-
Notifications
You must be signed in to change notification settings - Fork 81
/
Copy pathlinear_model.html
2584 lines (2354 loc) · 219 KB
/
linear_model.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
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
881
882
883
884
885
886
887
888
889
890
891
892
893
894
895
896
897
898
899
900
901
902
903
904
905
906
907
908
909
910
911
912
913
914
915
916
917
918
919
920
921
922
923
924
925
926
927
928
929
930
931
932
933
934
935
936
937
938
939
940
941
942
943
944
945
946
947
948
949
950
951
952
953
954
955
956
957
958
959
960
961
962
963
964
965
966
967
968
969
970
971
972
973
974
975
976
977
978
979
980
981
982
983
984
985
986
987
988
989
990
991
992
993
994
995
996
997
998
999
1000
<!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="1.1. Linear Models" />
<meta property="og:type" content="website" />
<meta property="og:url" content="https://scikit-learn/stable/modules/linear_model.html" />
<meta property="og:site_name" content="scikit-learn" />
<meta property="og:description" content="The following are a set of methods intended for regression in which the target value is expected to be a linear combination of the features. In mathematical notation, if\hat{y} is the predicted val..." />
<meta property="og:image" content="https://scikit-learn/stable/_images/sphx_glr_plot_ols_001.png" />
<meta property="og:image:alt" content="scikit-learn" />
<meta name="description" content="The following are a set of methods intended for regression in which the target value is expected to be a linear combination of the features. In mathematical notation, if\hat{y} is the predicted val..." />
<title>1.1. Linear Models — scikit-learn 1.6.1 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=8878045cc6db502f8baf" rel="stylesheet" />
<link href="../_static/styles/pydata-sphinx-theme.css?digest=8878045cc6db502f8baf" 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=e3ca86de" />
<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=d67e4bb0" />
<!-- So that users can add custom icons -->
<script src="../_static/scripts/fontawesome.js?digest=8878045cc6db502f8baf"></script>
<!-- Pre-loaded scripts that we'll load fully later -->
<link rel="preload" as="script" href="../_static/scripts/bootstrap.js?digest=8878045cc6db502f8baf" />
<link rel="preload" as="script" href="../_static/scripts/pydata-sphinx-theme.js?digest=8878045cc6db502f8baf" />
<script src="../_static/documentation_options.js?v=d6a008b6"></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/linear_model';</script>
<script>
DOCUMENTATION_OPTIONS.theme_version = '0.16.1';
DOCUMENTATION_OPTIONS.theme_switcher_json_url = 'https://scikit-learn.org/dev/_static/versions.json';
DOCUMENTATION_OPTIONS.theme_switcher_version_match = '1.6.1';
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>
<script src="../_static/scripts/sg_plotly_resize.js?v=eeb41cab"></script>
<link rel="canonical" href="https://scikit-learn.org/stable/modules/linear_model.html" />
<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="1.2. Linear and Quadratic Discriminant Analysis" href="lda_qda.html" />
<link rel="prev" title="1. Supervised learning" href="../supervised_learning.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-external" href="https://scikit-learn.org/dev/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-external" href="https://scikit-learn.org/dev/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 current active"><a class="current reference internal" href="#">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"><a class="reference internal" href="isotonic.html">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>
<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">1.1. </span>Linear Models</span></li>
</ul>
</nav>
</div>
</div>
</div>
</div>
<div id="searchbox"></div>
<article class="bd-article">
<section id="linear-models">
<span id="linear-model"></span><h1><span class="section-number">1.1. </span>Linear Models<a class="headerlink" href="#linear-models" title="Link to this heading">#</a></h1>
<p>The following are a set of methods intended for regression in which
the target value is expected to be a linear combination of the features.
