<!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="Plot classification boundaries with different SVM Kernels" />
<meta property="og:type" content="website" />
<meta property="og:url" content="https://scikit-learn/stable/auto_examples/svm/plot_svm_kernels.html" />
<meta property="og:site_name" content="scikit-learn" />
<meta property="og:description" content="This example shows how different kernels in a SVC(Support Vector Classifier) influence the classification boundaries in a binary, two-dimensional classification problem. SVCs aim to find a hyperpla..." />
<meta property="og:image" content="https://scikit-learn.org/stable/_static/scikit-learn-logo-small.png" />
<meta property="og:image:alt" content="scikit-learn" />
<meta name="description" content="This example shows how different kernels in a SVC(Support Vector Classifier) influence the classification boundaries in a binary, two-dimensional classification problem. SVCs aim to find a hyperpla..." />
<title>Plot classification boundaries with different SVM Kernels — scikit-learn 1.7.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=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=8f2a1f02" />
<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=2c9f8f05" />
<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=8f525996" />
<!-- 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=473747f4"></script>
<script src="../../_static/doctools.js?v=9bcbadda"></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=96e329c5"></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 = 'auto_examples/svm/plot_svm_kernels';</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 = '1.7.dev0';
DOCUMENTATION_OPTIONS.show_version_warning_banner =
true;
</script>
<script src="../../_static/scripts/dropdown.js?v=d6825577"></script>
<script src="../../_static/scripts/version-switcher.js?v=a6dd8357"></script>
<script src="../../_static/scripts/sg_plotly_resize.js?v=2167d4db"></script>
<link rel="canonical" href="https://scikit-learn.org/stable/auto_examples/svm/plot_svm_kernels.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="Plot different SVM classifiers in the iris dataset" href="plot_iris_svc.html" />
<link rel="prev" title="One-class SVM with non-linear kernel (RBF)" href="plot_oneclass.html" />
<meta name="viewport" content="width=device-width, initial-scale=1"/>
<meta name="docsearch:language" content="en"/>
<meta name="docsearch:version" content="1.7" />
</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 ">
<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 current active">
<a class="nav-link nav-internal" href="../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 ">
<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 current active">
<a class="nav-link nav-internal" href="../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 has-children"><a class="reference internal" href="../release_highlights/index.html">Release Highlights</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="../release_highlights/plot_release_highlights_1_6_0.html">Release Highlights for scikit-learn 1.6</a></li>
<li class="toctree-l2"><a class="reference internal" href="../release_highlights/plot_release_highlights_1_5_0.html">Release Highlights for scikit-learn 1.5</a></li>
<li class="toctree-l2"><a class="reference internal" href="../release_highlights/plot_release_highlights_1_4_0.html">Release Highlights for scikit-learn 1.4</a></li>
<li class="toctree-l2"><a class="reference internal" href="../release_highlights/plot_release_highlights_1_3_0.html">Release Highlights for scikit-learn 1.3</a></li>
<li class="toctree-l2"><a class="reference internal" href="../release_highlights/plot_release_highlights_1_2_0.html">Release Highlights for scikit-learn 1.2</a></li>
<li class="toctree-l2"><a class="reference internal" href="../release_highlights/plot_release_highlights_1_1_0.html">Release Highlights for scikit-learn 1.1</a></li>
<li class="toctree-l2"><a class="reference internal" href="../release_highlights/plot_release_highlights_1_0_0.html">Release Highlights for scikit-learn 1.0</a></li>
<li class="toctree-l2"><a class="reference internal" href="../release_highlights/plot_release_highlights_0_24_0.html">Release Highlights for scikit-learn 0.24</a></li>
<li class="toctree-l2"><a class="reference internal" href="../release_highlights/plot_release_highlights_0_23_0.html">Release Highlights for scikit-learn 0.23</a></li>
<li class="toctree-l2"><a class="reference internal" href="../release_highlights/plot_release_highlights_0_22_0.html">Release Highlights for scikit-learn 0.22</a></li>
</ul>
</details></li>
<li class="toctree-l1 has-children"><a class="reference internal" href="../bicluster/index.html">Biclustering</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="../bicluster/plot_spectral_biclustering.html">A demo of the Spectral Biclustering algorithm</a></li>
<li class="toctree-l2"><a class="reference internal" href="../bicluster/plot_spectral_coclustering.html">A demo of the Spectral Co-Clustering algorithm</a></li>
<li class="toctree-l2"><a class="reference internal" href="../bicluster/plot_bicluster_newsgroups.html">Biclustering documents with the Spectral Co-clustering algorithm</a></li>
</ul>
</details></li>
<li class="toctree-l1 has-children"><a class="reference internal" href="../calibration/index.html">Calibration</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="../calibration/plot_compare_calibration.html">Comparison of Calibration of Classifiers</a></li>
<li class="toctree-l2"><a class="reference internal" href="../calibration/plot_calibration_curve.html">Probability Calibration curves</a></li>
<li class="toctree-l2"><a class="reference internal" href="../calibration/plot_calibration_multiclass.html">Probability Calibration for 3-class classification</a></li>
<li class="toctree-l2"><a class="reference internal" href="../calibration/plot_calibration.html">Probability calibration of classifiers</a></li>
</ul>
</details></li>
<li class="toctree-l1 has-children"><a class="reference internal" href="../classification/index.html">Classification</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="../classification/plot_classifier_comparison.html">Classifier comparison</a></li>
<li class="toctree-l2"><a class="reference internal" href="../classification/plot_lda_qda.html">Linear and Quadratic Discriminant Analysis with covariance ellipsoid</a></li>
<li class="toctree-l2"><a class="reference internal" href="../classification/plot_lda.html">Normal, Ledoit-Wolf and OAS Linear Discriminant Analysis for classification</a></li>
<li class="toctree-l2"><a class="reference internal" href="../classification/plot_classification_probability.html">Plot classification probability</a></li>
<li class="toctree-l2"><a class="reference internal" href="../classification/plot_digits_classification.html">Recognizing hand-written digits</a></li>
</ul>
</details></li>
<li class="toctree-l1 has-children"><a class="reference internal" href="../cluster/index.html">Clustering</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="../cluster/plot_kmeans_digits.html">A demo of K-Means clustering on the handwritten digits data</a></li>
<li class="toctree-l2"><a class="reference internal" href="../cluster/plot_coin_ward_segmentation.html">A demo of structured Ward hierarchical clustering on an image of coins</a></li>
<li class="toctree-l2"><a class="reference internal" href="../cluster/plot_mean_shift.html">A demo of the mean-shift clustering algorithm</a></li>
<li class="toctree-l2"><a class="reference internal" href="../cluster/plot_adjusted_for_chance_measures.html">Adjustment for chance in clustering performance evaluation</a></li>
<li class="toctree-l2"><a class="reference internal" href="../cluster/plot_agglomerative_clustering.html">Agglomerative clustering with and without structure</a></li>
<li class="toctree-l2"><a class="reference internal" href="../cluster/plot_agglomerative_clustering_metrics.html">Agglomerative clustering with different metrics</a></li>
<li class="toctree-l2"><a class="reference internal" href="../cluster/plot_kmeans_plusplus.html">An example of K-Means++ initialization</a></li>
<li class="toctree-l2"><a class="reference internal" href="../cluster/plot_bisect_kmeans.html">Bisecting K-Means and Regular K-Means Performance Comparison</a></li>
<li class="toctree-l2"><a class="reference internal" href="../cluster/plot_birch_vs_minibatchkmeans.html">Compare BIRCH and MiniBatchKMeans</a></li>
<li class="toctree-l2"><a class="reference internal" href="../cluster/plot_cluster_comparison.html">Comparing different clustering algorithms on toy datasets</a></li>
<li class="toctree-l2"><a class="reference internal" href="../cluster/plot_linkage_comparison.html">Comparing different hierarchical linkage methods on toy datasets</a></li>
<li class="toctree-l2"><a class="reference internal" href="../cluster/plot_mini_batch_kmeans.html">Comparison of the K-Means and MiniBatchKMeans clustering algorithms</a></li>
<li class="toctree-l2"><a class="reference internal" href="../cluster/plot_dbscan.html">Demo of DBSCAN clustering algorithm</a></li>
<li class="toctree-l2"><a class="reference internal" href="../cluster/plot_hdbscan.html">Demo of HDBSCAN clustering algorithm</a></li>
<li class="toctree-l2"><a class="reference internal" href="../cluster/plot_optics.html">Demo of OPTICS clustering algorithm</a></li>
<li class="toctree-l2"><a class="reference internal" href="../cluster/plot_affinity_propagation.html">Demo of affinity propagation clustering algorithm</a></li>
<li class="toctree-l2"><a class="reference internal" href="../cluster/plot_kmeans_assumptions.html">Demonstration of k-means assumptions</a></li>
<li class="toctree-l2"><a class="reference internal" href="../cluster/plot_kmeans_stability_low_dim_dense.html">Empirical evaluation of the impact of k-means initialization</a></li>
<li class="toctree-l2"><a class="reference internal" href="../cluster/plot_digits_agglomeration.html">Feature agglomeration</a></li>
<li class="toctree-l2"><a class="reference internal" href="../cluster/plot_feature_agglomeration_vs_univariate_selection.html">Feature agglomeration vs. univariate selection</a></li>
<li class="toctree-l2"><a class="reference internal" href="../cluster/plot_ward_structured_vs_unstructured.html">Hierarchical clustering: structured vs unstructured ward</a></li>
<li class="toctree-l2"><a class="reference internal" href="../cluster/plot_inductive_clustering.html">Inductive Clustering</a></li>
<li class="toctree-l2"><a class="reference internal" href="../cluster/plot_dict_face_patches.html">Online learning of a dictionary of parts of faces</a></li>
<li class="toctree-l2"><a class="reference internal" href="../cluster/plot_agglomerative_dendrogram.html">Plot Hierarchical Clustering Dendrogram</a></li>
<li class="toctree-l2"><a class="reference internal" href="../cluster/plot_coin_segmentation.html">Segmenting the picture of greek coins in regions</a></li>
<li class="toctree-l2"><a class="reference internal" href="../cluster/plot_kmeans_silhouette_analysis.html">Selecting the number of clusters with silhouette analysis on KMeans clustering</a></li>
<li class="toctree-l2"><a class="reference internal" href="../cluster/plot_segmentation_toy.html">Spectral clustering for image segmentation</a></li>
<li class="toctree-l2"><a class="reference internal" href="../cluster/plot_digits_linkage.html">Various Agglomerative Clustering on a 2D embedding of digits</a></li>
<li class="toctree-l2"><a class="reference internal" href="../cluster/plot_face_compress.html">Vector Quantization Example</a></li>
</ul>
</details></li>
<li class="toctree-l1 has-children"><a class="reference internal" href="../covariance/index.html">Covariance estimation</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="../covariance/plot_lw_vs_oas.html">Ledoit-Wolf vs OAS estimation</a></li>
<li class="toctree-l2"><a class="reference internal" href="../covariance/plot_mahalanobis_distances.html">Robust covariance estimation and Mahalanobis distances relevance</a></li>
<li class="toctree-l2"><a class="reference internal" href="../covariance/plot_robust_vs_empirical_covariance.html">Robust vs Empirical covariance estimate</a></li>
