-
Notifications
You must be signed in to change notification settings - Fork 81
/
Copy pathisotonic.html
808 lines (577 loc) · 32.6 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
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
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
<!DOCTYPE html>
<html lang="en" data-content_root="../" >
<head>
<meta charset="utf-8" />
<meta name="viewport" content="width=device-width, initial-scale=1.0" /><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..." />
<title>1.15. Isotonic regression — scikit-learn 1.6.dev0 documentation</title>
<script data-cfasync="false">
document.documentElement.dataset.mode = localStorage.getItem("mode") || "";
document.documentElement.dataset.theme = localStorage.getItem("theme") || "";
</script>
<!--
this give us a css class that will be invisible only if js is disabled
-->
<noscript>
<style>
.pst-js-only { display: none !important; }
</style>
</noscript>
<!-- Loaded before other Sphinx assets -->
<link href="../_static/styles/theme.css?digest=26a4bc78f4c0ddb94549" rel="stylesheet" />
<link href="../_static/styles/pydata-sphinx-theme.css?digest=26a4bc78f4c0ddb94549" rel="stylesheet" />
<link rel="stylesheet" type="text/css" href="../_static/pygments.css?v=a746c00c" />
<link rel="stylesheet" type="text/css" href="../_static/copybutton.css?v=76b2166b" />
<link rel="stylesheet" type="text/css" href="../_static/plot_directive.css" />
<link rel="stylesheet" type="text/css" href="https://fonts.googleapis.com/css?family=Vibur" />
<link rel="stylesheet" type="text/css" href="../_static/jupyterlite_sphinx.css?v=ca70e7f1" />
<link rel="stylesheet" type="text/css" href="../_static/sg_gallery.css?v=d2d258e8" />
<link rel="stylesheet" type="text/css" href="../_static/sg_gallery-binder.css?v=f4aeca0c" />
<link rel="stylesheet" type="text/css" href="../_static/sg_gallery-dataframe.css?v=2082cf3c" />
<link rel="stylesheet" type="text/css" href="../_static/sg_gallery-rendered-html.css?v=1277b6f3" />
<link rel="stylesheet" type="text/css" href="../_static/sphinx-design.min.css?v=95c83b7e" />
<link rel="stylesheet" type="text/css" href="../_static/styles/colors.css?v=cc94ab7d" />
<link rel="stylesheet" type="text/css" href="../_static/styles/custom.css?v=e340c087" />
<!-- So that users can add custom icons -->
<script src="../_static/scripts/fontawesome.js?digest=26a4bc78f4c0ddb94549"></script>
<!-- Pre-loaded scripts that we'll load fully later -->
<link rel="preload" as="script" href="../_static/scripts/bootstrap.js?digest=26a4bc78f4c0ddb94549" />
<link rel="preload" as="script" href="../_static/scripts/pydata-sphinx-theme.js?digest=26a4bc78f4c0ddb94549" />
<script src="../_static/documentation_options.js?v=d875d36e"></script>
<script src="../_static/doctools.js?v=9a2dae69"></script>
<script src="../_static/sphinx_highlight.js?v=dc90522c"></script>
<script src="../_static/clipboard.min.js?v=a7894cd8"></script>
<script src="../_static/copybutton.js?v=97f0b27d"></script>
<script src="../_static/jupyterlite_sphinx.js?v=d6bdf5f8"></script>
<script src="../_static/design-tabs.js?v=f930bc37"></script>
<script data-domain="scikit-learn.org" defer="defer" src="https://views.scientific-python.org/js/script.js"></script>
<script async="async" src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-chtml.js"></script>
<script>DOCUMENTATION_OPTIONS.pagename = 'modules/isotonic';</script>
<script>
DOCUMENTATION_OPTIONS.theme_version = '0.16.0';
DOCUMENTATION_OPTIONS.theme_switcher_json_url = 'https://scikit-learn.org/dev/_static/versions.json';
DOCUMENTATION_OPTIONS.theme_switcher_version_match = '1.6.dev0';
DOCUMENTATION_OPTIONS.show_version_warning_banner = true;
</script>
<script src="../_static/scripts/dropdown.js?v=e2048168"></script>
<script src="../_static/scripts/version-switcher.js?v=a6dd8357"></script>
<link rel="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="1.16. Probability calibration" href="calibration.html" />
<link rel="prev" title="1.14. Semi-supervised learning" href="semi_supervised.html" />
<meta name="viewport" content="width=device-width, initial-scale=1"/>
<meta name="docsearch:language" content="en"/>
<meta name="docsearch:version" content="1.6" />
</head>
<body data-bs-spy="scroll" data-bs-target=".bd-toc-nav" data-offset="180" data-bs-root-margin="0px 0px -60%" data-default-mode="">
<div id="pst-skip-link" class="skip-link d-print-none"><a href="#main-content">Skip to main content</a></div>
<div id="pst-scroll-pixel-helper"></div>
<button type="button" class="btn rounded-pill" id="pst-back-to-top">
<i class="fa-solid fa-arrow-up"></i>Back to top</button>
<dialog id="pst-search-dialog">
<form class="bd-search d-flex align-items-center"
action="../search.html"
method="get">
<i class="fa-solid fa-magnifying-glass"></i>
<input type="search"
class="form-control"
name="q"
placeholder="Search the docs ..."
