-
Notifications
You must be signed in to change notification settings - Fork 81
/
Copy pathplot_bias_variance.html
458 lines (384 loc) · 42.9 KB
/
plot_bias_variance.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
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
<!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>Single estimator versus bagging: bias-variance decomposition — scikit-learn 0.19.2 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: '0.19.2',
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>
<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="Plot the decision surfaces of ensembles of trees on the iris dataset" href="plot_forest_iris.html" />
<link rel="prev" title="Gradient Boosting Out-of-Bag estimates" href="plot_gradient_boosting_oob.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/ensemble/plot_bias_variance.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="../../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="../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_gradient_boosting_oob.html"
accesskey="P">Previous
<br/>
<span class="smallrellink">
Gradient Boos...
</span>
<span class="hiddenrellink">
Gradient Boosting Out-of-Bag estimates
</span>
</a>
</div>
<div class="spacer">
</div>
<div class="rellink">
<a href="plot_forest_iris.html"
accesskey="N">Next
<br/>
<span class="smallrellink">
Plot the deci...
</span>
<span class="hiddenrellink">
Plot the decision surfaces of ensembles of trees on the iris dataset
</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 v0.19.2</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="#">Single estimator versus bagging: bias-variance decomposition</a><ul>
<li><a class="reference internal" href="#references">References</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="single-estimator-versus-bagging-bias-variance-decomposition">
<span id="sphx-glr-auto-examples-ensemble-plot-bias-variance-py"></span><h1>Single estimator versus bagging: bias-variance decomposition<a class="headerlink" href="#single-estimator-versus-bagging-bias-variance-decomposition" title="Permalink to this headline">¶</a></h1>
<p>This example illustrates and compares the bias-variance decomposition of the
expected mean squared error of a single estimator against a bagging ensemble.</p>
<p>In regression, the expected mean squared error of an estimator can be
decomposed in terms of bias, variance and noise. On average over datasets of
the regression problem, the bias term measures the average amount by which the
predictions of the estimator differ from the predictions of the best possible
estimator for the problem (i.e., the Bayes model). The variance term measures
the variability of the predictions of the estimator when fit over different
instances LS of the problem. Finally, the noise measures the irreducible part
of the error which is due the variability in the data.</p>
<p>The upper left figure illustrates the predictions (in dark red) of a single
decision tree trained over a random dataset LS (the blue dots) of a toy 1d
regression problem. It also illustrates the predictions (in light red) of other
single decision trees trained over other (and different) randomly drawn
instances LS of the problem. Intuitively, the variance term here corresponds to
the width of the beam of predictions (in light red) of the individual
estimators. The larger the variance, the more sensitive are the predictions for
<cite>x</cite> to small changes in the training set. The bias term corresponds to the
difference between the average prediction of the estimator (in cyan) and the
best possible model (in dark blue). On this problem, we can thus observe that
the bias is quite low (both the cyan and the blue curves are close to each
other) while the variance is large (the red beam is rather wide).</p>
<p>The lower left figure plots the pointwise decomposition of the expected mean
squared error of a single decision tree. It confirms that the bias term (in
blue) is low while the variance is large (in green). It also illustrates the
noise part of the error which, as expected, appears to be constant and around
<cite>0.01</cite>.</p>
<p>The right figures correspond to the same plots but using instead a bagging
ensemble of decision trees. In both figures, we can observe that the bias term
is larger than in the previous case. In the upper right figure, the difference
between the average prediction (in cyan) and the best possible model is larger
(e.g., notice the offset around <cite>x=2</cite>). In the lower right figure, the bias
curve is also slightly higher than in the lower left figure. In terms of
variance however, the beam of predictions is narrower, which suggests that the
