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<li><a class="reference internal" href="#">2.8. Density Estimation</a><ul>
<li><a class="reference internal" href="#density-estimation-histograms">2.8.1. Density Estimation: Histograms</a></li>
<li><a class="reference internal" href="#kernel-density-estimation">2.8.2. Kernel Density Estimation</a></li>
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<section id="density-estimation">
<span id="id1"></span><h1><span class="section-number">2.8. </span>Density Estimation<a class="headerlink" href="#density-estimation" title="Permalink to this heading">¶</a></h1>
<p>Density estimation walks the line between unsupervised learning, feature
engineering, and data modeling. Some of the most popular and useful
density estimation techniques are mixture models such as
Gaussian Mixtures (<a class="reference internal" href="generated/sklearn.mixture.GaussianMixture.html#sklearn.mixture.GaussianMixture" title="sklearn.mixture.GaussianMixture"><code class="xref py py-class docutils literal notranslate"><span class="pre">GaussianMixture</span></code></a>), and
neighbor-based approaches such as the kernel density estimate
(<a class="reference internal" href="generated/sklearn.neighbors.KernelDensity.html#sklearn.neighbors.KernelDensity" title="sklearn.neighbors.KernelDensity"><code class="xref py py-class docutils literal notranslate"><span class="pre">KernelDensity</span></code></a>).
Gaussian Mixtures are discussed more fully in the context of
<a class="reference internal" href="clustering.html#clustering"><span class="std std-ref">clustering</span></a>, because the technique is also useful as
an unsupervised clustering scheme.</p>
<p>Density estimation is a very simple concept, and most people are already
familiar with one common density estimation technique: the histogram.</p>
<section id="density-estimation-histograms">
<h2><span class="section-number">2.8.1. </span>Density Estimation: Histograms<a class="headerlink" href="#density-estimation-histograms" title="Permalink to this heading">¶</a></h2>
<p>A histogram is a simple visualization of data where bins are defined, and the
number of data points within each bin is tallied. An example of a histogram
can be seen in the upper-left panel of the following figure:</p>
<p class="centered">
<strong><a class="reference external" href="../auto_examples/neighbors/plot_kde_1d.html"><img alt="hist_to_kde" src="../_images/sphx_glr_plot_kde_1d_001.png" style="width: 512.0px; height: 384.0px;" /></a></strong></p><p>A major problem with histograms, however, is that the choice of binning can
have a disproportionate effect on the resulting visualization. Consider the
upper-right panel of the above figure. It shows a histogram over the same
data, with the bins shifted right. The results of the two visualizations look
entirely different, and might lead to different interpretations of the data.</p>
<p>Intuitively, one can also think of a histogram as a stack of blocks, one block
per point. By stacking the blocks in the appropriate grid space, we recover
the histogram. But what if, instead of stacking the blocks on a regular grid,
we center each block on the point it represents, and sum the total height at
each location? This idea leads to the lower-left visualization. It is perhaps
not as clean as a histogram, but the fact that the data drive the block
locations mean that it is a much better representation of the underlying
data.</p>
<p>This visualization is an example of a <em>kernel density estimation</em>, in this case
with a top-hat kernel (i.e. a square block at each point). We can recover a
smoother distribution by using a smoother kernel. The bottom-right plot shows
a Gaussian kernel density estimate, in which each point contributes a Gaussian
curve to the total. The result is a smooth density estimate which is derived
from the data, and functions as a powerful non-parametric model of the
distribution of points.</p>
</section>
<section id="kernel-density-estimation">
<span id="kernel-density"></span><h2><span class="section-number">2.8.2. </span>Kernel Density Estimation<a class="headerlink" href="#kernel-density-estimation" title="Permalink to this heading">¶</a></h2>
<p>Kernel density estimation in scikit-learn is implemented in the
<a class="reference internal" href="generated/sklearn.neighbors.KernelDensity.html#sklearn.neighbors.KernelDensity" title="sklearn.neighbors.KernelDensity"><code class="xref py py-class docutils literal notranslate"><span class="pre">KernelDensity</span></code></a> estimator, which uses the
