{"id":203741,"date":"2023-06-16T12:18:56","date_gmt":"2023-06-16T12:18:56","guid":{"rendered":"https:\/\/datascientest.com\/?p=203741"},"modified":"2023-11-09T13:51:57","modified_gmt":"2023-11-09T13:51:57","slug":"reduction-de-dimension","status":"publish","type":"post","link":"https:\/\/datascientest.com\/reduction-de-dimension","title":{"rendered":"R\u00e9duction de dimension : Comment \u00e7a fonctionne ?"},"content":{"rendered":"\t\t<div data-elementor-type=\"wp-post\" data-elementor-id=\"203741\" class=\"elementor elementor-203741\" data-elementor-post-type=\"post\">\n\t\t\t\t\t\t\t<div class=\"elementor-element elementor-element-70cdf24 e-flex e-con-boxed e-con\" data-id=\"70cdf24\" data-element_type=\"container\" data-settings=\"{&quot;content_width&quot;:&quot;boxed&quot;}\">\n\t\t\t\t\t<div class=\"e-con-inner\">\n\t\t\t\t<div class=\"elementor-element elementor-element-a15607c elementor-widget elementor-widget-heading\" data-id=\"a15607c\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<style>\/*! elementor - v3.15.0 - 02-08-2023 *\/\n.elementor-heading-title{padding:0;margin:0;line-height:1}.elementor-widget-heading .elementor-heading-title[class*=elementor-size-]>a{color:inherit;font-size:inherit;line-height:inherit}.elementor-widget-heading .elementor-heading-title.elementor-size-small{font-size:15px}.elementor-widget-heading .elementor-heading-title.elementor-size-medium{font-size:19px}.elementor-widget-heading .elementor-heading-title.elementor-size-large{font-size:29px}.elementor-widget-heading .elementor-heading-title.elementor-size-xl{font-size:39px}.elementor-widget-heading .elementor-heading-title.elementor-size-xxl{font-size:59px}<\/style><h2 class=\"elementor-heading-title elementor-size-default\">L\u2019analyse de donn\u00e9es peut \u00eatre consid\u00e9r\u00e9e comme une science exp\u00e9rimentale, o\u00f9 les propri\u00e9t\u00e9s sont d\u00e9montr\u00e9es apr\u00e8s avoir \u00e9t\u00e9 observ\u00e9es, et o\u00f9 des codes sont \u00e9tablis de mani\u00e8re heuristique pour interpr\u00e9ter les r\u00e9sultats. Dans le domaine de l\u2019apprentissage automatique (machine learning) et de l\u2019analyse des donn\u00e9es, la r\u00e9duction de dimension joue un r\u00f4le essentiel pour simplifier les ensembles de donn\u00e9es complexes.<\/h2>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div class=\"elementor-element elementor-element-fca9932 e-flex e-con-boxed e-con\" data-id=\"fca9932\" data-element_type=\"container\" data-settings=\"{&quot;content_width&quot;:&quot;boxed&quot;}\">\n\t\t\t\t\t<div class=\"e-con-inner\">\n\t\t\t\t<div class=\"elementor-element elementor-element-8608eef elementor-widget elementor-widget-text-editor\" data-id=\"8608eef\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<style>\/*! elementor - v3.15.0 - 02-08-2023 *\/\n.elementor-widget-text-editor.elementor-drop-cap-view-stacked .elementor-drop-cap{background-color:#69727d;color:#fff}.elementor-widget-text-editor.elementor-drop-cap-view-framed .elementor-drop-cap{color:#69727d;border:3px solid;background-color:transparent}.elementor-widget-text-editor:not(.elementor-drop-cap-view-default) .elementor-drop-cap{margin-top:8px}.elementor-widget-text-editor:not(.elementor-drop-cap-view-default) .elementor-drop-cap-letter{width:1em;height:1em}.elementor-widget-text-editor .elementor-drop-cap{float:left;text-align:center;line-height:1;font-size:50px}.elementor-widget-text-editor .elementor-drop-cap-letter{display:inline-block}<\/style>\t\t\t\t<p>L\u2019id\u00e9e de base de la r\u00e9duction de dimension consiste \u00e0 <strong>repr\u00e9senter des donn\u00e9es multidimensionnelles dans un espace de dimension bien inf\u00e9rieure<\/strong> tout en conservant les informations importantes et pertinentes.\u00a0<\/p><p><span style=\"font-weight: 400;\">Avant de se plonger dans plus de d\u00e9tails math\u00e9matiques, il est important de comprendre certains termes cl\u00e9s.