-
Notifications
You must be signed in to change notification settings - Fork 81
/
Copy pathplot_sparse_cov.html
402 lines (357 loc) · 34.7 KB
/
plot_sparse_cov.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
<!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>Sparse inverse covariance estimation — scikit-learn 0.20.4 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: '0.20.4',
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="Shrinkage covariance estimation: LedoitWolf vs OAS and max-likelihood" href="plot_covariance_estimation.html" />
<link rel="prev" title="Ledoit-Wolf vs OAS estimation" href="plot_lw_vs_oas.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="https://scikit-learn.org/stable/auto_examples/covariance/plot_sparse_cov.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="../../modules/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/index.html">Development</a></li>
<li><a href="../../roadmap.html">Roadmap</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="../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="plot_lw_vs_oas.html"
accesskey="P">Previous
<br/>
<span class="smallrellink">
Ledoit-Wolf v...
</span>
<span class="hiddenrellink">
Ledoit-Wolf vs OAS estimation
</span>
</a>
</div>
<div class="spacer">
</div>
<div class="rellink">
<a href="plot_covariance_estimation.html"
accesskey="N">Next
<br/>
<span class="smallrellink">
Shrinkage cov...
</span>
<span class="hiddenrellink">
Shrinkage covariance estimation: LedoitWolf vs OAS and max-likelihood
</span>
</a>
</div>
<!-- Ad a link to the 'up' page -->
<div class="spacer">
</div>
<div class="rellink">
<a href="../index.html">
Up
<br/>
<span class="smallrellink">
Examples
</span>
<span class="hiddenrellink">
Examples
</span>
</a>
</div>
</div>
<p class="doc-version"><b>scikit-learn v0.20.4</b><br/>
<a href="http://scikit-learn.org/dev/versions.html">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="#">Sparse inverse covariance estimation</a></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="sphx-glr-download-link-note admonition note">
<p class="first admonition-title">Note</p>
<p class="last">Click <a class="reference internal" href="#sphx-glr-download-auto-examples-covariance-plot-sparse-cov-py"><span class="std std-ref">here</span></a> to download the full example code</p>
</div>
<div class="sphx-glr-example-title section" id="sparse-inverse-covariance-estimation">
<span id="sphx-glr-auto-examples-covariance-plot-sparse-cov-py"></span><h1>Sparse inverse covariance estimation<a class="headerlink" href="#sparse-inverse-covariance-estimation" title="Permalink to this headline">¶</a></h1>
<p>Using the GraphicalLasso estimator to learn a covariance and sparse precision
from a small number of samples.</p>
<p>To estimate a probabilistic model (e.g. a Gaussian model), estimating the
precision matrix, that is the inverse covariance matrix, is as important
as estimating the covariance matrix. Indeed a Gaussian model is
parametrized by the precision matrix.</p>
<p>To be in favorable recovery conditions, we sample the data from a model
with a sparse inverse covariance matrix. In addition, we ensure that the
data is not too much correlated (limiting the largest coefficient of the
precision matrix) and that there a no small coefficients in the
precision matrix that cannot be recovered. In addition, with a small
number of observations, it is easier to recover a correlation matrix
rather than a covariance, thus we scale the time series.</p>
<p>Here, the number of samples is slightly larger than the number of
dimensions, thus the empirical covariance is still invertible. However,
as the observations are strongly correlated, the empirical covariance
matrix is ill-conditioned and as a result its inverse –the empirical
precision matrix– is very far from the ground truth.</p>
<p>If we use l2 shrinkage, as with the Ledoit-Wolf estimator, as the number
of samples is small, we need to shrink a lot. As a result, the
Ledoit-Wolf precision is fairly close to the ground truth precision, that
is not far from being diagonal, but the off-diagonal structure is lost.</p>
<p>The l1-penalized estimator can recover part of this off-diagonal
structure. It learns a sparse precision. It is not able to
recover the exact sparsity pattern: it detects too many non-zero
coefficients. However, the highest non-zero coefficients of the l1
estimated correspond to the non-zero coefficients in the ground truth.