Ball Tree or KD Tree for efficient queries (see <a class="reference internal" href="neighbors.html#neighbors"><span class="std std-ref">Nearest Neighbors</span></a> for
a discussion of these). Though the above example
uses a 1D data set for simplicity, kernel density estimation can be
performed in any number of dimensions, though in practice the curse of
dimensionality causes its performance to degrade in high dimensions.</p>
<p>In the following figure, 100 points are drawn from a bimodal distribution,
and the kernel density estimates are shown for three choices of kernels:</p>
<p class="centered">
<strong><a class="reference external" href="../auto_examples/neighbors/plot_kde_1d.html"><img alt="kde_1d_distribution" src="../_images/sphx_glr_plot_kde_1d_003.png" style="width: 512.0px; height: 384.0px;" /></a></strong></p><p>It’s clear how the kernel shape affects the smoothness of the resulting
distribution. The scikit-learn kernel density estimator can be used as
follows:</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">sklearn.neighbors</span> <span class="kn">import</span> <span class="n">KernelDensity</span>
<span class="gp">>>> </span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="gp">>>> </span><span class="n">X</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="o">-</span><span class="mi">2</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="o">-</span><span class="mi">3</span><span class="p">,</span> <span class="o">-</span><span class="mi">2</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">]])</span>
<span class="gp">>>> </span><span class="n">kde</span> <span class="o">=</span> <span class="n">KernelDensity</span><span class="p">(</span><span class="n">kernel</span><span class="o">=</span><span class="s1">'gaussian'</span><span class="p">,</span> <span class="n">bandwidth</span><span class="o">=</span><span class="mf">0.2</span><span class="p">)</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">kde</span><span class="o">.</span><span class="n">score_samples</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
<span class="go">array([-0.41075698, -0.41075698, -0.41076071, -0.41075698, -0.41075698,</span>
<span class="go"> -0.41076071])</span>
</pre></div>
</div>
<p>Here we have used <code class="docutils literal notranslate"><span class="pre">kernel='gaussian'</span></code>, as seen above.
Mathematically, a kernel is a positive function <span class="math notranslate nohighlight">\(K(x;h)\)</span>
which is controlled by the bandwidth parameter <span class="math notranslate nohighlight">\(h\)</span>.
Given this kernel form, the density estimate at a point <span class="math notranslate nohighlight">\(y\)</span> within
a group of points <span class="math notranslate nohighlight">\(x_i; i=1\cdots N\)</span> is given by:</p>
<div class="math notranslate nohighlight">
\[\rho_K(y) = \sum_{i=1}^{N} K(y - x_i; h)\]</div>
<p>The bandwidth here acts as a smoothing parameter, controlling the tradeoff
between bias and variance in the result. A large bandwidth leads to a very
smooth (i.e. high-bias) density distribution. A small bandwidth leads
to an unsmooth (i.e. high-variance) density distribution.</p>
<p>The parameter <code class="docutils literal notranslate"><span class="pre">bandwidth</span></code> controls this smoothing. One can either set
manually this parameter or use Scott’s and Silvermann’s estimation
methods.</p>
<p><a class="reference internal" href="generated/sklearn.neighbors.KernelDensity.html#sklearn.neighbors.KernelDensity" title="sklearn.neighbors.KernelDensity"><code class="xref py py-class docutils literal notranslate"><span class="pre">KernelDensity</span></code></a> implements several common kernel
forms, which are shown in the following figure:</p>
<p class="centered">
<strong><a class="reference external" href="../auto_examples/neighbors/plot_kde_1d.html"><img alt="kde_kernels" src="../_images/sphx_glr_plot_kde_1d_002.png" style="width: 512.0px; height: 384.0px;" /></a></strong></p><p><details>
<summary class="btn btn-light">
<strong>kernels’ mathematical expressions</strong>
<span class="tooltiptext">Click for more details</span>
</summary>
<div class="card"></p>
<p>The form of these kernels is as follows:</p>
<ul>
<li><p>Gaussian kernel (<code class="docutils literal notranslate"><span class="pre">kernel</span> <span class="pre">=</span> <span class="pre">'gaussian'</span></code>)</p>
<p><span class="math notranslate nohighlight">\(K(x; h) \propto \exp(- \frac{x^2}{2h^2} )\)</span></p>
</li>
