forked from scikit-learn/scikit-learn.github.io
-
Notifications
You must be signed in to change notification settings - Fork 0
/
Copy pathdensity.html
394 lines (362 loc) · 26.3 KB
/
density.html
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
<!DOCTYPE html>
<!--[if IE 8]><html class="no-js lt-ie9" lang="en" > <![endif]-->
<!--[if gt IE 8]><!--> <html class="no-js" lang="en" > <!--<![endif]-->
<head>
<meta charset="utf-8">
<meta name="generator" content="Docutils 0.19: https://docutils.sourceforge.io/" />
<meta property="og:title" content="2.8. Density Estimation" />
<meta property="og:type" content="website" />
<meta property="og:url" content="https://scikit-learn/stable/modules/density.html" />
<meta property="og:site_name" content="scikit-learn" />
<meta property="og:description" content="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..." />
<meta property="og:image" content="https://scikit-learn/stable/modules/auto_examples/neighbors/images/sphx_glr_plot_kde_1d_001.png" />
<meta property="og:image:alt" content="hist_to_kde" />
<meta name="description" content="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..." />
<meta name="viewport" content="width=device-width, initial-scale=1.0">
<title>2.8. Density Estimation — scikit-learn 1.4.dev0 documentation</title>
<link rel="canonical" href="http://scikit-learn.org/stable/modules/density.html" />
<link rel="shortcut icon" href="../_static/favicon.ico"/>
<link rel="stylesheet" href="../_static/css/vendor/bootstrap.min.css" type="text/css" />
<link rel="stylesheet" href="../_static/pygments.css" type="text/css" />
<link rel="stylesheet" href="../_static/css/theme.css" type="text/css" />
<link rel="stylesheet" href="../_static/copybutton.css" type="text/css" />
<link rel="stylesheet" href="../_static/plot_directive.css" type="text/css" />
<link rel="stylesheet" href="https://fonts.googleapis.com/css?family=Vibur" type="text/css" />
<link rel="stylesheet" href="../_static/jupyterlite_sphinx.css" type="text/css" />
<link rel="stylesheet" href="../_static/sg_gallery.css" type="text/css" />
<link rel="stylesheet" href="../_static/sg_gallery-binder.css" type="text/css" />
<link rel="stylesheet" href="../_static/sg_gallery-dataframe.css" type="text/css" />
<link rel="stylesheet" href="../_static/sg_gallery-rendered-html.css" type="text/css" />
<link rel="stylesheet" href="../_static/css/theme.css" type="text/css" />
<script id="documentation_options" data-url_root="../" src="../_static/documentation_options.js"></script>
<script src="../_static/js/vendor/jquery-3.6.3.slim.min.js"></script>
</head>
<body>
<nav id="navbar" class="sk-docs-navbar navbar navbar-expand-md navbar-light bg-light py-0">
<div class="container-fluid sk-docs-container px-0">
<a class="navbar-brand py-0" href="../index.html">
<img
class="sk-brand-img"
src="../_static/scikit-learn-logo-small.png"
alt="logo"/>
</a>
<button
id="sk-navbar-toggler"
class="navbar-toggler"
type="button"
data-toggle="collapse"
data-target="#navbarSupportedContent"
aria-controls="navbarSupportedContent"
aria-expanded="false"
aria-label="Toggle navigation"
>
<span class="navbar-toggler-icon"></span>
</button>
<div class="sk-navbar-collapse collapse navbar-collapse" id="navbarSupportedContent">
<ul class="navbar-nav mr-auto">
<li class="nav-item">
<a class="sk-nav-link nav-link" href="../install.html">Install</a>
</li>
<li class="nav-item">
<a class="sk-nav-link nav-link" href="../user_guide.html">User Guide</a>
</li>
<li class="nav-item">
<a class="sk-nav-link nav-link" href="classes.html">API</a>
</li>
<li class="nav-item">
<a class="sk-nav-link nav-link" href="../auto_examples/index.html">Examples</a>
</li>
<li class="nav-item">