Finally, the coefficients of the l1 precision estimate are biased toward
zero: because of the penalty, they are all smaller than the corresponding
ground truth value, as can be seen on the figure.</p>
<p>Note that, the color range of the precision matrices is tweaked to
improve readability of the figure. The full range of values of the
empirical precision is not displayed.</p>
<p>The alpha parameter of the GraphicalLasso setting the sparsity of the model is
set by internal cross-validation in the GraphicalLassoCV. As can be
seen on figure 2, the grid to compute the cross-validation score is
iteratively refined in the neighborhood of the maximum.</p>
<ul class="sphx-glr-horizontal">
<li><img alt="../../_images/sphx_glr_plot_sparse_cov_001.png" class="sphx-glr-multi-img first" src="../../_images/sphx_glr_plot_sparse_cov_001.png" />
</li>
<li><img alt="../../_images/sphx_glr_plot_sparse_cov_002.png" class="sphx-glr-multi-img first" src="../../_images/sphx_glr_plot_sparse_cov_002.png" />
</li>
</ul>
<p class="sphx-glr-script-out">Out:</p>
<div class="sphx-glr-script-out highlight-none"><div class="highlight"><pre><span></span>
</pre></div>
</div>
<div class="line-block">
<div class="line"><br /></div>
</div>
<div class="highlight-default"><div class="highlight"><pre><span></span><span class="nb">print</span><span class="p">(</span><span class="vm">__doc__</span><span class="p">)</span>
<span class="c1"># author: Gael Varoquaux <[email protected]></span>
<span class="c1"># License: BSD 3 clause</span>
<span class="c1"># Copyright: INRIA</span>
<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="kn">from</span> <span class="nn">scipy</span> <span class="k">import</span> <span class="n">linalg</span>
<span class="kn">from</span> <span class="nn">sklearn.datasets</span> <span class="k">import</span> <a href="../../modules/generated/sklearn.datasets.make_sparse_spd_matrix.html#sklearn.datasets.make_sparse_spd_matrix" title="View documentation for sklearn.datasets.make_sparse_spd_matrix"><span class="n">make_sparse_spd_matrix</span></a>
<span class="kn">from</span> <span class="nn">sklearn.covariance</span> <span class="k">import</span> <a href="../../modules/generated/sklearn.covariance.GraphicalLassoCV.html#sklearn.covariance.GraphicalLassoCV" title="View documentation for sklearn.covariance.GraphicalLassoCV"><span class="n">GraphicalLassoCV</span></a><span class="p">,</span> <a href="../../modules/generated/sklearn.covariance.ledoit_wolf.html#sklearn.covariance.ledoit_wolf" title="View documentation for sklearn.covariance.ledoit_wolf"><span class="n">ledoit_wolf</span></a>
<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
<span class="c1"># #############################################################################</span>
<span class="c1"># Generate the data</span>
<span class="n">n_samples</span> <span class="o">=</span> <span class="mi">60</span>
<span class="n">n_features</span> <span class="o">=</span> <span class="mi">20</span>
<span class="n">prng</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">RandomState</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
<span class="n">prec</span> <span class="o">=</span> <a href="../../modules/generated/sklearn.datasets.make_sparse_spd_matrix.html#sklearn.datasets.make_sparse_spd_matrix" title="View documentation for sklearn.datasets.make_sparse_spd_matrix"><span class="n">make_sparse_spd_matrix</span></a><span class="p">(</span><span class="n">n_features</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=.</span><span class="mi">98</span><span class="p">,</span>
<span class="n">smallest_coef</span><span class="o">=.</span><span class="mi">4</span><span class="p">,</span>
<span class="n">largest_coef</span><span class="o">=.</span><span class="mi">7</span><span class="p">,</span>
<span class="n">random_state</span><span class="o">=</span><span class="n">prng</span><span class="p">)</span>
<span class="n">cov</span> <span class="o">=</span> <a href="https://docs.scipy.org/doc/scipy/reference/generated/scipy.linalg.inv.html#scipy.linalg.inv" title="View documentation for scipy.linalg.inv"><span class="n">linalg</span><span class="o">.</span><span class="n">inv</span></a><span class="p">(</span><span class="n">prec</span><span class="p">)</span>
