forked from scikit-learn/scikit-learn.github.io
-
Notifications
You must be signed in to change notification settings - Fork 0
/
Copy pathmetrics.html
425 lines (390 loc) · 28.6 KB
/
metrics.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
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN"
"http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd">
<html xmlns="http://www.w3.org/1999/xhtml">
<head>
<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
<title>4.8. Pairwise metrics, Affinities and Kernels — scikit-learn 0.20.dev0 documentation</title>
<!-- htmltitle is before nature.css - we use this hack to load bootstrap first -->
<meta name="viewport" content="width=device-width, initial-scale=1.0" />
<link rel="stylesheet" href="../_static/css/bootstrap.min.css" media="screen" />
<link rel="stylesheet" href="../_static/css/bootstrap-responsive.css"/>
<link rel="stylesheet" href="../_static/nature.css" type="text/css" />
<link rel="stylesheet" href="../_static/pygments.css" type="text/css" />
<link rel="stylesheet" href="../_static/gallery.css" type="text/css" />
<script type="text/javascript">
var DOCUMENTATION_OPTIONS = {
URL_ROOT: '../',
VERSION: '0.20.dev0',
COLLAPSE_INDEX: false,
FILE_SUFFIX: '.html',
HAS_SOURCE: true,
SOURCELINK_SUFFIX: '.txt'
};
</script>
<script type="text/javascript" src="../_static/jquery.js"></script>
<script type="text/javascript" src="../_static/underscore.js"></script>
<script type="text/javascript" src="../_static/doctools.js"></script>
<script type="text/javascript" src="../_static/js/copybutton.js"></script>
<script type="text/javascript" src="../_static/js/extra.js"></script>
<script type="text/javascript" src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.0/MathJax.js?config=TeX-AMS_SVG"></script>
<link rel="shortcut icon" href="../_static/favicon.ico"/>
<link rel="author" title="About these documents" href="../about.html" />
<link rel="search" title="Search" href="../search.html" />
<link rel="next" title="4.9. Transforming the prediction target (y)" href="preprocessing_targets.html" />
<link rel="prev" title="4.7. Kernel Approximation" href="kernel_approximation.html" />
<meta name="viewport" content="width=device-width, initial-scale=1.0" />
<script src="../_static/js/bootstrap.min.js" type="text/javascript"></script>
<script>
VERSION_SUBDIR = (function(groups) {
return groups ? groups[1] : null;
})(location.href.match(/^https?:\/\/scikit-learn.org\/([^\/]+)/));
</script>
<link rel="canonical" href="http://scikit-learn.org/stable/modules/metrics.html" />
<script type="text/javascript">
$("div.buttonNext, div.buttonPrevious").hover(
function () {
$(this).css('background-color', '#FF9C34');
},
function () {
$(this).css('background-color', '#A7D6E2');
}
);
function showMenu() {
var topNav = document.getElementById("scikit-navbar");
if (topNav.className === "navbar") {
topNav.className += " responsive";
} else {
topNav.className = "navbar";
}
};
</script>
</head>
<body>
<div class="header-wrapper">
<div class="header">
<p class="logo"><a href="../index.html">
<img src="../_static/scikit-learn-logo-small.png" alt="Logo"/>
</a>
</p><div class="navbar" id="scikit-navbar">
<ul>
<li><a href="../index.html">Home</a></li>
<li><a href="../install.html">Installation</a></li>
<li class="btn-li"><div class="btn-group">
<a href="../documentation.html">Documentation</a>
<a class="btn dropdown-toggle" data-toggle="dropdown">
<span class="caret"></span>
</a>
<ul class="dropdown-menu">
<li class="link-title">Scikit-learn <script>document.write(DOCUMENTATION_OPTIONS.VERSION + (VERSION_SUBDIR ? " (" + VERSION_SUBDIR + ")" : ""));</script></li>
<li><a href="../tutorial/index.html">Tutorials</a></li>
<li><a href="../user_guide.html">User guide</a></li>
<li><a href="classes.html">API</a></li>
<li><a href="../glossary.html">Glossary</a></li>
<li><a href="../faq.html">FAQ</a></li>
<li><a href="../developers/contributing.html">Contributing</a></li>
<li class="divider"></li>
<script>if (VERSION_SUBDIR != "stable") document.write('<li><a href="http://scikit-learn.org/stable/documentation.html">Stable version</a></li>')</script>
<script>if (VERSION_SUBDIR != "dev") document.write('<li><a href="http://scikit-learn.org/dev/documentation.html">Development version</a></li>')</script>
<li><a href="http://scikit-learn.org/dev/versions.html">All available versions</a></li>
<li><a href="../_downloads/scikit-learn-docs.pdf">PDF documentation</a></li>
</ul>
</div>
</li>
<li><a href="../auto_examples/index.html">Examples</a></li>
</ul>
<a href="javascript:void(0);" onclick="showMenu()">
<div class="nav-icon">
<div class="hamburger-line"></div>
<div class="hamburger-line"></div>
<div class="hamburger-line"></div>
</div>
</a>
<div class="search_form">
<div class="gcse-search" id="cse" style="width: 100%;"></div>
</div>
</div> <!-- end navbar --></div>
</div>
<!-- GitHub "fork me" ribbon -->
<a href="https://github.com/scikit-learn/scikit-learn">
<img class="fork-me"
style="position: absolute; top: 0; right: 0; border: 0;"
src="../_static/img/forkme.png"
alt="Fork me on GitHub" />
</a>
<div class="content-wrapper">
<div class="sphinxsidebar">
<div class="sphinxsidebarwrapper">
<div class="rel">
<div class="rellink">
<a href="kernel_approximation.html"