<a class="sk-nav-link nav-link" target="_blank" rel="noopener noreferrer" href="https://blog.scikit-learn.org/">Community</a>
</li>
<li class="nav-item">
<a class="sk-nav-link nav-link nav-more-item-mobile-items" href="../getting_started.html" >Getting Started</a>
</li>
<li class="nav-item">
<a class="sk-nav-link nav-link nav-more-item-mobile-items" href="../tutorial/index.html" >Tutorial</a>
</li>
<li class="nav-item">
<a class="sk-nav-link nav-link nav-more-item-mobile-items" href="../whats_new/v1.4.html" >What's new</a>
</li>
<li class="nav-item">
<a class="sk-nav-link nav-link nav-more-item-mobile-items" href="../glossary.html" >Glossary</a>
</li>
<li class="nav-item">
<a class="sk-nav-link nav-link nav-more-item-mobile-items" href="../developers/index.html" >Development</a>
</li>
<li class="nav-item">
<a class="sk-nav-link nav-link nav-more-item-mobile-items" href="../faq.html" >FAQ</a>
</li>
<li class="nav-item">
<a class="sk-nav-link nav-link nav-more-item-mobile-items" href="../support.html" >Support</a>
</li>
<li class="nav-item">
<a class="sk-nav-link nav-link nav-more-item-mobile-items" href="../related_projects.html" >Related packages</a>
</li>
<li class="nav-item">
<a class="sk-nav-link nav-link nav-more-item-mobile-items" href="../roadmap.html" >Roadmap</a>
</li>
<li class="nav-item">
<a class="sk-nav-link nav-link nav-more-item-mobile-items" href="../governance.html" >Governance</a>
</li>
<li class="nav-item">
<a class="sk-nav-link nav-link nav-more-item-mobile-items" href="../about.html" >About us</a>
</li>
<li class="nav-item">
<a class="sk-nav-link nav-link nav-more-item-mobile-items" href="https://github.com/scikit-learn/scikit-learn" >GitHub</a>
</li>
<li class="nav-item">
<a class="sk-nav-link nav-link nav-more-item-mobile-items" href="https://scikit-learn.org/dev/versions.html" >Other Versions and Download</a>
</li>
<li class="nav-item dropdown nav-more-item-dropdown">
<a class="sk-nav-link nav-link dropdown-toggle" href="#" id="navbarDropdown" role="button" data-toggle="dropdown" aria-haspopup="true" aria-expanded="false">More</a>
<div class="dropdown-menu" aria-labelledby="navbarDropdown">
<a class="sk-nav-dropdown-item dropdown-item" href="../getting_started.html" >Getting Started</a>
<a class="sk-nav-dropdown-item dropdown-item" href="../tutorial/index.html" >Tutorial</a>
<a class="sk-nav-dropdown-item dropdown-item" href="../whats_new/v1.4.html" >What's new</a>
<a class="sk-nav-dropdown-item dropdown-item" href="../glossary.html" >Glossary</a>
<a class="sk-nav-dropdown-item dropdown-item" href="../developers/index.html" >Development</a>
<a class="sk-nav-dropdown-item dropdown-item" href="../faq.html" >FAQ</a>
<a class="sk-nav-dropdown-item dropdown-item" href="../support.html" >Support</a>
<a class="sk-nav-dropdown-item dropdown-item" href="../related_projects.html" >Related packages</a>
<a class="sk-nav-dropdown-item dropdown-item" href="../roadmap.html" >Roadmap</a>
<a class="sk-nav-dropdown-item dropdown-item" href="../governance.html" >Governance</a>
<a class="sk-nav-dropdown-item dropdown-item" href="../about.html" >About us</a>
<a class="sk-nav-dropdown-item dropdown-item" href="https://github.com/scikit-learn/scikit-learn" >GitHub</a>
<a class="sk-nav-dropdown-item dropdown-item" href="https://scikit-learn.org/dev/versions.html" >Other Versions and Download</a>
</div>
</li>
</ul>
<div id="searchbox" role="search">
<div class="searchformwrapper">
<form class="search" action="../search.html" method="get">
<input class="sk-search-text-input" type="text" name="q" aria-labelledby="searchlabel" />
<input class="sk-search-text-btn" type="submit" value="Go" />
</form>
</div>
</div>
</div>
</div>
</nav>