<span class="n">d</span> <span class="o">=</span> <a href="https://docs.scipy.org/doc/numpy/reference/generated/numpy.sqrt.html#numpy.sqrt" title="View documentation for numpy.sqrt"><span class="n">np</span><span class="o">.</span><span class="n">sqrt</span></a><span class="p">(</span><a href="https://docs.scipy.org/doc/numpy/reference/generated/numpy.diag.html#numpy.diag" title="View documentation for numpy.diag"><span class="n">np</span><span class="o">.</span><span class="n">diag</span></a><span class="p">(</span><span class="n">cov</span><span class="p">))</span>
<span class="n">cov</span> <span class="o">/=</span> <span class="n">d</span>
<span class="n">cov</span> <span class="o">/=</span> <span class="n">d</span><span class="p">[:,</span> <a href="https://docs.scipy.org/doc/numpy/reference/constants.html#numpy.newaxis" title="View documentation for numpy.newaxis"><span class="n">np</span><span class="o">.</span><span class="n">newaxis</span></a><span class="p">]</span>
<span class="n">prec</span> <span class="o">*=</span> <span class="n">d</span>
<span class="n">prec</span> <span class="o">*=</span> <span class="n">d</span><span class="p">[:,</span> <a href="https://docs.scipy.org/doc/numpy/reference/constants.html#numpy.newaxis" title="View documentation for numpy.newaxis"><span class="n">np</span><span class="o">.</span><span class="n">newaxis</span></a><span class="p">]</span>
<span class="n">X</span> <span class="o">=</span> <span class="n">prng</span><span class="o">.</span><span class="n">multivariate_normal</span><span class="p">(</span><a href="https://docs.scipy.org/doc/numpy/reference/generated/numpy.zeros.html#numpy.zeros" title="View documentation for numpy.zeros"><span class="n">np</span><span class="o">.</span><span class="n">zeros</span></a><span class="p">(</span><span class="n">n_features</span><span class="p">),</span> <span class="n">cov</span><span class="p">,</span> <span class="n">size</span><span class="o">=</span><span class="n">n_samples</span><span class="p">)</span>
<span class="n">X</span> <span class="o">-=</span> <span class="n">X</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">axis</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
<span class="n">X</span> <span class="o">/=</span> <span class="n">X</span><span class="o">.</span><span class="n">std</span><span class="p">(</span><span class="n">axis</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
<span class="c1"># #############################################################################</span>
<span class="c1"># Estimate the covariance</span>
<span class="n">emp_cov</span> <span class="o">=</span> <a href="https://docs.scipy.org/doc/numpy/reference/generated/numpy.dot.html#numpy.dot" title="View documentation for numpy.dot"><span class="n">np</span><span class="o">.</span><span class="n">dot</span></a><span class="p">(</span><span class="n">X</span><span class="o">.</span><span class="n">T</span><span class="p">,</span> <span class="n">X</span><span class="p">)</span> <span class="o">/</span> <span class="n">n_samples</span>
<span class="n">model</span> <span class="o">=</span> <a href="../../modules/generated/sklearn.covariance.GraphicalLassoCV.html#sklearn.covariance.GraphicalLassoCV" title="View documentation for sklearn.covariance.GraphicalLassoCV"><span class="n">GraphicalLassoCV</span></a><span class="p">(</span><span class="n">cv</span><span class="o">=</span><span class="mi">5</span><span class="p">)</span>
<span class="n">model</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">cov_</span> <span class="o">=</span> <span class="n">model</span><span class="o">.</span><span class="n">covariance_</span>
<span class="n">prec_</span> <span class="o">=</span> <span class="n">model</span><span class="o">.</span><span class="n">precision_</span>
<span class="n">lw_cov_</span><span class="p">,</span> <span class="n">_</span> <span class="o">=</span> <a href="../../modules/generated/sklearn.covariance.ledoit_wolf.html#sklearn.covariance.ledoit_wolf" title="View documentation for sklearn.covariance.ledoit_wolf"><span class="n">ledoit_wolf</span></a><span class="p">(</span><span class="n">X</span><span class="p">)</span>
<span class="n">lw_prec_</span> <span class="o">=</span> <a href="https://docs.scipy.org/doc/scipy/reference/generated/scipy.linalg.inv.html#scipy.linalg.inv" title="View documentation for scipy.linalg.inv"><span class="n">linalg</span><span class="o">.</span><span class="n">inv</span></a><span class="p">(</span><span class="n">lw_cov_</span><span class="p">)</span>
<span class="c1"># #############################################################################</span>
<span class="c1"># Plot the results</span>