<li class="toctree-l2"><a class="reference internal" href="../covariance/plot_covariance_estimation.html">Shrinkage covariance estimation: LedoitWolf vs OAS and max-likelihood</a></li>
<li class="toctree-l2"><a class="reference internal" href="../covariance/plot_sparse_cov.html">Sparse inverse covariance estimation</a></li>
</ul>
</details></li>
<li class="toctree-l1 has-children"><a class="reference internal" href="../cross_decomposition/index.html">Cross decomposition</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_decomposition/plot_compare_cross_decomposition.html">Compare cross decomposition methods</a></li>
<li class="toctree-l2"><a class="reference internal" href="../cross_decomposition/plot_pcr_vs_pls.html">Principal Component Regression vs Partial Least Squares Regression</a></li>
</ul>
</details></li>
<li class="toctree-l1 has-children"><a class="reference internal" href="../datasets/index.html">Dataset examples</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/plot_random_multilabel_dataset.html">Plot randomly generated multilabel dataset</a></li>
</ul>
</details></li>
<li class="toctree-l1 has-children"><a class="reference internal" href="../tree/index.html">Decision Trees</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="../tree/plot_tree_regression.html">Decision Tree Regression</a></li>
<li class="toctree-l2"><a class="reference internal" href="../tree/plot_iris_dtc.html">Plot the decision surface of decision trees trained on the iris dataset</a></li>
<li class="toctree-l2"><a class="reference internal" href="../tree/plot_cost_complexity_pruning.html">Post pruning decision trees with cost complexity pruning</a></li>
<li class="toctree-l2"><a class="reference internal" href="../tree/plot_unveil_tree_structure.html">Understanding the decision tree structure</a></li>
</ul>
</details></li>
<li class="toctree-l1 has-children"><a class="reference internal" href="../decomposition/index.html">Decomposition</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="../decomposition/plot_ica_blind_source_separation.html">Blind source separation using FastICA</a></li>
<li class="toctree-l2"><a class="reference internal" href="../decomposition/plot_pca_vs_lda.html">Comparison of LDA and PCA 2D projection of Iris dataset</a></li>
<li class="toctree-l2"><a class="reference internal" href="../decomposition/plot_faces_decomposition.html">Faces dataset decompositions</a></li>
<li class="toctree-l2"><a class="reference internal" href="../decomposition/plot_varimax_fa.html">Factor Analysis (with rotation) to visualize patterns</a></li>
<li class="toctree-l2"><a class="reference internal" href="../decomposition/plot_ica_vs_pca.html">FastICA on 2D point clouds</a></li>
<li class="toctree-l2"><a class="reference internal" href="../decomposition/plot_image_denoising.html">Image denoising using dictionary learning</a></li>
<li class="toctree-l2"><a class="reference internal" href="../decomposition/plot_incremental_pca.html">Incremental PCA</a></li>
<li class="toctree-l2"><a class="reference internal" href="../decomposition/plot_kernel_pca.html">Kernel PCA</a></li>
<li class="toctree-l2"><a class="reference internal" href="../decomposition/plot_pca_vs_fa_model_selection.html">Model selection with Probabilistic PCA and Factor Analysis (FA)</a></li>
<li class="toctree-l2"><a class="reference internal" href="../decomposition/plot_pca_iris.html">Principal Component Analysis (PCA) on Iris Dataset</a></li>
<li class="toctree-l2"><a class="reference internal" href="../decomposition/plot_sparse_coding.html">Sparse coding with a precomputed dictionary</a></li>
</ul>
</details></li>
<li class="toctree-l1 has-children"><a class="reference internal" href="../developing_estimators/index.html">Developing Estimators</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="../developing_estimators/sklearn_is_fitted.html"><code class="docutils literal notranslate"><span class="pre">__sklearn_is_fitted__</span></code> as Developer API</a></li>
</ul>
</details></li>
<li class="toctree-l1 has-children"><a class="reference internal" href="../ensemble/index.html">Ensemble methods</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="../ensemble/plot_gradient_boosting_categorical.html">Categorical Feature Support in Gradient Boosting</a></li>
<li class="toctree-l2"><a class="reference internal" href="../ensemble/plot_stack_predictors.html">Combine predictors using stacking</a></li>
<li class="toctree-l2"><a class="reference internal" href="../ensemble/plot_forest_hist_grad_boosting_comparison.html">Comparing Random Forests and Histogram Gradient Boosting models</a></li>
<li class="toctree-l2"><a class="reference internal" href="../ensemble/plot_random_forest_regression_multioutput.html">Comparing random forests and the multi-output meta estimator</a></li>
<li class="toctree-l2"><a class="reference internal" href="../ensemble/plot_adaboost_regression.html">Decision Tree Regression with AdaBoost</a></li>
<li class="toctree-l2"><a class="reference internal" href="../ensemble/plot_gradient_boosting_early_stopping.html">Early stopping in Gradient Boosting</a></li>
<li class="toctree-l2"><a class="reference internal" href="../ensemble/plot_forest_importances.html">Feature importances with a forest of trees</a></li>
<li class="toctree-l2"><a class="reference internal" href="../ensemble/plot_feature_transformation.html">Feature transformations with ensembles of trees</a></li>
<li class="toctree-l2"><a class="reference internal" href="../ensemble/plot_hgbt_regression.html">Features in Histogram Gradient Boosting Trees</a></li>
<li class="toctree-l2"><a class="reference internal" href="../ensemble/plot_gradient_boosting_oob.html">Gradient Boosting Out-of-Bag estimates</a></li>
<li class="toctree-l2"><a class="reference internal" href="../ensemble/plot_gradient_boosting_regression.html">Gradient Boosting regression</a></li>
<li class="toctree-l2"><a class="reference internal" href="../ensemble/plot_gradient_boosting_regularization.html">Gradient Boosting regularization</a></li>
<li class="toctree-l2"><a class="reference internal" href="../ensemble/plot_random_forest_embedding.html">Hashing feature transformation using Totally Random Trees</a></li>
<li class="toctree-l2"><a class="reference internal" href="../ensemble/plot_isolation_forest.html">IsolationForest example</a></li>
<li class="toctree-l2"><a class="reference internal" href="../ensemble/plot_monotonic_constraints.html">Monotonic Constraints</a></li>
<li class="toctree-l2"><a class="reference internal" href="../ensemble/plot_adaboost_multiclass.html">Multi-class AdaBoosted Decision Trees</a></li>
<li class="toctree-l2"><a class="reference internal" href="../ensemble/plot_ensemble_oob.html">OOB Errors for Random Forests</a></li>
<li class="toctree-l2"><a class="reference internal" href="../ensemble/plot_voting_probas.html">Plot class probabilities calculated by the VotingClassifier</a></li>
<li class="toctree-l2"><a class="reference internal" href="../ensemble/plot_voting_regressor.html">Plot individual and voting regression predictions</a></li>
<li class="toctree-l2"><a class="reference internal" href="../ensemble/plot_voting_decision_regions.html">Plot the decision boundaries of a VotingClassifier</a></li>
<li class="toctree-l2"><a class="reference internal" href="../ensemble/plot_forest_iris.html">Plot the decision surfaces of ensembles of trees on the iris dataset</a></li>
<li class="toctree-l2"><a class="reference internal" href="../ensemble/plot_gradient_boosting_quantile.html">Prediction Intervals for Gradient Boosting Regression</a></li>
<li class="toctree-l2"><a class="reference internal" href="../ensemble/plot_bias_variance.html">Single estimator versus bagging: bias-variance decomposition</a></li>
<li class="toctree-l2"><a class="reference internal" href="../ensemble/plot_adaboost_twoclass.html">Two-class AdaBoost</a></li>
</ul>
</details></li>
<li class="toctree-l1 has-children"><a class="reference internal" href="../applications/index.html">Examples based on real world datasets</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="../applications/plot_tomography_l1_reconstruction.html">Compressive sensing: tomography reconstruction with L1 prior (Lasso)</a></li>
<li class="toctree-l2"><a class="reference internal" href="../applications/plot_face_recognition.html">Faces recognition example using eigenfaces and SVMs</a></li>
<li class="toctree-l2"><a class="reference internal" href="../applications/plot_digits_denoising.html">Image denoising using kernel PCA</a></li>
<li class="toctree-l2"><a class="reference internal" href="../applications/plot_time_series_lagged_features.html">Lagged features for time series forecasting</a></li>
<li class="toctree-l2"><a class="reference internal" href="../applications/plot_model_complexity_influence.html">Model Complexity Influence</a></li>
<li class="toctree-l2"><a class="reference internal" href="../applications/plot_out_of_core_classification.html">Out-of-core classification of text documents</a></li>
<li class="toctree-l2"><a class="reference internal" href="../applications/plot_outlier_detection_wine.html">Outlier detection on a real data set</a></li>
<li class="toctree-l2"><a class="reference internal" href="../applications/plot_prediction_latency.html">Prediction Latency</a></li>
<li class="toctree-l2"><a class="reference internal" href="../applications/plot_species_distribution_modeling.html">Species distribution modeling</a></li>
<li class="toctree-l2"><a class="reference internal" href="../applications/plot_cyclical_feature_engineering.html">Time-related feature engineering</a></li>
<li class="toctree-l2"><a class="reference internal" href="../applications/plot_topics_extraction_with_nmf_lda.html">Topic extraction with Non-negative Matrix Factorization and Latent Dirichlet Allocation</a></li>
<li class="toctree-l2"><a class="reference internal" href="../applications/plot_stock_market.html">Visualizing the stock market structure</a></li>
<li class="toctree-l2"><a class="reference internal" href="../applications/wikipedia_principal_eigenvector.html">Wikipedia principal eigenvector</a></li>
</ul>
</details></li>
<li class="toctree-l1 has-children"><a class="reference internal" href="../feature_selection/index.html">Feature Selection</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="../feature_selection/plot_f_test_vs_mi.html">Comparison of F-test and mutual information</a></li>
<li class="toctree-l2"><a class="reference internal" href="../feature_selection/plot_select_from_model_diabetes.html">Model-based and sequential feature selection</a></li>
<li class="toctree-l2"><a class="reference internal" href="../feature_selection/plot_feature_selection_pipeline.html">Pipeline ANOVA SVM</a></li>
<li class="toctree-l2"><a class="reference internal" href="../feature_selection/plot_rfe_digits.html">Recursive feature elimination</a></li>
<li class="toctree-l2"><a class="reference internal" href="../feature_selection/plot_rfe_with_cross_validation.html">Recursive feature elimination with cross-validation</a></li>
<li class="toctree-l2"><a class="reference internal" href="../feature_selection/plot_feature_selection.html">Univariate Feature Selection</a></li>
</ul>
</details></li>
<li class="toctree-l1 has-children"><a class="reference internal" href="../frozen/index.html">Frozen Estimators</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="../frozen/plot_frozen_examples.html">Examples of Using <code class="docutils literal notranslate"><span class="pre">FrozenEstimator</span></code></a></li>
</ul>
</details></li>
<li class="toctree-l1 has-children"><a class="reference internal" href="../mixture/index.html">Gaussian Mixture Models</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/plot_concentration_prior.html">Concentration Prior Type Analysis of Variation Bayesian Gaussian Mixture</a></li>
<li class="toctree-l2"><a class="reference internal" href="../mixture/plot_gmm_pdf.html">Density Estimation for a Gaussian mixture</a></li>
<li class="toctree-l2"><a class="reference internal" href="../mixture/plot_gmm_init.html">GMM Initialization Methods</a></li>
<li class="toctree-l2"><a class="reference internal" href="../mixture/plot_gmm_covariances.html">GMM covariances</a></li>
<li class="toctree-l2"><a class="reference internal" href="../mixture/plot_gmm.html">Gaussian Mixture Model Ellipsoids</a></li>