accesskey="P">Previous
<br/>
<span class="smallrellink">
4.7. Kernel A...
</span>
<span class="hiddenrellink">
4.7. Kernel Approximation
</span>
</a>
</div>
<div class="spacer">
</div>
<div class="rellink">
<a href="preprocessing_targets.html"
accesskey="N">Next
<br/>
<span class="smallrellink">
4.9. Transfor...
</span>
<span class="hiddenrellink">
4.9. Transforming the prediction target (y)
</span>
</a>
</div>
<!-- Ad a link to the 'up' page -->
<div class="spacer">
</div>
<div class="rellink">
<a href="../data_transforms.html">
Up
<br/>
<span class="smallrellink">
4. Dataset tr...
</span>
<span class="hiddenrellink">
4. Dataset transformations
</span>
</a>
</div>
</div>
<p class="doc-version"><b>scikit-learn v0.20.dev0</b><br/>
<a href="http://scikit-learn.org/stable/support.html#documentation-resources">Other versions</a></p>
<p class="citing">Please <b><a href="../about.html#citing-scikit-learn" style="font-size: 110%;">cite us </a></b>if you use the software.</p>
<ul>
<li><a class="reference internal" href="#">4.8. Pairwise metrics, Affinities and Kernels</a><ul>
<li><a class="reference internal" href="#cosine-similarity">4.8.1. Cosine similarity</a></li>
<li><a class="reference internal" href="#linear-kernel">4.8.2. Linear kernel</a></li>
<li><a class="reference internal" href="#polynomial-kernel">4.8.3. Polynomial kernel</a></li>
<li><a class="reference internal" href="#sigmoid-kernel">4.8.4. Sigmoid kernel</a></li>
<li><a class="reference internal" href="#rbf-kernel">4.8.5. RBF kernel</a></li>
<li><a class="reference internal" href="#laplacian-kernel">4.8.6. Laplacian kernel</a></li>
<li><a class="reference internal" href="#chi-squared-kernel">4.8.7. Chi-squared kernel</a></li>
</ul>
</li>
</ul>
</div>
</div>
<input type="checkbox" id="nav-trigger" class="nav-trigger" checked />
<label for="nav-trigger"></label>
<div class="content">
<div class="documentwrapper">
<div class="bodywrapper">
<div class="body" role="main">
<div class="section" id="pairwise-metrics-affinities-and-kernels">
<span id="metrics"></span><h1>4.8. Pairwise metrics, Affinities and Kernels<a class="headerlink" href="#pairwise-metrics-affinities-and-kernels" title="Permalink to this headline">¶</a></h1>
<p>The <a class="reference internal" href="classes.html#module-sklearn.metrics.pairwise" title="sklearn.metrics.pairwise"><code class="xref py py-mod docutils literal"><span class="pre">sklearn.metrics.pairwise</span></code></a> submodule implements utilities to evaluate
pairwise distances or affinity of sets of samples.</p>
<p>This module contains both distance metrics and kernels. A brief summary is
given on the two here.</p>
<p>Distance metrics are functions <code class="docutils literal"><span class="pre">d(a,</span> <span class="pre">b)</span></code> such that <code class="docutils literal"><span class="pre">d(a,</span> <span class="pre">b)</span> <span class="pre"><</span> <span class="pre">d(a,</span> <span class="pre">c)</span></code>
if objects <code class="docutils literal"><span class="pre">a</span></code> and <code class="docutils literal"><span class="pre">b</span></code> are considered “more similar” than objects <code class="docutils literal"><span class="pre">a</span></code>
and <code class="docutils literal"><span class="pre">c</span></code>. Two objects exactly alike would have a distance of zero.