<a href="https://matplotlib.org/api/_as_gen/matplotlib.pyplot.figure.html#matplotlib.pyplot.figure" title="View documentation for matplotlib.pyplot.figure"><span class="n">plt</span><span class="o">.</span><span class="n">figure</span></a><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span> <span class="mi">6</span><span class="p">))</span>
<a href="https://matplotlib.org/api/_as_gen/matplotlib.pyplot.subplots_adjust.html#matplotlib.pyplot.subplots_adjust" title="View documentation for matplotlib.pyplot.subplots_adjust"><span class="n">plt</span><span class="o">.</span><span class="n">subplots_adjust</span></a><span class="p">(</span><span class="n">left</span><span class="o">=</span><span class="mf">0.02</span><span class="p">,</span> <span class="n">right</span><span class="o">=</span><span class="mf">0.98</span><span class="p">)</span>
<span class="c1"># plot the covariances</span>
<span class="n">covs</span> <span class="o">=</span> <span class="p">[(</span><span class="s1">'Empirical'</span><span class="p">,</span> <span class="n">emp_cov</span><span class="p">),</span> <span class="p">(</span><span class="s1">'Ledoit-Wolf'</span><span class="p">,</span> <span class="n">lw_cov_</span><span class="p">),</span>
<span class="p">(</span><span class="s1">'GraphicalLassoCV'</span><span class="p">,</span> <span class="n">cov_</span><span class="p">),</span> <span class="p">(</span><span class="s1">'True'</span><span class="p">,</span> <span class="n">cov</span><span class="p">)]</span>
<span class="n">vmax</span> <span class="o">=</span> <span class="n">cov_</span><span class="o">.</span><span class="n">max</span><span class="p">()</span>
<span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="n">this_cov</span><span class="p">)</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">covs</span><span class="p">):</span>
<a href="https://matplotlib.org/api/_as_gen/matplotlib.pyplot.subplot.html#matplotlib.pyplot.subplot" title="View documentation for matplotlib.pyplot.subplot"><span class="n">plt</span><span class="o">.</span><span class="n">subplot</span></a><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
<a href="https://matplotlib.org/api/_as_gen/matplotlib.pyplot.imshow.html#matplotlib.pyplot.imshow" title="View documentation for matplotlib.pyplot.imshow"><span class="n">plt</span><span class="o">.</span><span class="n">imshow</span></a><span class="p">(</span><span class="n">this_cov</span><span class="p">,</span> <span class="n">interpolation</span><span class="o">=</span><span class="s1">'nearest'</span><span class="p">,</span> <span class="n">vmin</span><span class="o">=-</span><span class="n">vmax</span><span class="p">,</span> <span class="n">vmax</span><span class="o">=</span><span class="n">vmax</span><span class="p">,</span>
<span class="n">cmap</span><span class="o">=</span><span class="n">plt</span><span class="o">.</span><span class="n">cm</span><span class="o">.</span><span class="n">RdBu_r</span><span class="p">)</span>
<a href="https://matplotlib.org/api/_as_gen/matplotlib.pyplot.xticks.html#matplotlib.pyplot.xticks" title="View documentation for matplotlib.pyplot.xticks"><span class="n">plt</span><span class="o">.</span><span class="n">xticks</span></a><span class="p">(())</span>
<a href="https://matplotlib.org/api/_as_gen/matplotlib.pyplot.yticks.html#matplotlib.pyplot.yticks" title="View documentation for matplotlib.pyplot.yticks"><span class="n">plt</span><span class="o">.</span><span class="n">yticks</span></a><span class="p">(())</span>
<a href="https://matplotlib.org/api/_as_gen/matplotlib.pyplot.title.html#matplotlib.pyplot.title" title="View documentation for matplotlib.pyplot.title"><span class="n">plt</span><span class="o">.</span><span class="n">title</span></a><span class="p">(</span><span class="s1">'</span><span class="si">%s</span><span class="s1"> covariance'</span> <span class="o">%</span> <span class="n">name</span><span class="p">)</span>
<span class="c1"># plot the precisions</span>
<span class="n">precs</span> <span class="o">=</span> <span class="p">[(</span><span class="s1">'Empirical'</span><span class="p">,</span> <a href="https://docs.scipy.org/doc/scipy/reference/generated/scipy.linalg.inv.html#scipy.linalg.inv" title="View documentation for scipy.linalg.inv"><span class="n">linalg</span><span class="o">.</span><span class="n">inv</span></a><span class="p">(</span><span class="n">emp_cov</span><span class="p">)),</span> <span class="p">(</span><span class="s1">'Ledoit-Wolf'</span><span class="p">,</span> <span class="n">lw_prec_</span><span class="p">),</span>