<li><p>Tophat kernel (<code class="docutils literal notranslate"><span class="pre">kernel</span> <span class="pre">=</span> <span class="pre">'tophat'</span></code>)</p>
<p><span class="math notranslate nohighlight">\(K(x; h) \propto 1\)</span> if <span class="math notranslate nohighlight">\(x < h\)</span></p>
</li>
<li><p>Epanechnikov kernel (<code class="docutils literal notranslate"><span class="pre">kernel</span> <span class="pre">=</span> <span class="pre">'epanechnikov'</span></code>)</p>
<p><span class="math notranslate nohighlight">\(K(x; h) \propto 1 - \frac{x^2}{h^2}\)</span></p>
</li>
<li><p>Exponential kernel (<code class="docutils literal notranslate"><span class="pre">kernel</span> <span class="pre">=</span> <span class="pre">'exponential'</span></code>)</p>
<p><span class="math notranslate nohighlight">\(K(x; h) \propto \exp(-x/h)\)</span></p>
</li>
<li><p>Linear kernel (<code class="docutils literal notranslate"><span class="pre">kernel</span> <span class="pre">=</span> <span class="pre">'linear'</span></code>)</p>
<p><span class="math notranslate nohighlight">\(K(x; h) \propto 1 - x/h\)</span> if <span class="math notranslate nohighlight">\(x < h\)</span></p>
</li>
<li><p>Cosine kernel (<code class="docutils literal notranslate"><span class="pre">kernel</span> <span class="pre">=</span> <span class="pre">'cosine'</span></code>)</p>
<p><span class="math notranslate nohighlight">\(K(x; h) \propto \cos(\frac{\pi x}{2h})\)</span> if <span class="math notranslate nohighlight">\(x < h\)</span></p>
</li>
</ul>
<p></div>
</details></p>
<p>The kernel density estimator can be used with any of the valid distance
metrics (see <a class="reference internal" href="generated/sklearn.metrics.DistanceMetric.html#sklearn.metrics.DistanceMetric" title="sklearn.metrics.DistanceMetric"><code class="xref py py-class docutils literal notranslate"><span class="pre">DistanceMetric</span></code></a> for a list of
available metrics), though the results are properly normalized only
for the Euclidean metric. One particularly useful metric is the
<a class="reference external" href="https://en.wikipedia.org/wiki/Haversine_formula">Haversine distance</a>
which measures the angular distance between points on a sphere. Here
is an example of using a kernel density estimate for a visualization
of geospatial data, in this case the distribution of observations of two
different species on the South American continent:</p>
<p class="centered">
<strong><a class="reference external" href="../auto_examples/neighbors/plot_species_kde.html"><img alt="species_kde" src="../_images/sphx_glr_plot_species_kde_001.png" style="width: 512.0px; height: 384.0px;" /></a></strong></p><p>One other useful application of kernel density estimation is to learn a
non-parametric generative model of a dataset in order to efficiently
draw new samples from this generative model.
Here is an example of using this process to
create a new set of hand-written digits, using a Gaussian kernel learned
on a PCA projection of the data:</p>
<p class="centered">
<strong><a class="reference external" href="../auto_examples/neighbors/plot_digits_kde_sampling.html"><img alt="digits_kde" src="../_images/sphx_glr_plot_digits_kde_sampling_001.png" style="width: 512.0px; height: 384.0px;" /></a></strong></p><p>The “new” data consists of linear combinations of the input data, with weights
probabilistically drawn given the KDE model.</p>
<aside class="topic">
<p class="topic-title">Examples:</p>
<ul class="simple">
<li><p><a class="reference internal" href="../auto_examples/neighbors/plot_kde_1d.html#sphx-glr-auto-examples-neighbors-plot-kde-1d-py"><span class="std std-ref">Simple 1D Kernel Density Estimation</span></a>: computation of simple kernel
density estimates in one dimension.</p></li>
<li><p><a class="reference internal" href="../auto_examples/neighbors/plot_digits_kde_sampling.html#sphx-glr-auto-examples-neighbors-plot-digits-kde-sampling-py"><span class="std std-ref">Kernel Density Estimation</span></a>: an example of using
Kernel Density estimation to learn a generative model of the hand-written
digits data, and drawing new samples from this model.</p></li>
<li><p><a class="reference internal" href="../auto_examples/neighbors/plot_species_kde.html#sphx-glr-auto-examples-neighbors-plot-species-kde-py"><span class="std std-ref">Kernel Density Estimate of Species Distributions</span></a>: an example of Kernel Density
estimation using the Haversine distance metric to visualize geospatial data</p></li>
</ul>
</aside>
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