One of the most popular examples is Euclidean distance.
To be a ‘true’ metric, it must obey the following four conditions:</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="mf">1.</span> <span class="n">d</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span> <span class="o">>=</span> <span class="mi">0</span><span class="p">,</span> <span class="k">for</span> <span class="nb">all</span> <span class="n">a</span> <span class="ow">and</span> <span class="n">b</span>
<span class="mf">2.</span> <span class="n">d</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span> <span class="o">==</span> <span class="mi">0</span><span class="p">,</span> <span class="k">if</span> <span class="ow">and</span> <span class="n">only</span> <span class="k">if</span> <span class="n">a</span> <span class="o">=</span> <span class="n">b</span><span class="p">,</span> <span class="n">positive</span> <span class="n">definiteness</span>
<span class="mf">3.</span> <span class="n">d</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span> <span class="o">==</span> <span class="n">d</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">a</span><span class="p">),</span> <span class="n">symmetry</span>
<span class="mf">4.</span> <span class="n">d</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">c</span><span class="p">)</span> <span class="o"><=</span> <span class="n">d</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span> <span class="o">+</span> <span class="n">d</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">c</span><span class="p">),</span> <span class="n">the</span> <span class="n">triangle</span> <span class="n">inequality</span>
</pre></div>
</div>
<p>Kernels are measures of similarity, i.e. <code class="docutils literal"><span class="pre">s(a,</span> <span class="pre">b)</span> <span class="pre">></span> <span class="pre">s(a,</span> <span class="pre">c)</span></code>
if objects <code class="docutils literal"><span class="pre">a</span></code> and <code class="docutils literal"><span class="pre">b</span></code> are considered “more similar” than objects
<code class="docutils literal"><span class="pre">a</span></code> and <code class="docutils literal"><span class="pre">c</span></code>. A kernel must also be positive semi-definite.</p>
<p>There are a number of ways to convert between a distance metric and a
similarity measure, such as a kernel. Let <code class="docutils literal"><span class="pre">D</span></code> be the distance, and <code class="docutils literal"><span class="pre">S</span></code> be
the kernel:</p>
<blockquote>
<div><ol class="arabic simple">
<li><code class="docutils literal"><span class="pre">S</span> <span class="pre">=</span> <span class="pre">np.exp(-D</span> <span class="pre">*</span> <span class="pre">gamma)</span></code>, where one heuristic for choosing
<code class="docutils literal"><span class="pre">gamma</span></code> is <code class="docutils literal"><span class="pre">1</span> <span class="pre">/</span> <span class="pre">num_features</span></code></li>
<li><code class="docutils literal"><span class="pre">S</span> <span class="pre">=</span> <span class="pre">1.</span> <span class="pre">/</span> <span class="pre">(D</span> <span class="pre">/</span> <span class="pre">np.max(D))</span></code></li>
</ol>
</div></blockquote>
<div class="section" id="cosine-similarity">
<span id="id1"></span><h2>4.8.1. Cosine similarity<a class="headerlink" href="#cosine-similarity" title="Permalink to this headline">¶</a></h2>
<p><a class="reference internal" href="generated/sklearn.metrics.pairwise.cosine_similarity.html#sklearn.metrics.pairwise.cosine_similarity" title="sklearn.metrics.pairwise.cosine_similarity"><code class="xref py py-func docutils literal"><span class="pre">cosine_similarity</span></code></a> computes the L2-normalized dot product of vectors.
That is, if <span class="math">\(x\)</span> and <span class="math">\(y\)</span> are row vectors,
their cosine similarity <span class="math">\(k\)</span> is defined as:</p>
<div class="math">
\[k(x, y) = \frac{x y^\top}{\|x\| \|y\|}\]</div>
<p>This is called cosine similarity, because Euclidean (L2) normalization
projects the vectors onto the unit sphere,
and their dot product is then the cosine of the angle between the points
denoted by the vectors.</p>
<p>This kernel is a popular choice for computing the similarity of documents
represented as tf-idf vectors.