In mathematical notation, if <span class="math notranslate nohighlight">\(\hat{y}\)</span> is the predicted
value.</p>
<div class="math notranslate nohighlight">
\[\hat{y}(w, x) = w_0 + w_1 x_1 + ... + w_p x_p\]</div>
<p>Across the module, we designate the vector <span class="math notranslate nohighlight">\(w = (w_1,
..., w_p)\)</span> as <code class="docutils literal notranslate"><span class="pre">coef_</span></code> and <span class="math notranslate nohighlight">\(w_0\)</span> as <code class="docutils literal notranslate"><span class="pre">intercept_</span></code>.</p>
<p>To perform classification with generalized linear models, see
<a class="reference internal" href="#logistic-regression"><span class="std std-ref">Logistic regression</span></a>.</p>
<section id="ordinary-least-squares">
<span id="id1"></span><h2><span class="section-number">1.1.1. </span>Ordinary Least Squares<a class="headerlink" href="#ordinary-least-squares" title="Link to this heading">#</a></h2>
<p><a class="reference internal" href="generated/sklearn.linear_model.LinearRegression.html#sklearn.linear_model.LinearRegression" title="sklearn.linear_model.LinearRegression"><code class="xref py py-class docutils literal notranslate"><span class="pre">LinearRegression</span></code></a> fits a linear model with coefficients
<span class="math notranslate nohighlight">\(w = (w_1, ..., w_p)\)</span> to minimize the residual sum
of squares between the observed targets in the dataset, and the
targets predicted by the linear approximation. Mathematically it
solves a problem of the form:</p>
<div class="math notranslate nohighlight">
\[\min_{w} || X w - y||_2^2\]</div>
<figure class="align-center">
<a class="reference external image-reference" href="../auto_examples/linear_model/plot_ols.html"><img alt="../_images/sphx_glr_plot_ols_001.png" src="../_images/sphx_glr_plot_ols_001.png" style="width: 500.0px; height: 250.0px;" />
</a>
</figure>
<p><a class="reference internal" href="generated/sklearn.linear_model.LinearRegression.html#sklearn.linear_model.LinearRegression" title="sklearn.linear_model.LinearRegression"><code class="xref py py-class docutils literal notranslate"><span class="pre">LinearRegression</span></code></a> will take in its <code class="docutils literal notranslate"><span class="pre">fit</span></code> method arrays <code class="docutils literal notranslate"><span class="pre">X</span></code>, <code class="docutils literal notranslate"><span class="pre">y</span></code>
and will store the coefficients <span class="math notranslate nohighlight">\(w\)</span> of the linear model in its
<code class="docutils literal notranslate"><span class="pre">coef_</span></code> member:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">sklearn</span> <span class="kn">import</span> <span class="n">linear_model</span>
<span class="gp">>>> </span><span class="n">reg</span> <span class="o">=</span> <span class="n">linear_model</span><span class="o">.</span><span class="n">LinearRegression</span><span class="p">()</span>
<span class="gp">>>> </span><span class="n">reg</span><span class="o">.</span><span class="n">fit</span><span class="p">([[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">]],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">])</span>
<span class="go">LinearRegression()</span>
<span class="gp">>>> </span><span class="n">reg</span><span class="o">.</span><span class="n">coef_</span>
<span class="go">array([0.5, 0.5])</span>
</pre></div>
</div>
<p>The coefficient estimates for Ordinary Least Squares rely on the
independence of the features. When features are correlated and the
columns of the design matrix <span class="math notranslate nohighlight">\(X\)</span> have an approximately linear
dependence, the design matrix becomes close to singular
and as a result, the least-squares estimate becomes highly sensitive
to random errors in the observed target, producing a large
variance. This situation of <em>multicollinearity</em> can arise, for
example, when data are collected without an experimental design.</p>
<p class="rubric">Examples</p>
<ul class="simple">
<li><p><a class="reference internal" href="../auto_examples/linear_model/plot_ols.html#sphx-glr-auto-examples-linear-model-plot-ols-py"><span class="std std-ref">Ordinary Least Squares Example</span></a></p></li>
</ul>
<section id="non-negative-least-squares">
<h3><span class="section-number">1.1.1.1. </span>Non-Negative Least Squares<a class="headerlink" href="#non-negative-least-squares" title="Link to this heading">#</a></h3>
<p>It is possible to constrain all the coefficients to be non-negative, which may
be useful when they represent some physical or naturally non-negative
quantities (e.g., frequency counts or prices of goods).