aria-label="Search the docs ..."
autocomplete="off"
autocorrect="off"
autocapitalize="off"
spellcheck="false"/>
<span class="search-button__kbd-shortcut"><kbd class="kbd-shortcut__modifier">Ctrl</kbd>+<kbd>K</kbd></span>
</form>
</dialog>
<div class="pst-async-banner-revealer d-none">
<aside id="bd-header-version-warning" class="d-none d-print-none" aria-label="Version warning"></aside>
</div>
<header class="bd-header navbar navbar-expand-lg bd-navbar d-print-none">
<div class="bd-header__inner bd-page-width">
<button class="pst-navbar-icon sidebar-toggle primary-toggle" aria-label="Site navigation">
<span class="fa-solid fa-bars"></span>
</button>
<div class=" navbar-header-items__start">
<div class="navbar-item">
<a class="navbar-brand logo" href="../index.html">
<img src="../_static/scikit-learn-logo-small.png" class="logo__image only-light" alt="scikit-learn homepage"/>
<img src="../_static/scikit-learn-logo-small.png" class="logo__image only-dark pst-js-only" alt="scikit-learn homepage"/>
</a></div>
</div>
<div class=" navbar-header-items">
<div class="me-auto navbar-header-items__center">
<div class="navbar-item">
<nav>
<ul class="bd-navbar-elements navbar-nav">
<li class="nav-item ">
<a class="nav-link nav-internal" href="../install.html">
Install
</a>
</li>
<li class="nav-item current active">
<a class="nav-link nav-internal" href="../user_guide.html">
User Guide
</a>
</li>
<li class="nav-item ">
<a class="nav-link nav-internal" href="../api/index.html">
API
</a>
</li>
<li class="nav-item ">
<a class="nav-link nav-internal" href="../auto_examples/index.html">
Examples
</a>
</li>
<li class="nav-item ">
<a class="nav-link nav-external" href="https://blog.scikit-learn.org/">
Community
</a>
</li>
<li class="nav-item dropdown">
<button class="btn dropdown-toggle nav-item" type="button" data-bs-toggle="dropdown" aria-expanded="false" aria-controls="pst-nav-more-links">
More
</button>
<ul id="pst-nav-more-links" class="dropdown-menu">
<li class=" ">
<a class="nav-link dropdown-item nav-internal" href="../getting_started.html">
Getting Started
</a>
</li>
<li class=" ">
<a class="nav-link dropdown-item nav-internal" href="../whats_new.html">
Release History
</a>
</li>
<li class=" ">
<a class="nav-link dropdown-item nav-internal" href="../glossary.html">
Glossary
</a>
</li>
<li class=" ">
<a class="nav-link dropdown-item nav-internal" href="../developers/index.html">
Development
</a>
</li>
<li class=" ">
<a class="nav-link dropdown-item nav-internal" href="../faq.html">
FAQ
</a>
</li>
<li class=" ">
<a class="nav-link dropdown-item nav-internal" href="../support.html">
Support
</a>
</li>
<li class=" ">
<a class="nav-link dropdown-item nav-internal" href="../related_projects.html">
Related Projects
</a>
</li>
<li class=" ">
<a class="nav-link dropdown-item nav-internal" href="../roadmap.html">
Roadmap
</a>
</li>
<li class=" ">
<a class="nav-link dropdown-item nav-internal" href="../governance.html">
Governance
</a>
</li>
<li class=" ">
<a class="nav-link dropdown-item nav-internal" href="../about.html">
About us
</a>
</li>
</ul>
</li>
</ul>
</nav></div>
</div>
<div class="navbar-header-items__end">
<div class="navbar-item navbar-persistent--container">
<button class="btn btn-sm pst-navbar-icon search-button search-button__button pst-js-only" title="Search" aria-label="Search" data-bs-placement="bottom" data-bs-toggle="tooltip">
<i class="fa-solid fa-magnifying-glass fa-lg"></i>
</button>
</div>
<div class="navbar-item">
<button class="btn btn-sm nav-link pst-navbar-icon theme-switch-button pst-js-only" aria-label="Color mode" data-bs-title="Color mode" data-bs-placement="bottom" data-bs-toggle="tooltip">