<li class="toctree-l2"><a class="reference internal" href="../mixture/plot_gmm_selection.html">Gaussian Mixture Model Selection</a></li>
<li class="toctree-l2"><a class="reference internal" href="../mixture/plot_gmm_sin.html">Gaussian Mixture Model Sine Curve</a></li>
</ul>
</details></li>
<li class="toctree-l1 has-children"><a class="reference internal" href="../gaussian_process/index.html">Gaussian Process for Machine 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="../gaussian_process/plot_gpr_noisy.html">Ability of Gaussian process regression (GPR) to estimate data noise-level</a></li>
<li class="toctree-l2"><a class="reference internal" href="../gaussian_process/plot_compare_gpr_krr.html">Comparison of kernel ridge and Gaussian process regression</a></li>
<li class="toctree-l2"><a class="reference internal" href="../gaussian_process/plot_gpr_co2.html">Forecasting of CO2 level on Mona Loa dataset using Gaussian process regression (GPR)</a></li>
<li class="toctree-l2"><a class="reference internal" href="../gaussian_process/plot_gpr_noisy_targets.html">Gaussian Processes regression: basic introductory example</a></li>
<li class="toctree-l2"><a class="reference internal" href="../gaussian_process/plot_gpc_iris.html">Gaussian process classification (GPC) on iris dataset</a></li>
<li class="toctree-l2"><a class="reference internal" href="../gaussian_process/plot_gpr_on_structured_data.html">Gaussian processes on discrete data structures</a></li>
<li class="toctree-l2"><a class="reference internal" href="../gaussian_process/plot_gpc_xor.html">Illustration of Gaussian process classification (GPC) on the XOR dataset</a></li>
<li class="toctree-l2"><a class="reference internal" href="../gaussian_process/plot_gpr_prior_posterior.html">Illustration of prior and posterior Gaussian process for different kernels</a></li>
<li class="toctree-l2"><a class="reference internal" href="../gaussian_process/plot_gpc_isoprobability.html">Iso-probability lines for Gaussian Processes classification (GPC)</a></li>
<li class="toctree-l2"><a class="reference internal" href="../gaussian_process/plot_gpc.html">Probabilistic predictions with Gaussian process classification (GPC)</a></li>
</ul>
</details></li>
<li class="toctree-l1 has-children"><a class="reference internal" href="../linear_model/index.html">Generalized Linear Models</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="../linear_model/plot_ard.html">Comparing Linear Bayesian Regressors</a></li>
<li class="toctree-l2"><a class="reference internal" href="../linear_model/plot_bayesian_ridge_curvefit.html">Curve Fitting with Bayesian Ridge Regression</a></li>
<li class="toctree-l2"><a class="reference internal" href="../linear_model/plot_logistic_multinomial.html">Decision Boundaries of Multinomial and One-vs-Rest Logistic Regression</a></li>
<li class="toctree-l2"><a class="reference internal" href="../linear_model/plot_sgd_early_stopping.html">Early stopping of Stochastic Gradient Descent</a></li>
<li class="toctree-l2"><a class="reference internal" href="../linear_model/plot_elastic_net_precomputed_gram_matrix_with_weighted_samples.html">Fitting an Elastic Net with a precomputed Gram Matrix and Weighted Samples</a></li>
<li class="toctree-l2"><a class="reference internal" href="../linear_model/plot_huber_vs_ridge.html">HuberRegressor vs Ridge on dataset with strong outliers</a></li>
<li class="toctree-l2"><a class="reference internal" href="../linear_model/plot_multi_task_lasso_support.html">Joint feature selection with multi-task Lasso</a></li>
<li class="toctree-l2"><a class="reference internal" href="../linear_model/plot_logistic_l1_l2_sparsity.html">L1 Penalty and Sparsity in Logistic Regression</a></li>
<li class="toctree-l2"><a class="reference internal" href="../linear_model/plot_lasso_and_elasticnet.html">L1-based models for Sparse Signals</a></li>
<li class="toctree-l2"><a class="reference internal" href="../linear_model/plot_lasso_lars_ic.html">Lasso model selection via information criteria</a></li>
<li class="toctree-l2"><a class="reference internal" href="../linear_model/plot_lasso_model_selection.html">Lasso model selection: AIC-BIC / cross-validation</a></li>
<li class="toctree-l2"><a class="reference internal" href="../linear_model/plot_lasso_dense_vs_sparse_data.html">Lasso on dense and sparse data</a></li>
<li class="toctree-l2"><a class="reference internal" href="../linear_model/plot_lasso_lasso_lars_elasticnet_path.html">Lasso, Lasso-LARS, and Elastic Net paths</a></li>
<li class="toctree-l2"><a class="reference internal" href="../linear_model/plot_logistic.html">Logistic function</a></li>
<li class="toctree-l2"><a class="reference internal" href="../linear_model/plot_sparse_logistic_regression_mnist.html">MNIST classification using multinomial logistic + L1</a></li>
<li class="toctree-l2"><a class="reference internal" href="../linear_model/plot_sparse_logistic_regression_20newsgroups.html">Multiclass sparse logistic regression on 20newgroups</a></li>
<li class="toctree-l2"><a class="reference internal" href="../linear_model/plot_nnls.html">Non-negative least squares</a></li>
<li class="toctree-l2"><a class="reference internal" href="../linear_model/plot_sgdocsvm_vs_ocsvm.html">One-Class SVM versus One-Class SVM using Stochastic Gradient Descent</a></li>
<li class="toctree-l2"><a class="reference internal" href="../linear_model/plot_ols_ridge.html">Ordinary Least Squares and Ridge Regression</a></li>
<li class="toctree-l2"><a class="reference internal" href="../linear_model/plot_omp.html">Orthogonal Matching Pursuit</a></li>
<li class="toctree-l2"><a class="reference internal" href="../linear_model/plot_ridge_path.html">Plot Ridge coefficients as a function of the regularization</a></li>
<li class="toctree-l2"><a class="reference internal" href="../linear_model/plot_sgd_iris.html">Plot multi-class SGD on the iris dataset</a></li>
<li class="toctree-l2"><a class="reference internal" href="../linear_model/plot_poisson_regression_non_normal_loss.html">Poisson regression and non-normal loss</a></li>
<li class="toctree-l2"><a class="reference internal" href="../linear_model/plot_polynomial_interpolation.html">Polynomial and Spline interpolation</a></li>
<li class="toctree-l2"><a class="reference internal" href="../linear_model/plot_quantile_regression.html">Quantile regression</a></li>
<li class="toctree-l2"><a class="reference internal" href="../linear_model/plot_logistic_path.html">Regularization path of L1- Logistic Regression</a></li>
<li class="toctree-l2"><a class="reference internal" href="../linear_model/plot_ridge_coeffs.html">Ridge coefficients as a function of the L2 Regularization</a></li>
<li class="toctree-l2"><a class="reference internal" href="../linear_model/plot_robust_fit.html">Robust linear estimator fitting</a></li>
<li class="toctree-l2"><a class="reference internal" href="../linear_model/plot_ransac.html">Robust linear model estimation using RANSAC</a></li>
<li class="toctree-l2"><a class="reference internal" href="../linear_model/plot_sgd_separating_hyperplane.html">SGD: Maximum margin separating hyperplane</a></li>
<li class="toctree-l2"><a class="reference internal" href="../linear_model/plot_sgd_penalties.html">SGD: Penalties</a></li>
<li class="toctree-l2"><a class="reference internal" href="../linear_model/plot_sgd_weighted_samples.html">SGD: Weighted samples</a></li>
<li class="toctree-l2"><a class="reference internal" href="../linear_model/plot_sgd_loss_functions.html">SGD: convex loss functions</a></li>
<li class="toctree-l2"><a class="reference internal" href="../linear_model/plot_theilsen.html">Theil-Sen Regression</a></li>
<li class="toctree-l2"><a class="reference internal" href="../linear_model/plot_tweedie_regression_insurance_claims.html">Tweedie regression on insurance claims</a></li>
</ul>
</details></li>
<li class="toctree-l1 has-children"><a class="reference internal" href="../inspection/index.html">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="../inspection/plot_linear_model_coefficient_interpretation.html">Common pitfalls in the interpretation of coefficients of linear models</a></li>
<li class="toctree-l2"><a class="reference internal" href="../inspection/plot_causal_interpretation.html">Failure of Machine Learning to infer causal effects</a></li>
<li class="toctree-l2"><a class="reference internal" href="../inspection/plot_partial_dependence.html">Partial Dependence and Individual Conditional Expectation Plots</a></li>
<li class="toctree-l2"><a class="reference internal" href="../inspection/plot_permutation_importance.html">Permutation Importance vs Random Forest Feature Importance (MDI)</a></li>
<li class="toctree-l2"><a class="reference internal" href="../inspection/plot_permutation_importance_multicollinear.html">Permutation Importance with Multicollinear or Correlated Features</a></li>
</ul>
</details></li>
<li class="toctree-l1 has-children"><a class="reference internal" href="../kernel_approximation/index.html">Kernel Approximation</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="../kernel_approximation/plot_scalable_poly_kernels.html">Scalable learning with polynomial kernel approximation</a></li>
</ul>
</details></li>
<li class="toctree-l1 has-children"><a class="reference internal" href="../manifold/index.html">Manifold 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="../manifold/plot_compare_methods.html">Comparison of Manifold Learning methods</a></li>
<li class="toctree-l2"><a class="reference internal" href="../manifold/plot_manifold_sphere.html">Manifold Learning methods on a severed sphere</a></li>
<li class="toctree-l2"><a class="reference internal" href="../manifold/plot_lle_digits.html">Manifold learning on handwritten digits: Locally Linear Embedding, Isomap…</a></li>
<li class="toctree-l2"><a class="reference internal" href="../manifold/plot_mds.html">Multi-dimensional scaling</a></li>
<li class="toctree-l2"><a class="reference internal" href="../manifold/plot_swissroll.html">Swiss Roll And Swiss-Hole Reduction</a></li>
<li class="toctree-l2"><a class="reference internal" href="../manifold/plot_t_sne_perplexity.html">t-SNE: The effect of various perplexity values on the shape</a></li>
</ul>
</details></li>
<li class="toctree-l1 has-children"><a class="reference internal" href="../miscellaneous/index.html">Miscellaneous</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="../miscellaneous/plot_partial_dependence_visualization_api.html">Advanced Plotting With Partial Dependence</a></li>
<li class="toctree-l2"><a class="reference internal" href="../miscellaneous/plot_anomaly_comparison.html">Comparing anomaly detection algorithms for outlier detection on toy datasets</a></li>
<li class="toctree-l2"><a class="reference internal" href="../miscellaneous/plot_kernel_ridge_regression.html">Comparison of kernel ridge regression and SVR</a></li>
<li class="toctree-l2"><a class="reference internal" href="../miscellaneous/plot_pipeline_display.html">Displaying Pipelines</a></li>
<li class="toctree-l2"><a class="reference internal" href="../miscellaneous/plot_estimator_representation.html">Displaying estimators and complex pipelines</a></li>
<li class="toctree-l2"><a class="reference internal" href="../miscellaneous/plot_outlier_detection_bench.html">Evaluation of outlier detection estimators</a></li>
<li class="toctree-l2"><a class="reference internal" href="../miscellaneous/plot_kernel_approximation.html">Explicit feature map approximation for RBF kernels</a></li>
<li class="toctree-l2"><a class="reference internal" href="../miscellaneous/plot_multioutput_face_completion.html">Face completion with a multi-output estimators</a></li>
<li class="toctree-l2"><a class="reference internal" href="../miscellaneous/plot_set_output.html">Introducing the <code class="docutils literal notranslate"><span class="pre">set_output</span></code> API</a></li>
<li class="toctree-l2"><a class="reference internal" href="../miscellaneous/plot_isotonic_regression.html">Isotonic Regression</a></li>
<li class="toctree-l2"><a class="reference internal" href="../miscellaneous/plot_metadata_routing.html">Metadata Routing</a></li>
<li class="toctree-l2"><a class="reference internal" href="../miscellaneous/plot_multilabel.html">Multilabel classification</a></li>
