756
[[Variables]]
amplitude: 78.8171374 +/- 1.21910939 (1.55%) (init = 100)
center: 47.0751649 +/- 0.07576660 (0.16%) (init = 50)
sigma: 4.93298753 +/- 0.07984021 (1.62%) (init = 5)
slope: 0.01839006 +/- 7.1957e-04 (3.91%) (init = 0)
intercept: 4.39234411 +/- 0.04420227 (1.01%) (init = 0)
fwhm: 11.6162977 +/- 0.18800933 (1.62%) == '2.3548200*sigma'
height: 6.37412722 +/- 0.08603873 (1.35%) == '0.3989423*amplitude/max(1e-15, sigma)'
[[Correlations]] (unreported correlations are < 0.100)
C(slope, intercept) = -0.8421
C(amplitude, sigma) = +0.6371
C(amplitude, intercept) = -0.3373
C(sigma, intercept) = -0.2149
C(center, slope) = -0.1026
</pre></div>
</div>
<div class="output stream highlight-none notranslate"><div class="highlight"><pre><span></span>## Confidence Report:
99.73% 95.45% 68.27% _BEST_ 68.27% 95.45% 99.73%
amplitude: -3.62610 -2.41983 -1.21237 78.81714 +1.22111 +2.45479 +3.70515
center : -0.22849 -0.15214 -0.07584 47.07516 +0.07587 +0.15225 +0.22873
sigma : -0.23335 -0.15640 -0.07870 4.93299 +0.08000 +0.16158 +0.24509
slope : -0.00217 -0.00144 -0.00072 0.01839 +0.00072 +0.00144 +0.00217
intercept: -0.13326 -0.08860 -0.04423 4.39234 +0.04421 +0.08854 +0.13312
</pre></div>
</div>
</div>
</div>
<p>The reports show that we obtained a pretty good fit, and that the automated
estimates of the uncertainties are actually pretty good – agreeing to the
second decimal place. But we also see that some of the uncertainties do become
noticeably asymmetric at high <span class="math notranslate nohighlight">\(n-\sigma\)</span> levels.</p>
<p>We’ll plot this data and fit, and then further explore these uncertainties
using <code class="xref py py-func docutils literal notranslate"><span class="pre">conf_interval2d()</span></code>. Please note that <code class="xref py py-func docutils literal notranslate"><span class="pre">conf_interval2d()</span></code> shows the
confidence intervals obtained considering the standard errors, not those obtained
by ‘conf_interval(out, out, sigmas=sigma_levels)’.</p>
<div class="jupyter_cell jupyter_container docutils container">
<div class="cell_input code_cell docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1">#########################</span>
<span class="c1"># plot initial fit</span>
<span class="n">colors</span> <span class="o">=</span> <span class="p">(</span><span class="s1">'#2030b0'</span><span class="p">,</span> <span class="s1">'#b02030'</span><span class="p">,</span> <span class="s1">'#207070'</span><span class="p">)</span>
<span class="n">fig</span><span class="p">,</span> <span class="n">axes</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</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">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">15</span><span class="p">,</span> <span class="mf">9.5</span><span class="p">))</span>
<span class="n">axes</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="o">.</span><span class="n">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="s1">'o'</span><span class="p">,</span> <span class="n">markersize</span><span class="o">=</span><span class="mi">3</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s1">'data'</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="n">colors</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
<span class="n">axes</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="o">.</span><span class="n">plot</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">best_fit</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s1">'fit'</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="n">colors</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
<span class="n">axes</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="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s1">'x'</span><span class="p">)</span>
<span class="n">axes</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="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">'y'</span><span class="p">)</span>
<span class="n">axes</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="o">.</span><span class="n">legend</span><span class="p">()</span>
<span class="n">aix</span><span class="p">,</span> <span class="n">aiy</span> <span class="o">=</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span>
<span class="n">nsamples</span> <span class="o">=</span> <span class="mi">50</span>
<span class="n">explicitly_calculate_sigma</span> <span class="o">=</span> <span class="kc">True</span>
<span class="k">for</span> <span class="n">pairs</span> <span class="ow">in</span> <span class="p">((</span><span class="s1">'sigma'</span><span class="p">,</span> <span class="s1">'amplitude'</span><span class="p">),</span> <span class="p">(</span><span class="s1">'intercept'</span><span class="p">,</span> <span class="s1">'amplitude'</span><span class="p">),</span>
<span class="p">(</span><span class="s1">'slope'</span><span class="p">,</span> <span class="s1">'intercept'</span><span class="p">),</span> <span class="p">(</span><span class="s1">'slope'</span><span class="p">,</span> <span class="s1">'center'</span><span class="p">),</span> <span class="p">(</span><span class="s1">'sigma'</span><span class="p">,</span> <span class="s1">'center'</span><span class="p">)):</span>
<span class="n">xpar</span><span class="p">,</span> <span class="n">ypar</span> <span class="o">=</span> <span class="n">pairs</span>
<span class="k">if</span> <span class="n">explicitly_calculate_sigma</span><span class="p">:</span>
<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"Generating chi-square map for </span><span class="si">{</span><span class="n">pairs</span><span class="si">}</span><span class="s2">"</span><span class="p">)</span>
<span class="n">c_x</span><span class="p">,</span> <span class="n">c_y</span><span class="p">,</span> <span class="n">chi2_mat</span> <span class="o">=</span> <span class="n">conf_interval2d</span><span class="p">(</span><span class="n">out</span><span class="p">,</span> <span class="n">out</span><span class="p">,</span> <span class="n">xpar</span><span class="p">,</span> <span class="n">ypar</span><span class="p">,</span>
<span class="n">nsamples</span><span class="p">,</span> <span class="n">nsamples</span><span class="p">,</span> <span class="n">nsigma</span><span class="o">=</span><span class="mf">3.5</span><span class="p">,</span>
<span class="n">chi2_out</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
<span class="c1"># explicitly calculate sigma matrix: sigma increases chi_square</span>
<span class="c1"># from chi_square_best</span>
<span class="c1"># to chi_square + sigma**2 * reduced_chi_square</span>
<span class="c1"># so: sigma = sqrt((chi2-chi2_best)/ reduced_chi_square)</span>
<span class="n">chi2_min</span> <span class="o">=</span> <span class="n">chi2_mat</span><span class="o">.</span><span class="n">min</span><span class="p">()</span>