<a class="reference internal" href="generated/sklearn.linear_model.LinearRegression.html#sklearn.linear_model.LinearRegression" title="sklearn.linear_model.LinearRegression"><code class="xref py py-class docutils literal notranslate"><span class="pre">LinearRegression</span></code></a> accepts a boolean <code class="docutils literal notranslate"><span class="pre">positive</span></code>
parameter: when set to <code class="docutils literal notranslate"><span class="pre">True</span></code> <a class="reference external" href="https://en.wikipedia.org/wiki/Non-negative_least_squares">Non-Negative Least Squares</a> are then applied.</p>
<p class="rubric">Examples</p>
<ul class="simple">
<li><p><a class="reference internal" href="../auto_examples/linear_model/plot_nnls.html#sphx-glr-auto-examples-linear-model-plot-nnls-py"><span class="std std-ref">Non-negative least squares</span></a></p></li>
</ul>
</section>
<section id="ordinary-least-squares-complexity">
<h3><span class="section-number">1.1.1.2. </span>Ordinary Least Squares Complexity<a class="headerlink" href="#ordinary-least-squares-complexity" title="Link to this heading">#</a></h3>
<p>The least squares solution is computed using the singular value
decomposition of X. If X is a matrix of shape <code class="docutils literal notranslate"><span class="pre">(n_samples,</span> <span class="pre">n_features)</span></code>
this method has a cost of
<span class="math notranslate nohighlight">\(O(n_{\text{samples}} n_{\text{features}}^2)\)</span>, assuming that
<span class="math notranslate nohighlight">\(n_{\text{samples}} \geq n_{\text{features}}\)</span>.</p>
</section>
</section>
<section id="ridge-regression-and-classification">
<span id="ridge-regression"></span><h2><span class="section-number">1.1.2. </span>Ridge regression and classification<a class="headerlink" href="#ridge-regression-and-classification" title="Link to this heading">#</a></h2>
<section id="regression">
<h3><span class="section-number">1.1.2.1. </span>Regression<a class="headerlink" href="#regression" title="Link to this heading">#</a></h3>
<p><a class="reference internal" href="generated/sklearn.linear_model.Ridge.html#sklearn.linear_model.Ridge" title="sklearn.linear_model.Ridge"><code class="xref py py-class docutils literal notranslate"><span class="pre">Ridge</span></code></a> regression addresses some of the problems of
<a class="reference internal" href="#ordinary-least-squares"><span class="std std-ref">Ordinary Least Squares</span></a> by imposing a penalty on the size of the
coefficients. The ridge coefficients minimize a penalized residual sum
of squares:</p>
<div class="math notranslate nohighlight">
\[\min_{w} || X w - y||_2^2 + \alpha ||w||_2^2\]</div>
<p>The complexity parameter <span class="math notranslate nohighlight">\(\alpha \geq 0\)</span> controls the amount
of shrinkage: the larger the value of <span class="math notranslate nohighlight">\(\alpha\)</span>, the greater the amount
of shrinkage and thus the coefficients become more robust to collinearity.</p>
<figure class="align-center">
<a class="reference external image-reference" href="../auto_examples/linear_model/plot_ridge_path.html"><img alt="../_images/sphx_glr_plot_ridge_path_001.png" src="../_images/sphx_glr_plot_ridge_path_001.png" style="width: 320.0px; height: 240.0px;" />
</a>
</figure>
<p>As with other linear models, <a class="reference internal" href="generated/sklearn.linear_model.Ridge.html#sklearn.linear_model.Ridge" title="sklearn.linear_model.Ridge"><code class="xref py py-class docutils literal notranslate"><span class="pre">Ridge</span></code></a> will take in its <code class="docutils literal notranslate"><span class="pre">fit</span></code> method
arrays <code class="docutils literal notranslate"><span class="pre">X</span></code>, <code class="docutils literal notranslate"><span class="pre">y</span></code> and will store the coefficients <span class="math notranslate nohighlight">\(w\)</span> of the linear model in
its <code class="docutils literal notranslate"><span class="pre">coef_</span></code> member:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">sklearn</span> <span class="kn">import</span> <span class="n">linear_model</span>
<span class="gp">>>> </span><span class="n">reg</span> <span class="o">=</span> <span class="n">linear_model</span><span class="o">.</span><span class="n">Ridge</span><span class="p">(</span><span class="n">alpha</span><span class="o">=</span><span class="mf">.5</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">reg</span><span class="o">.</span><span class="n">fit</span><span class="p">([[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mf">.1</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span>
<span class="go">Ridge(alpha=0.5)</span>
<span class="gp">>>> </span><span class="n">reg</span><span class="o">.</span><span class="n">coef_</span>
<span class="go">array([0.34545455, 0.34545455])</span>
<span class="gp">>>> </span><span class="n">reg</span><span class="o">.</span><span class="n">intercept_</span>
<span class="go">0.13636...</span>
</pre></div>
</div>
<p>Note that the class <a class="reference internal" href="generated/sklearn.linear_model.Ridge.html#sklearn.linear_model.Ridge" title="sklearn.linear_model.Ridge"><code class="xref py py-class docutils literal notranslate"><span class="pre">Ridge</span></code></a> allows for the user to specify that the
solver be automatically chosen by setting <code class="docutils literal notranslate"><span class="pre">solver="auto"</span></code>. When this option
is specified, <a class="reference internal" href="generated/sklearn.linear_model.Ridge.html#sklearn.linear_model.Ridge" title="sklearn.linear_model.Ridge"><code class="xref py py-class docutils literal notranslate"><span class="pre">Ridge</span></code></a> will choose between the <code class="docutils literal notranslate"><span class="pre">"lbfgs"</span></code>, <code class="docutils literal notranslate"><span class="pre">"cholesky"</span></code>,
