-
Notifications
You must be signed in to change notification settings - Fork 81
/
Copy pathisotonic.html
279 lines (247 loc) · 15.2 KB
/
isotonic.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
<!DOCTYPE html>
<!-- data-theme below is forced to be "light" but should be changed if we use pydata-theme-sphinx in the future -->
<!--[if IE 8]><html class="no-js lt-ie9" lang="en" data-content_root="../" data-theme="light"> <![endif]-->
<!--[if gt IE 8]><!--> <html class="no-js" lang="en" data-content_root="../" data-theme="light"> <!--<![endif]-->
<head>
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1" />
<meta property="og:title" content="1.15. Isotonic regression" />
<meta property="og:type" content="website" />
<meta property="og:url" content="https://scikit-learn/stable/modules/isotonic.html" />
<meta property="og:site_name" content="scikit-learn" />
<meta property="og:description" content="The class IsotonicRegression fits a non-decreasing real function to 1-dimensional data. It solves the following problem:\min \sum_i w_i (y_i - \hat{y}_i)^2 subject to\hat{y}_i \le \hat{y}_j wheneve..." />
<meta property="og:image" content="https://scikit-learn/stable/_images/sphx_glr_plot_isotonic_regression_001.png" />
<meta property="og:image:alt" content="scikit-learn" />
<meta name="description" content="The class IsotonicRegression fits a non-decreasing real function to 1-dimensional data. It solves the following problem:\min \sum_i w_i (y_i - \hat{y}_i)^2 subject to\hat{y}_i \le \hat{y}_j wheneve..." />
<meta name="viewport" content="width=device-width, initial-scale=1.0">
<title>1.15. Isotonic regression — scikit-learn 1.5.dev0 documentation</title>
<link rel="canonical" href="http://scikit-learn.org/stable/modules/isotonic.html" />
<link rel="shortcut icon" href="../_static/favicon.ico"/>
<link rel="stylesheet" href="../_static/css/vendor/bootstrap.min.css" type="text/css" />
<link rel="stylesheet" href="../_static/pygments.css" type="text/css" />
<link rel="stylesheet" href="../_static/css/theme.css" type="text/css" />
<link rel="stylesheet" href="../_static/copybutton.css" type="text/css" />
<link rel="stylesheet" href="../_static/plot_directive.css" type="text/css" />
<link rel="stylesheet" href="../https://fonts.googleapis.com/css?family=Vibur" type="text/css" />
<link rel="stylesheet" href="../_static/jupyterlite_sphinx.css" type="text/css" />
<link rel="stylesheet" href="../_static/sg_gallery.css" type="text/css" />
<link rel="stylesheet" href="../_static/sg_gallery-binder.css" type="text/css" />
<link rel="stylesheet" href="../_static/sg_gallery-dataframe.css" type="text/css" />
<link rel="stylesheet" href="../_static/sg_gallery-rendered-html.css" type="text/css" />
<link rel="stylesheet" href="../_static/css/theme.css" type="text/css" />
<script id="documentation_options" data-url_root="../" src="../_static/documentation_options.js"></script>
<script src="../_static/js/vendor/jquery-3.6.3.slim.min.js"></script>
<script src="../_static/js/details-permalink.js"></script>
</head>
<body>
<nav id="navbar" class="sk-docs-navbar navbar navbar-expand-md navbar-light bg-light py-0">
<div class="container-fluid sk-docs-container px-0">
<a class="navbar-brand py-0" href="../index.html">
<img
class="sk-brand-img"
src="../_static/scikit-learn-logo-small.png"
alt="logo"/>
</a>
<button
id="sk-navbar-toggler"
class="navbar-toggler"
type="button"
data-toggle="collapse"
data-target="#navbarSupportedContent"
aria-controls="navbarSupportedContent"
aria-expanded="false"
aria-label="Toggle navigation"
>