<span class="n">sigma_mat</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">chi2_mat</span><span class="o">-</span><span class="n">chi2_min</span><span class="p">)</span><span class="o">/</span><span class="n">out</span><span class="o">.</span><span class="n">redchi</span><span class="p">)</span>
<span class="k">else</span><span class="p">:</span>
<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"Generating sigma map for </span><span class="si">{</span><span class="n">pairs</span><span class="si">}</span><span class="s2">"</span><span class="p">)</span>
<span class="c1"># or, you could just calculate the matrix of probabilities as:</span>
<span class="n">c_x</span><span class="p">,</span> <span class="n">c_y</span><span class="p">,</span> <span class="n">sigma_mat</span> <span class="o">=</span> <span class="n">conf_interval2d</span><span class="p">(</span><span class="n">out</span><span class="p">,</span> <span class="n">out</span><span class="p">,</span> <span class="n">xpar</span><span class="p">,</span> <span class="n">ypar</span><span class="p">,</span>
<span class="n">nsamples</span><span class="p">,</span> <span class="n">nsamples</span><span class="p">,</span> <span class="n">nsigma</span><span class="o">=</span><span class="mf">3.5</span><span class="p">)</span>
<span class="n">aix</span> <span class="o">+=</span> <span class="mi">1</span>
<span class="k">if</span> <span class="n">aix</span> <span class="o">==</span> <span class="mi">2</span><span class="p">:</span>
<span class="n">aix</span> <span class="o">=</span> <span class="mi">0</span>
<span class="n">aiy</span> <span class="o">+=</span> <span class="mi">1</span>
<span class="n">ax</span> <span class="o">=</span> <span class="n">axes</span><span class="p">[</span><span class="n">aix</span><span class="p">,</span> <span class="n">aiy</span><span class="p">]</span>
<span class="n">cnt</span> <span class="o">=</span> <span class="n">ax</span><span class="o">.</span><span class="n">contour</span><span class="p">(</span><span class="n">c_x</span><span class="p">,</span> <span class="n">c_y</span><span class="p">,</span> <span class="n">sigma_mat</span><span class="p">,</span> <span class="n">levels</span><span class="o">=</span><span class="n">sigma_levels</span><span class="p">,</span> <span class="n">colors</span><span class="o">=</span><span class="n">colors</span><span class="p">,</span>
<span class="n">linestyles</span><span class="o">=</span><span class="s1">'-'</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">clabel</span><span class="p">(</span><span class="n">cnt</span><span class="p">,</span> <span class="n">inline</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span> <span class="n">fmt</span><span class="o">=</span><span class="sa">r</span><span class="s2">"$\sigma=</span><span class="si">%.0f</span><span class="s2">$"</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">13</span><span class="p">)</span>
<span class="c1"># draw boxes for estimated uncertaties:</span>
<span class="c1"># dotted : scaled stderr from initial fit</span>
<span class="c1"># dashed : values found from conf_interval()</span>
<span class="n">xv</span> <span class="o">=</span> <span class="n">out</span><span class="o">.</span><span class="n">params</span><span class="p">[</span><span class="n">xpar</span><span class="p">]</span><span class="o">.</span><span class="n">value</span>
<span class="n">xs</span> <span class="o">=</span> <span class="n">out</span><span class="o">.</span><span class="n">params</span><span class="p">[</span><span class="n">xpar</span><span class="p">]</span><span class="o">.</span><span class="n">stderr</span>
<span class="n">yv</span> <span class="o">=</span> <span class="n">out</span><span class="o">.</span><span class="n">params</span><span class="p">[</span><span class="n">ypar</span><span class="p">]</span><span class="o">.</span><span class="n">value</span>
<span class="n">ys</span> <span class="o">=</span> <span class="n">out</span><span class="o">.</span><span class="n">params</span><span class="p">[</span><span class="n">ypar</span><span class="p">]</span><span class="o">.</span><span class="n">stderr</span>
<span class="n">cix</span> <span class="o">=</span> <span class="n">ci</span><span class="p">[</span><span class="n">xpar</span><span class="p">]</span>
<span class="n">ciy</span> <span class="o">=</span> <span class="n">ci</span><span class="p">[</span><span class="n">ypar</span><span class="p">]</span>
<span class="n">nc</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">sigma_levels</span><span class="p">)</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">sigma_levels</span><span class="p">:</span>
<span class="c1"># dotted line: scaled stderr</span>
<span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">((</span><span class="n">xv</span><span class="o">-</span><span class="n">i</span><span class="o">*</span><span class="n">xs</span><span class="p">,</span> <span class="n">xv</span><span class="o">+</span><span class="n">i</span><span class="o">*</span><span class="n">xs</span><span class="p">,</span> <span class="n">xv</span><span class="o">+</span><span class="n">i</span><span class="o">*</span><span class="n">xs</span><span class="p">,</span> <span class="n">xv</span><span class="o">-</span><span class="n">i</span><span class="o">*</span><span class="n">xs</span><span class="p">,</span> <span class="n">xv</span><span class="o">-</span><span class="n">i</span><span class="o">*</span><span class="n">xs</span><span class="p">),</span>
<span class="p">(</span><span class="n">yv</span><span class="o">-</span><span class="n">i</span><span class="o">*</span><span class="n">ys</span><span class="p">,</span> <span class="n">yv</span><span class="o">-</span><span class="n">i</span><span class="o">*</span><span class="n">ys</span><span class="p">,</span> <span class="n">yv</span><span class="o">+</span><span class="n">i</span><span class="o">*</span><span class="n">ys</span><span class="p">,</span> <span class="n">yv</span><span class="o">+</span><span class="n">i</span><span class="o">*</span><span class="n">ys</span><span class="p">,</span> <span class="n">yv</span><span class="o">-</span><span class="n">i</span><span class="o">*</span><span class="n">ys</span><span class="p">),</span>
<span class="n">linestyle</span><span class="o">=</span><span class="s1">'dotted'</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="n">colors</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="c1"># dashed line: refined uncertainties from conf_interval</span>
<span class="n">xsp</span><span class="p">,</span> <span class="n">xsm</span> <span class="o">=</span> <span class="n">cix</span><span class="p">[</span><span class="n">nc</span><span class="o">+</span><span class="n">i</span><span class="p">][</span><span class="mi">1</span><span class="p">],</span> <span class="n">cix</span><span class="p">[</span><span class="n">nc</span><span class="o">-</span><span class="n">i</span><span class="p">][</span><span class="mi">1</span><span class="p">]</span>