variance is lower. Indeed, as the lower right figure confirms, the variance
term (in green) is lower than for single decision trees. Overall, the bias-
variance decomposition is therefore no longer the same. The tradeoff is better
for bagging: averaging several decision trees fit on bootstrap copies of the
dataset slightly increases the bias term but allows for a larger reduction of
the variance, which results in a lower overall mean squared error (compare the
red curves int the lower figures). The script output also confirms this
intuition. The total error of the bagging ensemble is lower than the total
error of a single decision tree, and this difference indeed mainly stems from a
reduced variance.</p>
<p>For further details on bias-variance decomposition, see section 7.3 of <a class="footnote-reference" href="#id2" id="id1">[1]</a>.</p>
<div class="section" id="references">
<h2>References<a class="headerlink" href="#references" title="Permalink to this headline">¶</a></h2>
<table class="docutils footnote" frame="void" id="id2" rules="none">
<colgroup><col class="label" /><col /></colgroup>
<tbody valign="top">
<tr><td class="label"><a class="fn-backref" href="#id1">[1]</a></td><td>T. Hastie, R. Tibshirani and J. Friedman,
“Elements of Statistical Learning”, Springer, 2009.</td></tr>
</tbody>
</table>
<img alt="../../_images/sphx_glr_plot_bias_variance_001.png" class="align-center" src="../../_images/sphx_glr_plot_bias_variance_001.png" />
<p class="sphx-glr-script-out">Out:</p>
<div class="sphx-glr-script-out highlight-default"><div class="highlight"><pre><span></span><span class="n">Tree</span><span class="p">:</span> <span class="mf">0.0255</span> <span class="p">(</span><span class="n">error</span><span class="p">)</span> <span class="o">=</span> <span class="mf">0.0003</span> <span class="p">(</span><span class="n">bias</span><span class="o">^</span><span class="mi">2</span><span class="p">)</span> <span class="o">+</span> <span class="mf">0.0152</span> <span class="p">(</span><span class="n">var</span><span class="p">)</span> <span class="o">+</span> <span class="mf">0.0098</span> <span class="p">(</span><span class="n">noise</span><span class="p">)</span>
<span class="n">Bagging</span><span class="p">(</span><span class="n">Tree</span><span class="p">):</span> <span class="mf">0.0196</span> <span class="p">(</span><span class="n">error</span><span class="p">)</span> <span class="o">=</span> <span class="mf">0.0004</span> <span class="p">(</span><span class="n">bias</span><span class="o">^</span><span class="mi">2</span><span class="p">)</span> <span class="o">+</span> <span class="mf">0.0092</span> <span class="p">(</span><span class="n">var</span><span class="p">)</span> <span class="o">+</span> <span class="mf">0.0098</span> <span class="p">(</span><span class="n">noise</span><span class="p">)</span>
</pre></div>
</div>
<div class="line-block">
<div class="line"><br /></div>
</div>
<div class="highlight-python"><div class="highlight"><pre><span></span><span class="k">print</span><span class="p">(</span><span class="vm">__doc__</span><span class="p">)</span>
<span class="c1"># Author: Gilles Louppe <[email protected]></span>
<span class="c1"># License: BSD 3 clause</span>
<span class="kn">import</span> <span class="nn">numpy</span> <span class="kn">as</span> <span class="nn">np</span>
<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="kn">as</span> <span class="nn">plt</span>
<span class="kn">from</span> <span class="nn">sklearn.ensemble</span> <span class="kn">import</span> <a href="../../modules/generated/sklearn.ensemble.BaggingRegressor.html#sklearn.ensemble.BaggingRegressor" title="View documentation for sklearn.ensemble.BaggingRegressor"><span class="n">BaggingRegressor</span></a>
<span class="kn">from</span> <span class="nn">sklearn.tree</span> <span class="kn">import</span> <a href="../../modules/generated/sklearn.tree.DecisionTreeRegressor.html#sklearn.tree.DecisionTreeRegressor" title="View documentation for sklearn.tree.DecisionTreeRegressor"><span class="n">DecisionTreeRegressor</span></a>
<span class="c1"># Settings</span>
<span class="n">n_repeat</span> <span class="o">=</span> <span class="mi">50</span> <span class="c1"># Number of iterations for computing expectations</span>
<span class="n">n_train</span> <span class="o">=</span> <span class="mi">50</span> <span class="c1"># Size of the training set</span>
<span class="n">n_test</span> <span class="o">=</span> <span class="mi">1000</span> <span class="c1"># Size of the test set</span>
<span class="n">noise</span> <span class="o">=</span> <span class="mf">0.1</span> <span class="c1"># Standard deviation of the noise</span>