<a class="reference internal" href="generated/sklearn.metrics.pairwise.cosine_similarity.html#sklearn.metrics.pairwise.cosine_similarity" title="sklearn.metrics.pairwise.cosine_similarity"><code class="xref py py-func docutils literal"><span class="pre">cosine_similarity</span></code></a> accepts <code class="docutils literal"><span class="pre">scipy.sparse</span></code> matrices.
(Note that the tf-idf functionality in <code class="docutils literal"><span class="pre">sklearn.feature_extraction.text</span></code>
can produce normalized vectors, in which case <a class="reference internal" href="generated/sklearn.metrics.pairwise.cosine_similarity.html#sklearn.metrics.pairwise.cosine_similarity" title="sklearn.metrics.pairwise.cosine_similarity"><code class="xref py py-func docutils literal"><span class="pre">cosine_similarity</span></code></a>
is equivalent to <a class="reference internal" href="generated/sklearn.metrics.pairwise.linear_kernel.html#sklearn.metrics.pairwise.linear_kernel" title="sklearn.metrics.pairwise.linear_kernel"><code class="xref py py-func docutils literal"><span class="pre">linear_kernel</span></code></a>, only slower.)</p>
<div class="topic">
<p class="topic-title first">References:</p>
<ul class="simple">
<li>C.D. Manning, P. Raghavan and H. Schütze (2008). Introduction to
Information Retrieval. Cambridge University Press.
<a class="reference external" href="http://nlp.stanford.edu/IR-book/html/htmledition/the-vector-space-model-for-scoring-1.html">http://nlp.stanford.edu/IR-book/html/htmledition/the-vector-space-model-for-scoring-1.html</a></li>
</ul>
</div>
</div>
<div class="section" id="linear-kernel">
<span id="id2"></span><h2>4.8.2. Linear kernel<a class="headerlink" href="#linear-kernel" title="Permalink to this headline">¶</a></h2>
<p>The function <a class="reference internal" href="generated/sklearn.metrics.pairwise.linear_kernel.html#sklearn.metrics.pairwise.linear_kernel" title="sklearn.metrics.pairwise.linear_kernel"><code class="xref py py-func docutils literal"><span class="pre">linear_kernel</span></code></a> computes the linear kernel, that is, a
special case of <a class="reference internal" href="generated/sklearn.metrics.pairwise.polynomial_kernel.html#sklearn.metrics.pairwise.polynomial_kernel" title="sklearn.metrics.pairwise.polynomial_kernel"><code class="xref py py-func docutils literal"><span class="pre">polynomial_kernel</span></code></a> with <code class="docutils literal"><span class="pre">degree=1</span></code> and <code class="docutils literal"><span class="pre">coef0=0</span></code> (homogeneous).
If <code class="docutils literal"><span class="pre">x</span></code> and <code class="docutils literal"><span class="pre">y</span></code> are column vectors, their linear kernel is:</p>
<div class="math">
\[k(x, y) = x^\top y\]</div>
</div>
<div class="section" id="polynomial-kernel">
<span id="id3"></span><h2>4.8.3. Polynomial kernel<a class="headerlink" href="#polynomial-kernel" title="Permalink to this headline">¶</a></h2>
<p>The function <a class="reference internal" href="generated/sklearn.metrics.pairwise.polynomial_kernel.html#sklearn.metrics.pairwise.polynomial_kernel" title="sklearn.metrics.pairwise.polynomial_kernel"><code class="xref py py-func docutils literal"><span class="pre">polynomial_kernel</span></code></a> computes the degree-d polynomial kernel
between two vectors. The polynomial kernel represents the similarity between two
vectors. Conceptually, the polynomial kernels considers not only the similarity
between vectors under the same dimension, but also across dimensions. When used
in machine learning algorithms, this allows to account for feature interaction.</p>
<p>The polynomial kernel is defined as:</p>
<div class="math">
\[k(x, y) = (\gamma x^\top y +c_0)^d\]</div>
<p>where:</p>
<blockquote>
<div><ul class="simple">