<span class="n">ysp</span><span class="p">,</span> <span class="n">ysm</span> <span class="o">=</span> <span class="n">ciy</span><span class="p">[</span><span class="n">nc</span><span class="o">+</span><span class="n">i</span><span class="p">][</span><span class="mi">1</span><span class="p">],</span> <span class="n">ciy</span><span class="p">[</span><span class="n">nc</span><span class="o">-</span><span class="n">i</span><span class="p">][</span><span class="mi">1</span><span class="p">]</span>
<span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">((</span><span class="n">xsm</span><span class="p">,</span> <span class="n">xsp</span><span class="p">,</span> <span class="n">xsp</span><span class="p">,</span> <span class="n">xsm</span><span class="p">,</span> <span class="n">xsm</span><span class="p">),</span> <span class="p">(</span><span class="n">ysm</span><span class="p">,</span> <span class="n">ysm</span><span class="p">,</span> <span class="n">ysp</span><span class="p">,</span> <span class="n">ysp</span><span class="p">,</span> <span class="n">ysm</span><span class="p">),</span>
<span class="n">linestyle</span><span class="o">=</span><span class="s1">'dashed'</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="n">colors</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="n">ax</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="n">xpar</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="n">ypar</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">grid</span><span class="p">(</span><span class="kc">True</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s1">'#d0d0d0'</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
<span class="c1"># <end examples/doc_confidence_chi2_maps.py></span>
</pre></div>
</div>
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<div class="output stream highlight-none notranslate"><div class="highlight"><pre><span></span>Generating chi-square map for ('sigma', 'amplitude')
</pre></div>
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<div class="output stream highlight-none notranslate"><div class="highlight"><pre><span></span>Generating chi-square map for ('intercept', 'amplitude')
</pre></div>
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<div class="output stream highlight-none notranslate"><div class="highlight"><pre><span></span>Generating chi-square map for ('slope', 'intercept')
</pre></div>
</div>
<div class="output stream highlight-none notranslate"><div class="highlight"><pre><span></span>Generating chi-square map for ('slope', 'center')
</pre></div>
</div>
<div class="output stream highlight-none notranslate"><div class="highlight"><pre><span></span>Generating chi-square map for ('sigma', 'center')
</pre></div>
</div>
<img alt="_images/confidence_5_5.png" src="_images/confidence_5_5.png" />
</div>
</div>
<p>Here we made contours for the <span class="math notranslate nohighlight">\(n-\sigma\)</span> levels from the 2-D array of
<span class="math notranslate nohighlight">\(\chi^2\)</span> by noting that the <span class="math notranslate nohighlight">\(n-\sigma\)</span> level will have
<span class="math notranslate nohighlight">\(\chi^2\)</span> increased by <span class="math notranslate nohighlight">\(n^2\chi_\nu^2\)</span> where <span class="math notranslate nohighlight">\(\chi_\nu^2\)</span> is
reduced chi-square.</p>
<p>The dotted boxes show both the scaled values of the standard errors from the
initial fit, and the dashed boxes show the confidence levels from
<code class="xref py py-meth docutils literal notranslate"><span class="pre">conf_interval()</span></code>. You can see that the notion of increasing
<span class="math notranslate nohighlight">\(\chi^2\)</span> by <span class="math notranslate nohighlight">\(\chi_\nu^2\)</span> works very well, and that there is a small
asymmetry in the uncertainties for the <code class="docutils literal notranslate"><span class="pre">amplitude</span></code> and <code class="docutils literal notranslate"><span class="pre">sigma</span></code> parameters.</p>
</section>
<section id="an-advanced-example-for-evaluating-confidence-intervals">
<span id="label-confidence-advanced"></span><h2>An advanced example for evaluating confidence intervals<a class="headerlink" href="#an-advanced-example-for-evaluating-confidence-intervals" title="Link to this heading">¶</a></h2>
<p>Now we look at a problem where calculating the error from approximated
covariance can lead to misleading result – the same double exponential
problem shown in <a class="reference internal" href="fitting.html#label-emcee"><span class="std std-ref">Minimizer.emcee() - calculating the posterior probability distribution of parameters</span></a>. In fact such a problem is particularly
hard for the Levenberg-Marquardt method, so we first estimate the results
using the slower but robust Nelder-Mead method. We can then compare the
uncertainties computed (if the <code class="docutils literal notranslate"><span class="pre">numdifftools</span></code> package is installed) with
those estimated using Levenberg-Marquardt around the previously found
solution. We can also compare to the results of using <code class="docutils literal notranslate"><span class="pre">emcee</span></code>.</p>
<div class="jupyter_cell docutils container">
<div class="cell_output docutils container">
</div>
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<div class="jupyter_cell jupyter_container docutils container">
<div class="cell_input code_cell docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># <examples/doc_confidence_advanced.py></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">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">lmfit</span>
<span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">10</span><span class="p">,</span> <span class="mi">250</span><span class="p">)</span>
<span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">seed</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
<span class="n">y</span> <span class="o">=</span> <span class="mf">3.0</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">x</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span> <span class="o">-</span> <span class="mf">5.0</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="p">(</span><span class="n">x</span><span class="o">-</span><span class="mf">0.1</span><span class="p">)</span><span class="o">/</span><span class="mf">10.</span><span class="p">)</span> <span class="o">+</span> <span class="mf">0.1</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="n">x</span><span class="o">.</span><span class="n">size</span><span class="p">)</span>