\u00a0<\/span><\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div class=\"elementor-element elementor-element-19a41bd e-flex e-con-boxed e-con\" data-id=\"19a41bd\" data-element_type=\"container\" data-settings=\"{&quot;content_width&quot;:&quot;boxed&quot;}\">\n\t\t\t\t\t<div class=\"e-con-inner\">\n\t\t\t\t<div class=\"elementor-element elementor-element-cced4f3 elementor-widget elementor-widget-heading\" data-id=\"cced4f3\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<h3 class=\"elementor-heading-title elementor-size-default\">Le vocabulaire de la r\u00e9duction de dimension<\/h3>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-894a55f elementor-widget elementor-widget-text-editor\" data-id=\"894a55f\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<ul><li><strong>L\u2019espace de dimension<\/strong><span style=\"font-weight: 400;\"> : nombre de caract\u00e9ristiques ou variables qui d\u00e9crivent un objet dans un dataset. Par exemple, une base de donn\u00e9es rassemblant des caract\u00e9ristiques sur des personnes peut comporter des variables telles que l\u2019\u00e2ge, le revenu, l\u2019adresse, le sexe, et le niveau d\u2019\u00e9ducation. Ceci d\u00e9finit un espace de dimension cinq.\u00a0<\/span><\/li><li><strong>La projection<\/strong><span style=\"font-weight: 400;\"> consiste \u00e0 transformer les donn\u00e9es d\u2019un espace de dimension \u00e9lev\u00e9e vers un espace de dimension r\u00e9duite en utilisant des techniques de r\u00e9duction de dimension. Cette transformation permet de repr\u00e9senter les donn\u00e9es de mani\u00e8re plus compacte tout en conservant les informations importantes.\u00a0<\/span><\/li><li><i><\/i><strong>L\u2019orthogonalit\u00e9<\/strong><span style=\"font-weight: 400;\"> est une propri\u00e9t\u00e9 math\u00e9matique dans laquelle deux vecteurs ou deux espaces sont perpendiculaires les uns aux autres. Dans le contexte de la r\u00e9duction de dimension, l\u2019orthogonalit\u00e9 est souvent utilis\u00e9e pour d\u00e9crire des variables ou des composantes qui sont d\u00e9corr\u00e9l\u00e9es les unes des autres.<\/span><\/li><\/ul>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-9047759 elementor-widget elementor-widget-text-editor\" data-id=\"9047759\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">Si les donn\u00e9es sont repr\u00e9sent\u00e9es dans un tableau, la <\/span><b>r\u00e9duction de dimensionnalit\u00e9<\/b><span style=\"font-weight: 400;\"> passe par une diminution du nombre de colonnes. <\/span><span style=\"font-weight: 400;\">Si notre <a href=\"https:\/\/datascientest.com\/dataset-definition\">dataset<\/a> est compos\u00e9 de plus de 3 variables, plus la dimension augmente, plus il est difficile de le visualiser. Ainsi, ce n\u2019est pas possible de visualiser un <\/span><b>jeu de donn\u00e9es<\/b><span style=\"font-weight: 400;\"> \u00e0 13 attributs mais cela est possible avec seulement 2 ou 3 attributs. L\u2019objectif est donc de projeter ces donn\u00e9es sur les 2 ou 3 axes les plus importants en termes de contenu d\u2019information afin de garder la repr\u00e9sentation des donn\u00e9es la plus proche de celles en grande dimension.\u00a0<\/span><\/p><p><b><i>Un exemple tr\u00e8s simple : <\/i><\/b><span style=\"font-weight: 400;\">Supposons que nous ayons un jeu de donn\u00e9es contenant des informations sur des \u00e9tudiants, comprenant les 5 variables suivantes et chaque \u00e9tudiant est repr\u00e9sent\u00e9 par une ligne dans le jeu de donn\u00e9es, chaque variable correspondant \u00e0 une colonne.<\/span><\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-a1645a0 elementor-widget elementor-widget-html\" data-id=\"a1645a0\" data-element_type=\"widget\" data-widget_type=\"html.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<style type=\"text\/css\">\r\n.tg  {border-collapse:collapse;border-spacing:0;}\r\n.tg td{border-color:black;border-style:solid;border-width:1px;font-family:Arial, sans-serif;font-size:14px;\r\n  overflow:hidden;padding:10px 5px;word-break:normal;}\r\n.tg th{border-color:black;border-style:solid;border-width:1px;font-family:Arial, sans-serif;font-size:14px;\r\n  font-weight:normal;overflow:hidden;padding:10px 5px;word-break:normal;}\r\n.tg .tg-80go{font-family:\"Trebuchet MS\", Helvetica, sans-serif !important;text-align:left;vertical-align:top}\r\n.tg .tg-4lje{font-family:\"Trebuchet MS\", Helvetica, sans-serif !important;text-align:center;vertical-align:top}\r\n.tg .tg-7t5w{background-color:#6200c9;color:#efefef;font-family:\"Trebuchet MS\", Helvetica, sans-serif !important;font-size:20px;\r\n  font-weight:bold;text-align:center;vertical-align:top}\r\n.tg .tg-pm1a{background-color:#6200c9;color:#efefef;font-family:\"Trebuchet