<i class="theme-switch fa-solid fa-sun fa-lg" data-mode="light" title="Light"></i>
<i class="theme-switch fa-solid fa-moon fa-lg" data-mode="dark" title="Dark"></i>
<i class="theme-switch fa-solid fa-circle-half-stroke fa-lg" data-mode="auto" title="System Settings"></i>
</button></div>
<div class="navbar-item"><ul class="navbar-icon-links"
aria-label="Icon Links">
<li class="nav-item">
<a href="https://github.com/scikit-learn/scikit-learn" title="GitHub" class="nav-link pst-navbar-icon" rel="noopener" target="_blank" data-bs-toggle="tooltip" data-bs-placement="bottom"><i class="fa-brands fa-square-github fa-lg" aria-hidden="true"></i>
<span class="sr-only">GitHub</span></a>
</li>
</ul></div>
<div class="navbar-item">
<div class="version-switcher__container dropdown pst-js-only">
<button id="pst-version-switcher-button-2"
type="button"
class="version-switcher__button btn btn-sm dropdown-toggle"
data-bs-toggle="dropdown"
aria-haspopup="listbox"
aria-controls="pst-version-switcher-list-2"
aria-label="Version switcher list"
>
Choose version <!-- this text may get changed later by javascript -->
<span class="caret"></span>
</button>
<div id="pst-version-switcher-list-2"
class="version-switcher__menu dropdown-menu list-group-flush py-0"
role="listbox" aria-labelledby="pst-version-switcher-button-2">
<!-- dropdown will be populated by javascript on page load -->
</div>
</div></div>
</div>
</div>
<div class="navbar-persistent--mobile">
<button class="btn btn-sm pst-navbar-icon search-button search-button__button pst-js-only" title="Search" aria-label="Search" data-bs-placement="bottom" data-bs-toggle="tooltip">
<i class="fa-solid fa-magnifying-glass fa-lg"></i>
</button>
</div>
<button class="pst-navbar-icon sidebar-toggle secondary-toggle" aria-label="On this page">
<span class="fa-solid fa-outdent"></span>
</button>
</div>
</header>
<div class="bd-container">
<div class="bd-container__inner bd-page-width">
<dialog id="pst-primary-sidebar-modal"></dialog>
<div id="pst-primary-sidebar" class="bd-sidebar-primary bd-sidebar">
<div class="sidebar-header-items sidebar-primary__section">
<div class="sidebar-header-items__center">
<div class="navbar-item">
<nav>
<ul class="bd-navbar-elements navbar-nav">
<li class="nav-item ">
<a class="nav-link nav-internal" href="../install.html">
Install
</a>
</li>
<li class="nav-item current active">
<a class="nav-link nav-internal" href="../user_guide.html">
User Guide
</a>
</li>
<li class="nav-item ">
<a class="nav-link nav-internal" href="../api/index.html">
API
</a>
</li>
<li class="nav-item ">
<a class="nav-link nav-internal" href="../auto_examples/index.html">
Examples
</a>
</li>
<li class="nav-item ">
<a class="nav-link nav-external" href="https://blog.scikit-learn.org/">
Community
</a>
</li>
<li class="nav-item ">
<a class="nav-link nav-internal" href="../getting_started.html">
Getting Started
</a>
</li>
<li class="nav-item ">
<a class="nav-link nav-internal" href="../whats_new.html">
Release History
</a>
</li>
<li class="nav-item ">
<a class="nav-link nav-internal" href="../glossary.html">
Glossary
</a>
</li>
<li class="nav-item ">
<a class="nav-link nav-internal" href="../developers/index.html">
Development
</a>
</li>
<li class="nav-item ">
<a class="nav-link nav-internal" href="../faq.html">
FAQ
</a>
</li>
<li class="nav-item ">
<a class="nav-link nav-internal" href="../support.html">
Support
</a>
</li>
<li class="nav-item ">
<a class="nav-link nav-internal" href="../related_projects.html">
Related Projects
</a>
</li>
<li class="nav-item ">
<a class="nav-link nav-internal" href="../roadmap.html">
Roadmap
</a>
</li>
<li class="nav-item ">
<a class="nav-link nav-internal" href="../governance.html">
