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‎tutorials/1_models-quick-fit.html

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<ul class="current nav bd-sidenav">
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<li class="toctree-l1 current active"><a class="current reference internal" href="#">Modeling 1: Make a quick fit using astropy.modeling</a></li>
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<li class="toctree-l1"><a class="reference internal" href="2_user-defined-model.html">Modeling 2: Create a User Defined Model using astropy.modeling</a></li>
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@@ -690,7 +688,7 @@ <h2>2) Fit a Polynomial model: Choose fitter wisely<a class="headerlink" href="#
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<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Text(0, 0.5, &#39;$y_1$&#39;)
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<img alt="_images/29eeec4d3dad2b705ea5f7b2791844247afecb976c6f267ff16f1118872200e5.png" src="_images/29eeec4d3dad2b705ea5f7b2791844247afecb976c6f267ff16f1118872200e5.png" />
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<img alt="_images/3634bb8b4c277819f7ce4c233fc84552a8f897def45ba7df53cedd5578cf8fde.png" src="_images/3634bb8b4c277819f7ce4c233fc84552a8f897def45ba7df53cedd5578cf8fde.png" />
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<p>To fit this data let’s remember the three steps: model, fitter and perform fit.</p>
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Degree: 3
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Parameters:
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c0 c1 c2 c3
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------------------ ----------------- ------------------- ------------------
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-9.172425941582441 12.59165151659485 -6.4125256413095135 1.0733596272176298
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------------------ ------------------ ------------------ ------------------
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-8.414565124986794 11.315131284061778 -5.936420715916029 1.0350472068170264
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</pre></div>
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Model set size: 1
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Degree: 3
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Parameters:
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c0 c1 c2 c3
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------------------ -------------------- ------------------ ------------------
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-2.181910827369914 -0.09737921440192401 -0.641962352228048 0.3102451837171443
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c0 c1 c2 c3
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------------------- -------------------- ------------------ ------------------
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0.20564283089638824 -0.15716654941300878 -1.239243988919935 0.4355616696000767
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</pre></div>
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<div class="cell_output docutils container">
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Reduced Chi Squared with LinearLSQFitter: 1.976453546686968
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Reduced Chi Squared with LinearLSQFitter: 1.7254506507627694
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</pre></div>
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<div class="cell_output docutils container">
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Reduced Chi Squared with SimplexLSQFitter: 3.6404719596242217
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Reduced Chi Squared with SimplexLSQFitter: 4.580753153741815
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</pre></div>
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<div class="cell_output docutils container">
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<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>&lt;matplotlib.legend.Legend at 0x7f043d6b6c60&gt;
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<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>&lt;matplotlib.legend.Legend at 0x7ff08227ba10&gt;
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</pre></div>
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<img alt="_images/5daacb87eb9a956a0a0d1bf47cf76b22fef5b552f8d494da052155270d4be995.png" src="_images/5daacb87eb9a956a0a0d1bf47cf76b22fef5b552f8d494da052155270d4be995.png" />
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<img alt="_images/b2cb6953351ca9c5f39fbc468440cdd51731a2b59186590a4e26b1d805a046f3.png" src="_images/b2cb6953351ca9c5f39fbc468440cdd51731a2b59186590a4e26b1d805a046f3.png" />
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<p>Results are as espected, the fit performed with the linear fitter is better than the second, non linear one.</p>
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<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Text(0, 0.5, &#39;$y_2$&#39;)
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<img alt="_images/a439ff8bd1722f8eb99d8ab93605e4bbd374fe7ac8a966b770601e8129461f6d.png" src="_images/a439ff8bd1722f8eb99d8ab93605e4bbd374fe7ac8a966b770601e8129461f6d.png" />
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<img alt="_images/9424b3add19b739bc9cbfc66cf567663995d37f79f0e0ac2bab63e87ff87f7d1.png" src="_images/9424b3add19b739bc9cbfc66cf567663995d37f79f0e0ac2bab63e87ff87f7d1.png" />
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<p>Let’s do our three steps to make the fit we want. For this fit we’re going to use a non-linear fitter, <code class="docutils literal notranslate"><span class="pre">LevMarLSQFitter</span></code>, because the model we need (<code class="docutils literal notranslate"><span class="pre">Gaussian1D</span></code>) is non-linear in the parameters.</p>
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Outputs: (&#39;y&#39;,)
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Model set size: 1
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Parameters:
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amplitude mean stddev