MS\", Helvetica, sans-serif !important;font-size:20px;\r\n  font-weight:bold;text-align:left;vertical-align:top}\r\n<\/style>\r\n<table class=\"tg\" style=\"undefined;table-layout: fixed; width: 600px\">\r\n<colgroup>\r\n<col style=\"width: 75px\">\r\n<col style=\"width: 50px\">\r\n<col style=\"width: 50px\">\r\n<col style=\"width: 50px\">\r\n<col style=\"width: 50px\">\r\n<col style=\"width: 75px\">\r\n<\/colgroup>\r\n<thead>\r\n  <tr>\r\n    <th class=\"tg-pm1a\">Etudiant<\/th>\r\n    <th class=\"tg-7t5w\">\u00c2ge<\/th>\r\n    <th class=\"tg-7t5w\">Taille<\/th>\r\n    <th class=\"tg-7t5w\">Poids<\/th>\r\n    <th class=\"tg-7t5w\">Maths<\/th>\r\n    <th class=\"tg-7t5w\">Sciences<\/th>\r\n  <\/tr>\r\n<\/thead>\r\n<tbody>\r\n  <tr>\r\n    <td class=\"tg-80go\">Etudiant 1<\/td>\r\n    <td class=\"tg-4lje\"><span style=\"font-weight:400;font-style:normal;text-decoration:none;color:#000;background-color:transparent\">18<\/span><\/td>\r\n    <td class=\"tg-4lje\"><span style=\"font-weight:400;font-style:normal;text-decoration:none;color:#000;background-color:transparent\">165<\/span><\/td>\r\n    <td class=\"tg-4lje\"><span style=\"font-weight:400;font-style:normal;text-decoration:none;color:#000;background-color:transparent\">60<\/span><\/td>\r\n    <td class=\"tg-4lje\"><span style=\"font-weight:400;font-style:normal;text-decoration:none;color:#000;background-color:transparent\">85<\/span><\/td>\r\n    <td class=\"tg-4lje\"><span style=\"font-weight:400;font-style:normal;text-decoration:none;color:#000;background-color:transparent\">90<\/span><\/td>\r\n  <\/tr>\r\n  <tr>\r\n    <td class=\"tg-80go\">Etudiant 2<\/td>\r\n    <td class=\"tg-4lje\"><span style=\"font-weight:400;font-style:normal;text-decoration:none;color:#000;background-color:transparent\">20<\/span><\/td>\r\n    <td class=\"tg-4lje\"><span style=\"font-weight:400;font-style:normal;text-decoration:none;color:#000;background-color:transparent\">170<\/span><\/td>\r\n    <td class=\"tg-4lje\"><span style=\"font-weight:400;font-style:normal;text-decoration:none;color:#000;background-color:transparent\">65<\/span><\/td>\r\n    <td class=\"tg-4lje\"><span style=\"font-weight:400;font-style:normal;text-decoration:none;color:#000;background-color:transparent\">75<\/span><\/td>\r\n    <td class=\"tg-4lje\"><span style=\"font-weight:400;font-style:normal;text-decoration:none;color:#000;background-color:transparent\">80<\/span><\/td>\r\n  <\/tr>\r\n  <tr>\r\n    <td class=\"tg-80go\">Etudiant 3<\/td>\r\n    <td class=\"tg-4lje\"><span style=\"font-weight:400;font-style:normal;text-decoration:none;color:#000;background-color:transparent\">19<\/span><\/td>\r\n    <td class=\"tg-4lje\"><span style=\"font-weight:400;font-style:normal;text-decoration:none;color:#000;background-color:transparent\">175<\/span><\/td>\r\n    <td class=\"tg-4lje\"><span style=\"font-weight:400;font-style:normal;text-decoration:none;color:#000;background-color:transparent\">70<\/span><\/td>\r\n    <td class=\"tg-4lje\"><span style=\"font-weight:400;font-style:normal;text-decoration:none;color:#000;background-color:transparent\">90<\/span><\/td>\r\n    <td class=\"tg-4lje\"><span style=\"font-weight:400;font-style:normal;text-decoration:none;color:#000;background-color:transparent\">95<\/span><\/td>\r\n  <\/tr>\r\n  <tr>\r\n    <td class=\"tg-80go\">Etudiant 4<\/td>\r\n    <td class=\"tg-4lje\"><span style=\"font-weight:400;font-style:normal;text-decoration:none;color:#000;background-color:transparent\">22<\/span><\/td>\r\n    <td class=\"tg-4lje\"><span style=\"font-weight:400;font-style:normal;text-decoration:none;color:#000;background-color:transparent\">180<\/span><\/td>\r\n    <td class=\"tg-4lje\"><span style=\"font-weight:400;font-style:normal;text-decoration:none;color:#000;background-color:transparent\">75<\/span><\/td>\r\n    <td class=\"tg-4lje\"><span style=\"font-weight:400;font-style:normal;text-decoration:none;color:#000;background-color:transparent\">80<\/span><\/td>\r\n    <td class=\"tg-4lje\"><span style=\"font-weight:400;font-style:normal;text-decoration:none;color:#000;background-color:transparent\">70<\/span><\/td>\r\n  <\/tr>\r\n<\/tbody>\r\n<\/table>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-03d682d elementor-widget elementor-widget-text-editor\" data-id=\"03d682d\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">Apr\u00e8s application de l\u2019ACP pour r\u00e9duire la dimension, nous pourrions obtenir un nouvel ensemble de donn\u00e9es avec moins de variables, par exemple, en conservant uniquement les deux premi\u00e8res composantes principales. On obtiendrait le jeu de donn\u00e9es suivant :<\/span><\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-9d98952 elementor-widget elementor-widget-html\" data-id=\"9d98952\" data-element_type=\"widget\" data-widget_type=\"html.