<span class="n">p</span> <span class="o">=</span> <span class="n">lmfit</span><span class="o">.</span><span class="n">create_params</span><span class="p">(</span><span class="n">a1</span><span class="o">=</span><span class="mi">4</span><span class="p">,</span> <span class="n">a2</span><span class="o">=</span><span class="mi">4</span><span class="p">,</span> <span class="n">t1</span><span class="o">=</span><span class="mi">3</span><span class="p">,</span> <span class="n">t2</span><span class="o">=</span><span class="mi">3</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">residual</span><span class="p">(</span><span class="n">p</span><span class="p">):</span>
<span class="k">return</span> <span class="n">p</span><span class="p">[</span><span class="s1">'a1'</span><span class="p">]</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">x</span><span class="o">/</span><span class="n">p</span><span class="p">[</span><span class="s1">'t1'</span><span class="p">])</span> <span class="o">+</span> <span class="n">p</span><span class="p">[</span><span class="s1">'a2'</span><span class="p">]</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="p">(</span><span class="n">x</span><span class="o">-</span><span class="mf">0.1</span><span class="p">)</span><span class="o">/</span><span class="n">p</span><span class="p">[</span><span class="s1">'t2'</span><span class="p">])</span> <span class="o">-</span> <span class="n">y</span>
<span class="c1"># create Minimizer</span>
<span class="n">mini</span> <span class="o">=</span> <span class="n">lmfit</span><span class="o">.</span><span class="n">Minimizer</span><span class="p">(</span><span class="n">residual</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">nan_policy</span><span class="o">=</span><span class="s1">'propagate'</span><span class="p">)</span>
<span class="c1"># first solve with Nelder-Mead algorithm</span>
<span class="n">out1</span> <span class="o">=</span> <span class="n">mini</span><span class="o">.</span><span class="n">minimize</span><span class="p">(</span><span class="n">method</span><span class="o">=</span><span class="s1">'Nelder'</span><span class="p">)</span>
<span class="c1"># then solve with Levenberg-Marquardt using the</span>
<span class="c1"># Nelder-Mead solution as a starting point</span>
<span class="n">out2</span> <span class="o">=</span> <span class="n">mini</span><span class="o">.</span><span class="n">minimize</span><span class="p">(</span><span class="n">method</span><span class="o">=</span><span class="s1">'leastsq'</span><span class="p">,</span> <span class="n">params</span><span class="o">=</span><span class="n">out1</span><span class="o">.</span><span class="n">params</span><span class="p">)</span>
<span class="n">lmfit</span><span class="o">.</span><span class="n">report_fit</span><span class="p">(</span><span class="n">out2</span><span class="o">.</span><span class="n">params</span><span class="p">,</span> <span class="n">min_correl</span><span class="o">=</span><span class="mf">0.5</span><span class="p">)</span>
<span class="n">ci</span><span class="p">,</span> <span class="n">trace</span> <span class="o">=</span> <span class="n">lmfit</span><span class="o">.</span><span class="n">conf_interval</span><span class="p">(</span><span class="n">mini</span><span class="p">,</span> <span class="n">out2</span><span class="p">,</span> <span class="n">sigmas</span><span class="o">=</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="n">trace</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
<span class="n">lmfit</span><span class="o">.</span><span class="n">printfuncs</span><span class="o">.</span><span class="n">report_ci</span><span class="p">(</span><span class="n">ci</span><span class="p">)</span>
<span class="c1"># plot data and best fit</span>
<span class="n">plt</span><span class="o">.</span><span class="n">figure</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">x</span><span class="p">,</span> <span class="n">y</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">x</span><span class="p">,</span> <span class="n">residual</span><span class="p">(</span><span class="n">out2</span><span class="o">.</span><span class="n">params</span><span class="p">)</span> <span class="o">+</span> <span class="n">y</span><span class="p">,</span> <span class="s1">'-'</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
<span class="c1"># plot confidence intervals (a1 vs t2 and a2 vs t2)</span>
<span class="n">fig</span><span class="p">,</span> <span class="n">axes</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</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="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mf">12.8</span><span class="p">,</span> <span class="mf">4.8</span><span class="p">))</span>
<span class="n">cx</span><span class="p">,</span> <span class="n">cy</span><span class="p">,</span> <span class="n">grid</span> <span class="o">=</span> <span class="n">lmfit</span><span class="o">.</span><span class="n">conf_interval2d</span><span class="p">(</span><span class="n">mini</span><span class="p">,</span> <span class="n">out2</span><span class="p">,</span> <span class="s1">'a1'</span><span class="p">,</span> <span class="s1">'t2'</span><span class="p">,</span> <span class="mi">30</span><span class="p">,</span> <span class="mi">30</span><span class="p">)</span>
<span class="n">ctp</span> <span class="o">=</span> <span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">contourf</span><span class="p">(</span><span class="n">cx</span><span class="p">,</span> <span class="n">cy</span><span class="p">,</span> <span class="n">grid</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">11</span><span class="p">))</span>
<span class="n">fig</span><span class="o">.</span><span class="n">colorbar</span><span class="p">(</span><span class="n">ctp</span><span class="p">,</span> <span class="n">ax</span><span class="o">=</span><span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
<span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s1">'a1'</span><span class="p">)</span>
<span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">'t2'</span><span class="p">)</span>
<span class="n">cx</span><span class="p">,</span> <span class="n">cy</span><span class="p">,</span> <span class="n">grid</span> <span class="o">=</span> <span class="n">lmfit</span><span class="o">.</span><span class="n">conf_interval2d</span><span class="p">(</span><span class="n">mini</span><span class="p">,</span> <span class="n">out2</span><span class="p">,</span> <span class="s1">'a2'</span><span class="p">,</span> <span class="s1">'t2'</span><span class="p">,</span> <span class="mi">30</span><span class="p">,</span> <span class="mi">30</span><span class="p">)</span>
<span class="n">ctp</span> <span class="o">=</span> <span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">contourf</span><span class="p">(</span><span class="n">cx</span><span class="p">,</span> <span class="n">cy</span><span class="p">,</span> <span class="n">grid</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">11</span><span class="p">))</span>