and <code class="docutils literal notranslate"><span class="pre">"sparse_cg"</span></code> solvers. <a class="reference internal" href="generated/sklearn.linear_model.Ridge.html#sklearn.linear_model.Ridge" title="sklearn.linear_model.Ridge"><code class="xref py py-class docutils literal notranslate"><span class="pre">Ridge</span></code></a> will begin checking the conditions
shown in the following table from top to bottom. If the condition is true,
the corresponding solver is chosen.</p>
<div class="pst-scrollable-table-container"><table class="table">
<tbody>
<tr class="row-odd"><td><p><strong>Solver</strong></p></td>
<td><p><strong>Condition</strong></p></td>
</tr>
<tr class="row-even"><td><p>‘lbfgs’</p></td>
<td><p>The <code class="docutils literal notranslate"><span class="pre">positive=True</span></code> option is specified.</p></td>
</tr>
<tr class="row-odd"><td><p>‘cholesky’</p></td>
<td><p>The input array X is not sparse.</p></td>
</tr>
<tr class="row-even"><td><p>‘sparse_cg’</p></td>
<td><p>None of the above conditions are fulfilled.</p></td>
</tr>
</tbody>
</table>
</div>
</section>
<section id="classification">
<h3><span class="section-number">1.1.2.2. </span>Classification<a class="headerlink" href="#classification" title="Link to this heading">#</a></h3>
<p>The <a class="reference internal" href="generated/sklearn.linear_model.Ridge.html#sklearn.linear_model.Ridge" title="sklearn.linear_model.Ridge"><code class="xref py py-class docutils literal notranslate"><span class="pre">Ridge</span></code></a> regressor has a classifier variant:
<a class="reference internal" href="generated/sklearn.linear_model.RidgeClassifier.html#sklearn.linear_model.RidgeClassifier" title="sklearn.linear_model.RidgeClassifier"><code class="xref py py-class docutils literal notranslate"><span class="pre">RidgeClassifier</span></code></a>. This classifier first converts binary targets to
<code class="docutils literal notranslate"><span class="pre">{-1,</span> <span class="pre">1}</span></code> and then treats the problem as a regression task, optimizing the
same objective as above. The predicted class corresponds to the sign of the
regressor’s prediction. For multiclass classification, the problem is
treated as multi-output regression, and the predicted class corresponds to
the output with the highest value.</p>
<p>It might seem questionable to use a (penalized) Least Squares loss to fit a
classification model instead of the more traditional logistic or hinge
losses. However, in practice, all those models can lead to similar
cross-validation scores in terms of accuracy or precision/recall, while the
penalized least squares loss used by the <a class="reference internal" href="generated/sklearn.linear_model.RidgeClassifier.html#sklearn.linear_model.RidgeClassifier" title="sklearn.linear_model.RidgeClassifier"><code class="xref py py-class docutils literal notranslate"><span class="pre">RidgeClassifier</span></code></a> allows for
a very different choice of the numerical solvers with distinct computational
performance profiles.</p>
<p>The <a class="reference internal" href="generated/sklearn.linear_model.RidgeClassifier.html#sklearn.linear_model.RidgeClassifier" title="sklearn.linear_model.RidgeClassifier"><code class="xref py py-class docutils literal notranslate"><span class="pre">RidgeClassifier</span></code></a> can be significantly faster than e.g.
<a class="reference internal" href="generated/sklearn.linear_model.LogisticRegression.html#sklearn.linear_model.LogisticRegression" title="sklearn.linear_model.LogisticRegression"><code class="xref py py-class docutils literal notranslate"><span class="pre">LogisticRegression</span></code></a> with a high number of classes because it can
compute the projection matrix <span class="math notranslate nohighlight">\((X^T X)^{-1} X^T\)</span> only once.</p>
<p>This classifier is sometimes referred to as a <a class="reference external" href="https://en.wikipedia.org/wiki/Least-squares_support-vector_machine">Least Squares Support Vector
Machines</a> with
a linear kernel.</p>
<p class="rubric">Examples</p>
<ul class="simple">
<li><p><a class="reference internal" href="../auto_examples/linear_model/plot_ridge_path.html#sphx-glr-auto-examples-linear-model-plot-ridge-path-py"><span class="std std-ref">Plot Ridge coefficients as a function of the regularization</span></a></p></li>
<li><p><a class="reference internal" href="../auto_examples/text/plot_document_classification_20newsgroups.html#sphx-glr-auto-examples-text-plot-document-classification-20newsgroups-py"><span class="std std-ref">Classification of text documents using sparse features</span></a></p></li>
<li><p><a class="reference internal" href="../auto_examples/inspection/plot_linear_model_coefficient_interpretation.html#sphx-glr-auto-examples-inspection-plot-linear-model-coefficient-interpretation-py"><span class="std std-ref">Common pitfalls in the interpretation of coefficients of linear models</span></a></p></li>
</ul>
</section>
<section id="ridge-complexity">
<h3><span class="section-number">1.1.2.3. </span>Ridge Complexity<a class="headerlink" href="#ridge-complexity" title="Link to this heading">#</a></h3>
<p>This method has the same order of complexity as
<a class="reference internal" href="#ordinary-least-squares"><span class="std std-ref">Ordinary Least Squares</span></a>.</p>
</section>
<section id="setting-the-regularization-parameter-leave-one-out-cross-validation">
<h3><span class="section-number">1.1.2.4. </span>Setting the regularization parameter: leave-one-out Cross-Validation<a class="headerlink" href="#setting-the-regularization-parameter-leave-one-out-cross-validation" title="Link to this heading">#</a></h3>
<p><a class="reference internal" href="generated/sklearn.linear_model.RidgeCV.html#sklearn.linear_model.RidgeCV" title="sklearn.linear_model.RidgeCV"><code class="xref py py-class docutils literal notranslate"><span class="pre">RidgeCV</span></code></a> and <a class="reference internal" href="generated/sklearn.linear_model.RidgeClassifierCV.html#sklearn.linear_model.RidgeClassifierCV" title="sklearn.linear_model.RidgeClassifierCV"><code class="xref py py-class docutils literal notranslate"><span class="pre">RidgeClassifierCV</span></code></a> implement ridge
regression/classification with built-in cross-validation of the alpha parameter.