Governance
</a>
</li>
<li class="nav-item ">
<a class="nav-link nav-internal" href="../about.html">
About us
</a>
</li>
</ul>
</nav></div>
</div>
<div class="sidebar-header-items__end">
<div class="navbar-item">
<button class="btn btn-sm nav-link pst-navbar-icon theme-switch-button pst-js-only" aria-label="Color mode" data-bs-title="Color mode" data-bs-placement="bottom" data-bs-toggle="tooltip">
<i class="theme-switch fa-solid fa-sun fa-lg" data-mode="light" title="Light"></i>
<i class="theme-switch fa-solid fa-moon fa-lg" data-mode="dark" title="Dark"></i>
<i class="theme-switch fa-solid fa-circle-half-stroke fa-lg" data-mode="auto" title="System Settings"></i>
</button></div>
<div class="navbar-item"><ul class="navbar-icon-links"
aria-label="Icon Links">
<li class="nav-item">
<a href="https://github.com/scikit-learn/scikit-learn" title="GitHub" class="nav-link pst-navbar-icon" rel="noopener" target="_blank" data-bs-toggle="tooltip" data-bs-placement="bottom"><i class="fa-brands fa-square-github fa-lg" aria-hidden="true"></i>
<span class="sr-only">GitHub</span></a>
</li>
</ul></div>
<div class="navbar-item">
<div class="version-switcher__container dropdown pst-js-only">
<button id="pst-version-switcher-button-3"
type="button"
class="version-switcher__button btn btn-sm dropdown-toggle"
data-bs-toggle="dropdown"
aria-haspopup="listbox"
aria-controls="pst-version-switcher-list-3"
aria-label="Version switcher list"
>
Choose version <!-- this text may get changed later by javascript -->
<span class="caret"></span>
</button>
<div id="pst-version-switcher-list-3"
class="version-switcher__menu dropdown-menu list-group-flush py-0"
role="listbox" aria-labelledby="pst-version-switcher-button-3">
<!-- dropdown will be populated by javascript on page load -->
</div>
</div></div>
</div>
</div>
<div class="sidebar-primary-items__start sidebar-primary__section">
<div class="sidebar-primary-item">
<nav class="bd-docs-nav bd-links"
aria-label="Section Navigation">
<p class="bd-links__title" role="heading" aria-level="1">Section Navigation</p>
<div class="bd-toc-item navbar-nav"><ul class="current nav bd-sidenav">
<li class="toctree-l1 current active has-children"><a class="reference internal" href="../supervised_learning.html">1. Supervised learning</a><details open="open"><summary><span class="toctree-toggle" role="presentation"><i class="fa-solid fa-chevron-down"></i></span></summary><ul class="current">
<li class="toctree-l2"><a class="reference internal" href="linear_model.html">1.1. Linear Models</a></li>
<li class="toctree-l2"><a class="reference internal" href="lda_qda.html">1.2. Linear and Quadratic Discriminant Analysis</a></li>
<li class="toctree-l2"><a class="reference internal" href="kernel_ridge.html">1.3. Kernel ridge regression</a></li>
<li class="toctree-l2"><a class="reference internal" href="svm.html">1.4. Support Vector Machines</a></li>
<li class="toctree-l2"><a class="reference internal" href="sgd.html">1.5. Stochastic Gradient Descent</a></li>
<li class="toctree-l2"><a class="reference internal" href="neighbors.html">1.6. Nearest Neighbors</a></li>
<li class="toctree-l2"><a class="reference internal" href="gaussian_process.html">1.7. Gaussian Processes</a></li>
<li class="toctree-l2"><a class="reference internal" href="cross_decomposition.html">1.8. Cross decomposition</a></li>
<li class="toctree-l2"><a class="reference internal" href="naive_bayes.html">1.9. Naive Bayes</a></li>
<li class="toctree-l2"><a class="reference internal" href="tree.html">1.10. Decision Trees</a></li>
<li class="toctree-l2"><a class="reference internal" href="ensemble.html">1.11. Ensembles: Gradient boosting, random forests, bagging, voting, stacking</a></li>