<span class="n">fig</span><span class="o">.</span><span class="n">colorbar</span><span class="p">(</span><span class="n">ctp</span><span class="p">,</span> <span class="n">ax</span><span class="o">=</span><span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
<span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s1">'a2'</span><span class="p">)</span>
<span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">'t2'</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
<span class="c1"># plot dependence between two parameters</span>
<span class="n">fig</span><span class="p">,</span> <span class="n">axes</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</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="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mf">12.8</span><span class="p">,</span> <span class="mf">4.8</span><span class="p">))</span>
<span class="n">cx1</span><span class="p">,</span> <span class="n">cy1</span><span class="p">,</span> <span class="n">prob</span> <span class="o">=</span> <span class="n">trace</span><span class="p">[</span><span class="s1">'a1'</span><span class="p">][</span><span class="s1">'a1'</span><span class="p">],</span> <span class="n">trace</span><span class="p">[</span><span class="s1">'a1'</span><span class="p">][</span><span class="s1">'t2'</span><span class="p">],</span> <span class="n">trace</span><span class="p">[</span><span class="s1">'a1'</span><span class="p">][</span><span class="s1">'prob'</span><span class="p">]</span>
<span class="n">cx2</span><span class="p">,</span> <span class="n">cy2</span><span class="p">,</span> <span class="n">prob2</span> <span class="o">=</span> <span class="n">trace</span><span class="p">[</span><span class="s1">'t2'</span><span class="p">][</span><span class="s1">'t2'</span><span class="p">],</span> <span class="n">trace</span><span class="p">[</span><span class="s1">'t2'</span><span class="p">][</span><span class="s1">'a1'</span><span class="p">],</span> <span class="n">trace</span><span class="p">[</span><span class="s1">'t2'</span><span class="p">][</span><span class="s1">'prob'</span><span class="p">]</span>
<span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">cx1</span><span class="p">,</span> <span class="n">cy1</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="n">prob</span><span class="p">,</span> <span class="n">s</span><span class="o">=</span><span class="mi">30</span><span class="p">)</span>
<span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s1">'a1'</span><span class="p">)</span>
<span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">'t2'</span><span class="p">)</span>
<span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">cx2</span><span class="p">,</span> <span class="n">cy2</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="n">prob2</span><span class="p">,</span> <span class="n">s</span><span class="o">=</span><span class="mi">30</span><span class="p">)</span>
<span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s1">'t2'</span><span class="p">)</span>
<span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">'a1'</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
<span class="c1"># <end examples/doc_confidence_advanced.py></span>
</pre></div>
</div>
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<p>which will report:</p>
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<div class="output stream highlight-none notranslate"><div class="highlight"><pre><span></span>[[Variables]]
a1: 2.98622095 +/- 0.14867027 (4.98%) (init = 2.986237)
a2: -4.33526363 +/- 0.11527574 (2.66%) (init = -4.335256)
t1: 1.30994276 +/- 0.13121215 (10.02%) (init = 1.309932)
t2: 11.8240337 +/- 0.46316956 (3.92%) (init = 11.82408)
[[Correlations]] (unreported correlations are < 0.500)
C(a2, t2) = +0.9871
C(a2, t1) = -0.9246
C(t1, t2) = -0.8805
C(a1, t1) = -0.5988
95.45% 68.27% _BEST_ 68.27% 95.45%
a1: -0.27285 -0.14165 2.98622 +0.16354 +0.36343
a2: -0.30440 -0.13219 -4.33526 +0.10689 +0.19684
t1: -0.23392 -0.12494 1.30994 +0.14660 +0.32369
t2: -1.01937 -0.48813 11.82403 +0.46045 +0.90439
</pre></div>
</div>
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</div>
<p>Again we called <code class="xref py py-func docutils literal notranslate"><span class="pre">conf_interval()</span></code>, this time with tracing and only for
1- and 2-<span class="math notranslate nohighlight">\(\sigma\)</span>. Comparing these two different estimates, we see
that the estimate for <code class="docutils literal notranslate"><span class="pre">a1</span></code> is reasonably well approximated from the
covariance matrix, but the estimates for <code class="docutils literal notranslate"><span class="pre">a2</span></code> and especially for <code class="docutils literal notranslate"><span class="pre">t1</span></code>, and
<code class="docutils literal notranslate"><span class="pre">t2</span></code> are very asymmetric and that going from 1 <span class="math notranslate nohighlight">\(\sigma\)</span> (68%
confidence) to 2 <span class="math notranslate nohighlight">\(\sigma\)</span> (95% confidence) is not very predictable.</p>
<p>Plots of the confidence region are shown in the figures below for <code class="docutils literal notranslate"><span class="pre">a1</span></code> and
<code class="docutils literal notranslate"><span class="pre">t2</span></code> (left), and <code class="docutils literal notranslate"><span class="pre">a2</span></code> and <code class="docutils literal notranslate"><span class="pre">t2</span></code> (right):</p>
<div class="jupyter_cell docutils container">
<div class="cell_output docutils container">
<img alt="_images/confidence_9_0.svg" src="_images/confidence_9_0.svg" /></div>
</div>
<p>Neither of these plots is very much like an ellipse, which is implicitly
assumed by the approach using the covariance matrix. The plots actually
look quite a bit like those found with MCMC and shown in the “corner plot”
in <a class="reference internal" href="fitting.html#label-emcee"><span class="std std-ref">Minimizer.emcee() - calculating the posterior probability distribution of parameters</span></a>. In fact, comparing the confidence interval results
here with the results for the 1- and 2-<span class="math notranslate nohighlight">\(\sigma\)</span> error estimated with
<code class="docutils literal notranslate"><span class="pre">emcee</span></code>, we can see that the agreement is pretty good and that the
asymmetry in the parameter distributions are reflected well in the
asymmetry of the uncertainties.</p>
<p>The trace returned as the optional second argument from
<code class="xref py py-func docutils literal notranslate"><span class="pre">conf_interval()</span></code> contains a dictionary for each variable parameter.
