Kernel Density Estimation
=========================


.. _kernel_density_notebook:

`Link to Notebook GitHub <https://github.com/statsmodels/statsmodels/blob/master/examples/notebooks/kernel_density.ipynb>`_

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   <div class="highlight"><pre><span class="kn">import</span> <span class="nn">numpy</span> <span class="kn">as</span> <span class="nn">np</span>
   <span class="kn">from</span> <span class="nn">scipy</span> <span class="kn">import</span> <span class="n">stats</span>
   <span class="kn">import</span> <span class="nn">statsmodels.api</span> <span class="kn">as</span> <span class="nn">sm</span>
   <span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="kn">as</span> <span class="nn">plt</span>
   <span class="kn">from</span> <span class="nn">statsmodels.distributions.mixture_rvs</span> <span class="kn">import</span> <span class="n">mixture_rvs</span>
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   <h4 id="A-univariate-example.">A univariate example.<a class="anchor-link" href="#A-univariate-example.">&#182;</a></h4>
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   <div class="highlight"><pre><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">12345</span><span class="p">)</span>
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   <div class="highlight"><pre><span class="n">obs_dist1</span> <span class="o">=</span> <span class="n">mixture_rvs</span><span class="p">([</span><span class="o">.</span><span class="mi">25</span><span class="p">,</span><span class="o">.</span><span class="mi">75</span><span class="p">],</span> <span class="n">size</span><span class="o">=</span><span class="mi">10000</span><span class="p">,</span> <span class="n">dist</span><span class="o">=</span><span class="p">[</span><span class="n">stats</span><span class="o">.</span><span class="n">norm</span><span class="p">,</span> <span class="n">stats</span><span class="o">.</span><span class="n">norm</span><span class="p">],</span>
                   <span class="n">kwargs</span> <span class="o">=</span> <span class="p">(</span><span class="nb">dict</span><span class="p">(</span><span class="n">loc</span><span class="o">=-</span><span class="mi">1</span><span class="p">,</span><span class="n">scale</span><span class="o">=.</span><span class="mi">5</span><span class="p">),</span><span class="nb">dict</span><span class="p">(</span><span class="n">loc</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span><span class="n">scale</span><span class="o">=.</span><span class="mi">5</span><span class="p">)))</span>
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   <div class="highlight"><pre><span class="n">kde</span> <span class="o">=</span> <span class="n">sm</span><span class="o">.</span><span class="n">nonparametric</span><span class="o">.</span><span class="n">KDEUnivariate</span><span class="p">(</span><span class="n">obs_dist1</span><span class="p">)</span>
   <span class="n">kde</span><span class="o">.</span><span class="n">fit</span><span class="p">()</span>
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   <div class="highlight"><pre><span class="n">fig</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">12</span><span class="p">,</span><span class="mi">8</span><span class="p">))</span>
   <span class="n">ax</span> <span class="o">=</span> <span class="n">fig</span><span class="o">.</span><span class="n">add_subplot</span><span class="p">(</span><span class="mi">111</span><span class="p">)</span>
   <span class="n">ax</span><span class="o">.</span><span class="n">hist</span><span class="p">(</span><span class="n">obs_dist1</span><span class="p">,</span> <span class="n">bins</span><span class="o">=</span><span class="mi">50</span><span class="p">,</span> <span class="n">normed</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s">&#39;red&#39;</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">kde</span><span class="o">.</span><span class="n">support</span><span class="p">,</span> <span class="n">kde</span><span class="o">.</span><span class="n">density</span><span class="p">,</span> <span class="n">lw</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s">&#39;black&#39;</span><span class="p">);</span>
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   In&nbsp;[6]:
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   <div class="highlight"><pre><span class="n">obs_dist2</span> <span class="o">=</span> <span class="n">mixture_rvs</span><span class="p">([</span><span class="o">.</span><span class="mi">25</span><span class="p">,</span><span class="o">.</span><span class="mi">75</span><span class="p">],</span> <span class="n">size</span><span class="o">=</span><span class="mi">10000</span><span class="p">,</span> <span class="n">dist</span><span class="o">=</span><span class="p">[</span><span class="n">stats</span><span class="o">.</span><span class="n">norm</span><span class="p">,</span> <span class="n">stats</span><span class="o">.</span><span class="n">beta</span><span class="p">],</span>
               <span class="n">kwargs</span> <span class="o">=</span> <span class="p">(</span><span class="nb">dict</span><span class="p">(</span><span class="n">loc</span><span class="o">=-</span><span class="mi">1</span><span class="p">,</span><span class="n">scale</span><span class="o">=.</span><span class="mi">5</span><span class="p">),</span><span class="nb">dict</span><span class="p">(</span><span class="n">loc</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span><span class="n">scale</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span><span class="n">args</span><span class="o">=</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="o">.</span><span class="mi">5</span><span class="p">))))</span>
   
   <span class="n">kde2</span> <span class="o">=</span> <span class="n">sm</span><span class="o">.</span><span class="n">nonparametric</span><span class="o">.</span><span class="n">KDEUnivariate</span><span class="p">(</span><span class="n">obs_dist2</span><span class="p">)</span>
   <span class="n">kde2</span><span class="o">.</span><span class="n">fit</span><span class="p">()</span>
   </pre></div>
   