<a href="http://docs.scipy.org/doc/numpy-1.8.1/reference/generated/numpy.random.seed.html#numpy.random.seed" title="View documentation for numpy.random.seed"><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">seed</span></a><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
<span class="c1"># Change this for exploring the bias-variance decomposition of other</span>
<span class="c1"># estimators. This should work well for estimators with high variance (e.g.,</span>
<span class="c1"># decision trees or KNN), but poorly for estimators with low variance (e.g.,</span>
<span class="c1"># linear models).</span>
<span class="n">estimators</span> <span class="o">=</span> <span class="p">[(</span><span class="s2">"Tree"</span><span class="p">,</span> <a href="../../modules/generated/sklearn.tree.DecisionTreeRegressor.html#sklearn.tree.DecisionTreeRegressor" title="View documentation for sklearn.tree.DecisionTreeRegressor"><span class="n">DecisionTreeRegressor</span></a><span class="p">()),</span>
<span class="p">(</span><span class="s2">"Bagging(Tree)"</span><span class="p">,</span> <a href="../../modules/generated/sklearn.ensemble.BaggingRegressor.html#sklearn.ensemble.BaggingRegressor" title="View documentation for sklearn.ensemble.BaggingRegressor"><span class="n">BaggingRegressor</span></a><span class="p">(</span><a href="../../modules/generated/sklearn.tree.DecisionTreeRegressor.html#sklearn.tree.DecisionTreeRegressor" title="View documentation for sklearn.tree.DecisionTreeRegressor"><span class="n">DecisionTreeRegressor</span></a><span class="p">()))]</span>
<span class="n">n_estimators</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">estimators</span><span class="p">)</span>
<span class="c1"># Generate data</span>
<span class="k">def</span> <span class="nf">f</span><span class="p">(</span><span class="n">x</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">ravel</span><span class="p">()</span>
<span class="k">return</span> <a href="http://docs.scipy.org/doc/numpy-1.8.1/reference/generated/numpy.exp.html#numpy.exp" title="View documentation for numpy.exp"><span class="n">np</span><span class="o">.</span><span class="n">exp</span></a><span class="p">(</span><span class="o">-</span><span class="n">x</span> <span class="o">**</span> <span class="mi">2</span><span class="p">)</span> <span class="o">+</span> <span class="mf">1.5</span> <span class="o">*</span> <a href="http://docs.scipy.org/doc/numpy-1.8.1/reference/generated/numpy.exp.html#numpy.exp" title="View documentation for numpy.exp"><span class="n">np</span><span class="o">.</span><span class="n">exp</span></a><span class="p">(</span><span class="o">-</span><span class="p">(</span><span class="n">x</span> <span class="o">-</span> <span class="mi">2</span><span class="p">)</span> <span class="o">**</span> <span class="mi">2</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">generate</span><span class="p">(</span><span class="n">n_samples</span><span class="p">,</span> <span class="n">noise</span><span class="p">,</span> <span class="n">n_repeat</span><span class="o">=</span><span class="mi">1</span><span class="p">):</span>
<span class="n">X</span> <span class="o">=</span> <a href="http://docs.scipy.org/doc/numpy-1.8.1/reference/generated/numpy.random.rand.html#numpy.random.rand" title="View documentation for numpy.random.rand"><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">rand</span></a><span class="p">(</span><span class="n">n_samples</span><span class="p">)</span> <span class="o">*</span> <span class="mi">10</span> <span class="o">-</span> <span class="mi">5</span>
<span class="n">X</span> <span class="o">=</span> <a href="http://docs.scipy.org/doc/numpy-1.8.1/reference/generated/numpy.sort.html#numpy.sort" title="View documentation for numpy.sort"><span class="n">np</span><span class="o">.</span><span class="n">sort</span></a><span class="p">(</span><span class="n">X</span><span class="p">)</span>
<span class="k">if</span> <span class="n">n_repeat</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
<span class="n">y</span> <span class="o">=</span> <span class="n">f</span><span class="p">(</span><span class="n">X</span><span class="p">)</span> <span class="o">+</span> <a href="http://docs.scipy.org/doc/numpy-1.8.1/reference/generated/numpy.random.normal.html#numpy.random.normal" title="View documentation for numpy.random.normal"><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">normal</span></a><span class="p">(</span><span class="mf">0.0</span><span class="p">,</span> <span class="n">noise</span><span class="p">,</span> <span class="n">n_samples</span><span class="p">)</span>
<span class="k">else</span><span class="p">:</span>