They work in the same way as <a class="reference internal" href="generated/sklearn.model_selection.GridSearchCV.html#sklearn.model_selection.GridSearchCV" title="sklearn.model_selection.GridSearchCV"><code class="xref py py-class docutils literal notranslate"><span class="pre">GridSearchCV</span></code></a> except
that it defaults to efficient Leave-One-Out <a class="reference internal" href="../glossary.html#term-cross-validation"><span class="xref std std-term">cross-validation</span></a>.
When using the default <a class="reference internal" href="../glossary.html#term-cross-validation"><span class="xref std std-term">cross-validation</span></a>, alpha cannot be 0 due to the
formulation used to calculate Leave-One-Out error. See <a class="reference internal" href="#rl2007" id="id3"><span>[RL2007]</span></a> for details.</p>
<p>Usage example:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="gp">>>> </span><span class="kn">from</span> <span class="nn">sklearn</span> <span class="kn">import</span> <span class="n">linear_model</span>
<span class="gp">>>> </span><span class="n">reg</span> <span class="o">=</span> <span class="n">linear_model</span><span class="o">.</span><span class="n">RidgeCV</span><span class="p">(</span><span class="n">alphas</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">logspace</span><span class="p">(</span><span class="o">-</span><span class="mi">6</span><span class="p">,</span> <span class="mi">6</span><span class="p">,</span> <span class="mi">13</span><span class="p">))</span>
<span class="gp">>>> </span><span class="n">reg</span><span class="o">.</span><span class="n">fit</span><span class="p">([[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mf">.1</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span>
<span class="go">RidgeCV(alphas=array([1.e-06, 1.e-05, 1.e-04, 1.e-03, 1.e-02, 1.e-01, 1.e+00, 1.e+01,</span>
<span class="go"> 1.e+02, 1.e+03, 1.e+04, 1.e+05, 1.e+06]))</span>
<span class="gp">>>> </span><span class="n">reg</span><span class="o">.</span><span class="n">alpha_</span>
<span class="go">0.01</span>
</pre></div>
</div>
<p>Specifying the value of the <a class="reference internal" href="../glossary.html#term-cv"><span class="xref std std-term">cv</span></a> attribute will trigger the use of
cross-validation with <a class="reference internal" href="generated/sklearn.model_selection.GridSearchCV.html#sklearn.model_selection.GridSearchCV" title="sklearn.model_selection.GridSearchCV"><code class="xref py py-class docutils literal notranslate"><span class="pre">GridSearchCV</span></code></a>, for
example <code class="docutils literal notranslate"><span class="pre">cv=10</span></code> for 10-fold cross-validation, rather than Leave-One-Out
Cross-Validation.</p>
<details class="sd-sphinx-override sd-dropdown sd-card sd-mb-3" id="references">
<summary class="sd-summary-title sd-card-header">
<span class="sd-summary-text">References<a class="headerlink" href="#references" title="Link to this dropdown">#</a></span><span class="sd-summary-state-marker sd-summary-chevron-right"><svg version="1.1" width="1.5em" height="1.5em" class="sd-octicon sd-octicon-chevron-right" viewBox="0 0 24 24" aria-hidden="true"><path d="M8.72 18.78a.75.75 0 0 1 0-1.06L14.44 12 8.72 6.28a.751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018l6.25 6.25a.75.75 0 0 1 0 1.06l-6.25 6.25a.75.75 0 0 1-1.06 0Z"></path></svg></span></summary><div class="sd-summary-content sd-card-body docutils">
<div role="list" class="citation-list">
<div class="citation" id="rl2007" role="doc-biblioentry">
<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id3">RL2007</a><span class="fn-bracket">]</span></span>
<p class="sd-card-text">“Notes on Regularized Least Squares”, Rifkin & Lippert (<a class="reference external" href="http://cbcl.mit.edu/publications/ps/MIT-CSAIL-TR-2007-025.pdf">technical report</a>,
<a class="reference external" href="https://www.mit.edu/~9.520/spring07/Classes/rlsslides.pdf">course slides</a>).</p>
</div>
</div>
</div>
</details></section>
</section>
<section id="lasso">
<span id="id4"></span><h2><span class="section-number">1.1.3. </span>Lasso<a class="headerlink" href="#lasso" title="Link to this heading">#</a></h2>
<p>The <a class="reference internal" href="generated/sklearn.linear_model.Lasso.html#sklearn.linear_model.Lasso" title="sklearn.linear_model.Lasso"><code class="xref py py-class docutils literal notranslate"><span class="pre">Lasso</span></code></a> is a linear model that estimates sparse coefficients.