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<style type=\"text\/css\">\r\n.tg  {border-collapse:collapse;border-spacing:0;}\r\n.tg td{border-color:black;border-style:solid;border-width:1px;font-family:Arial, sans-serif;font-size:14px;\r\n  overflow:hidden;padding:10px 5px;word-break:normal;}\r\n.tg th{border-color:black;border-style:solid;border-width:1px;font-family:Arial, sans-serif;font-size:14px;\r\n  font-weight:normal;overflow:hidden;padding:10px 5px;word-break:normal;}\r\n.tg .tg-80go{font-family:\"Trebuchet MS\", Helvetica, sans-serif !important;text-align:left;vertical-align:top}\r\n.tg .tg-4lje{font-family:\"Trebuchet MS\", Helvetica, sans-serif !important;text-align:center;vertical-align:top}\r\n.tg .tg-7t5w{background-color:#6200c9;color:#efefef;font-family:\"Trebuchet MS\", Helvetica, sans-serif !important;font-size:20px;\r\n  font-weight:bold;text-align:center;vertical-align:top}\r\n.tg .tg-pm1a{background-color:#6200c9;color:#efefef;font-family:\"Trebuchet MS\", Helvetica, sans-serif !important;font-size:20px;\r\n  font-weight:bold;text-align:left;vertical-align:top}\r\n<\/style>\r\n<table class=\"tg\" style=\"undefined;table-layout: fixed; width: 600px\">\r\n<colgroup>\r\n<col style=\"width: 75px\">\r\n<col style=\"width: 200px\">\r\n<col style=\"width: 200px\">\r\n<\/colgroup>\r\n<thead>\r\n  <tr>\r\n    <th class=\"tg-pm1a\">Etudiant<\/th>\r\n    <th class=\"tg-7t5w\">Composante Principale 1<\/th>\r\n    <th class=\"tg-7t5w\">Composante Principale 2<\/th>\r\n  <\/tr>\r\n<\/thead>\r\n<tbody>\r\n  <tr>\r\n    <td class=\"tg-80go\">Etudiant 1<\/td>\r\n    <td class=\"tg-4lje\"><span style=\"font-weight:400;font-style:normal;text-decoration:none;color:#000;background-color:transparent\">0.23<\/span><\/td>\r\n    <td class=\"tg-4lje\"><span style=\"font-weight:400;font-style:normal;text-decoration:none;color:#000;background-color:transparent\">-0.12<\/span><\/td>\r\n  <\/tr>\r\n  <tr>\r\n    <td class=\"tg-80go\">Etudiant 2<\/td>\r\n    <td class=\"tg-4lje\"><span style=\"font-weight:400;font-style:normal;text-decoration:none;color:#000;background-color:transparent\">-0.12<\/span><\/td>\r\n    <td class=\"tg-4lje\"><span style=\"font-weight:400;font-style:normal;text-decoration:none;color:#000;background-color:transparent\">0.10<\/span><\/td>\r\n  <\/tr>\r\n  <tr>\r\n    <td class=\"tg-80go\">Etudiant 3<\/td>\r\n    <td class=\"tg-4lje\"><span style=\"font-weight:400;font-style:normal;text-decoration:none;color:#000;background-color:transparent\">0.30<\/span><\/td>\r\n    <td class=\"tg-4lje\"><span style=\"font-weight:400;font-style:normal;text-decoration:none;color:#000;background-color:transparent\">0.25<\/span><\/td>\r\n  <\/tr>\r\n  <tr>\r\n    <td class=\"tg-80go\">Etudiant 4<\/td>\r\n    <td class=\"tg-4lje\"><span style=\"font-weight:400;font-style:normal;text-decoration:none;color:#000;background-color:transparent\">-0.41<\/span><\/td>\r\n    <td class=\"tg-4lje\"><span style=\"font-weight:400;font-style:normal;text-decoration:none;color:#000;background-color:transparent\">-0.23<\/span><\/td>\r\n  <\/tr>\r\n<\/tbody>\r\n<\/table>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-33dfb51 elementor-widget elementor-widget-text-editor\" data-id=\"33dfb51\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">Ainsi, cet ensemble de donn\u00e9es est facilement repr\u00e9sentable sur un graphique en 2 dimensions : <\/span><\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-4e98343 elementor-widget elementor-widget-image\" data-id=\"4e98343\" data-element_type=\"widget\" data-widget_type=\"image.