<span class="p">(</span><span class="s1">'GraphicalLasso'</span><span class="p">,</span> <span class="n">prec_</span><span class="p">),</span> <span class="p">(</span><span class="s1">'True'</span><span class="p">,</span> <span class="n">prec</span><span class="p">)]</span>
<span class="n">vmax</span> <span class="o">=</span> <span class="o">.</span><span class="mi">9</span> <span class="o">*</span> <span class="n">prec_</span><span class="o">.</span><span class="n">max</span><span class="p">()</span>
<span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="n">this_prec</span><span class="p">)</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">precs</span><span class="p">):</span>
<span class="n">ax</span> <span class="o">=</span> <a href="https://matplotlib.org/api/_as_gen/matplotlib.pyplot.subplot.html#matplotlib.pyplot.subplot" title="View documentation for matplotlib.pyplot.subplot"><span class="n">plt</span><span class="o">.</span><span class="n">subplot</span></a><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="n">i</span> <span class="o">+</span> <span class="mi">5</span><span class="p">)</span>
<a href="https://matplotlib.org/api/_as_gen/matplotlib.pyplot.imshow.html#matplotlib.pyplot.imshow" title="View documentation for matplotlib.pyplot.imshow"><span class="n">plt</span><span class="o">.</span><span class="n">imshow</span></a><span class="p">(</span><a href="https://docs.scipy.org/doc/numpy/reference/generated/numpy.ma.masked_equal.html#numpy.ma.masked_equal" title="View documentation for numpy.ma.masked_equal"><span class="n">np</span><span class="o">.</span><span class="n">ma</span><span class="o">.</span><span class="n">masked_equal</span></a><span class="p">(</span><span class="n">this_prec</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span>
<span class="n">interpolation</span><span class="o">=</span><span class="s1">'nearest'</span><span class="p">,</span> <span class="n">vmin</span><span class="o">=-</span><span class="n">vmax</span><span class="p">,</span> <span class="n">vmax</span><span class="o">=</span><span class="n">vmax</span><span class="p">,</span>
<span class="n">cmap</span><span class="o">=</span><span class="n">plt</span><span class="o">.</span><span class="n">cm</span><span class="o">.</span><span class="n">RdBu_r</span><span class="p">)</span>
<a href="https://matplotlib.org/api/_as_gen/matplotlib.pyplot.xticks.html#matplotlib.pyplot.xticks" title="View documentation for matplotlib.pyplot.xticks"><span class="n">plt</span><span class="o">.</span><span class="n">xticks</span></a><span class="p">(())</span>
<a href="https://matplotlib.org/api/_as_gen/matplotlib.pyplot.yticks.html#matplotlib.pyplot.yticks" title="View documentation for matplotlib.pyplot.yticks"><span class="n">plt</span><span class="o">.</span><span class="n">yticks</span></a><span class="p">(())</span>
<a href="https://matplotlib.org/api/_as_gen/matplotlib.pyplot.title.html#matplotlib.pyplot.title" title="View documentation for matplotlib.pyplot.title"><span class="n">plt</span><span class="o">.</span><span class="n">title</span></a><span class="p">(</span><span class="s1">'</span><span class="si">%s</span><span class="s1"> precision'</span> <span class="o">%</span> <span class="n">name</span><span class="p">)</span>
<span class="k">if</span> <span class="nb">hasattr</span><span class="p">(</span><span class="n">ax</span><span class="p">,</span> <span class="s1">'set_facecolor'</span><span class="p">):</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set_facecolor</span><span class="p">(</span><span class="s1">'.7'</span><span class="p">)</span>
<span class="k">else</span><span class="p">:</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set_axis_bgcolor</span><span class="p">(</span><span class="s1">'.7'</span><span class="p">)</span>
<span class="c1"># plot the model selection metric</span>
<a href="https://matplotlib.org/api/_as_gen/matplotlib.pyplot.figure.html#matplotlib.pyplot.figure" title="View documentation for matplotlib.pyplot.figure"><span class="n">plt</span><span class="o">.</span><span class="n">figure</span></a><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">3</span><span class="p">))</span>
<a href="https://matplotlib.org/api/_as_gen/matplotlib.pyplot.axes.html#matplotlib.pyplot.axes" title="View documentation for matplotlib.pyplot.axes"><span class="n">plt</span><span class="o">.</span><span class="n">axes</span></a><span class="p">([</span><span class="o">.</span><span class="mi">2</span><span class="p">,</span> <span class="o">.</span><span class="mi">15</span><span class="p">,</span> <span class="o">.</span><span class="mi">75</span><span class="p">,</span> <span class="o">.</span><span class="mi">7</span><span class="p">])</span>