<li class="toctree-l2"><a class="reference internal" href="../miscellaneous/plot_roc_curve_visualization_api.html">ROC Curve with Visualization API</a></li>
<li class="toctree-l2"><a class="reference internal" href="../miscellaneous/plot_johnson_lindenstrauss_bound.html">The Johnson-Lindenstrauss bound for embedding with random projections</a></li>
<li class="toctree-l2"><a class="reference internal" href="../miscellaneous/plot_display_object_visualization.html">Visualizations with Display Objects</a></li>
</ul>
</details></li>
<li class="toctree-l1 has-children"><a class="reference internal" href="../impute/index.html">Missing Value Imputation</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="../impute/plot_missing_values.html">Imputing missing values before building an estimator</a></li>
<li class="toctree-l2"><a class="reference internal" href="../impute/plot_iterative_imputer_variants_comparison.html">Imputing missing values with variants of IterativeImputer</a></li>
</ul>
</details></li>
<li class="toctree-l1 has-children"><a class="reference internal" href="../model_selection/index.html">Model Selection</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="../model_selection/plot_grid_search_refit_callable.html">Balance model complexity and cross-validated score</a></li>
<li class="toctree-l2"><a class="reference internal" href="../model_selection/plot_likelihood_ratios.html">Class Likelihood Ratios to measure classification performance</a></li>
<li class="toctree-l2"><a class="reference internal" href="../model_selection/plot_randomized_search.html">Comparing randomized search and grid search for hyperparameter estimation</a></li>
<li class="toctree-l2"><a class="reference internal" href="../model_selection/plot_successive_halving_heatmap.html">Comparison between grid search and successive halving</a></li>
<li class="toctree-l2"><a class="reference internal" href="../model_selection/plot_confusion_matrix.html">Confusion matrix</a></li>
<li class="toctree-l2"><a class="reference internal" href="../model_selection/plot_grid_search_digits.html">Custom refit strategy of a grid search with cross-validation</a></li>
<li class="toctree-l2"><a class="reference internal" href="../model_selection/plot_multi_metric_evaluation.html">Demonstration of multi-metric evaluation on cross_val_score and GridSearchCV</a></li>
<li class="toctree-l2"><a class="reference internal" href="../model_selection/plot_det.html">Detection error tradeoff (DET) curve</a></li>
<li class="toctree-l2"><a class="reference internal" href="../model_selection/plot_train_error_vs_test_error.html">Effect of model regularization on training and test error</a></li>
<li class="toctree-l2"><a class="reference internal" href="../model_selection/plot_roc.html">Multiclass Receiver Operating Characteristic (ROC)</a></li>
<li class="toctree-l2"><a class="reference internal" href="../model_selection/plot_nested_cross_validation_iris.html">Nested versus non-nested cross-validation</a></li>
<li class="toctree-l2"><a class="reference internal" href="../model_selection/plot_cv_predict.html">Plotting Cross-Validated Predictions</a></li>
<li class="toctree-l2"><a class="reference internal" href="../model_selection/plot_learning_curve.html">Plotting Learning Curves and Checking Models’ Scalability</a></li>
<li class="toctree-l2"><a class="reference internal" href="../model_selection/plot_tuned_decision_threshold.html">Post-hoc tuning the cut-off point of decision function</a></li>
<li class="toctree-l2"><a class="reference internal" href="../model_selection/plot_cost_sensitive_learning.html">Post-tuning the decision threshold for cost-sensitive learning</a></li>
<li class="toctree-l2"><a class="reference internal" href="../model_selection/plot_precision_recall.html">Precision-Recall</a></li>
<li class="toctree-l2"><a class="reference internal" href="../model_selection/plot_roc_crossval.html">Receiver Operating Characteristic (ROC) with cross validation</a></li>
<li class="toctree-l2"><a class="reference internal" href="../model_selection/plot_grid_search_text_feature_extraction.html">Sample pipeline for text feature extraction and evaluation</a></li>
<li class="toctree-l2"><a class="reference internal" href="../model_selection/plot_grid_search_stats.html">Statistical comparison of models using grid search</a></li>
<li class="toctree-l2"><a class="reference internal" href="../model_selection/plot_successive_halving_iterations.html">Successive Halving Iterations</a></li>
<li class="toctree-l2"><a class="reference internal" href="../model_selection/plot_permutation_tests_for_classification.html">Test with permutations the significance of a classification score</a></li>
<li class="toctree-l2"><a class="reference internal" href="../model_selection/plot_underfitting_overfitting.html">Underfitting vs. Overfitting</a></li>
<li class="toctree-l2"><a class="reference internal" href="../model_selection/plot_cv_indices.html">Visualizing cross-validation behavior in scikit-learn</a></li>
</ul>
</details></li>
<li class="toctree-l1 has-children"><a class="reference internal" href="../multiclass/index.html">Multiclass methods</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="../multiclass/plot_multiclass_overview.html">Overview of multiclass training meta-estimators</a></li>
</ul>
</details></li>
<li class="toctree-l1 has-children"><a class="reference internal" href="../multioutput/index.html">Multioutput methods</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="../multioutput/plot_classifier_chain_yeast.html">Multilabel classification using a classifier chain</a></li>
</ul>
</details></li>
<li class="toctree-l1 has-children"><a class="reference internal" href="../neighbors/index.html">Nearest Neighbors</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="../neighbors/approximate_nearest_neighbors.html">Approximate nearest neighbors in TSNE</a></li>
<li class="toctree-l2"><a class="reference internal" href="../neighbors/plot_caching_nearest_neighbors.html">Caching nearest neighbors</a></li>
<li class="toctree-l2"><a class="reference internal" href="../neighbors/plot_nca_classification.html">Comparing Nearest Neighbors with and without Neighborhood Components Analysis</a></li>
<li class="toctree-l2"><a class="reference internal" href="../neighbors/plot_nca_dim_reduction.html">Dimensionality Reduction with Neighborhood Components Analysis</a></li>
<li class="toctree-l2"><a class="reference internal" href="../neighbors/plot_species_kde.html">Kernel Density Estimate of Species Distributions</a></li>
<li class="toctree-l2"><a class="reference internal" href="../neighbors/plot_digits_kde_sampling.html">Kernel Density Estimation</a></li>
<li class="toctree-l2"><a class="reference internal" href="../neighbors/plot_nearest_centroid.html">Nearest Centroid Classification</a></li>
<li class="toctree-l2"><a class="reference internal" href="../neighbors/plot_classification.html">Nearest Neighbors Classification</a></li>
<li class="toctree-l2"><a class="reference internal" href="../neighbors/plot_regression.html">Nearest Neighbors regression</a></li>
<li class="toctree-l2"><a class="reference internal" href="../neighbors/plot_nca_illustration.html">Neighborhood Components Analysis Illustration</a></li>
<li class="toctree-l2"><a class="reference internal" href="../neighbors/plot_lof_novelty_detection.html">Novelty detection with Local Outlier Factor (LOF)</a></li>
<li class="toctree-l2"><a class="reference internal" href="../neighbors/plot_lof_outlier_detection.html">Outlier detection with Local Outlier Factor (LOF)</a></li>
<li class="toctree-l2"><a class="reference internal" href="../neighbors/plot_kde_1d.html">Simple 1D Kernel Density Estimation</a></li>
</ul>
</details></li>
<li class="toctree-l1 has-children"><a class="reference internal" href="../neural_networks/index.html">Neural Networks</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="../neural_networks/plot_mlp_training_curves.html">Compare Stochastic learning strategies for MLPClassifier</a></li>
<li class="toctree-l2"><a class="reference internal" href="../neural_networks/plot_rbm_logistic_classification.html">Restricted Boltzmann Machine features for digit classification</a></li>
<li class="toctree-l2"><a class="reference internal" href="../neural_networks/plot_mlp_alpha.html">Varying regularization in Multi-layer Perceptron</a></li>
<li class="toctree-l2"><a class="reference internal" href="../neural_networks/plot_mnist_filters.html">Visualization of MLP weights on MNIST</a></li>
</ul>
</details></li>
<li class="toctree-l1 has-children"><a class="reference internal" href="../compose/index.html">Pipelines and composite estimators</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/plot_column_transformer.html">Column Transformer with Heterogeneous Data Sources</a></li>
<li class="toctree-l2"><a class="reference internal" href="../compose/plot_column_transformer_mixed_types.html">Column Transformer with Mixed Types</a></li>
<li class="toctree-l2"><a class="reference internal" href="../compose/plot_feature_union.html">Concatenating multiple feature extraction methods</a></li>
<li class="toctree-l2"><a class="reference internal" href="../compose/plot_transformed_target.html">Effect of transforming the targets in regression model</a></li>
<li class="toctree-l2"><a class="reference internal" href="../compose/plot_digits_pipe.html">Pipelining: chaining a PCA and a logistic regression</a></li>
<li class="toctree-l2"><a class="reference internal" href="../compose/plot_compare_reduction.html">Selecting dimensionality reduction with Pipeline and GridSearchCV</a></li>
</ul>
</details></li>
<li class="toctree-l1 has-children"><a class="reference internal" href="../preprocessing/index.html">Preprocessing</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="../preprocessing/plot_all_scaling.html">Compare the effect of different scalers on data with outliers</a></li>
<li class="toctree-l2"><a class="reference internal" href="../preprocessing/plot_target_encoder.html">Comparing Target Encoder with Other Encoders</a></li>
<li class="toctree-l2"><a class="reference internal" href="../preprocessing/plot_discretization_strategies.html">Demonstrating the different strategies of KBinsDiscretizer</a></li>
<li class="toctree-l2"><a class="reference internal" href="../preprocessing/plot_discretization_classification.html">Feature discretization</a></li>
<li class="toctree-l2"><a class="reference internal" href="../preprocessing/plot_scaling_importance.html">Importance of Feature Scaling</a></li>
<li class="toctree-l2"><a class="reference internal" href="../preprocessing/plot_map_data_to_normal.html">Map data to a normal distribution</a></li>
<li class="toctree-l2"><a class="reference internal" href="../preprocessing/plot_target_encoder_cross_val.html">Target Encoder’s Internal Cross fitting</a></li>
<li class="toctree-l2"><a class="reference internal" href="../preprocessing/plot_discretization.html">Using KBinsDiscretizer to discretize continuous features</a></li>
</ul>
</details></li>
<li class="toctree-l1 has-children"><a class="reference internal" href="../semi_supervised/index.html">Semi Supervised Classification</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="../semi_supervised/plot_semi_supervised_versus_svm_iris.html">Decision boundary of semi-supervised classifiers versus SVM on the Iris dataset</a></li>
<li class="toctree-l2"><a class="reference internal" href="../semi_supervised/plot_self_training_varying_threshold.html">Effect of varying threshold for self-training</a></li>
<li class="toctree-l2"><a class="reference internal" href="../semi_supervised/plot_label_propagation_digits_active_learning.html">Label Propagation digits active learning</a></li>
<li class="toctree-l2"><a class="reference internal" href="../semi_supervised/plot_label_propagation_digits.html">Label Propagation digits: Demonstrating performance</a></li>
<li class="toctree-l2"><a class="reference internal" href="../semi_supervised/plot_label_propagation_structure.html">Label Propagation learning a complex structure</a></li>
<li class="toctree-l2"><a class="reference internal" href="../semi_supervised/plot_semi_supervised_newsgroups.html">Semi-supervised Classification on a Text Dataset</a></li>
</ul>