The values are dictionaries with arrays of values for each variable, and an
array of corresponding probabilities for the corresponding cumulative
variables. This can be used to show the dependence between two
parameters:</p>
<div class="jupyter_cell jupyter_container docutils container">
<div class="cell_input code_cell docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">fig</span><span class="p">,</span> <span class="n">axes</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</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="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mf">12.8</span><span class="p">,</span> <span class="mf">4.8</span><span class="p">))</span>
<span class="n">cx1</span><span class="p">,</span> <span class="n">cy1</span><span class="p">,</span> <span class="n">prob</span> <span class="o">=</span> <span class="n">trace</span><span class="p">[</span><span class="s1">'a1'</span><span class="p">][</span><span class="s1">'a1'</span><span class="p">],</span> <span class="n">trace</span><span class="p">[</span><span class="s1">'a1'</span><span class="p">][</span><span class="s1">'t2'</span><span class="p">],</span> <span class="n">trace</span><span class="p">[</span><span class="s1">'a1'</span><span class="p">][</span><span class="s1">'prob'</span><span class="p">]</span>
<span class="n">cx2</span><span class="p">,</span> <span class="n">cy2</span><span class="p">,</span> <span class="n">prob2</span> <span class="o">=</span> <span class="n">trace</span><span class="p">[</span><span class="s1">'t2'</span><span class="p">][</span><span class="s1">'t2'</span><span class="p">],</span> <span class="n">trace</span><span class="p">[</span><span class="s1">'t2'</span><span class="p">][</span><span class="s1">'a1'</span><span class="p">],</span> <span class="n">trace</span><span class="p">[</span><span class="s1">'t2'</span><span class="p">][</span><span class="s1">'prob'</span><span class="p">]</span>
<span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">cx1</span><span class="p">,</span> <span class="n">cy1</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="n">prob</span><span class="p">,</span> <span class="n">s</span><span class="o">=</span><span class="mi">30</span><span class="p">)</span>
<span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s1">'a1'</span><span class="p">)</span>
<span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">'t2'</span><span class="p">)</span>
<span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">cx2</span><span class="p">,</span> <span class="n">cy2</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="n">prob2</span><span class="p">,</span> <span class="n">s</span><span class="o">=</span><span class="mi">30</span><span class="p">)</span>
<span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s1">'t2'</span><span class="p">)</span>
<span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">'a1'</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
</div>
</div>
<p>which shows the trace of values:</p>
<div class="jupyter_cell docutils container">
<div class="cell_output docutils container">
<img alt="_images/confidence_11_0.svg" src="_images/confidence_11_0.svg" /></div>
</div>
<p>As an alternative/complement to the confidence intervals, the <code class="xref py py-meth docutils literal notranslate"><span class="pre">Minimizer.emcee()</span></code>
method uses Markov Chain Monte Carlo to sample the posterior probability distribution.
These distributions demonstrate the range of solutions that the data supports and we
refer to <a class="reference internal" href="fitting.html#label-emcee"><span class="std std-ref">Minimizer.emcee() - calculating the posterior probability distribution of parameters</span></a> where this methodology was used on the same problem.</p>
<p>Credible intervals (the Bayesian equivalent of the frequentist confidence
interval) can be obtained with this method. MCMC can be used for model
selection, to determine outliers, to marginalize over nuisance parameters, etcetera.
For example, you may have fractionally underestimated the uncertainties on a
dataset. MCMC can be used to estimate the true level of uncertainty on each
data point. A tutorial on the possibilities offered by MCMC can be found at <a class="footnote-reference brackets" href="#id2" id="id1" role="doc-noteref"><span class="fn-bracket">[</span>1<span class="fn-bracket">]</span></a>.</p>
<aside class="footnote-list brackets">
<aside class="footnote brackets" id="id2" role="doc-footnote">
<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id1">1</a><span class="fn-bracket">]</span></span>
<p><a class="reference external" href="https://jakevdp.github.io/blog/2014/03/11/frequentism-and-bayesianism-a-practical-intro/">https://jakevdp.github.io/blog/2014/03/11/frequentism-and-bayesianism-a-practical-intro/</a></p>
</aside>
</aside>
</section>
<section id="confidence-interval-functions">
<h2>Confidence Interval Functions<a class="headerlink" href="#confidence-interval-functions" title="Link to this heading">¶</a></h2>
<dl class="py function">
<dt class="sig sig-object py" id="lmfit.conf_interval">
<span class="sig-name descname"><span class="pre">conf_interval</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">minimizer</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">result</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">p_names</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">sigmas</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">trace</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">False</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">maxiter</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">200</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">verbose</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">False</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">prob_func</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">min_rel_change</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">1e-05</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#lmfit.conf_interval" title="Link to this definition">¶</a></dt>
<dd><p>Calculate the confidence interval (CI) for parameters.</p>
<p>The parameter for which the CI is calculated will be varied, while the
remaining parameters are re-optimized to minimize the chi-square. The
resulting chi-square is used to calculate the probability with a given
statistic (e.g., F-test). This function uses a 1d-rootfinder from SciPy
to find the values resulting in the searched confidence region.</p>
<dl class="field-list simple">
<dt class="field-odd">Parameters<span class="colon">:</span></dt>
<dd class="field-odd"><ul class="simple">
<li><p><strong>minimizer</strong> (<a class="reference internal" href="fitting.html#lmfit.minimizer.Minimizer" title="lmfit.minimizer.Minimizer"><em>Minimizer</em></a>) – The minimizer to use, holding objective function.</p></li>
<li><p><strong>result</strong> (<a class="reference internal" href="fitting.html#lmfit.minimizer.MinimizerResult" title="lmfit.minimizer.MinimizerResult"><em>MinimizerResult</em></a>) – The result of running minimize().</p></li>
<li><p><strong>p_names</strong> (<a class="reference external" href="https://docs.python.org/3/library/stdtypes.html#list" title="(in Python v3.12)"><em>list</em></a><em>, </em><em>optional</em>) – Names of the parameters for which the CI is calculated. If None
(default), the CI is calculated for every parameter.</p></li>
<li><p><strong>sigmas</strong> (<a class="reference external" href="https://docs.python.org/3/library/stdtypes.html#list" title="(in Python v3.12)"><em>list</em></a><em>, </em><em>optional</em>) – The sigma-levels to find (default is [1, 2, 3]). See Notes below.</p></li>
<li><p><strong>trace</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#bool" title="(in Python v3.12)"><em>bool</em></a><em>, </em><em>optional</em>) – Defaults to False; if True, each result of a probability
calculation is saved along with the parameter. This can be used to
plot so-called “profile traces”.</p></li>
<li><p><strong>maxiter</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#int" title="(in Python v3.12)"><em>int</em></a><em>, </em><em>optional</em>) – Maximum of iteration to find an upper limit (default is 200).</p></li>
<li><p><strong>verbose</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#bool" title="(in Python v3.12)"><em>bool</em></a><em>, </em><em>optional</em>) – Print extra debugging information (default is False).</p></li>
<li><p><strong>prob_func</strong> (<em>None</em><em> or </em><em>callable</em><em>, </em><em>optional</em>) – Function to calculate the probability from the optimized chi-square.