<span class="navbar-toggler-icon"></span>
</button>
<div class="sk-navbar-collapse collapse navbar-collapse" id="navbarSupportedContent">
<ul class="navbar-nav mr-auto">
<li class="nav-item">
<a class="sk-nav-link nav-link" href="../install.html">Install</a>
</li>
<li class="nav-item">
<a class="sk-nav-link nav-link" href="../user_guide.html">User Guide</a>
</li>
<li class="nav-item">
<a class="sk-nav-link nav-link" href="classes.html">API</a>
</li>
<li class="nav-item">
<a class="sk-nav-link nav-link" href="../auto_examples/index.html">Examples</a>
</li>
<li class="nav-item">
<a class="sk-nav-link nav-link" target="_blank" rel="noopener noreferrer" href="https://blog.scikit-learn.org/">Community</a>
</li>
<li class="nav-item">
<a class="sk-nav-link nav-link nav-more-item-mobile-items" href="../getting_started.html" >Getting Started</a>
</li>
<li class="nav-item">
<a class="sk-nav-link nav-link nav-more-item-mobile-items" href="../tutorial/index.html" >Tutorial</a>
</li>
<li class="nav-item">
<a class="sk-nav-link nav-link nav-more-item-mobile-items" href="../whats_new/v1.5.html" >What's new</a>
</li>
<li class="nav-item">
<a class="sk-nav-link nav-link nav-more-item-mobile-items" href="../glossary.html" >Glossary</a>
</li>
<li class="nav-item">
<a class="sk-nav-link nav-link nav-more-item-mobile-items" href="../developers/index.html" >Development</a>
</li>
<li class="nav-item">
<a class="sk-nav-link nav-link nav-more-item-mobile-items" href="../faq.html" >FAQ</a>
</li>
<li class="nav-item">
<a class="sk-nav-link nav-link nav-more-item-mobile-items" href="../support.html" >Support</a>
</li>
<li class="nav-item">
<a class="sk-nav-link nav-link nav-more-item-mobile-items" href="../related_projects.html" >Related packages</a>
</li>
<li class="nav-item">
<a class="sk-nav-link nav-link nav-more-item-mobile-items" href="../roadmap.html" >Roadmap</a>
</li>
<li class="nav-item">
<a class="sk-nav-link nav-link nav-more-item-mobile-items" href="../governance.html" >Governance</a>
</li>
<li class="nav-item">
<a class="sk-nav-link nav-link nav-more-item-mobile-items" href="../about.html" >About us</a>
</li>
<li class="nav-item">
<a class="sk-nav-link nav-link nav-more-item-mobile-items" href="https://github.com/scikit-learn/scikit-learn" >GitHub</a>
</li>
<li class="nav-item">
<a class="sk-nav-link nav-link nav-more-item-mobile-items" href="https://scikit-learn.org/dev/versions.html" >Other Versions and Download</a>
</li>
<li class="nav-item dropdown nav-more-item-dropdown">
<a class="sk-nav-link nav-link dropdown-toggle" href="#" id="navbarDropdown" role="button" data-toggle="dropdown" aria-haspopup="true" aria-expanded="false">More</a>
<div class="dropdown-menu" aria-labelledby="navbarDropdown">
<a class="sk-nav-dropdown-item dropdown-item" href="../getting_started.html" >Getting Started</a>
<a class="sk-nav-dropdown-item dropdown-item" href="../tutorial/index.html" >Tutorial</a>
<a class="sk-nav-dropdown-item dropdown-item" href="../whats_new/v1.5.html" >What's new</a>
<a class="sk-nav-dropdown-item dropdown-item" href="../glossary.html" >Glossary</a>
<a class="sk-nav-dropdown-item dropdown-item" href="../developers/index.html" >Development</a>
<a class="sk-nav-dropdown-item dropdown-item" href="../faq.html" >FAQ</a>
<a class="sk-nav-dropdown-item dropdown-item" href="../support.html" >Support</a>
<a class="sk-nav-dropdown-item dropdown-item" href="../related_projects.html" >Related packages</a>
<a class="sk-nav-dropdown-item dropdown-item" href="../roadmap.html" >Roadmap</a>