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   In&nbsp;[7]:
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   <div class="highlight"><pre><span class="n">fig</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">12</span><span class="p">,</span><span class="mi">8</span><span class="p">))</span>
   <span class="n">ax</span> <span class="o">=</span> <span class="n">fig</span><span class="o">.</span><span class="n">add_subplot</span><span class="p">(</span><span class="mi">111</span><span class="p">)</span>
   <span class="n">ax</span><span class="o">.</span><span class="n">hist</span><span class="p">(</span><span class="n">obs_dist2</span><span class="p">,</span> <span class="n">bins</span><span class="o">=</span><span class="mi">50</span><span class="p">,</span> <span class="n">normed</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s">&#39;red&#39;</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">kde2</span><span class="o">.</span><span class="n">support</span><span class="p">,</span> <span class="n">kde2</span><span class="o">.</span><span class="n">density</span><span class="p">,</span> <span class="n">lw</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s">&#39;black&#39;</span><span class="p">);</span>
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   <p>The fitted KDE object is a full non-parametric distribution.</p>
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   <div class="highlight"><pre><span class="n">obs_dist3</span> <span class="o">=</span> <span class="n">mixture_rvs</span><span class="p">([</span><span class="o">.</span><span class="mi">25</span><span class="p">,</span><span class="o">.</span><span class="mi">75</span><span class="p">],</span> <span class="n">size</span><span class="o">=</span><span class="mi">1000</span><span class="p">,</span> <span class="n">dist</span><span class="o">=</span><span class="p">[</span><span class="n">stats</span><span class="o">.</span><span class="n">norm</span><span class="p">,</span> <span class="n">stats</span><span class="o">.</span><span class="n">norm</span><span class="p">],</span>
                   <span class="n">kwargs</span> <span class="o">=</span> <span class="p">(</span><span class="nb">dict</span><span class="p">(</span><span class="n">loc</span><span class="o">=-</span><span class="mi">1</span><span class="p">,</span><span class="n">scale</span><span class="o">=.</span><span class="mi">5</span><span class="p">),</span><span class="nb">dict</span><span class="p">(</span><span class="n">loc</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span><span class="n">scale</span><span class="o">=.</span><span class="mi">5</span><span class="p">)))</span>
   <span class="n">kde3</span> <span class="o">=</span> <span class="n">sm</span><span class="o">.</span><span class="n">nonparametric</span><span class="o">.</span><span class="n">KDEUnivariate</span><span class="p">(</span><span class="n">obs_dist3</span><span class="p">)</span>
   <span class="n">kde3</span><span class="o">.</span><span class="n">fit</span><span class="p">()</span>
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   <div class="highlight"><pre><span class="n">kde3</span><span class="o">.</span><span class="n">entropy</span>
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   <pre>
   1.3343301918419768
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   <div class="highlight"><pre><span class="n">kde3</span><span class="o">.</span><span class="n">evaluate</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
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   array([ 0.17992591])
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   <h4 id="CDF">CDF<a class="anchor-link" href="#CDF">&#182;</a></h4>
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   <div class="highlight"><pre><span class="n">fig</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">12</span><span class="p">,</span><span class="mi">8</span><span class="p">))</span>
   <span class="n">ax</span> <span class="o">=</span> <span class="n">fig</span><span class="o">.</span><span class="n">add_subplot</span><span class="p">(</span><span class="mi">111</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">kde3</span><span class="o">.</span><span class="n">support</span><span class="p">,</span> <span class="n">kde3</span><span class="o">.</span><span class="n">cdf</span><span class="p">);</span>
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   <h4 id="Cumulative-Hazard-Function">Cumulative Hazard Function<a class="anchor-link" href="#Cumulative-Hazard-Function">&#182;</a></h4>
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   In&nbsp;[12]:
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   <div class="highlight"><pre><span class="n">fig</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">12</span><span class="p">,</span><span class="mi">8</span><span class="p">))</span>
   <span class="n">ax</span> <span class="o">=</span> <span class="n">fig</span><span class="o">.</span><span class="n">add_subplot</span><span class="p">(</span><span class="mi">111</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">kde3</span><span class="o">.</span><span class="n">support</span><span class="p">,</span> <span class="n">kde3</span><span class="o">.</span><span class="n">cumhazard</span><span class="p">);</span>
   </pre></div>
   
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   YII=
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   <h4 id="Inverse-CDF">Inverse CDF<a class="anchor-link" href="#Inverse-CDF">&#182;</a></h4>
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   In&nbsp;[13]:
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   <div class="highlight"><pre><span class="n">fig</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">12</span><span class="p">,</span><span class="mi">8</span><span class="p">))</span>
   <span class="n">ax</span> <span class="o">=</span> <span class="n">fig</span><span class="o">.</span><span class="n">add_subplot</span><span class="p">(</span><span class="mi">111</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">kde3</span><span class="o">.</span><span class="n">support</span><span class="p">,</span> <span class="n">kde3</span><span class="o">.</span><span class="n">icdf</span><span class="p">);</span>
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   <h4 id="Survival-Function">Survival Function<a class="anchor-link" href="#Survival-Function">&#182;</a></h4>
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   In&nbsp;[14]:
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   <div class="highlight"><pre><span class="n">fig</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">12</span><span class="p">,</span><span class="mi">8</span><span class="p">))</span>
   <span class="n">ax</span> <span class="o">=</span> <span class="n">fig</span><span class="o">.</span><span class="n">add_subplot</span><span class="p">(</span><span class="mi">111</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">kde3</span><span class="o">.</span><span class="n">support</span><span class="p">,</span> <span class="n">kde3</span><span class="o">.</span><span class="n">sf</span><span class="p">);</span>
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