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<style>\/*! elementor - v3.15.0 - 02-08-2023 *\/\n.elementor-widget-image{text-align:center}.elementor-widget-image a{display:inline-block}.elementor-widget-image a img[src$=\".svg\"]{width:48px}.elementor-widget-image img{vertical-align:middle;display:inline-block}<\/style>\t\t\t\t\t\t\t\t\t\t\t\t<img decoding=\"async\" width=\"619\" height=\"394\" src=\"https:\/\/datascientest.com\/wp-content\/uploads\/2023\/06\/image5-3.png\" class=\"attachment-full size-full wp-image-203753\" alt=\"\" srcset=\"https:\/\/datascientest.com\/wp-content\/uploads\/2023\/06\/image5-3.png 619w, https:\/\/datascientest.com\/wp-content\/uploads\/2023\/06\/image5-3-300x191.png 300w\" sizes=\"(max-width: 619px) 100vw, 619px\" \/>\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-eb98d6b elementor-align-center button-principale elementor-widget elementor-widget-button\" data-id=\"eb98d6b\" data-element_type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm elementor-animation-grow\" href=\"https:\/\/datascientest.com\/formations-data\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t<span class=\"elementor-button-text\">Se former \u00e0 la r\u00e9duction de dimension<\/span>\n\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div class=\"elementor-element elementor-element-dffa30d e-flex e-con-boxed e-con\" data-id=\"dffa30d\" data-element_type=\"container\" data-settings=\"{&quot;content_width&quot;:&quot;boxed&quot;}\">\n\t\t\t\t\t<div class=\"e-con-inner\">\n\t\t\t\t<div class=\"elementor-element elementor-element-28a4846 elementor-widget elementor-widget-heading\" data-id=\"28a4846\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<h3 class=\"elementor-heading-title elementor-size-default\">En savoir plus sur l'Analyse en Composantes Principales (ACP)<\/h3>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-dd25e21 elementor-widget elementor-widget-text-editor\" data-id=\"dd25e21\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<span style=\"font-weight: 400;\">L\u2019une des m\u00e9thodes les plus couramment utilis\u00e9es est l\u2019Analyse en Composantes Principales (ACP). L\u2019ACP cherche \u00e0 trouver les directions principales des donn\u00e9es qui expliquent la plus grande partie de la variance observ\u00e9e. Pour cela, nous utilisons la notion de covariance.<\/span>\n\n<span style=\"font-weight: 400;\">Repr\u00e9sentons notre ensemble de donn\u00e9es, compos\u00e9 de n observations (\u00e9chantillons) et p variables (caract\u00e9ristiques), par une matrice X de taille n x p. Avant d\u2019appliquer l\u2019ACP, il est souvent recommand\u00e9 de standardiser les variables pour qu\u2019elles aient une moyenne nulle et une variance unitaire, afin que les variables avec des \u00e9chelles diff\u00e9rentes ne dominent pas l\u2019analyse.\u00a0<\/span>\n<ol>\n \t<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">On calcule la matrice de covariance de X, que l\u2019on note C. Cette matrice mesure les relations lin\u00e9aires entre les variables. Elle est calcul\u00e9e gr\u00e2ce \u00e0 la formule suivante : <strong>C = 1\/n<var>X<sup>T<\/sup><\/var> X<\/strong><\/span><\/li>\n \t<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Nous cherchons les vecteurs propres de la matrice de covariance C. Les vecteurs propres sont des directions dans l\u2019espace des variables qui d\u00e9crivent la variance des donn\u00e9es. Les vecteurs propres sont normalis\u00e9s afin que leur norme soit \u00e9gale \u00e0 1.\u00a0\u00a0<\/span><\/li>\n \t<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Nous ordonnons ces vecteurs propres par ordre d\u00e9croissant de leur valeur propre associ\u00e9e :<\/span><span style=\"font-weight: 400;\"> <strong><var>\u03bd<sub>1<\/sub><\/var><\/strong> associ\u00e9 \u00e0 <strong><var>\u03bb<sub>1<\/sub><\/var><\/strong>, <strong><var>\u03bd<sub>2<\/sub><\/var><\/strong> associ\u00e9 \u00e0 <strong><var>\u03bb<sub>2<\/sub><\/var><\/strong> et <strong><var>\u03bb<sub>1<\/sub><\/var>\u2265<var>\u03bb<sub>2<\/sub><\/var><\/strong> et ainsi de suite. Plus la valeur propre est \u00e9lev\u00e9e, plus la variance expliqu\u00e9e par la direction correspondante est grande.\u00a0<\/span><\/li>\n \t<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">On choisit un nombre k de composantes principales (souvent 1, 2 ou 3) \u00e0 conserver et on s\u00e9lectionne les k premiers vecteurs propres (<var>\u03bd<sub>1<\/sub><\/var>, <var>\u03bd<sub>2<\/sub><\/var>, &#8230; , <var>\u03bd<sub>k<\/sub><\/var>). Ces vecteurs propres forment une base orthonormale dans le nouvel espace de dimension r\u00e9duite. Ainsi, les nouvelles variables cr\u00e9\u00e9es sont d\u00e9corr\u00e9l\u00e9es. On note W la matrice form\u00e9e par ces k premiers vecteurs propres.