Default is None and uses the built-in function <cite>f_compare</cite>
(i.e., F-test).</p></li>
<li><p><strong>min_rel_change</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#float" title="(in Python v3.12)"><em>float</em></a><em>, </em><em>optional</em>) – Minimum relative change in probability (default is 1e-5).</p></li>
</ul>
</dd>
<dt class="field-even">Returns<span class="colon">:</span></dt>
<dd class="field-even"><p><ul class="simple">
<li><p><strong>output</strong> (<em>dict</em>) – A dictionary containing a list of <code class="docutils literal notranslate"><span class="pre">(sigma,</span> <span class="pre">vals)</span></code>-tuples for
each parameter.</p></li>
<li><p><strong>trace_dict</strong> (<em>dict, optional</em>) – Only if trace is True. Is a dictionary, the key is the parameter
which was fixed. The values are again a dict with the names as
keys, but with an additional key ‘prob’. Each contains an array
of the corresponding values.</p></li>
</ul>
</p>
</dd>
</dl>
<div class="admonition seealso">
<p class="admonition-title">See also</p>
<p><a class="reference internal" href="#lmfit.conf_interval2d" title="lmfit.conf_interval2d"><code class="xref py py-obj docutils literal notranslate"><span class="pre">conf_interval2d</span></code></a></p>
</div>
<p class="rubric">Notes</p>
<p>The values for <cite>sigma</cite> are taken as the number of standard deviations
for a normal distribution and converted to probabilities. That is, the
default <code class="docutils literal notranslate"><span class="pre">sigma=[1,</span> <span class="pre">2,</span> <span class="pre">3]</span></code> will use probabilities of 0.6827, 0.9545,
and 0.9973. If any of the sigma values is less than 1, that will be
interpreted as a probability. That is, a value of 1 and 0.6827 will
give the same results, within precision.</p>
<p class="rubric">Examples</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">lmfit.printfuncs</span> <span class="kn">import</span> <span class="o">*</span>
<span class="gp">>>> </span><span class="n">mini</span> <span class="o">=</span> <span class="n">minimize</span><span class="p">(</span><span class="n">some_func</span><span class="p">,</span> <span class="n">params</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">mini</span><span class="o">.</span><span class="n">leastsq</span><span class="p">()</span>
<span class="go">True</span>
<span class="gp">>>> </span><span class="n">report_errors</span><span class="p">(</span><span class="n">params</span><span class="p">)</span>
<span class="gp">... </span><span class="c1">#report</span>
<span class="gp">>>> </span><span class="n">ci</span> <span class="o">=</span> <span class="n">conf_interval</span><span class="p">(</span><span class="n">mini</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">report_ci</span><span class="p">(</span><span class="n">ci</span><span class="p">)</span>
<span class="gp">... </span><span class="c1">#report</span>
</pre></div>
</div>
<p>Now with quantiles for the sigmas and using the trace.</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">ci</span><span class="p">,</span> <span class="n">trace</span> <span class="o">=</span> <span class="n">conf_interval</span><span class="p">(</span><span class="n">mini</span><span class="p">,</span> <span class="n">sigmas</span><span class="o">=</span><span class="p">[</span><span class="mf">0.5</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">trace</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">fixed</span> <span class="o">=</span> <span class="n">trace</span><span class="p">[</span><span class="s1">'para1'</span><span class="p">][</span><span class="s1">'para1'</span><span class="p">]</span>
<span class="gp">>>> </span><span class="n">free</span> <span class="o">=</span> <span class="n">trace</span><span class="p">[</span><span class="s1">'para1'</span><span class="p">][</span><span class="s1">'not_para1'</span><span class="p">]</span>
<span class="gp">>>> </span><span class="n">prob</span> <span class="o">=</span> <span class="n">trace</span><span class="p">[</span><span class="s1">'para1'</span><span class="p">][</span><span class="s1">'prob'</span><span class="p">]</span>
</pre></div>
</div>
<p>This makes it possible to plot the dependence between free and fixed
parameters.</p>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="lmfit.conf_interval2d">
<span class="sig-name descname"><span class="pre">conf_interval2d</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">minimizer</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">result</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">x_name</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">y_name</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">nx</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">10</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">ny</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">10</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">limits</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">prob_func</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">nsigma</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">5</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">chi2_out</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">False</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#lmfit.conf_interval2d" title="Link to this definition">¶</a></dt>
<dd><p>Calculate confidence regions for two fixed parameters.</p>
<p>The method itself is explained in <cite>conf_interval</cite>: here we are fixing
two parameters.</p>
<dl class="field-list simple">
<dt class="field-odd">Parameters<span class="colon">:</span></dt>
<dd class="field-odd"><ul class="simple">
<li><p><strong>minimizer</strong> (<a class="reference internal" href="fitting.html#lmfit.minimizer.Minimizer" title="lmfit.minimizer.Minimizer"><em>Minimizer</em></a>) – The minimizer to use, holding objective function.</p></li>
<li><p><strong>result</strong> (<a class="reference internal" href="fitting.html#lmfit.minimizer.MinimizerResult" title="lmfit.minimizer.MinimizerResult"><em>MinimizerResult</em></a>) – The result of running minimize().</p></li>
<li><p><strong>x_name</strong> (<a class="reference external" href="https://docs.python.org/3/library/stdtypes.html#str" title="(in Python v3.12)"><em>str</em></a>) – The name of the parameter which will be the x direction.</p></li>
<li><p><strong>y_name</strong> (<a class="reference external" href="https://docs.python.org/3/library/stdtypes.html#str" title="(in Python v3.12)"><em>str</em></a>) – The name of the parameter which will be the y direction.</p></li>
<li><p><strong>nx</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#int" title="(in Python v3.12)"><em>int</em></a><em>, </em><em>optional</em>) – Number of points in the x direction (default is 10).</p></li>
<li><p><strong>ny</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#int" title="(in Python v3.12)"><em>int</em></a><em>, </em><em>optional</em>) – Number of points in the y direction (default is 10).</p></li>
<li><p><strong>limits</strong> (<a class="reference external" href="https://docs.python.org/3/library/stdtypes.html#tuple" title="(in Python v3.12)"><em>tuple</em></a><em>, </em><em>optional</em>) – Should have the form <code class="docutils literal notranslate"><span class="pre">((x_upper,</span> <span class="pre">x_lower),</span> <span class="pre">(y_upper,</span> <span class="pre">y_lower))</span></code>.