<li class="toctree-l2"><a class="reference internal" href="multiclass.html">1.12. Multiclass and multioutput algorithms</a></li>
<li class="toctree-l2"><a class="reference internal" href="feature_selection.html">1.13. Feature selection</a></li>
<li class="toctree-l2"><a class="reference internal" href="semi_supervised.html">1.14. Semi-supervised learning</a></li>
<li class="toctree-l2 current active"><a class="current reference internal" href="#">1.15. Isotonic regression</a></li>
<li class="toctree-l2"><a class="reference internal" href="calibration.html">1.16. Probability calibration</a></li>
<li class="toctree-l2"><a class="reference internal" href="neural_networks_supervised.html">1.17. Neural network models (supervised)</a></li>
</ul>
</details></li>
<li class="toctree-l1 has-children"><a class="reference internal" href="../unsupervised_learning.html">2. Unsupervised learning</a><details><summary><span class="toctree-toggle" role="presentation"><i class="fa-solid fa-chevron-down"></i></span></summary><ul>
<li class="toctree-l2"><a class="reference internal" href="mixture.html">2.1. Gaussian mixture models</a></li>
<li class="toctree-l2"><a class="reference internal" href="manifold.html">2.2. Manifold learning</a></li>
<li class="toctree-l2"><a class="reference internal" href="clustering.html">2.3. Clustering</a></li>
<li class="toctree-l2"><a class="reference internal" href="biclustering.html">2.4. Biclustering</a></li>
<li class="toctree-l2"><a class="reference internal" href="decomposition.html">2.5. Decomposing signals in components (matrix factorization problems)</a></li>
<li class="toctree-l2"><a class="reference internal" href="covariance.html">2.6. Covariance estimation</a></li>
<li class="toctree-l2"><a class="reference internal" href="outlier_detection.html">2.7. Novelty and Outlier Detection</a></li>
<li class="toctree-l2"><a class="reference internal" href="density.html">2.8. Density Estimation</a></li>
<li class="toctree-l2"><a class="reference internal" href="neural_networks_unsupervised.html">2.9. Neural network models (unsupervised)</a></li>
</ul>
</details></li>
<li class="toctree-l1 has-children"><a class="reference internal" href="../model_selection.html">3. Model selection and evaluation</a><details><summary><span class="toctree-toggle" role="presentation"><i class="fa-solid fa-chevron-down"></i></span></summary><ul>
<li class="toctree-l2"><a class="reference internal" href="cross_validation.html">3.1. Cross-validation: evaluating estimator performance</a></li>
<li class="toctree-l2"><a class="reference internal" href="grid_search.html">3.2. Tuning the hyper-parameters of an estimator</a></li>
<li class="toctree-l2"><a class="reference internal" href="classification_threshold.html">3.3. Tuning the decision threshold for class prediction</a></li>
<li class="toctree-l2"><a class="reference internal" href="model_evaluation.html">3.4. Metrics and scoring: quantifying the quality of predictions</a></li>
<li class="toctree-l2"><a class="reference internal" href="learning_curve.html">3.5. Validation curves: plotting scores to evaluate models</a></li>
</ul>
</details></li>
<li class="toctree-l1 has-children"><a class="reference internal" href="../inspection.html">4. Inspection</a><details><summary><span class="toctree-toggle" role="presentation"><i class="fa-solid fa-chevron-down"></i></span></summary><ul>
<li class="toctree-l2"><a class="reference internal" href="partial_dependence.html">4.1. Partial Dependence and Individual Conditional Expectation plots</a></li>
<li class="toctree-l2"><a class="reference internal" href="permutation_importance.html">4.2. Permutation feature importance</a></li>