<\/span><\/li>\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">On veut maintenant projeter nos donn\u00e9es dans ce nouvel espace, dont une base est compos\u00e9e des vecteurs (<var>\u03bd<sub>1<\/sub><\/var>, <var>\u03bd<sub>2<\/sub><\/var>, &#8230; , <var>\u03bd<sub>k<\/sub><\/var>). Pour projeter les donn\u00e9es originales dans cet espace de dimension r\u00e9duite, nous utilisons la transformation lin\u00e9aire suivante : <strong>Y = XW<\/strong>, o\u00f9 Y correspond aux coordonn\u00e9es de nos donn\u00e9es suite \u00e0 la projection.<\/li>\n<\/ol>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-328d7f5 elementor-widget elementor-widget-text-editor\" data-id=\"328d7f5\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">Pour plus de d\u00e9tails sur l\u2019ACP, vous pouvez lire <a href=\"https:\/\/datascientest.com\/acp\">notre article d\u00e9di\u00e9<\/a> sp\u00e9cifiquement \u00e0 cette m\u00e9thode.<\/span><span style=\"font-weight: 400;\"><br \/><\/span><\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-882ac71 elementor-widget elementor-widget-text-editor\" data-id=\"882ac71\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">Il existe d\u2019autres m\u00e9thodes de r\u00e9duction de dimension, les plus connues restent :<\/span><\/p><ul><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\"><strong>LDA (linear discriminant analysis)<\/strong> :<\/span><span style=\"font-weight: 400;\"> l&rsquo;analyse discriminante lin\u00e9aire qui permet d\u2019identifier les directions d\u00e9corr\u00e9l\u00e9es les unes des autres. Elle vise \u00e0 trouver une projection lin\u00e9aire des donn\u00e9es qui maximise la s\u00e9paration entre les classes tout en minimisant la variance intraclasse.<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><strong>T-SNE<\/strong><strong> (t-Distributed Stochastic Neighbor Embedding)<\/strong> : T-SNE est une m\u00e9thode non lin\u00e9aire de r\u00e9duction de dimension permettant de repr\u00e9senter des donn\u00e9es de haute dimension dans un espace de dimension r\u00e9duite (typiquement 2D ou 3D). Pour plus de d\u00e9tails \u00e0 ce sujet, lisez cet article.\u00a0Le t-NSE est particuli\u00e8rement adapt\u00e9 \u00e0 la visualisation des structures complexes et des relations non lin\u00e9aires dans les donn\u00e9es.<\/li><\/ul>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div class=\"elementor-element elementor-element-2435554 e-flex e-con-boxed e-con\" data-id=\"2435554\" data-element_type=\"container\" data-settings=\"{&quot;content_width&quot;:&quot;boxed&quot;}\">\n\t\t\t\t\t<div class=\"e-con-inner\">\n\t\t\t\t<div class=\"elementor-element elementor-element-319b942 elementor-widget elementor-widget-heading\" data-id=\"319b942\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<h3 class=\"elementor-heading-title elementor-size-default\">Limites et difficult\u00e9s de la r\u00e9duction de dimension<\/h3>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-3f2c06c elementor-widget elementor-widget-text-editor\" data-id=\"3f2c06c\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">Il est important de noter que la r\u00e9duction de dimension est un compromis entre la simplification des donn\u00e9es et la perte d\u2019information. En r\u00e9duisant la dimension, il est possible de perdre certaines informations subtiles ou des d\u00e9tails sp\u00e9cifiques. Par cons\u00e9quent, il est crucial de trouver le bon \u00e9quilibre en s\u00e9lectionnant le nombre appropri\u00e9 de composantes principales capturant la majeure partie de l\u2019information pertinente.<\/span><\/p><p><span style=\"font-weight: 400;\">Chaque m\u00e9thode a ses avantages et ses limites, et le choix de la m\u00e9thode d\u00e9pend du contexte sp\u00e9cifique et des objectifs de l\u2019analyse de donn\u00e9es.