<a class="sk-nav-dropdown-item dropdown-item" href="../governance.html" >Governance</a>
<a class="sk-nav-dropdown-item dropdown-item" href="../about.html" >About us</a>
<a class="sk-nav-dropdown-item dropdown-item" href="https://github.com/scikit-learn/scikit-learn" >GitHub</a>
<a class="sk-nav-dropdown-item dropdown-item" href="https://scikit-learn.org/dev/versions.html" >Other Versions and Download</a>
</div>
</li>
</ul>
<div id="searchbox" role="search">
<div class="searchformwrapper">
<form class="search" action="../search.html" method="get">
<input class="sk-search-text-input" type="text" name="q" aria-labelledby="searchlabel" />
<input class="sk-search-text-btn" type="submit" value="Go" />
</form>
</div>
</div>
</div>
</div>
</nav>
<div class="d-flex" id="sk-doc-wrapper">
<input type="checkbox" name="sk-toggle-checkbox" id="sk-toggle-checkbox">
<label id="sk-sidemenu-toggle" class="sk-btn-toggle-toc btn sk-btn-primary" for="sk-toggle-checkbox">Toggle Menu</label>
<div id="sk-sidebar-wrapper" class="border-right">
<div class="sk-sidebar-toc-wrapper">
<div class="btn-group w-100 mb-2" role="group" aria-label="rellinks">
<a href="semi_supervised.html" role="button" class="btn sk-btn-rellink py-1" sk-rellink-tooltip="1.14. Semi-supervised learning">Prev</a><a href="../supervised_learning.html" role="button" class="btn sk-btn-rellink py-1" sk-rellink-tooltip="1. Supervised learning">Up</a>
<a href="calibration.html" role="button" class="btn sk-btn-rellink py-1" sk-rellink-tooltip="1.16. Probability calibration">Next</a>
</div>
<div class="alert alert-danger p-1 mb-2" role="alert">
<p class="text-center mb-0">
<strong>scikit-learn 1.5.dev0</strong><br/>
<a href="http://scikit-learn.org/dev/versions.html">Other versions</a>
</p>
</div>
<div class="alert alert-warning p-1 mb-2" role="alert">
<p class="text-center mb-0">
Please <a class="font-weight-bold" href="../about.html#citing-scikit-learn"><string>cite us</string></a> if you use the software.
</p>
</div>
<div class="sk-sidebar-toc">
<ul>
<li><a class="reference internal" href="#">1.15. Isotonic regression</a></li>
</ul>
</div>
</div>
</div>
<div id="sk-page-content-wrapper">
<div class="sk-page-content container-fluid body px-md-3" role="main">
<section id="isotonic-regression">
<span id="isotonic"></span><h1><span class="section-number">1.15. </span>Isotonic regression<a class="headerlink" href="#isotonic-regression" title="Link to this heading">¶</a></h1>
<p>The class <a class="reference internal" href="generated/sklearn.isotonic.IsotonicRegression.html#sklearn.isotonic.IsotonicRegression" title="sklearn.isotonic.IsotonicRegression"><code class="xref py py-class docutils literal notranslate"><span class="pre">IsotonicRegression</span></code></a> fits a non-decreasing real function to
1-dimensional data. It solves the following problem:</p>
<div class="math notranslate nohighlight">
\[\min \sum_i w_i (y_i - \hat{y}_i)^2\]</div>
<p>subject to <span class="math notranslate nohighlight">\(\hat{y}_i \le \hat{y}_j\)</span> whenever <span class="math notranslate nohighlight">\(X_i \le X_j\)</span>,
where the weights <span class="math notranslate nohighlight">\(w_i\)</span> are strictly positive, and both <code class="docutils literal notranslate"><span class="pre">X</span></code> and <code class="docutils literal notranslate"><span class="pre">y</span></code> are
arbitrary real quantities.</p>
<p>The <code class="docutils literal notranslate"><span class="pre">increasing</span></code> parameter changes the constraint to