</details></li>
<li class="toctree-l1 current active has-children"><a class="reference internal" href="index.html">Support Vector Machines</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="plot_oneclass.html">One-class SVM with non-linear kernel (RBF)</a></li>
<li class="toctree-l2 current active"><a class="current reference internal" href="#">Plot classification boundaries with different SVM Kernels</a></li>
<li class="toctree-l2"><a class="reference internal" href="plot_iris_svc.html">Plot different SVM classifiers in the iris dataset</a></li>
<li class="toctree-l2"><a class="reference internal" href="plot_linearsvc_support_vectors.html">Plot the support vectors in LinearSVC</a></li>
<li class="toctree-l2"><a class="reference internal" href="plot_rbf_parameters.html">RBF SVM parameters</a></li>
<li class="toctree-l2"><a class="reference internal" href="plot_svm_margin.html">SVM Margins Example</a></li>
<li class="toctree-l2"><a class="reference internal" href="plot_svm_tie_breaking.html">SVM Tie Breaking Example</a></li>
<li class="toctree-l2"><a class="reference internal" href="plot_custom_kernel.html">SVM with custom kernel</a></li>
<li class="toctree-l2"><a class="reference internal" href="plot_svm_anova.html">SVM-Anova: SVM with univariate feature selection</a></li>
<li class="toctree-l2"><a class="reference internal" href="plot_separating_hyperplane.html">SVM: Maximum margin separating hyperplane</a></li>
<li class="toctree-l2"><a class="reference internal" href="plot_separating_hyperplane_unbalanced.html">SVM: Separating hyperplane for unbalanced classes</a></li>
<li class="toctree-l2"><a class="reference internal" href="plot_weighted_samples.html">SVM: Weighted samples</a></li>
<li class="toctree-l2"><a class="reference internal" href="plot_svm_scale_c.html">Scaling the regularization parameter for SVCs</a></li>
<li class="toctree-l2"><a class="reference internal" href="plot_svm_regression.html">Support Vector Regression (SVR) using linear and non-linear kernels</a></li>
</ul>
</details></li>
<li class="toctree-l1 has-children"><a class="reference internal" href="../text/index.html">Working with text documents</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="../text/plot_document_classification_20newsgroups.html">Classification of text documents using sparse features</a></li>
<li class="toctree-l2"><a class="reference internal" href="../text/plot_document_clustering.html">Clustering text documents using k-means</a></li>
<li class="toctree-l2"><a class="reference internal" href="../text/plot_hashing_vs_dict_vectorizer.html">FeatureHasher and DictVectorizer Comparison</a></li>
</ul>
</details></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="../index.html" class="nav-link">Examples</a></li>
<li class="breadcrumb-item"><a href="index.html" class="nav-link">Support Vector Machines</a></li>
<li class="breadcrumb-item active" aria-current="page"><span class="ellipsis">Plot classification boundaries with different SVM Kernels</span></li>
</ul>
</nav>
</div>
</div>
</div>
</div>
<div id="searchbox"></div>
<article class="bd-article">
<div class="sphx-glr-download-link-note admonition note">
<p class="admonition-title">Note</p>
<p><a class="reference internal" href="#sphx-glr-download-auto-examples-svm-plot-svm-kernels-py"><span class="std std-ref">Go to the end</span></a>
to download the full example code. or to run this example in your browser via JupyterLite or Binder</p>
</div>
<section class="sphx-glr-example-title" id="plot-classification-boundaries-with-different-svm-kernels">
<span id="sphx-glr-auto-examples-svm-plot-svm-kernels-py"></span><h1>Plot classification boundaries with different SVM Kernels<a class="headerlink" href="#plot-classification-boundaries-with-different-svm-kernels" title="Link to this heading">#</a></h1>
<p>This example shows how different kernels in a <a class="reference internal" href="../../modules/generated/sklearn.svm.SVC.html#sklearn.svm.SVC" title="sklearn.svm.SVC"><code class="xref py py-class docutils literal notranslate"><span class="pre">SVC</span></code></a> (Support Vector
Classifier) influence the classification boundaries in a binary, two-dimensional
classification problem.</p>
<p>SVCs aim to find a hyperplane that effectively separates the classes in their training
data by maximizing the margin between the outermost data points of each class. This is
achieved by finding the best weight vector <span class="math notranslate nohighlight">\(w\)</span> that defines the decision boundary
hyperplane and minimizes the sum of hinge losses for misclassified samples, as measured
by the <a class="reference internal" href="../../modules/generated/sklearn.metrics.hinge_loss.html#sklearn.metrics.hinge_loss" title="sklearn.metrics.hinge_loss"><code class="xref py py-func docutils literal notranslate"><span class="pre">hinge_loss</span></code></a> function. By default, regularization is
applied with the parameter <code class="docutils literal notranslate"><span class="pre">C=1</span></code>, which allows for a certain degree of misclassification
tolerance.</p>
<p>If the data is not linearly separable in the original feature space, a non-linear kernel
parameter can be set. Depending on the kernel, the process involves adding new features
or transforming existing features to enrich and potentially add meaning to the data.
When a kernel other than <code class="docutils literal notranslate"><span class="pre">"linear"</span></code> is set, the SVC applies the <a class="reference external" href="https://en.wikipedia.org/wiki/Kernel_method#Mathematics:_the_kernel_trick">kernel trick</a>, which
computes the similarity between pairs of data points using the kernel function without
explicitly transforming the entire dataset. The kernel trick surpasses the otherwise
necessary matrix transformation of the whole dataset by only considering the relations
between all pairs of data points. The kernel function maps two vectors (each pair of
observations) to their similarity using their dot product.</p>
<p>The hyperplane can then be calculated using the kernel function as if the dataset were
represented in a higher-dimensional space. Using a kernel function instead of an
explicit matrix transformation improves performance, as the kernel function has a time
complexity of <span class="math notranslate nohighlight">\(O({n}^2)\)</span>, whereas matrix transformation scales according to the
specific transformation being applied.</p>
<p>In this example, we compare the most common kernel types of Support Vector Machines: the
linear kernel (<code class="docutils literal notranslate"><span class="pre">"linear"</span></code>), the polynomial kernel (<code class="docutils literal notranslate"><span class="pre">"poly"</span></code>), the radial basis function
kernel (<code class="docutils literal notranslate"><span class="pre">"rbf"</span></code>) and the sigmoid kernel (<code class="docutils literal notranslate"><span class="pre">"sigmoid"</span></code>).</p>
<div class="highlight-Python notranslate"><div class="highlight"><pre><span></span><span class="c1"># Authors: The scikit-learn developers</span>
<span class="c1"># SPDX-License-Identifier: BSD-3-Clause</span>
</pre></div>
</div>
<section id="creating-a-dataset">
<h2>Creating a dataset<a class="headerlink" href="#creating-a-dataset" title="Link to this heading">#</a></h2>
<p>We create a two-dimensional classification dataset with 16 samples and two classes. We
plot the samples with the colors matching their respective targets.</p>
<div class="highlight-Python notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span><span class="w"> </span><span class="nn">matplotlib.pyplot</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">plt</span>
<span class="kn">import</span><span class="w"> </span><span class="nn">numpy</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">np</span>
<span class="n">X</span> <span class="o">=</span> <a href="https://numpy.org/doc/stable/reference/generated/numpy.array.html#numpy.array" title="numpy.array" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-function"><span class="n">np</span><span class="o">.</span><span class="n">array</span></a><span class="p">(</span>
<span class="p">[</span>
<span class="p">[</span><span class="mf">0.4</span><span class="p">,</span> <span class="o">-</span><span class="mf">0.7</span><span class="p">],</span>
<span class="p">[</span><span class="o">-</span><span class="mf">1.5</span><span class="p">,</span> <span class="o">-</span><span class="mf">1.0</span><span class="p">],</span>
<span class="p">[</span><span class="o">-</span><span class="mf">1.4</span><span class="p">,</span> <span class="o">-</span><span class="mf">0.9</span><span class="p">],</span>
<span class="p">[</span><span class="o">-</span><span class="mf">1.3</span><span class="p">,</span> <span class="o">-</span><span class="mf">1.2</span><span class="p">],</span>
<span class="p">[</span><span class="o">-</span><span class="mf">1.1</span><span class="p">,</span> <span class="o">-</span><span class="mf">0.2</span><span class="p">],</span>
<span class="p">[</span><span class="o">-</span><span class="mf">1.2</span><span class="p">,</span> <span class="o">-</span><span class="mf">0.4</span><span class="p">],</span>
<span class="p">[</span><span class="o">-</span><span class="mf">0.5</span><span class="p">,</span> <span class="mf">1.2</span><span class="p">],</span>
<span class="p">[</span><span class="o">-</span><span class="mf">1.5</span><span class="p">,</span> <span class="mf">2.1</span><span class="p">],</span>
<span class="p">[</span><span class="mf">1.0</span><span class="p">,</span> <span class="mf">1.0</span><span class="p">],</span>
<span class="p">[</span><span class="mf">1.3</span><span class="p">,</span> <span class="mf">0.8</span><span class="p">],</span>
<span class="p">[</span><span class="mf">1.2</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">],</span>
<span class="p">[</span><span class="mf">0.2</span><span class="p">,</span> <span class="o">-</span><span class="mf">2.0</span><span class="p">],</span>
<span class="p">[</span><span class="mf">0.5</span><span class="p">,</span> <span class="o">-</span><span class="mf">2.4</span><span class="p">],</span>
<span class="p">[</span><span class="mf">0.2</span><span class="p">,</span> <span class="o">-</span><span class="mf">2.3</span><span class="p">],</span>
<span class="p">[</span><span class="mf">0.0</span><span class="p">,</span> <span class="o">-</span><span class="mf">2.7</span><span class="p">],</span>
<span class="p">[</span><span class="mf">1.3</span><span class="p">,</span> <span class="mf">2.1</span><span class="p">],</span>
<span class="p">]</span>
<span class="p">)</span>
<span class="n">y</span> <span class="o">=</span> <a href="https://numpy.org/doc/stable/reference/generated/numpy.array.html#numpy.array" title="numpy.array" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-function"><span class="n">np</span><span class="o">.</span><span class="n">array</span></a><span class="p">([</span><span class="mi">0</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="mi">0</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="mi">0</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">1</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="mi">1</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="mi">1</span><span class="p">])</span>
<span class="c1"># Plotting settings</span>
<span class="n">fig</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.subplots.html#matplotlib.pyplot.subplots" title="matplotlib.pyplot.subplots" class="sphx-glr-backref-module-matplotlib-pyplot sphx-glr-backref-type-py-function"><span class="n">plt</span><span class="o">.</span><span class="n">subplots</span></a><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">3</span><span class="p">))</span>
<span class="n">x_min</span><span class="p">,</span> <span class="n">x_max</span><span class="p">,</span> <span class="n">y_min</span><span class="p">,</span> <span class="n">y_max</span> <span class="o">=</span> <span class="o">-</span><span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="o">-</span><span class="mi">3</span><span class="p">,</span> <span class="mi">3</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set</span><span class="p">(</span><span class="n">xlim</span><span class="o">=</span><span class="p">(</span><span class="n">x_min</span><span class="p">,</span> <span class="n">x_max</span><span class="p">),</span> <span class="n">ylim</span><span class="o">=</span><span class="p">(</span><span class="n">y_min</span><span class="p">,</span> <span class="n">y_max</span><span class="p">))</span>