It is useful in some contexts due to its tendency to prefer solutions
with fewer non-zero coefficients, effectively reducing the number of
features upon which the given solution is dependent. For this reason,
Lasso and its variants are fundamental to the field of compressed sensing.
Under certain conditions, it can recover the exact set of non-zero
coefficients (see
<a class="reference internal" href="../auto_examples/applications/plot_tomography_l1_reconstruction.html#sphx-glr-auto-examples-applications-plot-tomography-l1-reconstruction-py"><span class="std std-ref">Compressive sensing: tomography reconstruction with L1 prior (Lasso)</span></a>).</p>
<p>Mathematically, it consists of a linear model with an added regularization term.
The objective function to minimize is:</p>
<div class="math notranslate nohighlight">
\[\min_{w} { \frac{1}{2n_{\text{samples}}} ||X w - y||_2 ^ 2 + \alpha ||w||_1}\]</div>
<p>The lasso estimate thus solves the minimization of the
least-squares penalty with <span class="math notranslate nohighlight">\(\alpha ||w||_1\)</span> added, where
<span class="math notranslate nohighlight">\(\alpha\)</span> is a constant and <span class="math notranslate nohighlight">\(||w||_1\)</span> is the <span class="math notranslate nohighlight">\(\ell_1\)</span>-norm of
the coefficient vector.</p>
<p>The implementation in the class <a class="reference internal" href="generated/sklearn.linear_model.Lasso.html#sklearn.linear_model.Lasso" title="sklearn.linear_model.Lasso"><code class="xref py py-class docutils literal notranslate"><span class="pre">Lasso</span></code></a> uses coordinate descent as
the algorithm to fit the coefficients. See <a class="reference internal" href="#least-angle-regression"><span class="std std-ref">Least Angle Regression</span></a>
for another implementation:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">sklearn</span> <span class="kn">import</span> <span class="n">linear_model</span>
<span class="gp">>>> </span><span class="n">reg</span> <span class="o">=</span> <span class="n">linear_model</span><span class="o">.</span><span class="n">Lasso</span><span class="p">(</span><span class="n">alpha</span><span class="o">=</span><span class="mf">0.1</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">reg</span><span class="o">.</span><span class="n">fit</span><span class="p">([[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span>
<span class="go">Lasso(alpha=0.1)</span>
<span class="gp">>>> </span><span class="n">reg</span><span class="o">.</span><span class="n">predict</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]])</span>
<span class="go">array([0.8])</span>
</pre></div>
</div>
<p>The function <a class="reference internal" href="generated/sklearn.linear_model.lasso_path.html#sklearn.linear_model.lasso_path" title="sklearn.linear_model.lasso_path"><code class="xref py py-func docutils literal notranslate"><span class="pre">lasso_path</span></code></a> is useful for lower-level tasks, as it
computes the coefficients along the full path of possible values.</p>
<p class="rubric">Examples</p>
<ul class="simple">
<li><p><a class="reference internal" href="../auto_examples/linear_model/plot_lasso_and_elasticnet.html#sphx-glr-auto-examples-linear-model-plot-lasso-and-elasticnet-py"><span class="std std-ref">L1-based models for Sparse Signals</span></a></p></li>
<li><p><a class="reference internal" href="../auto_examples/applications/plot_tomography_l1_reconstruction.html#sphx-glr-auto-examples-applications-plot-tomography-l1-reconstruction-py"><span class="std std-ref">Compressive sensing: tomography reconstruction with L1 prior (Lasso)</span></a></p></li>
<li><p><a class="reference internal" href="../auto_examples/inspection/plot_linear_model_coefficient_interpretation.html#sphx-glr-auto-examples-inspection-plot-linear-model-coefficient-interpretation-py"><span class="std std-ref">Common pitfalls in the interpretation of coefficients of linear models</span></a></p></li>
</ul>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p><strong>Feature selection with Lasso</strong></p>
<p>As the Lasso regression yields sparse models, it can
thus be used to perform feature selection, as detailed in
<a class="reference internal" href="feature_selection.html#l1-feature-selection"><span class="std std-ref">L1-based feature selection</span></a>.</p>
</div>
<details class="sd-sphinx-override sd-dropdown sd-card sd-mb-3" id="references-2">
<summary class="sd-summary-title sd-card-header">
<span class="sd-summary-text">References<a class="headerlink" href="#references-2" title="Link to this dropdown">#</a></span><span class="sd-summary-state-marker sd-summary-chevron-right"><svg version="1.1" width="1.5em" height="1.5em" class="sd-octicon sd-octicon-chevron-right" viewBox="0 0 24 24" aria-hidden="true"><path d="M8.72 18.78a.75.75 0 0 1 0-1.06L14.44 12 8.72 6.28a.751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018l6.25 6.25a.75.75 0 0 1 0 1.06l-6.25 6.25a.75.75 0 0 1-1.06 0Z"></path></svg></span></summary><div class="sd-summary-content sd-card-body docutils">