<\/span><\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-6f594ca elementor-widget elementor-widget-text-editor\" data-id=\"6f594ca\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">Si cet article vous a plu, vous pouvez d\u00e9couvrir plus en d\u00e9tail ces m\u00e9thodes dans nos formations <a href=\"https:\/\/datascientest.com\/formation-data-analyst\">Data Analyst<\/a> et <a href=\"https:\/\/datascientest.com\/formation-data-scientist\">Data Scientist<\/a>. <strong>N\u2019h\u00e9sitez pas \u00e0 les consulter ! <\/strong><\/span><\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-c327edf elementor-align-center button-principale elementor-widget elementor-widget-button\" data-id=\"c327edf\" data-element_type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm elementor-animation-grow\" href=\"https:\/\/datascientest.com\/formations-data\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t<span class=\"elementor-button-text\">D\u00e9couvrir nos formations en Data Science<\/span>\n\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div class=\"elementor-element elementor-element-a4a8bf7 e-flex e-con-boxed e-con\" data-id=\"a4a8bf7\" data-element_type=\"container\" data-settings=\"{&quot;content_width&quot;:&quot;boxed&quot;}\">\n\t\t\t\t\t<div class=\"e-con-inner\">\n\t\t\t\t<div class=\"elementor-element elementor-element-04dd1e9 elementor-widget elementor-widget-html\" data-id=\"04dd1e9\" data-element_type=\"widget\" data-widget_type=\"html.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<script type=\"application\/ld+json\">\r\n{\r\n  \"@context\": \"https:\/\/schema.org\",\r\n  \"@type\": \"FAQPage\",\r\n  \"mainEntity\": [{\r\n    \"@type\": \"Question\",\r\n    \"name\": \"Le vocabulaire de la r\u00e9duction de dimension\",\r\n    \"acceptedAnswer\": {\r\n      \"@type\": \"Answer\",\r\n      \"text\": \"L\u2019espace de dimension : nombre de caract\u00e9ristiques ou variables qui d\u00e9crivent un objet dans un dataset. Par exemple, une base de donn\u00e9es rassemblant des caract\u00e9ristiques sur des personnes peut comporter des variables telles que l\u2019\u00e2ge, le revenu, l\u2019adresse, le sexe, et le niveau d\u2019\u00e9ducation. Ceci d\u00e9finit un espace de dimension cinq. La projection consiste \u00e0 transformer les donn\u00e9es d\u2019un espace de dimension \u00e9lev\u00e9e vers un espace de dimension r\u00e9duite en utilisant des techniques de r\u00e9duction de dimension. Cette transformation permet de repr\u00e9senter les donn\u00e9es de mani\u00e8re plus compacte tout en conservant les informations importantes.\"\r\n    }\r\n  },{\r\n    \"@type\": \"Question\",\r\n    \"name\": \"En savoir plus sur l'Analyse en Composantes Principales (ACP)\",\r\n    \"acceptedAnswer\": {\r\n      \"@type\": \"Answer\",\r\n      \"text\": \"L\u2019une des m\u00e9thodes les plus couramment utilis\u00e9es est l\u2019Analyse en Composantes Principales (ACP). L\u2019ACP cherche \u00e0 trouver les directions principales des donn\u00e9es qui expliquent la plus grande partie de la variance observ\u00e9e. Pour cela, nous utilisons la notion de covariance. Repr\u00e9sentons notre ensemble de donn\u00e9es, compos\u00e9 de n observations (\u00e9chantillons) et p variables (caract\u00e9ristiques), par une matrice X de taille n x p. Avant d\u2019appliquer l\u2019ACP, il est souvent recommand\u00e9 de standardiser les variables pour qu\u2019elles aient une moyenne nulle et une variance unitaire, afin que les variables avec des \u00e9chelles diff\u00e9rentes ne dominent pas l\u2019analyse.\"\r\n    }\r\n  },{\r\n    \"@type\": \"Question\",\r\n    \"name\": \"Limites et difficult\u00e9s de la r\u00e9duction de dimension\",\r\n    \"acceptedAnswer\": {\r\n      \"@type\": \"Answer\",\r\n      \"text\": \"Il est important de noter que la r\u00e9duction de dimension est un compromis entre la simplification des donn\u00e9es et la perte d\u2019information. En r\u00e9duisant la dimension, il est possible de perdre certaines informations subtiles ou des d\u00e9tails sp\u00e9cifiques. Par cons\u00e9quent, il est crucial de trouver le bon \u00e9quilibre en s\u00e9lectionnant le nombre appropri\u00e9 de composantes principales capturant la majeure partie de l\u2019information pertinente.