<span class="n">y</span> <span class="o">=</span> <a href="http://docs.scipy.org/doc/numpy-1.8.1/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_samples</span><span class="p">,</span> <span class="n">n_repeat</span><span class="p">))</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n_repeat</span><span class="p">):</span>
<span class="n">y</span><span class="p">[:,</span> <span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">f</span><span class="p">(</span><span class="n">X</span><span class="p">)</span> <span class="o">+</span> <a href="http://docs.scipy.org/doc/numpy-1.8.1/reference/generated/numpy.random.normal.html#numpy.random.normal" title="View documentation for numpy.random.normal"><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">normal</span></a><span class="p">(</span><span class="mf">0.0</span><span class="p">,</span> <span class="n">noise</span><span class="p">,</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">reshape</span><span class="p">((</span><span class="n">n_samples</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span>
<span class="k">return</span> <span class="n">X</span><span class="p">,</span> <span class="n">y</span>
<span class="n">X_train</span> <span class="o">=</span> <span class="p">[]</span>
<span class="n">y_train</span> <span class="o">=</span> <span class="p">[]</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n_repeat</span><span class="p">):</span>
<span class="n">X</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <span class="n">generate</span><span class="p">(</span><span class="n">n_samples</span><span class="o">=</span><span class="n">n_train</span><span class="p">,</span> <span class="n">noise</span><span class="o">=</span><span class="n">noise</span><span class="p">)</span>
<span class="n">X_train</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
<span class="n">y_train</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">y</span><span class="p">)</span>
<span class="n">X_test</span><span class="p">,</span> <span class="n">y_test</span> <span class="o">=</span> <span class="n">generate</span><span class="p">(</span><span class="n">n_samples</span><span class="o">=</span><span class="n">n_test</span><span class="p">,</span> <span class="n">noise</span><span class="o">=</span><span class="n">noise</span><span class="p">,</span> <span class="n">n_repeat</span><span class="o">=</span><span class="n">n_repeat</span><span class="p">)</span>
<a href="http://matplotlib.org/api/_as_gen/matplotlib.figure.AxesStack.html#matplotlib.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">8</span><span class="p">))</span>
<span class="c1"># Loop over estimators to compare</span>
<span class="k">for</span> <span class="n">n</span><span class="p">,</span> <span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="n">estimator</span><span class="p">)</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">estimators</span><span class="p">):</span>
<span class="c1"># Compute predictions</span>
<span class="n">y_predict</span> <span class="o">=</span> <a href="http://docs.scipy.org/doc/numpy-1.8.1/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_test</span><span class="p">,</span> <span class="n">n_repeat</span><span class="p">))</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n_repeat</span><span class="p">):</span>
<span class="n">estimator</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_train</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">y_train</span><span class="p">[</span><span class="n">i</span><span class="p">])</span>
<span class="n">y_predict</span><span class="p">[:,</span> <span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">estimator</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X_test</span><span class="p">)</span>
<span class="c1"># Bias^2 + Variance + Noise decomposition of the mean squared error</span>
<span class="n">y_error</span> <span class="o">=</span> <a href="http://docs.scipy.org/doc/numpy-1.8.1/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_test</span><span class="p">)</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n_repeat</span><span class="p">):</span>
<span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n_repeat</span><span class="p">):</span>
<span class="n">y_error</span> <span class="o">+=</span> <span class="p">(</span><span class="n">y_test</span><span class="p">[:,</span> <span class="n">j</span><span class="p">]</span> <span class="o">-</span> <span class="n">y_predict</span><span class="p">[:,</span> <span class="n">i</span><span class="p">])</span> <span class="o">**</span> <span class="mi">2</span>
<span class="n">y_error</span> <span class="o">/=</span> <span class="p">(</span><span class="n">n_repeat</span> <span class="o">*</span> <span class="n">n_repeat</span><span class="p">)</span>