<a href="https://matplotlib.org/api/_as_gen/matplotlib.pyplot.plot.html#matplotlib.pyplot.plot" title="View documentation for matplotlib.pyplot.plot"><span class="n">plt</span><span class="o">.</span><span class="n">plot</span></a><span class="p">(</span><span class="n">model</span><span class="o">.</span><span class="n">cv_alphas_</span><span class="p">,</span> <a href="https://docs.scipy.org/doc/numpy/reference/generated/numpy.mean.html#numpy.mean" title="View documentation for numpy.mean"><span class="n">np</span><span class="o">.</span><span class="n">mean</span></a><span class="p">(</span><span class="n">model</span><span class="o">.</span><span class="n">grid_scores_</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">),</span> <span class="s1">'o-'</span><span class="p">)</span>
<a href="https://matplotlib.org/api/_as_gen/matplotlib.pyplot.axvline.html#matplotlib.pyplot.axvline" title="View documentation for matplotlib.pyplot.axvline"><span class="n">plt</span><span class="o">.</span><span class="n">axvline</span></a><span class="p">(</span><span class="n">model</span><span class="o">.</span><span class="n">alpha_</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s1">'.5'</span><span class="p">)</span>
<a href="https://matplotlib.org/api/_as_gen/matplotlib.pyplot.title.html#matplotlib.pyplot.title" title="View documentation for matplotlib.pyplot.title"><span class="n">plt</span><span class="o">.</span><span class="n">title</span></a><span class="p">(</span><span class="s1">'Model selection'</span><span class="p">)</span>
<a href="https://matplotlib.org/api/_as_gen/matplotlib.pyplot.ylabel.html#matplotlib.pyplot.ylabel" title="View documentation for matplotlib.pyplot.ylabel"><span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span></a><span class="p">(</span><span class="s1">'Cross-validation score'</span><span class="p">)</span>
<a href="https://matplotlib.org/api/_as_gen/matplotlib.pyplot.xlabel.html#matplotlib.pyplot.xlabel" title="View documentation for matplotlib.pyplot.xlabel"><span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span></a><span class="p">(</span><span class="s1">'alpha'</span><span class="p">)</span>
<a href="https://matplotlib.org/api/_as_gen/matplotlib.pyplot.show.html#matplotlib.pyplot.show" title="View documentation for matplotlib.pyplot.show"><span class="n">plt</span><span class="o">.</span><span class="n">show</span></a><span class="p">()</span>
</pre></div>
</div>
<p class="sphx-glr-timing"><strong>Total running time of the script:</strong> ( 0 minutes 0.615 seconds)</p>
<div class="sphx-glr-footer class sphx-glr-footer-example docutils container" id="sphx-glr-download-auto-examples-covariance-plot-sparse-cov-py">
<div class="sphx-glr-download docutils container">
<a class="reference download internal" href="../../_downloads/plot_sparse_cov.py" download=""><code class="xref download docutils literal"><span class="pre">Download</span> <span class="pre">Python</span> <span class="pre">source</span> <span class="pre">code:</span> <span class="pre">plot_sparse_cov.py</span></code></a></div>
<div class="sphx-glr-download docutils container">
<a class="reference download internal" href="../../_downloads/plot_sparse_cov.ipynb" download=""><code class="xref download docutils literal"><span class="pre">Download</span> <span class="pre">Jupyter</span> <span class="pre">notebook:</span> <span class="pre">plot_sparse_cov.ipynb</span></code></a></div>
</div>
<p class="sphx-glr-signature"><a class="reference external" href="https://sphinx-gallery.github.io">Gallery generated by Sphinx-Gallery</a></p>
</div>
</div>
</div>
</div>
<div class="clearer"></div>
</div>
</div>
<div class="footer">
© 2007 - 2018, scikit-learn developers (BSD License).
<a href="../../_sources/auto_examples/covariance/plot_sparse_cov.rst.txt" rel="nofollow">Show this page source</a>
</div>
<div class="rel">
<div class="buttonPrevious">
<a href="plot_lw_vs_oas.html">Previous
</a>
</div>
<div class="buttonNext">
<a href="plot_covariance_estimation.html">Next
</a>
</div>
</div>
<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>
(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>
<script src="https://scikit-learn.org/versionwarning.js"></script>
</body>
</html>