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<em>This is an excerpt from the <a href="http://shop.oreilly.com/product/0636920034919.do">Python Data Science Handbook</a> by Jake VanderPlas; Jupyter notebooks are available <a href="https://github.com/jakevdp/PythonDataScienceHandbook">on GitHub</a>.</em>
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<em>The text is released under the <a href="https://creativecommons.org/licenses/by-nc-nd/3.0/us/legalcode">CC-BY-NC-ND license</a>, and code is released under the <a href="https://opensource.org/licenses/MIT">MIT license</a>. If you find this content useful, please consider supporting the work by <a href="http://shop.oreilly.com/product/0636920034919.do">buying the book</a>!</em>
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            <h1>Visualizing Errors</h1>
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<p><!--NAVIGATION-->
&lt; <a href="04.02-simple-scatter-plots.html">Simple Scatter Plots</a> | <a href="index.html">Contents</a> | <a href="04.04-density-and-contour-plots.html">Density and Contour Plots</a> &gt;</p>
<p><a href="https://colab.research.google.com/github/jakevdp/PythonDataScienceHandbook/blob/master/notebooks/04.03-Errorbars.ipynb"><img align="left" src="https://colab.research.google.com/assets/colab-badge.svg" alt="Open in Colab" title="Open and Execute in Google Colaboratory"></a></p>

</div>
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<p>For any scientific measurement, accurate accounting for errors is nearly as important, if not more important, than accurate reporting of the number itself.
For example, imagine that I am using some astrophysical observations to estimate the Hubble Constant, the local measurement of the expansion rate of the Universe.
I know that the current literature suggests a value of around 71 (km/s)/Mpc, and I measure a value of 74 (km/s)/Mpc with my method. Are the values consistent? The only correct answer, given this information, is this: there is no way to know.</p>
<p>Suppose I augment this information with reported uncertainties: the current literature suggests a value of around 71 $\pm$ 2.5 (km/s)/Mpc, and my method has measured a value of 74 $\pm$ 5 (km/s)/Mpc. Now are the values consistent? That is a question that can be quantitatively answered.</p>
<p>In visualization of data and results, showing these errors effectively can make a plot convey much more complete information.</p>

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<h2 id="Basic-Errorbars">Basic Errorbars<a class="anchor-link" href="#Basic-Errorbars">&#182;</a></h2><p>A basic errorbar can be created with a single Matplotlib function call:</p>

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</div>
</div>
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<div class="prompt input_prompt">In&nbsp;[1]:</div>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="o">%</span><span class="k">matplotlib</span> inline
<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
<span class="n">plt</span><span class="o">.</span><span class="n">style</span><span class="o">.</span><span class="n">use</span><span class="p">(</span><span class="s1">&#39;seaborn-whitegrid&#39;</span><span class="p">)</span>
<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
</pre></div>

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</div>
</div>

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<div class="input">
<div class="prompt input_prompt">In&nbsp;[2]:</div>
<div class="inner_cell">
    <div class="input_area">
<div class=" highlight hl-ipython3"><pre><span></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">0</span><span class="p">,</span> <span class="mi">10</span><span class="p">,</span> <span class="mi">50</span><span class="p">)</span>
<span class="n">dy</span> <span class="o">=</span> <span class="mf">0.8</span>
<span class="n">y</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="o">+</span> <span class="n">dy</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="mi">50</span><span class="p">)</span>

<span class="n">plt</span><span class="o">.</span><span class="n">errorbar</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">yerr</span><span class="o">=</span><span class="n">dy</span><span class="p">,</span> <span class="n">fmt</span><span class="o">=</span><span class="s1">&#39;.k&#39;</span><span class="p">);</span>
</pre></div>

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"
>
</div>

</div>

</div>
</div>

</div>
<div class="cell border-box-sizing text_cell rendered"><div class="prompt input_prompt">
</div>
<div class="inner_cell">
<div class="text_cell_render border-box-sizing rendered_html">
<p>Here the <code>fmt</code> is a format code controlling the appearance of lines and points, and has the same syntax as the shorthand used in <code>plt.plot</code>, outlined in <a href="04.01-simple-line-plots.html">Simple Line Plots</a> and <a href="04.02-simple-scatter-plots.html">Simple Scatter Plots</a>.</p>
<p>In addition to these basic options, the <code>errorbar</code> function has many options to fine-tune the outputs.
Using these additional options you can easily customize the aesthetics of your errorbar plot.
I often find it helpful, especially in crowded plots, to make the errorbars lighter than the points themselves:</p>

</div>
</div>
</div>
<div class="cell border-box-sizing code_cell rendered">
<div class="input">
<div class="prompt input_prompt">In&nbsp;[3]:</div>
<div class="inner_cell">
    <div class="input_area">
<div class=" highlight hl-ipython3"><pre><span></span><span class="n">plt</span><span class="o">.</span><span class="n">errorbar</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">yerr</span><span class="o">=</span><span class="n">dy</span><span class="p">,</span> <span class="n">fmt</span><span class="o">=</span><span class="s1">&#39;o&#39;</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;black&#39;</span><span class="p">,</span>
             <span class="n">ecolor</span><span class="o">=</span><span class="s1">&#39;lightgray&#39;</span><span class="p">,</span> <span class="n">elinewidth</span><span class="o">=</span><span class="mi">3</span><span class="p">,</span> <span class="n">capsize</span><span class="o">=</span><span class="mi">0</span><span class="p">);</span>
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<p>In addition to these options, you can also specify horizontal errorbars (<code>xerr</code>), one-sided errorbars, and many other variants.
For more information on the options available, refer to the docstring of <code>plt.errorbar</code>.</p>