<span class="c1"># Plot samples by color and add legend</span>
<span class="n">scatter</span> <span class="o">=</span> <span class="n">ax</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">X</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">X</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">],</span> <span class="n">s</span><span class="o">=</span><span class="mi">150</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="n">y</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="n">y</span><span class="p">,</span> <span class="n">edgecolors</span><span class="o">=</span><span class="s2">"k"</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="o">*</span><span class="n">scatter</span><span class="o">.</span><span class="n">legend_elements</span><span class="p">(),</span> <span class="n">loc</span><span class="o">=</span><span class="s2">"upper right"</span><span class="p">,</span> <span class="n">title</span><span class="o">=</span><span class="s2">"Classes"</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s2">"Samples in two-dimensional feature space"</span><span class="p">)</span>
<span class="n">_</span> <span class="o">=</span> <a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.show.html#matplotlib.pyplot.show" title="matplotlib.pyplot.show" class="sphx-glr-backref-module-matplotlib-pyplot sphx-glr-backref-type-py-function"><span class="n">plt</span><span class="o">.</span><span class="n">show</span></a><span class="p">()</span>
</pre></div>
</div>
<img src="../../_images/sphx_glr_plot_svm_kernels_001.png" srcset="../../_images/sphx_glr_plot_svm_kernels_001.png" alt="Samples in two-dimensional feature space" class = "sphx-glr-single-img"/><p>We can see that the samples are not clearly separable by a straight line.</p>
</section>
<section id="training-svc-model-and-plotting-decision-boundaries">
<h2>Training SVC model and plotting decision boundaries<a class="headerlink" href="#training-svc-model-and-plotting-decision-boundaries" title="Link to this heading">#</a></h2>
<p>We define a function that fits a <a class="reference internal" href="../../modules/generated/sklearn.svm.SVC.html#sklearn.svm.SVC" title="sklearn.svm.SVC"><code class="xref py py-class docutils literal notranslate"><span class="pre">SVC</span></code></a> classifier,
allowing the <code class="docutils literal notranslate"><span class="pre">kernel</span></code> parameter as an input, and then plots the decision
boundaries learned by the model using
<a class="reference internal" href="../../modules/generated/sklearn.inspection.DecisionBoundaryDisplay.html#sklearn.inspection.DecisionBoundaryDisplay" title="sklearn.inspection.DecisionBoundaryDisplay"><code class="xref py py-class docutils literal notranslate"><span class="pre">DecisionBoundaryDisplay</span></code></a>.</p>
<p>Notice that for the sake of simplicity, the <code class="docutils literal notranslate"><span class="pre">C</span></code> parameter is set to its
default value (<code class="docutils literal notranslate"><span class="pre">C=1</span></code>) in this example and the <code class="docutils literal notranslate"><span class="pre">gamma</span></code> parameter is set to
<code class="docutils literal notranslate"><span class="pre">gamma=2</span></code> across all kernels, although it is automatically ignored for the
linear kernel. In a real classification task, where performance matters,
parameter tuning (by using <a class="reference internal" href="../../modules/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
instance) is highly recommended to capture different structures within the
data.</p>
<p>Setting <code class="docutils literal notranslate"><span class="pre">response_method="predict"</span></code> in
<a class="reference internal" href="../../modules/generated/sklearn.inspection.DecisionBoundaryDisplay.html#sklearn.inspection.DecisionBoundaryDisplay" title="sklearn.inspection.DecisionBoundaryDisplay"><code class="xref py py-class docutils literal notranslate"><span class="pre">DecisionBoundaryDisplay</span></code></a> colors the areas based
on their predicted class. Using <code class="docutils literal notranslate"><span class="pre">response_method="decision_function"</span></code> allows
us to also plot the decision boundary and the margins to both sides of it.
Finally the support vectors used during training (which always lay on the
margins) are identified by means of the <code class="docutils literal notranslate"><span class="pre">support_vectors_</span></code> attribute of
the trained SVCs, and plotted as well.</p>
<div class="highlight-Python notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span><span class="w"> </span><span class="nn">sklearn</span><span class="w"> </span><span class="kn">import</span> <span class="n">svm</span>
<span class="kn">from</span><span class="w"> </span><span class="nn">sklearn.inspection</span><span class="w"> </span><span class="kn">import</span> <span class="n">DecisionBoundaryDisplay</span>
<span class="k">def</span><span class="w"> </span><span class="nf">plot_training_data_with_decision_boundary</span><span class="p">(</span>
<span class="n">kernel</span><span class="p">,</span> <span class="n">ax</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">long_title</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span> <span class="n">support_vectors</span><span class="o">=</span><span class="kc">True</span>
<span class="p">):</span>
<span class="c1"># Train the SVC</span>
<span class="n">clf</span> <span class="o">=</span> <a href="../../modules/generated/sklearn.svm.SVC.html#sklearn.svm.SVC" title="sklearn.svm.SVC" class="sphx-glr-backref-module-sklearn-svm sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">svm</span><span class="o">.</span><span class="n">SVC</span></a><span class="p">(</span><span class="n">kernel</span><span class="o">=</span><span class="n">kernel</span><span class="p">,</span> <span class="n">gamma</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
<span class="c1"># Settings for plotting</span>
<span class="k">if</span> <span class="n">ax</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
<span class="n">_</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.subplots.html#matplotlib.pyplot.subplots" title="matplotlib.pyplot.subplots" class="sphx-glr-backref-module-matplotlib-pyplot sphx-glr-backref-type-py-function"><span class="n">plt</span><span class="o">.</span><span class="n">subplots</span></a><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">3</span><span class="p">))</span>
<span class="n">x_min</span><span class="p">,</span> <span class="n">x_max</span><span class="p">,</span> <span class="n">y_min</span><span class="p">,</span> <span class="n">y_max</span> <span class="o">=</span> <span class="o">-</span><span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="o">-</span><span class="mi">3</span><span class="p">,</span> <span class="mi">3</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set</span><span class="p">(</span><span class="n">xlim</span><span class="o">=</span><span class="p">(</span><span class="n">x_min</span><span class="p">,</span> <span class="n">x_max</span><span class="p">),</span> <span class="n">ylim</span><span class="o">=</span><span class="p">(</span><span class="n">y_min</span><span class="p">,</span> <span class="n">y_max</span><span class="p">))</span>
<span class="c1"># Plot decision boundary and margins</span>
<span class="n">common_params</span> <span class="o">=</span> <span class="p">{</span><span class="s2">"estimator"</span><span class="p">:</span> <span class="n">clf</span><span class="p">,</span> <span class="s2">"X"</span><span class="p">:</span> <span class="n">X</span><span class="p">,</span> <span class="s2">"ax"</span><span class="p">:</span> <span class="n">ax</span><span class="p">}</span>
<a href="../../modules/generated/sklearn.inspection.DecisionBoundaryDisplay.html#sklearn.inspection.DecisionBoundaryDisplay.from_estimator" title="sklearn.inspection.DecisionBoundaryDisplay.from_estimator" class="sphx-glr-backref-module-sklearn-inspection-DecisionBoundaryDisplay sphx-glr-backref-type-py-method"><span class="n">DecisionBoundaryDisplay</span><span class="o">.</span><span class="n">from_estimator</span></a><span class="p">(</span>
<span class="o">**</span><span class="n">common_params</span><span class="p">,</span>
<span class="n">response_method</span><span class="o">=</span><span class="s2">"predict"</span><span class="p">,</span>
<span class="n">plot_method</span><span class="o">=</span><span class="s2">"pcolormesh"</span><span class="p">,</span>
<span class="n">alpha</span><span class="o">=</span><span class="mf">0.3</span><span class="p">,</span>
<span class="p">)</span>
<a href="../../modules/generated/sklearn.inspection.DecisionBoundaryDisplay.html#sklearn.inspection.DecisionBoundaryDisplay.from_estimator" title="sklearn.inspection.DecisionBoundaryDisplay.from_estimator" class="sphx-glr-backref-module-sklearn-inspection-DecisionBoundaryDisplay sphx-glr-backref-type-py-method"><span class="n">DecisionBoundaryDisplay</span><span class="o">.</span><span class="n">from_estimator</span></a><span class="p">(</span>
<span class="o">**</span><span class="n">common_params</span><span class="p">,</span>
<span class="n">response_method</span><span class="o">=</span><span class="s2">"decision_function"</span><span class="p">,</span>
<span class="n">plot_method</span><span class="o">=</span><span class="s2">"contour"</span><span class="p">,</span>
<span class="n">levels</span><span class="o">=</span><span class="p">[</span><span class="o">-</span><span class="mi">1</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="n">colors</span><span class="o">=</span><span class="p">[</span><span class="s2">"k"</span><span class="p">,</span> <span class="s2">"k"</span><span class="p">,</span> <span class="s2">"k"</span><span class="p">],</span>
<span class="n">linestyles</span><span class="o">=</span><span class="p">[</span><span class="s2">"--"</span><span class="p">,</span> <span class="s2">"-"</span><span class="p">,</span> <span class="s2">"--"</span><span class="p">],</span>
<span class="p">)</span>
<span class="k">if</span> <span class="n">support_vectors</span><span class="p">:</span>
<span class="c1"># Plot bigger circles around samples that serve as support vectors</span>
<span class="n">ax</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span>
<span class="n">clf</span><span class="o">.</span><span class="n">support_vectors_</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">],</span>
<span class="n">clf</span><span class="o">.</span><span class="n">support_vectors_</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">],</span>
<span class="n">s</span><span class="o">=</span><span class="mi">150</span><span class="p">,</span>
<span class="n">facecolors</span><span class="o">=</span><span class="s2">"none"</span><span class="p">,</span>
<span class="n">edgecolors</span><span class="o">=</span><span class="s2">"k"</span><span class="p">,</span>
<span class="p">)</span>
<span class="c1"># Plot samples by color and add legend</span>
<span class="n">ax</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">X</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">X</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">],</span> <span class="n">c</span><span class="o">=</span><span class="n">y</span><span class="p">,</span> <span class="n">s</span><span class="o">=</span><span class="mi">30</span><span class="p">,</span> <span class="n">edgecolors</span><span class="o">=</span><span class="s2">"k"</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="o">*</span><span class="n">scatter</span><span class="o">.</span><span class="n">legend_elements</span><span class="p">(),</span> <span class="n">loc</span><span class="o">=</span><span class="s2">"upper right"</span><span class="p">,</span> <span class="n">title</span><span class="o">=</span><span class="s2">"Classes"</span><span class="p">)</span>