<p class="sd-card-text">The following two references explain the iterations
used in the coordinate descent solver of scikit-learn, as well as
the duality gap computation used for convergence control.</p>
<ul class="simple">
<li><p class="sd-card-text">“Regularization Path For Generalized linear Models by Coordinate Descent”,
Friedman, Hastie & Tibshirani, J Stat Softw, 2010 (<a class="reference external" href="https://www.jstatsoft.org/article/view/v033i01/v33i01.pdf">Paper</a>).</p></li>
<li><p class="sd-card-text">“An Interior-Point Method for Large-Scale L1-Regularized Least Squares,”
S. J. Kim, K. Koh, M. Lustig, S. Boyd and D. Gorinevsky,
in IEEE Journal of Selected Topics in Signal Processing, 2007
(<a class="reference external" href="https://web.stanford.edu/~boyd/papers/pdf/l1_ls.pdf">Paper</a>)</p></li>
</ul>
</div>
</details><section id="setting-regularization-parameter">
<h3><span class="section-number">1.1.3.1. </span>Setting regularization parameter<a class="headerlink" href="#setting-regularization-parameter" title="Link to this heading">#</a></h3>
<p>The <code class="docutils literal notranslate"><span class="pre">alpha</span></code> parameter controls the degree of sparsity of the estimated
coefficients.</p>
<section id="using-cross-validation">
<h4><span class="section-number">1.1.3.1.1. </span>Using cross-validation<a class="headerlink" href="#using-cross-validation" title="Link to this heading">#</a></h4>
<p>scikit-learn exposes objects that set the Lasso <code class="docutils literal notranslate"><span class="pre">alpha</span></code> parameter by
cross-validation: <a class="reference internal" href="generated/sklearn.linear_model.LassoCV.html#sklearn.linear_model.LassoCV" title="sklearn.linear_model.LassoCV"><code class="xref py py-class docutils literal notranslate"><span class="pre">LassoCV</span></code></a> and <a class="reference internal" href="generated/sklearn.linear_model.LassoLarsCV.html#sklearn.linear_model.LassoLarsCV" title="sklearn.linear_model.LassoLarsCV"><code class="xref py py-class docutils literal notranslate"><span class="pre">LassoLarsCV</span></code></a>.
<a class="reference internal" href="generated/sklearn.linear_model.LassoLarsCV.html#sklearn.linear_model.LassoLarsCV" title="sklearn.linear_model.LassoLarsCV"><code class="xref py py-class docutils literal notranslate"><span class="pre">LassoLarsCV</span></code></a> is based on the <a class="reference internal" href="#least-angle-regression"><span class="std std-ref">Least Angle Regression</span></a> algorithm
explained below.</p>
<p>For high-dimensional datasets with many collinear features,
<a class="reference internal" href="generated/sklearn.linear_model.LassoCV.html#sklearn.linear_model.LassoCV" title="sklearn.linear_model.LassoCV"><code class="xref py py-class docutils literal notranslate"><span class="pre">LassoCV</span></code></a> is most often preferable. However, <a class="reference internal" href="generated/sklearn.linear_model.LassoLarsCV.html#sklearn.linear_model.LassoLarsCV" title="sklearn.linear_model.LassoLarsCV"><code class="xref py py-class docutils literal notranslate"><span class="pre">LassoLarsCV</span></code></a> has
the advantage of exploring more relevant values of <code class="docutils literal notranslate"><span class="pre">alpha</span></code> parameter, and
if the number of samples is very small compared to the number of
features, it is often faster than <a class="reference internal" href="generated/sklearn.linear_model.LassoCV.html#sklearn.linear_model.LassoCV" title="sklearn.linear_model.LassoCV"><code class="xref py py-class docutils literal notranslate"><span class="pre">LassoCV</span></code></a>.</p>
<p class="centered">
<strong><a class="reference external" href="../auto_examples/linear_model/plot_lasso_model_selection.html"><img alt="lasso_cv_1" src="../_images/sphx_glr_plot_lasso_model_selection_002.png" style="width: 307.2px; height: 230.39999999999998px;" /></a> <a class="reference external" href="../auto_examples/linear_model/plot_lasso_model_selection.html"><img alt="lasso_cv_2" src="../_images/sphx_glr_plot_lasso_model_selection_003.png" style="width: 307.2px; height: 230.39999999999998px;" /></a></strong></p></section>
<section id="information-criteria-based-model-selection">
<span id="lasso-lars-ic"></span><h4><span class="section-number">1.1.3.1.2. </span>Information-criteria based model selection<a class="headerlink" href="#information-criteria-based-model-selection" title="Link to this heading">#</a></h4>
<p>Alternatively, the estimator <a class="reference internal" href="generated/sklearn.linear_model.LassoLarsIC.html#sklearn.linear_model.LassoLarsIC" title="sklearn.linear_model.LassoLarsIC"><code class="xref py py-class docutils literal notranslate"><span class="pre">LassoLarsIC</span></code></a> proposes to use the
Akaike information criterion (AIC) and the Bayes Information criterion (BIC).