<div class="d-flex" id="sk-doc-wrapper">
<input type="checkbox" name="sk-toggle-checkbox" id="sk-toggle-checkbox">
<label id="sk-sidemenu-toggle" class="sk-btn-toggle-toc btn sk-btn-primary" for="sk-toggle-checkbox">Toggle Menu</label>
<div id="sk-sidebar-wrapper" class="border-right">
<div class="sk-sidebar-toc-wrapper">
<div class="btn-group w-100 mb-2" role="group" aria-label="rellinks">
<a href="outlier_detection.html" role="button" class="btn sk-btn-rellink py-1" sk-rellink-tooltip="2.7. Novelty and Outlier Detection">Prev</a><a href="../unsupervised_learning.html" role="button" class="btn sk-btn-rellink py-1" sk-rellink-tooltip="2. Unsupervised learning">Up</a>
<a href="neural_networks_unsupervised.html" role="button" class="btn sk-btn-rellink py-1" sk-rellink-tooltip="2.9. Neural network models (unsupervised)">Next</a>
</div>
<div class="alert alert-danger p-1 mb-2" role="alert">
<p class="text-center mb-0">
<strong>scikit-learn 1.4.dev0</strong><br/>
<a href="http://scikit-learn.org/dev/versions.html">Other versions</a>
</p>
</div>
<div class="alert alert-warning p-1 mb-2" role="alert">
<p class="text-center mb-0">
Please <a class="font-weight-bold" href="../about.html#citing-scikit-learn"><string>cite us</string></a> if you use the software.
</p>
</div>
<div class="sk-sidebar-toc">
<ul>
<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>
</ul>
</li>
</ul>
</div>
</div>
</div>
<div id="sk-page-content-wrapper">
<div class="sk-page-content container-fluid body px-md-3" role="main">
<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>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>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>
</section>
</section>
</div>
<div class="container">
<footer class="sk-content-footer">
© 2007 - 2023, scikit-learn developers (BSD License).
<a href="../_sources/modules/density.rst.txt" rel="nofollow">Show this page source</a>
</footer>
</div>
</div>
</div>
<script src="../_static/js/vendor/bootstrap.min.js"></script>
<script>
window.ga=window.ga||function(){(ga.q=ga.q||[]).push(arguments)};ga.l=+new Date;
ga('create', 'UA-22606712-2', 'auto');
ga('set', 'anonymizeIp', true);
ga('send', 'pageview');
</script>
<script async src='https://www.google-analytics.com/analytics.js'></script>
<script defer data-domain="scikit-learn.org" src="https://views.scientific-python.org/js/script.js">
</script>
<script src="../_static/clipboard.min.js"></script>
<script src="../_static/copybutton.js"></script>
<script>
$(document).ready(function() {
/* Add a [>>>] button on the top-right corner of code samples to hide
* the >>> and ... prompts and the output and thus make the code
* copyable. */
var div = $('.highlight-python .highlight,' +
'.highlight-python3 .highlight,' +
'.highlight-pycon .highlight,' +
'.highlight-default .highlight')
var pre = div.find('pre');
// get the styles from the current theme
pre.parent().parent().css('position', 'relative');
// create and add the button to all the code blocks that contain >>>
div.each(function(index) {
var jthis = $(this);
// tracebacks (.gt) contain bare text elements that need to be
// wrapped in a span to work with .nextUntil() (see later)
jthis.find('pre:has(.gt)').contents().filter(function() {
return ((this.nodeType == 3) && (this.data.trim().length > 0));
}).wrap('<span>');
});
/*** Add permalink buttons next to glossary terms ***/
$('dl.glossary > dt[id]').append(function() {
return ('<a class="headerlink" href="#' +
this.getAttribute('id') +
'" title="Permalink to this term">¶</a>');
});
});
</script>
<script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-chtml.js"></script>
<script src="https://scikit-learn.org/versionwarning.js"></script>
</body>
</html>