<li><code class="docutils literal"><span class="pre">x</span></code>, <code class="docutils literal"><span class="pre">y</span></code> are the input vectors</li>
<li><code class="docutils literal"><span class="pre">d</span></code> is the kernel degree</li>
</ul>
</div></blockquote>
<p>If <span class="math">\(c_0 = 0\)</span> the kernel is said to be homogeneous.</p>
</div>
<div class="section" id="sigmoid-kernel">
<span id="id4"></span><h2>4.8.4. Sigmoid kernel<a class="headerlink" href="#sigmoid-kernel" title="Permalink to this headline">¶</a></h2>
<p>The function <a class="reference internal" href="generated/sklearn.metrics.pairwise.sigmoid_kernel.html#sklearn.metrics.pairwise.sigmoid_kernel" title="sklearn.metrics.pairwise.sigmoid_kernel"><code class="xref py py-func docutils literal"><span class="pre">sigmoid_kernel</span></code></a> computes the sigmoid kernel between two
vectors. The sigmoid kernel is also known as hyperbolic tangent, or Multilayer
Perceptron (because, in the neural network field, it is often used as neuron
activation function). It is defined as:</p>
<div class="math">
\[k(x, y) = \tanh( \gamma x^\top y + c_0)\]</div>
<p>where:</p>
<blockquote>
<div><ul class="simple">
<li><code class="docutils literal"><span class="pre">x</span></code>, <code class="docutils literal"><span class="pre">y</span></code> are the input vectors</li>
<li><span class="math">\(\gamma\)</span> is known as slope</li>
<li><span class="math">\(c_0\)</span> is known as intercept</li>
</ul>
</div></blockquote>
</div>
<div class="section" id="rbf-kernel">
<span id="id5"></span><h2>4.8.5. RBF kernel<a class="headerlink" href="#rbf-kernel" title="Permalink to this headline">¶</a></h2>
<p>The function <a class="reference internal" href="generated/sklearn.metrics.pairwise.rbf_kernel.html#sklearn.metrics.pairwise.rbf_kernel" title="sklearn.metrics.pairwise.rbf_kernel"><code class="xref py py-func docutils literal"><span class="pre">rbf_kernel</span></code></a> computes the radial basis function (RBF) kernel
between two vectors. This kernel is defined as:</p>
<div class="math">
\[k(x, y) = \exp( -\gamma \| x-y \|^2)\]</div>
<p>where <code class="docutils literal"><span class="pre">x</span></code> and <code class="docutils literal"><span class="pre">y</span></code> are the input vectors. If <span class="math">\(\gamma = \sigma^{-2}\)</span>
the kernel is known as the Gaussian kernel of variance <span class="math">\(\sigma^2\)</span>.</p>
</div>
<div class="section" id="laplacian-kernel">
<span id="id6"></span><h2>4.8.6. Laplacian kernel<a class="headerlink" href="#laplacian-kernel" title="Permalink to this headline">¶</a></h2>
<p>The function <a class="reference internal" href="generated/sklearn.metrics.pairwise.laplacian_kernel.html#sklearn.metrics.pairwise.laplacian_kernel" title="sklearn.metrics.pairwise.laplacian_kernel"><code class="xref py py-func docutils literal"><span class="pre">laplacian_kernel</span></code></a> is a variant on the radial basis
function kernel defined as:</p>
<div class="math">
\[k(x, y) = \exp( -\gamma \| x-y \|_1)\]</div>
<p>where <code class="docutils literal"><span class="pre">x</span></code> and <code class="docutils literal"><span class="pre">y</span></code> are the input vectors and <span class="math">\(\|x-y\|_1\)</span> is the
Manhattan distance between the input vectors.</p>
<p>It has proven useful in ML applied to noiseless data.
See e.g. <a class="reference external" href="http://onlinelibrary.wiley.com/doi/10.1002/qua.24954/abstract/">Machine learning for quantum mechanics in a nutshell</a>.</p>
</div>
<div class="section" id="chi-squared-kernel">
<span id="chi2-kernel"></span><h2>4.8.7. Chi-squared kernel<a class="headerlink" href="#chi-squared-kernel" title="Permalink to this headline">¶</a></h2>
<p>The chi-squared kernel is a very popular choice for training non-linear SVMs in
computer vision applications.