<span class="k">if</span> <span class="n">long_title</span><span class="p">:</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="sa">f</span><span class="s2">" Decision boundaries of </span><span class="si">{</span><span class="n">kernel</span><span class="si">}</span><span class="s2"> kernel in SVC"</span><span class="p">)</span>
<span class="k">else</span><span class="p">:</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="n">kernel</span><span class="p">)</span>
<span class="k">if</span> <span class="n">ax</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.show.html#matplotlib.pyplot.show" title="matplotlib.pyplot.show" class="sphx-glr-backref-module-matplotlib-pyplot sphx-glr-backref-type-py-function"><span class="n">plt</span><span class="o">.</span><span class="n">show</span></a><span class="p">()</span>
</pre></div>
</div>
<section id="linear-kernel">
<h3>Linear kernel<a class="headerlink" href="#linear-kernel" title="Link to this heading">#</a></h3>
<p>Linear kernel is the dot product of the input samples:</p>
<div class="math notranslate nohighlight">
\[K(\mathbf{x}_1, \mathbf{x}_2) = \mathbf{x}_1^\top \mathbf{x}_2\]</div>
<p>It is then applied to any combination of two data points (samples) in the
dataset. The dot product of the two points determines the
<a class="reference internal" href="../../modules/generated/sklearn.metrics.pairwise.cosine_similarity.html#sklearn.metrics.pairwise.cosine_similarity" title="sklearn.metrics.pairwise.cosine_similarity"><code class="xref py py-func docutils literal notranslate"><span class="pre">cosine_similarity</span></code></a> between both points. The
higher the value, the more similar the points are.</p>
<div class="highlight-Python notranslate"><div class="highlight"><pre><span></span><span class="n">plot_training_data_with_decision_boundary</span><span class="p">(</span><span class="s2">"linear"</span><span class="p">)</span>
</pre></div>
</div>
<img src="../../_images/sphx_glr_plot_svm_kernels_002.png" srcset="../../_images/sphx_glr_plot_svm_kernels_002.png" alt="Decision boundaries of linear kernel in SVC" class = "sphx-glr-single-img"/><p>Training a <a class="reference internal" href="../../modules/generated/sklearn.svm.SVC.html#sklearn.svm.SVC" title="sklearn.svm.SVC"><code class="xref py py-class docutils literal notranslate"><span class="pre">SVC</span></code></a> on a linear kernel results in an
untransformed feature space, where the hyperplane and the margins are
straight lines. Due to the lack of expressivity of the linear kernel, the
trained classes do not perfectly capture the training data.</p>
</section>
<section id="polynomial-kernel">
<h3>Polynomial kernel<a class="headerlink" href="#polynomial-kernel" title="Link to this heading">#</a></h3>
<p>The polynomial kernel changes the notion of similarity. The kernel function
is defined as:</p>
<div class="math notranslate nohighlight">
\[K(\mathbf{x}_1, \mathbf{x}_2) = (\gamma \cdot \
\mathbf{x}_1^\top\mathbf{x}_2 + r)^d\]</div>
<p>where <span class="math notranslate nohighlight">\({d}\)</span> is the degree (<code class="docutils literal notranslate"><span class="pre">degree</span></code>) of the polynomial, <span class="math notranslate nohighlight">\({\gamma}\)</span>
(<code class="docutils literal notranslate"><span class="pre">gamma</span></code>) controls the influence of each individual training sample on the
decision boundary and <span class="math notranslate nohighlight">\({r}\)</span> is the bias term (<code class="docutils literal notranslate"><span class="pre">coef0</span></code>) that shifts the
data up or down. Here, we use the default value for the degree of the
polynomial in the kernel function (<code class="docutils literal notranslate"><span class="pre">degree=3</span></code>). When <code class="docutils literal notranslate"><span class="pre">coef0=0</span></code> (the default),
the data is only transformed, but no additional dimension is added. Using a
polynomial kernel is equivalent to creating
<a class="reference internal" href="../../modules/generated/sklearn.preprocessing.PolynomialFeatures.html#sklearn.preprocessing.PolynomialFeatures" title="sklearn.preprocessing.PolynomialFeatures"><code class="xref py py-class docutils literal notranslate"><span class="pre">PolynomialFeatures</span></code></a> and then fitting a
<a class="reference internal" href="../../modules/generated/sklearn.svm.SVC.html#sklearn.svm.SVC" title="sklearn.svm.SVC"><code class="xref py py-class docutils literal notranslate"><span class="pre">SVC</span></code></a> with a linear kernel on the transformed data,
although this alternative approach would be computationally expensive for most
datasets.</p>
<div class="highlight-Python notranslate"><div class="highlight"><pre><span></span><span class="n">plot_training_data_with_decision_boundary</span><span class="p">(</span><span class="s2">"poly"</span><span class="p">)</span>
</pre></div>
</div>
<img src="../../_images/sphx_glr_plot_svm_kernels_003.png" srcset="../../_images/sphx_glr_plot_svm_kernels_003.png" alt="Decision boundaries of poly kernel in SVC" class = "sphx-glr-single-img"/><p>The polynomial kernel with <code class="docutils literal notranslate"><span class="pre">gamma=2`</span></code> adapts well to the training data,
causing the margins on both sides of the hyperplane to bend accordingly.</p>
</section>
<section id="rbf-kernel">
<h3>RBF kernel<a class="headerlink" href="#rbf-kernel" title="Link to this heading">#</a></h3>
<p>The radial basis function (RBF) kernel, also known as the Gaussian kernel, is
the default kernel for Support Vector Machines in scikit-learn. It measures
similarity between two data points in infinite dimensions and then approaches
classification by majority vote. The kernel function is defined as:</p>
<div class="math notranslate nohighlight">
\[K(\mathbf{x}_1, \mathbf{x}_2) = \exp\left(-\gamma \cdot
{\|\mathbf{x}_1 - \mathbf{x}_2\|^2}\right)\]</div>
<p>where <span class="math notranslate nohighlight">\({\gamma}\)</span> (<code class="docutils literal notranslate"><span class="pre">gamma</span></code>) controls the influence of each individual
training sample on the decision boundary.</p>
<p>The larger the euclidean distance between two points
<span class="math notranslate nohighlight">\(\|\mathbf{x}_1 - \mathbf{x}_2\|^2\)</span>
the closer the kernel function is to zero. This means that two points far away
are more likely to be dissimilar.</p>
<div class="highlight-Python notranslate"><div class="highlight"><pre><span></span><span class="n">plot_training_data_with_decision_boundary</span><span class="p">(</span><span class="s2">"rbf"</span><span class="p">)</span>
</pre></div>
</div>
<img src="../../_images/sphx_glr_plot_svm_kernels_004.png" srcset="../../_images/sphx_glr_plot_svm_kernels_004.png" alt="Decision boundaries of rbf kernel in SVC" class = "sphx-glr-single-img"/><p>In the plot we can see how the decision boundaries tend to contract around
data points that are close to each other.</p>
</section>
<section id="sigmoid-kernel">
<h3>Sigmoid kernel<a class="headerlink" href="#sigmoid-kernel" title="Link to this heading">#</a></h3>
<p>The sigmoid kernel function is defined as:</p>
<div class="math notranslate nohighlight">
\[K(\mathbf{x}_1, \mathbf{x}_2) = \tanh(\gamma \cdot
\mathbf{x}_1^\top\mathbf{x}_2 + r)\]</div>
<p>where the kernel coefficient <span class="math notranslate nohighlight">\({\gamma}\)</span> (<code class="docutils literal notranslate"><span class="pre">gamma</span></code>) controls the influence
of each individual training sample on the decision boundary and <span class="math notranslate nohighlight">\({r}\)</span> is
the bias term (<code class="docutils literal notranslate"><span class="pre">coef0</span></code>) that shifts the data up or down.</p>
<p>In the sigmoid kernel, the similarity between two data points is computed
using the hyperbolic tangent function (<span class="math notranslate nohighlight">\(\tanh\)</span>). The kernel function
scales and possibly shifts the dot product of the two points
(<span class="math notranslate nohighlight">\(\mathbf{x}_1\)</span> and <span class="math notranslate nohighlight">\(\mathbf{x}_2\)</span>).</p>
<div class="highlight-Python notranslate"><div class="highlight"><pre><span></span><span class="n">plot_training_data_with_decision_boundary</span><span class="p">(</span><span class="s2">"sigmoid"</span><span class="p">)</span>
</pre></div>
</div>
<img src="../../_images/sphx_glr_plot_svm_kernels_005.png" srcset="../../_images/sphx_glr_plot_svm_kernels_005.png" alt="Decision boundaries of sigmoid kernel in SVC" class = "sphx-glr-single-img"/><p>We can see that the decision boundaries obtained with the sigmoid kernel
appear curved and irregular. The decision boundary tries to separate the
classes by fitting a sigmoid-shaped curve, resulting in a complex boundary
that may not generalize well to unseen data. From this example it becomes
obvious, that the sigmoid kernel has very specific use cases, when dealing
with data that exhibits a sigmoidal shape. In this example, careful fine
tuning might find more generalizable decision boundaries. Because of it’s
specificity, the sigmoid kernel is less commonly used in practice compared to
other kernels.</p>
</section>
</section>
<section id="conclusion">
<h2>Conclusion<a class="headerlink" href="#conclusion" title="Link to this heading">#</a></h2>
<p>In this example, we have visualized the decision boundaries trained with the
provided dataset. The plots serve as an intuitive demonstration of how
different kernels utilize the training data to determine the classification
boundaries.</p>
<p>The hyperplanes and margins, although computed indirectly, can be imagined as
planes in the transformed feature space. However, in the plots, they are
represented relative to the original feature space, resulting in curved
decision boundaries for the polynomial, RBF, and sigmoid kernels.</p>
<p>Please note that the plots do not evaluate the individual kernel’s accuracy or
quality. They are intended to provide a visual understanding of how the
different kernels use the training data.</p>
<p>For a comprehensive evaluation, fine-tuning of <a class="reference internal" href="../../modules/generated/sklearn.svm.SVC.html#sklearn.svm.SVC" title="sklearn.svm.SVC"><code class="xref py py-class docutils literal notranslate"><span class="pre">SVC</span></code></a>
parameters using techniques such as
<a class="reference internal" href="../../modules/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> is recommended to capture the
underlying structures within the data.</p>
</section>
<section id="xor-dataset">
<h2>XOR dataset<a class="headerlink" href="#xor-dataset" title="Link to this heading">#</a></h2>
<p>A classical example of a dataset which is not linearly separable is the XOR
pattern. HEre we demonstrate how different kernels work on such a dataset.</p>
<div class="highlight-Python notranslate"><div class="highlight"><pre><span></span><span class="n">xx</span><span class="p">,</span> <span class="n">yy</span> <span class="o">=</span> <a href="https://numpy.org/doc/stable/reference/generated/numpy.meshgrid.html#numpy.meshgrid" title="numpy.meshgrid" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-function"><span class="n">np</span><span class="o">.</span><span class="n">meshgrid</span></a><span class="p">(</span><a href="https://numpy.org/doc/stable/reference/generated/numpy.linspace.html#numpy.linspace" title="numpy.linspace" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-function"><span class="n">np</span><span class="o">.</span><span class="n">linspace</span></a><span class="p">(</span><span class="o">-</span><span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">500</span><span class="p">),</span> <a href="https://numpy.org/doc/stable/reference/generated/numpy.linspace.html#numpy.linspace" title="numpy.linspace" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-function"><span class="n">np</span><span class="o">.</span><span class="n">linspace</span></a><span class="p">(</span><span class="o">-</span><span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">500</span><span class="p">))</span>