It is a computationally cheaper alternative to find the optimal value of alpha
as the regularization path is computed only once instead of k+1 times
when using k-fold cross-validation.</p>
<p>Indeed, these criteria are computed on the in-sample training set. In short,
they penalize the over-optimistic scores of the different Lasso models by
their flexibility (cf. to “Mathematical details” section below).</p>
<p>However, such criteria need a proper estimation of the degrees of freedom of
the solution, are derived for large samples (asymptotic results) and assume the
correct model is candidates under investigation. They also tend to break when
the problem is badly conditioned (e.g. more features than samples).</p>
<figure class="align-center">
<a class="reference external image-reference" href="../auto_examples/linear_model/plot_lasso_lars_ic.html"><img alt="../_images/sphx_glr_plot_lasso_lars_ic_001.png" src="../_images/sphx_glr_plot_lasso_lars_ic_001.png" style="width: 320.0px; height: 240.0px;" />
</a>
</figure>
<p class="rubric">Examples</p>
<ul class="simple">
<li><p><a class="reference internal" href="../auto_examples/linear_model/plot_lasso_model_selection.html#sphx-glr-auto-examples-linear-model-plot-lasso-model-selection-py"><span class="std std-ref">Lasso model selection: AIC-BIC / cross-validation</span></a></p></li>
<li><p><a class="reference internal" href="../auto_examples/linear_model/plot_lasso_lars_ic.html#sphx-glr-auto-examples-linear-model-plot-lasso-lars-ic-py"><span class="std std-ref">Lasso model selection via information criteria</span></a></p></li>
</ul>
</section>
<section id="aic-and-bic-criteria">
<span id="aic-bic"></span><h4><span class="section-number">1.1.3.1.3. </span>AIC and BIC criteria<a class="headerlink" href="#aic-and-bic-criteria" title="Link to this heading">#</a></h4>
<p>The definition of AIC (and thus BIC) might differ in the literature. In this
section, we give more information regarding the criterion computed in
scikit-learn.</p>
<details class="sd-sphinx-override sd-dropdown sd-card sd-mb-3" id="mathematical-details">
<summary class="sd-summary-title sd-card-header">
<span class="sd-summary-text">Mathematical details<a class="headerlink" href="#mathematical-details" title="Link to this dropdown">#</a></span><span class="sd-summary-state-marker sd-summary-chevron-right"><svg version="1.1" width="1.5em" height="1.5em" class="sd-octicon sd-octicon-chevron-right" viewBox="0 0 24 24" aria-hidden="true"><path d="M8.72 18.78a.75.75 0 0 1 0-1.06L14.44 12 8.72 6.28a.751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018l6.25 6.25a.75.75 0 0 1 0 1.06l-6.25 6.25a.75.75 0 0 1-1.06 0Z"></path></svg></span></summary><div class="sd-summary-content sd-card-body docutils">
<p class="sd-card-text">The AIC criterion is defined as:</p>
<div class="math notranslate nohighlight">
\[AIC = -2 \log(\hat{L}) + 2 d\]</div>
<p class="sd-card-text">where <span class="math notranslate nohighlight">\(\hat{L}\)</span> is the maximum likelihood of the model and
<span class="math notranslate nohighlight">\(d\)</span> is the number of parameters (as well referred to as degrees of
freedom in the previous section).</p>
<p class="sd-card-text">The definition of BIC replace the constant <span class="math notranslate nohighlight">\(2\)</span> by <span class="math notranslate nohighlight">\(\log(N)\)</span>:</p>
<div class="math notranslate nohighlight">