\"\r\n    }\r\n  }]\r\n}\r\n<\/script>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t<\/div>\n\t\t","protected":false},"excerpt":{"rendered":"<p>L\u2019analyse de donn\u00e9es peut \u00eatre consid\u00e9r\u00e9e comme une science exp\u00e9rimentale, o\u00f9 les propri\u00e9t\u00e9s sont d\u00e9montr\u00e9es apr\u00e8s avoir \u00e9t\u00e9 observ\u00e9es, et o\u00f9 des codes sont \u00e9tablis de mani\u00e8re heuristique pour interpr\u00e9ter les r\u00e9sultats. Dans le domaine de l\u2019apprentissage automatique (machine learning) et de l\u2019analyse des donn\u00e9es, la r\u00e9duction de dimension joue un r\u00f4le essentiel pour simplifier [&hellip;]<\/p>\n","protected":false},"author":77,"featured_media":203743,"comment_status":"open","ping_status":"open","sticky":false,"template":"elementor_theme","format":"standard","meta":{"content-type":"","inline_featured_image":false},"categories":[2497],"tags":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v21.7 (Yoast SEO v21.7) - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>R\u00e9duction de dimension : Comment \u00e7a fonctionne ?<\/title>\n<meta name=\"description\" content=\"Venez d\u00e9couvrir dans cet article les techniques math\u00e9matiques de r\u00e9duction de dimension telles que l\u2019Analyse en Composantes Principales (ACP) et l\u2019Analyse Discriminante Lin\u00e9aire (LDA), tr\u00e8s utilis\u00e9es aujourd\u2019hui en analyse de donn\u00e9es.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/datascientest.com\/reduction-de-dimension\" \/>\n<meta property=\"og:locale\" content=\"fr_FR\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"R\u00e9duction de dimension : Comment \u00e7a fonctionne ?\" \/>\n<meta property=\"og:url\" content=\"https:\/\/datascientest.com\/reduction-de-dimension\" \/>\n<meta property=\"og:site_name\" content=\"Formation Data Science | DataScientest.com\" \/>\n<meta property=\"article:published_time\" content=\"2023-06-16T12:18:56+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2023-11-09T13:51:57+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/datascientest.com\/wp-content\/uploads\/2023\/06\/Illu_BLOG__Reduction_dimension.png\" \/>\n\t<meta property=\"og:image:width\" content=\"1024\" \/>\n\t<meta property=\"og:image:height\" content=\"563\" \/>\n\t<meta property=\"og:image:type\" content=\"image\/png\" \/>\n<meta name=\"author\" content=\"Raphael Kassel\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"\u00c9crit par\" \/>\n\t<meta name=\"twitter:data1\" content=\"Raphael Kassel\" \/>\n\t<meta name=\"twitter:label2\" content=\"Dur\u00e9e de lecture estim\u00e9e\" \/>\n\t<meta name=\"twitter:data2\" content=\"6 minutes\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"WebPage\",\"@id\":\"https:\/\/datascientest.com\/reduction-de-dimension\",\"url\":\"https:\/\/datascientest.com\/reduction-de-dimension\",\"name\":\"R\u00e9duction de dimension : Comment \u00e7a fonctionne ?\",\"isPartOf\":{\"@id\":\"https:\/\/datascientest.com\/#website\"},\"datePublished\":\"2023-06-16T12:18:56+00:00\",\"dateModified\":\"2023-11-09T13:51:57+00:00\",\"author\":{\"@id\":\"https:\/\/datascientest.com\/#\/schema\/person\/6eca16f63daaafcbc96e15549b9d4ee1\"},\"description\":\"Venez d\u00e9couvrir dans cet article les techniques math\u00e9matiques de r\u00e9duction de dimension telles que l\u2019Analyse en Composantes Principales (ACP) et l\u2019Analyse Discriminante Lin\u00e9aire (LDA), tr\u00e8s utilis\u00e9es aujourd\u2019hui en analyse de donn\u00e9es.\",\"inLanguage\":\"fr-FR\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\/\/datascientest.com\/reduction-de-dimension\"]}]},{\"@type\":\"WebSite\",\"@id\":\"https:\/\/datascientest.com\/#website\",\"url\":\"https:\/\/datascientest.com\/\",\"name\":\"Formation Data Science | DataScientest.com\",\"description\":\"Leader de la formation en Data Science\",\"potentialAction\":[{\"@type\":\"SearchAction\",\"target\":{\"@type\":\"EntryPoint\",\"urlTemplate\":\"https:\/\/datascientest.com\/?s={search_term_string}\"},\"query-input\":\"required name=search_term_string\"}],\"inLanguage\":\"fr-FR\"},{\"@type\":\"Person\",\"@id\":\"https:\/\/datascientest.com\/#\/schema\/person\/6eca16f63daaafcbc96e15549b9d4ee1\",\"name\":\"Raphael Kassel\",\"image\":{\"@type\":\"ImageObject\",\"inLanguage\":\"fr-FR\",\"@id\":\"https:\/\/datascientest.com\/#\/schema\/person\/image\/\",\"url\":\"https:\/\/datascientest.com\/wp-content\/uploads\/2023\/11\/raph.jpeg\",\"contentUrl\":\"https:\/\/datascientest.com\/wp-content\/uploads\/2023\/11\/raph.jpeg\",\"caption\":\"Raphael Kassel\"},\"description\":\"Raphael Kassel est le Head of Data Programs chez DataScientest. 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