</ul>
</details></li>
<li class="toctree-l1"><a class="reference internal" href="../visualizations.html">5. Visualizations</a></li>
<li class="toctree-l1 has-children"><a class="reference internal" href="../data_transforms.html">6. Dataset transformations</a><details><summary><span class="toctree-toggle" role="presentation"><i class="fa-solid fa-chevron-down"></i></span></summary><ul>
<li class="toctree-l2"><a class="reference internal" href="compose.html">6.1. Pipelines and composite estimators</a></li>
<li class="toctree-l2"><a class="reference internal" href="feature_extraction.html">6.2. Feature extraction</a></li>
<li class="toctree-l2"><a class="reference internal" href="preprocessing.html">6.3. Preprocessing data</a></li>
<li class="toctree-l2"><a class="reference internal" href="impute.html">6.4. Imputation of missing values</a></li>
<li class="toctree-l2"><a class="reference internal" href="unsupervised_reduction.html">6.5. Unsupervised dimensionality reduction</a></li>
<li class="toctree-l2"><a class="reference internal" href="random_projection.html">6.6. Random Projection</a></li>
<li class="toctree-l2"><a class="reference internal" href="kernel_approximation.html">6.7. Kernel Approximation</a></li>
<li class="toctree-l2"><a class="reference internal" href="metrics.html">6.8. Pairwise metrics, Affinities and Kernels</a></li>
<li class="toctree-l2"><a class="reference internal" href="preprocessing_targets.html">6.9. Transforming the prediction target (<code class="docutils literal notranslate"><span class="pre">y</span></code>)</a></li>
</ul>
</details></li>
<li class="toctree-l1 has-children"><a class="reference internal" href="../datasets.html">7. Dataset loading utilities</a><details><summary><span class="toctree-toggle" role="presentation"><i class="fa-solid fa-chevron-down"></i></span></summary><ul>
<li class="toctree-l2"><a class="reference internal" href="../datasets/toy_dataset.html">7.1. Toy datasets</a></li>
<li class="toctree-l2"><a class="reference internal" href="../datasets/real_world.html">7.2. Real world datasets</a></li>
<li class="toctree-l2"><a class="reference internal" href="../datasets/sample_generators.html">7.3. Generated datasets</a></li>
<li class="toctree-l2"><a class="reference internal" href="../datasets/loading_other_datasets.html">7.4. Loading other datasets</a></li>
</ul>
</details></li>
<li class="toctree-l1 has-children"><a class="reference internal" href="../computing.html">8. Computing with scikit-learn</a><details><summary><span class="toctree-toggle" role="presentation"><i class="fa-solid fa-chevron-down"></i></span></summary><ul>
<li class="toctree-l2"><a class="reference internal" href="../computing/scaling_strategies.html">8.1. Strategies to scale computationally: bigger data</a></li>
<li class="toctree-l2"><a class="reference internal" href="../computing/computational_performance.html">8.2. Computational Performance</a></li>
<li class="toctree-l2"><a class="reference internal" href="../computing/parallelism.html">8.3. Parallelism, resource management, and configuration</a></li>
</ul>
</details></li>
<li class="toctree-l1"><a class="reference internal" href="../model_persistence.html">9. Model persistence</a></li>
<li class="toctree-l1"><a class="reference internal" href="../common_pitfalls.html">10. Common pitfalls and recommended practices</a></li>
<li class="toctree-l1 has-children"><a class="reference internal" href="../dispatching.html">11. Dispatching</a><details><summary><span class="toctree-toggle" role="presentation"><i class="fa-solid fa-chevron-down"></i></span></summary><ul>