It can be computed using <a class="reference internal" href="generated/sklearn.metrics.pairwise.chi2_kernel.html#sklearn.metrics.pairwise.chi2_kernel" title="sklearn.metrics.pairwise.chi2_kernel"><code class="xref py py-func docutils literal"><span class="pre">chi2_kernel</span></code></a> and then passed to an
<a class="reference internal" href="generated/sklearn.svm.SVC.html#sklearn.svm.SVC" title="sklearn.svm.SVC"><code class="xref py py-class docutils literal"><span class="pre">sklearn.svm.SVC</span></code></a> with <code class="docutils literal"><span class="pre">kernel="precomputed"</span></code>:</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">sklearn.svm</span> <span class="k">import</span> <span class="n">SVC</span>
<span class="gp">>>> </span><span class="kn">from</span> <span class="nn">sklearn.metrics.pairwise</span> <span class="k">import</span> <span class="n">chi2_kernel</span>
<span class="gp">>>> </span><span class="n">X</span> <span class="o">=</span> <span class="p">[[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</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">8</span><span class="p">],</span> <span class="p">[</span><span class="o">.</span><span class="mi">7</span><span class="p">,</span> <span class="o">.</span><span class="mi">3</span><span class="p">]]</span>
<span class="gp">>>> </span><span class="n">y</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span>
<span class="gp">>>> </span><span class="n">K</span> <span class="o">=</span> <span class="n">chi2_kernel</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">gamma</span><span class="o">=.</span><span class="mi">5</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">K</span>
<span class="go">array([[1. , 0.36787944, 0.89483932, 0.58364548],</span>
<span class="go"> [0.36787944, 1. , 0.51341712, 0.83822343],</span>
<span class="go"> [0.89483932, 0.51341712, 1. , 0.7768366 ],</span>
<span class="go"> [0.58364548, 0.83822343, 0.7768366 , 1. ]])</span>
<span class="gp">>>> </span><span class="n">svm</span> <span class="o">=</span> <span class="n">SVC</span><span class="p">(</span><span class="n">kernel</span><span class="o">=</span><span class="s1">'precomputed'</span><span class="p">)</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">K</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">svm</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">K</span><span class="p">)</span>
<span class="go">array([0, 1, 0, 1])</span>
</pre></div>
</div>
<p>It can also be directly used as the <code class="docutils literal"><span class="pre">kernel</span></code> argument:</p>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">svm</span> <span class="o">=</span> <span class="n">SVC</span><span class="p">(</span><span class="n">kernel</span><span class="o">=</span><span class="n">chi2_kernel</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="n">y</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">svm</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
<span class="go">array([0, 1, 0, 1])</span>
</pre></div>
</div>
<p>The chi squared kernel is given by</p>
<div class="math">
\[k(x, y) = \exp \left (-\gamma \sum_i \frac{(x[i] - y[i]) ^ 2}{x[i] + y[i]} \right )\]</div>
<p>The data is assumed to be non-negative, and is often normalized to have an L1-norm of one.
The normalization is rationalized with the connection to the chi squared distance,
which is a distance between discrete probability distributions.</p>
<p>The chi squared kernel is most commonly used on histograms (bags) of visual words.</p>
<div class="topic">
<p class="topic-title first">References:</p>
<ul class="simple">
<li>Zhang, J. and Marszalek, M. and Lazebnik, S. and Schmid, C.
Local features and kernels for classification of texture and object
categories: A comprehensive study
International Journal of Computer Vision 2007
<a class="reference external" href="http://research.microsoft.com/en-us/um/people/manik/projects/trade-off/papers/ZhangIJCV06.pdf">http://research.microsoft.com/en-us/um/people/manik/projects/trade-off/papers/ZhangIJCV06.pdf</a></li>
</ul>
</div>
</div>
</div>
</div>
</div>
</div>
<div class="clearer"></div>
</div>
</div>
<div class="footer">
© 2007 - 2018, scikit-learn developers (BSD License).
<a href="../_sources/modules/metrics.rst.txt" rel="nofollow">Show this page source</a>
</div>
<div class="rel">
<div class="buttonPrevious">
<a href="kernel_approximation.html">Previous
</a>
</div>
<div class="buttonNext">
<a href="preprocessing_targets.html">Next
</a>
</div>
</div>
<script type="text/javascript">
var _gaq = _gaq || [];
_gaq.push(['_setAccount', 'UA-22606712-2']);
_gaq.push(['_trackPageview']);
(function() {
var ga = document.createElement('script'); ga.type = 'text/javascript'; ga.async = true;
ga.src = ('https:' == document.location.protocol ? 'https://ssl' : 'http://www') + '.google-analytics.com/ga.js';
var s = document.getElementsByTagName('script')[0]; s.parentNode.insertBefore(ga, s);
})();
</script>
<script>
(function() {
var cx = '016639176250731907682:tjtqbvtvij0';
var gcse = document.createElement('script'); gcse.type = 'text/javascript'; gcse.async = true;
gcse.src = 'https://cse.google.com/cse.js?cx=' + cx;
var s = document.getElementsByTagName('script')[0]; s.parentNode.insertBefore(gcse, s);
})();
</script>
</body>
</html>