<span class="n">y_noise</span> <span class="o">=</span> <a href="http://docs.scipy.org/doc/numpy-1.8.1/reference/generated/numpy.var.html#numpy.var" title="View documentation for numpy.var"><span class="n">np</span><span class="o">.</span><span class="n">var</span></a><span class="p">(</span><span class="n">y_test</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="n">y_bias</span> <span class="o">=</span> <span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">X_test</span><span class="p">)</span> <span class="o">-</span> <a href="http://docs.scipy.org/doc/numpy-1.8.1/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">y_predict</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="o">**</span> <span class="mi">2</span>
<span class="n">y_var</span> <span class="o">=</span> <a href="http://docs.scipy.org/doc/numpy-1.8.1/reference/generated/numpy.var.html#numpy.var" title="View documentation for numpy.var"><span class="n">np</span><span class="o">.</span><span class="n">var</span></a><span class="p">(</span><span class="n">y_predict</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="k">print</span><span class="p">(</span><span class="s2">"{0}: {1:.4f} (error) = {2:.4f} (bias^2) "</span>
<span class="s2">" + {3:.4f} (var) + {4:.4f} (noise)"</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">name</span><span class="p">,</span>
<a href="http://docs.scipy.org/doc/numpy-1.8.1/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">y_error</span><span class="p">),</span>
<a href="http://docs.scipy.org/doc/numpy-1.8.1/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">y_bias</span><span class="p">),</span>
<a href="http://docs.scipy.org/doc/numpy-1.8.1/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">y_var</span><span class="p">),</span>
<a href="http://docs.scipy.org/doc/numpy-1.8.1/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">y_noise</span><span class="p">)))</span>
<span class="c1"># Plot figures</span>
<a href="http://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="n">n_estimators</span><span class="p">,</span> <span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
<a href="http://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">X_test</span><span class="p">,</span> <span class="n">f</span><span class="p">(</span><span class="n">X_test</span><span class="p">),</span> <span class="s2">"b"</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">"$f(x)$"</span><span class="p">)</span>
<a href="http://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">X_train</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">y_train</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="s2">".b"</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">"LS ~ $y = f(x)+noise$"</span><span class="p">)</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n_repeat</span><span class="p">):</span>
<span class="k">if</span> <span class="n">i</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
<a href="http://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">X_test</span><span class="p">,</span> <span class="n">y_predict</span><span class="p">[:,</span> <span class="n">i</span><span class="p">],</span> <span class="s2">"r"</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">"$\^y(x)$"</span><span class="p">)</span>
<span class="k">else</span><span class="p">:</span>
<a href="http://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">X_test</span><span class="p">,</span> <span class="n">y_predict</span><span class="p">[:,</span> <span class="n">i</span><span class="p">],</span> <span class="s2">"r"</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.05</span><span class="p">)</span>
<a href="http://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">X_test</span><span class="p">,</span> <a href="http://docs.scipy.org/doc/numpy-1.8.1/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">y_predict</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="s2">"c"</span><span class="p">,</span>
<span class="n">label</span><span class="o">=</span><span class="s2">"$\mathbb{E}_{LS} \^y(x)$"</span><span class="p">)</span>
<a href="http://matplotlib.org/api/_as_gen/matplotlib.pyplot.xlim.html#matplotlib.pyplot.xlim" title="View documentation for matplotlib.pyplot.xlim"><span class="n">plt</span><span class="o">.</span><span class="n">xlim</span></a><span class="p">([</span><span class="o">-</span><span class="mi">5</span><span class="p">,</span> <span class="mi">5</span><span class="p">])</span>
<a href="http://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="n">name</span><span class="p">)</span>
<span class="k">if</span> <span class="n">n</span> <span class="o">==</span> <span class="n">n_estimators</span> <span class="o">-</span> <span class="mi">1</span><span class="p">:</span>