<li class="toctree-l2"><a class="reference internal" href="array_api.html">11.1. Array API support (experimental)</a></li>
</ul>
</details></li>
<li class="toctree-l1"><a class="reference internal" href="../machine_learning_map.html">12. Choosing the right estimator</a></li>
<li class="toctree-l1"><a class="reference internal" href="../presentations.html">13. External Resources, Videos and Talks</a></li>
</ul>
</div>
</nav></div>
</div>
<div class="sidebar-primary-items__end sidebar-primary__section">
</div>
<div id="rtd-footer-container"></div>
</div>
<main id="main-content" class="bd-main" role="main">
<div class="bd-content">
<div class="bd-article-container">
<div class="bd-header-article d-print-none">
<div class="header-article-items header-article__inner">
<div class="header-article-items__start">
<div class="header-article-item">
<nav aria-label="Breadcrumb" class="d-print-none">
<ul class="bd-breadcrumbs">
<li class="breadcrumb-item breadcrumb-home">
<a href="../index.html" class="nav-link" aria-label="Home">
<i class="fa-solid fa-home"></i>
</a>
</li>
<li class="breadcrumb-item"><a href="../user_guide.html" class="nav-link">User Guide</a></li>
<li class="breadcrumb-item"><a href="../supervised_learning.html" class="nav-link"><span class="section-number">1. </span>Supervised learning</a></li>
<li class="breadcrumb-item active" aria-current="page"><span class="ellipsis"><span class="section-number">1.15. </span>Isotonic regression</span></li>
</ul>
</nav>
</div>
</div>
</div>
</div>
<div id="searchbox"></div>
<article class="bd-article">
<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>
<p class="rubric">Examples</p>
<ul class="simple">
<li><p><a class="reference internal" href="../auto_examples/miscellaneous/plot_isotonic_regression.html#sphx-glr-auto-examples-miscellaneous-plot-isotonic-regression-py"><span class="std std-ref">Isotonic Regression</span></a></p></li>
</ul>
</section>
</article>
<footer class="bd-footer-article">
<div class="footer-article-items footer-article__inner">
<div class="footer-article-item">
<div class="prev-next-area">
<a class="left-prev"
href="semi_supervised.html"
title="previous page">
<i class="fa-solid fa-angle-left"></i>
<div class="prev-next-info">
<p class="prev-next-subtitle">previous</p>
<p class="prev-next-title"><span class="section-number">1.14. </span>Semi-supervised learning</p>
</div>
</a>
<a class="right-next"
href="calibration.html"
title="next page">
<div class="prev-next-info">
<p class="prev-next-subtitle">next</p>
<p class="prev-next-title"><span class="section-number">1.16. </span>Probability calibration</p>
</div>
<i class="fa-solid fa-angle-right"></i>
</a>
</div></div>
</div>
</footer>
</div>
<dialog id="pst-secondary-sidebar-modal"></dialog>
<div id="pst-secondary-sidebar" class="bd-sidebar-secondary bd-toc"><div class="sidebar-secondary-items sidebar-secondary__inner">
<div class="sidebar-secondary-item">
<div class="tocsection sourcelink">
<a href="../_sources/modules/isotonic.rst.txt">
<i class="fa-solid fa-file-lines"></i> Show Source
</a>
</div>
</div>
</div></div>
</div>
<footer class="bd-footer-content">
</footer>
</main>
</div>
</div>
<!-- Scripts loaded after <body> so the DOM is not blocked -->
<script defer src="../_static/scripts/bootstrap.js?digest=26a4bc78f4c0ddb94549"></script>
<script defer src="../_static/scripts/pydata-sphinx-theme.js?digest=26a4bc78f4c0ddb94549"></script>
<footer class="bd-footer">
<div class="bd-footer__inner bd-page-width">
<div class="footer-items__start">
<div class="footer-item">
<p class="copyright">
© Copyright 2007 - 2024, scikit-learn developers (BSD License).
<br/>
</p>
</div>
</div>
</div>
</footer>
</body>
</html>