If not given, the default is nsigma*stderr in each direction.</p></li>
<li><p><strong>prob_func</strong> (<em>None</em><em> or </em><em>callable</em><em>, </em><em>deprecated</em>) – Starting with version 1.2, this argument is unused and has no effect.</p></li>
<li><p><strong>nsigma</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#float" title="(in Python v3.12)"><em>float</em></a><em> or </em><a class="reference external" href="https://docs.python.org/3/library/functions.html#int" title="(in Python v3.12)"><em>int</em></a><em>, </em><em>optional</em>) – Multiplier of stderr for limits (default is 5).</p></li>
<li><p><strong>chi2_out</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#bool" title="(in Python v3.12)"><em>bool</em></a>) – Whether to return chi-square at each coordinate instead of probability.</p></li>
</ul>
</dd>
<dt class="field-even">Returns<span class="colon">:</span></dt>
<dd class="field-even"><p><ul class="simple">
<li><p><strong>x</strong> (<em>numpy.ndarray</em>) – X-coordinates (same shape as <cite>nx</cite>).</p></li>
<li><p><strong>y</strong> (<em>numpy.ndarray</em>) – Y-coordinates (same shape as <cite>ny</cite>).</p></li>
<li><p><strong>grid</strong> (<em>numpy.ndarray</em>) – 2-D array (with shape <code class="docutils literal notranslate"><span class="pre">(nx,</span> <span class="pre">ny)</span></code>) containing the calculated
probabilities or chi-square.</p></li>
</ul>
</p>
</dd>
</dl>
<div class="admonition seealso">
<p class="admonition-title">See also</p>
<p><a class="reference internal" href="#lmfit.conf_interval" title="lmfit.conf_interval"><code class="xref py py-obj docutils literal notranslate"><span class="pre">conf_interval</span></code></a></p>
</div>
<p class="rubric">Examples</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">mini</span> <span class="o">=</span> <span class="n">Minimizer</span><span class="p">(</span><span class="n">some_func</span><span class="p">,</span> <span class="n">params</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">result</span> <span class="o">=</span> <span class="n">mini</span><span class="o">.</span><span class="n">leastsq</span><span class="p">()</span>
<span class="gp">>>> </span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">gr</span> <span class="o">=</span> <span class="n">conf_interval2d</span><span class="p">(</span><span class="n">mini</span><span class="p">,</span> <span class="n">result</span><span class="p">,</span> <span class="s1">'para1'</span><span class="p">,</span><span class="s1">'para2'</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">plt</span><span class="o">.</span><span class="n">contour</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">gr</span><span class="p">)</span>
</pre></div>
</div>
</dd></dl>
<dl class="py function">
<dt class="sig sig-object py" id="lmfit.ci_report">
<span class="sig-name descname"><span class="pre">ci_report</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">ci</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">with_offset</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">True</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">ndigits</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">5</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#lmfit.ci_report" title="Link to this definition">¶</a></dt>
<dd><p>Return text of a report for confidence intervals.</p>
<dl class="field-list simple">
<dt class="field-odd">Parameters<span class="colon">:</span></dt>
<dd class="field-odd"><ul class="simple">
<li><p><strong>ci</strong> (<a class="reference external" href="https://docs.python.org/3/library/stdtypes.html#dict" title="(in Python v3.12)"><em>dict</em></a>) – The result of <code class="xref py py-func docutils literal notranslate"><span class="pre">conf_interval()</span></code>: a dictionary
containing a list of <code class="docutils literal notranslate"><span class="pre">(sigma,</span> <span class="pre">vals)</span></code>-tuples for each parameter.</p></li>
<li><p><strong>with_offset</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#bool" title="(in Python v3.12)"><em>bool</em></a><em>, </em><em>optional</em>) – Whether to subtract best value from all other values (default is
True).</p></li>
<li><p><strong>ndigits</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#int" title="(in Python v3.12)"><em>int</em></a><em>, </em><em>optional</em>) – Number of significant digits to show (default is 5).</p></li>
</ul>
</dd>
<dt class="field-even">Returns<span class="colon">:</span></dt>
<dd class="field-even"><p>Text of formatted report on confidence intervals.</p>
</dd>
<dt class="field-odd">Return type<span class="colon">:</span></dt>
<dd class="field-odd"><p><a class="reference external" href="https://docs.python.org/3/library/stdtypes.html#str" title="(in Python v3.12)">str</a></p>
</dd>
</dl>
</dd></dl>
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