<a href="http://matplotlib.org/api/legend_api.html#matplotlib.legend" title="View documentation for matplotlib.pyplot.legend"><span class="n">plt</span><span class="o">.</span><span class="n">legend</span></a><span class="p">(</span><span class="n">loc</span><span class="o">=</span><span class="p">(</span><span class="mf">1.1</span><span class="p">,</span> <span class="o">.</span><span class="mi">5</span><span class="p">))</span>
<a href="http://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="n">n_estimators</span><span class="p">,</span> <span class="n">n_estimators</span> <span class="o">+</span> <span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
<a href="http://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">X_test</span><span class="p">,</span> <span class="n">y_error</span><span class="p">,</span> <span class="s2">"r"</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">"$error(x)$"</span><span class="p">)</span>
<a href="http://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">X_test</span><span class="p">,</span> <span class="n">y_bias</span><span class="p">,</span> <span class="s2">"b"</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">"$bias^2(x)$"</span><span class="p">),</span>
<a href="http://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">X_test</span><span class="p">,</span> <span class="n">y_var</span><span class="p">,</span> <span class="s2">"g"</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">"$variance(x)$"</span><span class="p">),</span>
<a href="http://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">X_test</span><span class="p">,</span> <span class="n">y_noise</span><span class="p">,</span> <span class="s2">"c"</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">"$noise(x)$"</span><span class="p">)</span>
<a href="http://matplotlib.org/api/_as_gen/matplotlib.pyplot.xlim.html#matplotlib.pyplot.xlim" title="View documentation for matplotlib.pyplot.xlim"><span class="n">plt</span><span class="o">.</span><span class="n">xlim</span></a><span class="p">([</span><span class="o">-</span><span class="mi">5</span><span class="p">,</span> <span class="mi">5</span><span class="p">])</span>
<a href="http://matplotlib.org/api/_as_gen/matplotlib.pyplot.ylim.html#matplotlib.pyplot.ylim" title="View documentation for matplotlib.pyplot.ylim"><span class="n">plt</span><span class="o">.</span><span class="n">ylim</span></a><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mf">0.1</span><span class="p">])</span>
<span class="k">if</span> <span class="n">n</span> <span class="o">==</span> <span class="n">n_estimators</span> <span class="o">-</span> <span class="mi">1</span><span class="p">:</span>
<a href="http://matplotlib.org/api/legend_api.html#matplotlib.legend" title="View documentation for matplotlib.pyplot.legend"><span class="n">plt</span><span class="o">.</span><span class="n">legend</span></a><span class="p">(</span><span class="n">loc</span><span class="o">=</span><span class="p">(</span><span class="mf">1.1</span><span class="p">,</span> <span class="o">.</span><span class="mi">5</span><span class="p">))</span>
<a href="http://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">right</span><span class="o">=.</span><span class="mi">75</span><span class="p">)</span>
<a href="http://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><strong>Total running time of the script:</strong> ( 0 minutes 1.047 seconds)</p>
<div class="sphx-glr-footer docutils container">
<div class="sphx-glr-download docutils container">
<a class="reference download internal" href="../../_downloads/plot_bias_variance.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_bias_variance.py</span></code></a></div>
<div class="sphx-glr-download docutils container">
<a class="reference download internal" href="../../_downloads/plot_bias_variance.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_bias_variance.ipynb</span></code></a></div>
</div>
<p class="sphx-glr-signature"><a class="reference external" href="https://sphinx-gallery.readthedocs.io">Generated by Sphinx-Gallery</a></p>
</div>
</div>
</div>
</div>
</div>
<div class="clearer"></div>
</div>
</div>
<div class="footer">
© 2007 - 2017, scikit-learn developers (BSD License).
<a href="../../_sources/auto_examples/ensemble/plot_bias_variance.rst.txt" rel="nofollow">Show this page source</a>
</div>
<div class="rel">
<div class="buttonPrevious">
<a href="plot_gradient_boosting_oob.html">Previous
</a>
</div>
<div class="buttonNext">
<a href="plot_forest_iris.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>
<script src="https://scikit-learn.org/versionwarning.js"></script>
</body>
</html>