<span class="math notranslate nohighlight">\(\hat{y}_i \ge \hat{y}_j\)</span> whenever <span class="math notranslate nohighlight">\(X_i \le X_j\)</span>. Setting it to
‘auto’ will automatically choose the constraint based on <a class="reference external" href="https://en.wikipedia.org/wiki/Spearman%27s_rank_correlation_coefficient">Spearman’s rank
correlation coefficient</a>.</p>
<p><a class="reference internal" href="generated/sklearn.isotonic.IsotonicRegression.html#sklearn.isotonic.IsotonicRegression" title="sklearn.isotonic.IsotonicRegression"><code class="xref py py-class docutils literal notranslate"><span class="pre">IsotonicRegression</span></code></a> produces a series of predictions
<span class="math notranslate nohighlight">\(\hat{y}_i\)</span> for the training data which are the closest to the targets
<span class="math notranslate nohighlight">\(y\)</span> in terms of mean squared error. These predictions are interpolated
for predicting to unseen data. The predictions of <a class="reference internal" href="generated/sklearn.isotonic.IsotonicRegression.html#sklearn.isotonic.IsotonicRegression" title="sklearn.isotonic.IsotonicRegression"><code class="xref py py-class docutils literal notranslate"><span class="pre">IsotonicRegression</span></code></a>
thus form a function that is piecewise linear:</p>
<figure class="align-center">
<a class="reference external image-reference" href="../auto_examples/miscellaneous/plot_isotonic_regression.html"><img alt="../_images/sphx_glr_plot_isotonic_regression_001.png" src="../_images/sphx_glr_plot_isotonic_regression_001.png" /></a>
</figure>
</section>
</div>
<div class="container">
<footer class="sk-content-footer">
© 2007 - 2024, scikit-learn developers (BSD License).
<a href="../_sources/modules/isotonic.rst.txt" rel="nofollow">Show this page source</a>
</footer>
</div>
</div>
</div>
<script src="../_static/js/vendor/bootstrap.min.js"></script>
<script>
window.ga=window.ga||function(){(ga.q=ga.q||[]).push(arguments)};ga.l=+new Date;
ga('create', 'UA-22606712-2', 'auto');
ga('set', 'anonymizeIp', true);
ga('send', 'pageview');
</script>
<script async src='https://www.google-analytics.com/analytics.js'></script>
<script defer data-domain="scikit-learn.org" src="https://views.scientific-python.org/js/script.js">
</script>
<script src="../_static/clipboard.min.js"></script>
<script src="../_static/copybutton.js"></script>
<script>
$(document).ready(function() {
/* Add a [>>>] button on the top-right corner of code samples to hide
* the >>> and ... prompts and the output and thus make the code
* copyable. */
var div = $('.highlight-python .highlight,' +
'.highlight-python3 .highlight,' +
'.highlight-pycon .highlight,' +
'.highlight-default .highlight')
var pre = div.find('pre');
// get the styles from the current theme
pre.parent().parent().css('position', 'relative');
// create and add the button to all the code blocks that contain >>>
div.each(function(index) {
var jthis = $(this);
// tracebacks (.gt) contain bare text elements that need to be
// wrapped in a span to work with .nextUntil() (see later)
jthis.find('pre:has(.gt)').contents().filter(function() {
return ((this.nodeType == 3) && (this.data.trim().length > 0));
}).wrap('<span>');
});
/*** Add permalink buttons next to glossary terms ***/
$('dl.glossary > dt[id]').append(function() {
return ('<a class="headerlink" href="#' +
this.getAttribute('id') +
'" title="Permalink to this term">¶</a>');
});
});
</script>
<script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-chtml.js"></script>
<script src="https://scikit-learn.org/versionwarning.js"></script>
</body>
</html>