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<h2 id="Continuous-Errors">Continuous Errors<a class="anchor-link" href="#Continuous-Errors">&#182;</a></h2><p>In some situations it is desirable to show errorbars on continuous quantities.
Though Matplotlib does not have a built-in convenience routine for this type of application, it's relatively easy to combine primitives like <code>plt.plot</code> and <code>plt.fill_between</code> for a useful result.</p>
<p>Here we'll perform a simple <em>Gaussian process regression</em>, using the Scikit-Learn API (see <a href="05.02-introducing-scikit-learn.html">Introducing Scikit-Learn</a> for details).
This is a method of fitting a very flexible non-parametric function to data with a continuous measure of the uncertainty.
We won't delve into the details of Gaussian process regression at this point, but will focus instead on how you might visualize such a continuous error measurement:</p>

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<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">from</span> <span class="nn">sklearn.gaussian_process</span> <span class="k">import</span> <span class="n">GaussianProcess</span>

<span class="c1"># define the model and draw some data</span>
<span class="n">model</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">x</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
<span class="n">xdata</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">6</span><span class="p">,</span> <span class="mi">8</span><span class="p">])</span>
<span class="n">ydata</span> <span class="o">=</span> <span class="n">model</span><span class="p">(</span><span class="n">xdata</span><span class="p">)</span>

<span class="c1"># Compute the Gaussian process fit</span>
<span class="n">gp</span> <span class="o">=</span> <span class="n">GaussianProcess</span><span class="p">(</span><span class="n">corr</span><span class="o">=</span><span class="s1">&#39;cubic&#39;</span><span class="p">,</span> <span class="n">theta0</span><span class="o">=</span><span class="mf">1e-2</span><span class="p">,</span> <span class="n">thetaL</span><span class="o">=</span><span class="mf">1e-4</span><span class="p">,</span> <span class="n">thetaU</span><span class="o">=</span><span class="mf">1E-1</span><span class="p">,</span>
                     <span class="n">random_start</span><span class="o">=</span><span class="mi">100</span><span class="p">)</span>
<span class="n">gp</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">xdata</span><span class="p">[:,</span> <span class="n">np</span><span class="o">.</span><span class="n">newaxis</span><span class="p">],</span> <span class="n">ydata</span><span class="p">)</span>

<span class="n">xfit</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">0</span><span class="p">,</span> <span class="mi">10</span><span class="p">,</span> <span class="mi">1000</span><span class="p">)</span>
<span class="n">yfit</span><span class="p">,</span> <span class="n">MSE</span> <span class="o">=</span> <span class="n">gp</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">xfit</span><span class="p">[:,</span> <span class="n">np</span><span class="o">.</span><span class="n">newaxis</span><span class="p">],</span> <span class="n">eval_MSE</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
<span class="n">dyfit</span> <span class="o">=</span> <span class="mi">2</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">MSE</span><span class="p">)</span>  <span class="c1"># 2*sigma ~ 95% confidence region</span>
</pre></div>

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<p>We now have <code>xfit</code>, <code>yfit</code>, and <code>dyfit</code>, which sample the continuous fit to our data.
We could pass these to the <code>plt.errorbar</code> function as above, but we don't really want to plot 1,000 points with 1,000 errorbars.
Instead, we can use the <code>plt.fill_between</code> function with a light color to visualize this continuous error:</p>

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<div class=" highlight hl-ipython3"><pre><span></span><span class="c1"># Visualize the result</span>
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">xdata</span><span class="p">,</span> <span class="n">ydata</span><span class="p">,</span> <span class="s1">&#39;or&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">xfit</span><span class="p">,</span> <span class="n">yfit</span><span class="p">,</span> <span class="s1">&#39;-&#39;</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;gray&#39;</span><span class="p">)</span>

<span class="n">plt</span><span class="o">.</span><span class="n">fill_between</span><span class="p">(</span><span class="n">xfit</span><span class="p">,</span> <span class="n">yfit</span> <span class="o">-</span> <span class="n">dyfit</span><span class="p">,</span> <span class="n">yfit</span> <span class="o">+</span> <span class="n">dyfit</span><span class="p">,</span>
                 <span class="n">color</span><span class="o">=</span><span class="s1">&#39;gray&#39;</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.2</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">xlim</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">10</span><span class="p">);</span>
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<p>Note what we've done here with the <code>fill_between</code> function: we pass an x value, then the lower y-bound, then the upper y-bound, and the result is that the area between these regions is filled.</p>
<p>The resulting figure gives a very intuitive view into what the Gaussian process regression algorithm is doing: in regions near a measured data point, the model is strongly constrained and this is reflected in the small model errors.
In regions far from a measured data point, the model is not strongly constrained, and the model errors increase.</p>
<p>For more information on the options available in <code>plt.fill_between()</code> (and the closely related <code>plt.fill()</code> function), see the function docstring or the Matplotlib documentation.</p>
<p>Finally, if this seems a bit too low level for your taste, refer to <a href="04.14-visualization-with-seaborn.html">Visualization With Seaborn</a>, where we discuss the Seaborn package, which has a more streamlined API for visualizing this type of continuous errorbar.</p>

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