<a href="https://numpy.org/doc/stable/reference/random/generated/numpy.random.seed.html#numpy.random.seed" title="numpy.random.seed" class="sphx-glr-backref-module-numpy-random sphx-glr-backref-type-py-function"><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">seed</span></a><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
<span class="n">X</span> <span class="o">=</span> <a href="https://numpy.org/doc/stable/reference/random/generated/numpy.random.randn.html#numpy.random.randn" title="numpy.random.randn" class="sphx-glr-backref-module-numpy-random sphx-glr-backref-type-py-function"><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">randn</span></a><span class="p">(</span><span class="mi">300</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
<span class="n">y</span> <span class="o">=</span> <a href="https://numpy.org/doc/stable/reference/generated/numpy.logical_xor.html#numpy.logical_xor" title="numpy.logical_xor" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-data"><span class="n">np</span><span class="o">.</span><span class="n">logical_xor</span></a><span class="p">(</span><span class="n">X</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">></span> <span class="mi">0</span><span class="p">,</span> <span class="n">X</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">]</span> <span class="o">></span> <span class="mi">0</span><span class="p">)</span>
<span class="n">_</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.subplots.html#matplotlib.pyplot.subplots" title="matplotlib.pyplot.subplots" class="sphx-glr-backref-module-matplotlib-pyplot sphx-glr-backref-type-py-function"><span class="n">plt</span><span class="o">.</span><span class="n">subplots</span></a><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">8</span><span class="p">,</span> <span class="mi">8</span><span class="p">))</span>
<span class="n">args</span> <span class="o">=</span> <span class="nb">dict</span><span class="p">(</span><span class="n">long_title</span><span class="o">=</span><span class="kc">False</span><span class="p">,</span> <span class="n">support_vectors</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>
<span class="n">plot_training_data_with_decision_boundary</span><span class="p">(</span><span class="s2">"linear"</span><span class="p">,</span> <span class="n">ax</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="o">**</span><span class="n">args</span><span class="p">)</span>
<span class="n">plot_training_data_with_decision_boundary</span><span class="p">(</span><span class="s2">"poly"</span><span class="p">,</span> <span class="n">ax</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="o">**</span><span class="n">args</span><span class="p">)</span>
<span class="n">plot_training_data_with_decision_boundary</span><span class="p">(</span><span class="s2">"rbf"</span><span class="p">,</span> <span class="n">ax</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="o">**</span><span class="n">args</span><span class="p">)</span>
<span class="n">plot_training_data_with_decision_boundary</span><span class="p">(</span><span class="s2">"sigmoid"</span><span class="p">,</span> <span class="n">ax</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="o">**</span><span class="n">args</span><span class="p">)</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.show.html#matplotlib.pyplot.show" title="matplotlib.pyplot.show" class="sphx-glr-backref-module-matplotlib-pyplot sphx-glr-backref-type-py-function"><span class="n">plt</span><span class="o">.</span><span class="n">show</span></a><span class="p">()</span>
</pre></div>
</div>
<img src="../../_images/sphx_glr_plot_svm_kernels_006.png" srcset="../../_images/sphx_glr_plot_svm_kernels_006.png" alt="linear, poly, rbf, sigmoid" class = "sphx-glr-single-img"/><p>As you can see from the plots above, only the <code class="docutils literal notranslate"><span class="pre">rbf</span></code> kernel can find a
reasonable decision boundary for the above dataset.</p>
<p class="sphx-glr-timing"><strong>Total running time of the script:</strong> (0 minutes 1.346 seconds)</p>
<div class="sphx-glr-footer sphx-glr-footer-example docutils container" id="sphx-glr-download-auto-examples-svm-plot-svm-kernels-py">
<div class="binder-badge docutils container">
<a class="reference external image-reference" href="https://mybinder.org/v2/gh/scikit-learn/scikit-learn/main?urlpath=lab/tree/notebooks/auto_examples/svm/plot_svm_kernels.ipynb"><img alt="Launch binder" src="../../_images/binder_badge_logo30.svg" style="width: 150px;" />
</a>
</div>
<div class="lite-badge docutils container">
<a class="reference external image-reference" href="../../lite/lab/index.html?path=auto_examples/svm/plot_svm_kernels.ipynb"><img alt="Launch JupyterLite" src="../../_images/jupyterlite_badge_logo30.svg" style="width: 150px;" />
</a>
</div>
<div class="sphx-glr-download sphx-glr-download-jupyter docutils container">
<p><a class="reference download internal" download="" href="../../_downloads/264f6891fa2130246a013d5f089e7b2e/plot_svm_kernels.ipynb"><code class="xref download docutils literal notranslate"><span class="pre">Download</span> <span class="pre">Jupyter</span> <span class="pre">notebook:</span> <span class="pre">plot_svm_kernels.ipynb</span></code></a></p>
</div>
<div class="sphx-glr-download sphx-glr-download-python docutils container">
<p><a class="reference download internal" download="" href="../../_downloads/8975399471ae75debd0b26fbe3013719/plot_svm_kernels.py"><code class="xref download docutils literal notranslate"><span class="pre">Download</span> <span class="pre">Python</span> <span class="pre">source</span> <span class="pre">code:</span> <span class="pre">plot_svm_kernels.py</span></code></a></p>
</div>
<div class="sphx-glr-download sphx-glr-download-zip docutils container">
<p><a class="reference download internal" download="" href="../../_downloads/d58abbe7bc4eabf74b338ceadfa1c679/plot_svm_kernels.zip"><code class="xref download docutils literal notranslate"><span class="pre">Download</span> <span class="pre">zipped:</span> <span class="pre">plot_svm_kernels.zip</span></code></a></p>
</div>
</div>
<p class="rubric">Related examples</p>
<div class="sphx-glr-thumbnails"><div class="sphx-glr-thumbcontainer" tooltip="This example illustrates the effect of the parameters gamma and C of the Radial Basis Function (RBF) kernel SVM."><img alt="" src="../../_images/sphx_glr_plot_rbf_parameters_thumb.png" />
<p><a class="reference internal" href="plot_rbf_parameters.html#sphx-glr-auto-examples-svm-plot-rbf-parameters-py"><span class="std std-ref">RBF SVM parameters</span></a></p>
<div class="sphx-glr-thumbnail-title">RBF SVM parameters</div>
</div><div class="sphx-glr-thumbcontainer" tooltip="Comparison of different linear SVM classifiers on a 2D projection of the iris dataset. We only consider the first 2 features of this dataset:"><img alt="" src="../../_images/sphx_glr_plot_iris_svc_thumb.png" />
<p><a class="reference internal" href="plot_iris_svc.html#sphx-glr-auto-examples-svm-plot-iris-svc-py"><span class="std std-ref">Plot different SVM classifiers in the iris dataset</span></a></p>
<div class="sphx-glr-thumbnail-title">Plot different SVM classifiers in the iris dataset</div>
</div><div class="sphx-glr-thumbcontainer" tooltip="A comparison for the decision boundaries generated on the iris dataset by Label Spreading, Self-training and SVM."><img alt="" src="../../_images/sphx_glr_plot_semi_supervised_versus_svm_iris_thumb.png" />
<p><a class="reference internal" href="../semi_supervised/plot_semi_supervised_versus_svm_iris.html#sphx-glr-auto-examples-semi-supervised-plot-semi-supervised-versus-svm-iris-py"><span class="std std-ref">Decision boundary of semi-supervised classifiers versus SVM on the Iris dataset</span></a></p>
<div class="sphx-glr-thumbnail-title">Decision boundary of semi-supervised classifiers versus SVM on the Iris dataset</div>
</div><div class="sphx-glr-thumbcontainer" tooltip="Plot the maximum margin separating hyperplane within a two-class separable dataset using a Support Vector Machine classifier with linear kernel."><img alt="" src="../../_images/sphx_glr_plot_separating_hyperplane_thumb.png" />
<p><a class="reference internal" href="plot_separating_hyperplane.html#sphx-glr-auto-examples-svm-plot-separating-hyperplane-py"><span class="std std-ref">SVM: Maximum margin separating hyperplane</span></a></p>
<div class="sphx-glr-thumbnail-title">SVM: Maximum margin separating hyperplane</div>
</div></div><p class="sphx-glr-signature"><a class="reference external" href="https://sphinx-gallery.github.io">Gallery generated by Sphinx-Gallery</a></p>
</section>
</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="plot_oneclass.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">One-class SVM with non-linear kernel (RBF)</p>
</div>
</a>
<a class="right-next"
href="plot_iris_svc.html"
title="next page">
<div class="prev-next-info">
<p class="prev-next-subtitle">next</p>
<p class="prev-next-title">Plot different SVM classifiers in the iris dataset</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
id="pst-page-navigation-heading-2"
class="page-toc tocsection onthispage">
<i class="fa-solid fa-list"></i> On this page
</div>
<nav class="bd-toc-nav page-toc" aria-labelledby="pst-page-navigation-heading-2">
<ul class="visible nav section-nav flex-column">
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#creating-a-dataset">Creating a dataset</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#training-svc-model-and-plotting-decision-boundaries">Training SVC model and plotting decision boundaries</a><ul class="nav section-nav flex-column">
<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#linear-kernel">Linear kernel</a></li>
<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#polynomial-kernel">Polynomial kernel</a></li>
<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#rbf-kernel">RBF kernel</a></li>
<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#sigmoid-kernel">Sigmoid kernel</a></li>
</ul>
</li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#conclusion">Conclusion</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#xor-dataset">XOR dataset</a></li>
</ul>
</nav></div>
<div class="sidebar-secondary-item">
<div role="note" aria-label="source link">
<h3>This Page</h3>
<ul class="this-page-menu">
<li><a href="../../_sources/auto_examples/svm/plot_svm_kernels.rst.txt"
rel="nofollow">Show Source</a></li>
</ul>
</div></div>
<div class="sidebar-secondary-item">
<div class="sphx-glr-sidebar-component">
<div class="sphx-glr-sidebar-item sphx-glr-download-python-sidebar" title="plot_svm_kernels.py">
<a download href="../../_downloads/8975399471ae75debd0b26fbe3013719/plot_svm_kernels.py">
<i class="fa-solid fa-download"></i>
Download source code
</a>
</div>
<div class="sphx-glr-sidebar-item sphx-glr-download-jupyter-sidebar" title="plot_svm_kernels.ipynb">
<a download href="../../_downloads/264f6891fa2130246a013d5f089e7b2e/plot_svm_kernels.ipynb">
<i class="fa-solid fa-download"></i>
Download Jupyter notebook
</a>
</div>
<div class="sphx-glr-sidebar-item sphx-glr-download-zip-sidebar" title="plot_svm_kernels.zip">
<a download href="../../_downloads/d58abbe7bc4eabf74b338ceadfa1c679/plot_svm_kernels.zip">
<i class="fa-solid fa-download"></i>
Download zipped
</a>
</div>
</div>
</div>
<div class="sidebar-secondary-item">
<div class="sphx-glr-sidebar-component">
<div class="sphx-glr-sidebar-item lite-badge-sidebar">
<a href="../../lite/lab/index.html?path=auto_examples/svm/plot_svm_kernels.ipynb">
<img src="../../_images/jupyterlite_badge_logo30.svg" alt="Launch JupyterLite">
</a>
</div>
<div class="sphx-glr-sidebar-item binder-badge-sidebar">
<a href="https://mybinder.org/v2/gh/scikit-learn/scikit-learn/main?urlpath=lab/tree/notebooks/auto_examples/svm/plot_svm_kernels.ipynb">
<img src="../../_images/binder_badge_logo30.svg" alt="Launch binder">
</a>
</div>
</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=8878045cc6db502f8baf"></script>
<script defer src="../../_static/scripts/pydata-sphinx-theme.js?digest=8878045cc6db502f8baf"></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 - 2025, scikit-learn developers (BSD License).
<br/>
</p>
</div>
</div>
</div>
</footer>
</body>
</html>