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----------------- ------------------ -----------------
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9.703971972909294 0.2905529841691312 10.24439667276053
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amplitude mean stddev
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------------------ ------------------- ------------------
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10.161667885872756 0.16429784623682972 10.000409076033547
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>[0.05013324 0.07443702 0.07476521]
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>[0.05133544 0.06625433 0.06645726]
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Amplitude: 9.703971972909294 +\- 0.22390454000753693
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Mean: 0.2905529841691312 +\- 0.27283148663688805
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Standard Deviation: 10.24439667276053 +\- 0.2734322676754956
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Amplitude: 10.161667885872756 +\- 0.2265732649327809
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Mean: 0.16429784623682972 +\- 0.2573991704994828
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Standard Deviation: 10.000409076033547 +\- 0.2577930649461142
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<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>&lt;&gt;:2: SyntaxWarning: invalid escape sequence &#39;\-&#39;
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&lt;&gt;:2: SyntaxWarning: invalid escape sequence &#39;\-&#39;
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&lt;&gt;:4: SyntaxWarning: invalid escape sequence &#39;\-&#39;
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&lt;&gt;:6: SyntaxWarning: invalid escape sequence &#39;\-&#39;
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/tmp/ipykernel_3578/1659586526.py:2: SyntaxWarning: invalid escape sequence &#39;\-&#39;
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/tmp/ipykernel_3529/1659586526.py:2: SyntaxWarning: invalid escape sequence &#39;\-&#39;
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&quot;Amplitude: {} +\- {}&quot;.format(best_fit_gauss.amplitude.value, np.sqrt(cov_diag[0]))
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/tmp/ipykernel_3578/1659586526.py:4: SyntaxWarning: invalid escape sequence &#39;\-&#39;
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/tmp/ipykernel_3529/1659586526.py:4: SyntaxWarning: invalid escape sequence &#39;\-&#39;
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print(&quot;Mean: {} +\- {}&quot;.format(best_fit_gauss.mean.value, np.sqrt(cov_diag[1])))
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/tmp/ipykernel_3578/1659586526.py:6: SyntaxWarning: invalid escape sequence &#39;\-&#39;
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/tmp/ipykernel_3529/1659586526.py:6: SyntaxWarning: invalid escape sequence &#39;\-&#39;
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&quot;Standard Deviation: {} +\- {}&quot;.format(
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>[ 9.70397395 0.29055502 10.24439249]
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>[10.16166902 0.1642994 10.00040685]
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Amplitude: 9.703973951984176 +\- 0.2085755106086411
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Mean: 0.2905550176310043 +\- 0.25415064931293285
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Standard Deviation: 10.24439248784542 +\- 0.25471029589649113
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Amplitude: 10.16166901860214 +\- 0.2180138118181192
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Mean: 0.1642993992949413 +\- 0.24767662180935193
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Standard Deviation: 10.00040684622568 +\- 0.24805566376335159
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<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>&lt;&gt;:1: SyntaxWarning: invalid escape sequence &#39;\-&#39;
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print(&quot;Amplitude: {} +\- {}&quot;.format(p_opt[0], np.sqrt(p_cov[0, 0])))
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Reduced Chi Squared using astropy.modeling: 0.8677576541316718
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Reduced Chi Squared using astropy.modeling: 0.9258750059366527
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Reduced Chi Squared using scipy: 0.8677576541512168
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Reduced Chi Squared using scipy: 0.9258750059343416
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<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>&lt;matplotlib.legend.Legend at 0x7f043df28680&gt;
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<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>&lt;matplotlib.legend.Legend at 0x7ff0821b5cd0&gt;
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<img alt="_images/6ba5eb18131d7de1839874150236563bd66c697a852a18b44ca6d275eb605649.png" src="_images/6ba5eb18131d7de1839874150236563bd66c697a852a18b44ca6d275eb605649.png" />
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<img alt="_images/9d65948539694d6a9224f22edf286e6e920acf5805cb94eeeb84f92f19db8cf0.png" src="_images/9d65948539694d6a9224f22edf286e6e920acf5805cb94eeeb84f92f19db8cf0.png" />
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<p><strong>Conclusion:</strong> Choose the method most convenient for every case you need to fit. We recomend <code class="docutils literal notranslate"><span class="pre">astropy.modeling</span></code> because is easier to write the name of the function you want to fit than to remember the expression every time we want to use it. Also, <code class="docutils literal notranslate"><span class="pre">astropy.modeling</span></code> becomes useful with more complicated models like <a class="reference external" href="http://docs.astropy.org/en/stable/modeling/#compound-models">two gaussians</a> plus a <a class="reference external" href="http://docs.astropy.org/en/stable/modeling/#blackbody-radiation">black body</a>, but that is another tutorial.</p>
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<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Text(0, 0.5, &#39;$y_3$&#39;)
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