{"id":475955,"date":"2023-08-09T07:24:43","date_gmt":"2023-08-09T07:24:43","guid":{"rendered":""},"modified":"2024-06-11T19:24:00","modified_gmt":"2024-06-11T19:24:00","slug":"auto-regressive-models","status":"publish","type":"wiki","link":"https:\/\/oneproxy.pro\/fr\/wiki\/auto-regressive-models\/","title":{"rendered":"Mod\u00e8les auto-r\u00e9gressifs"},"content":{"rendered":"<p>Les mod\u00e8les auto-r\u00e9gressifs sont une classe de mod\u00e8les statistiques largement utilis\u00e9s dans divers domaines, notamment le traitement du langage naturel, l&#039;analyse de s\u00e9ries chronologiques et la g\u00e9n\u00e9ration d&#039;images. Ces mod\u00e8les pr\u00e9disent une s\u00e9quence de valeurs bas\u00e9es sur des valeurs observ\u00e9es pr\u00e9c\u00e9demment, ce qui les rend bien adapt\u00e9s aux t\u00e2ches impliquant des donn\u00e9es s\u00e9quentielles. Les mod\u00e8les auto-r\u00e9gressifs se sont r\u00e9v\u00e9l\u00e9s tr\u00e8s efficaces pour g\u00e9n\u00e9rer des donn\u00e9es r\u00e9alistes et pr\u00e9dire les r\u00e9sultats futurs.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">L&#039;histoire de l&#039;origine des mod\u00e8les auto-r\u00e9gressifs et la premi\u00e8re mention de ceux-ci<\/h2>\n\n\n\n<p>Le concept d\u2019auto-r\u00e9gression remonte au d\u00e9but du XXe si\u00e8cle, avec les travaux pionniers r\u00e9alis\u00e9s par le statisticien britannique Yule en 1927. Cependant, ce sont les travaux du math\u00e9maticien Norbert Wiener dans les ann\u00e9es 1940 qui ont jet\u00e9 les bases des mod\u00e8les auto-r\u00e9gressifs modernes. Les recherches de Wiener sur les processus stochastiques et la pr\u00e9diction ont jet\u00e9 les bases du d\u00e9veloppement des mod\u00e8les auto-r\u00e9gressifs tels que nous les connaissons aujourd&#039;hui.<\/p>\n\n\n\n<p>Le terme \u00ab auto-r\u00e9gressif \u00bb a \u00e9t\u00e9 introduit pour la premi\u00e8re fois dans le domaine \u00e9conomique par Ragnar Frisch \u00e0 la fin des ann\u00e9es 1920. Frisch a utilis\u00e9 ce terme pour d\u00e9crire un mod\u00e8le qui r\u00e9gresse une variable par rapport \u00e0 ses propres valeurs d\u00e9cal\u00e9es, capturant ainsi la d\u00e9pendance d&#039;une variable par rapport \u00e0 son propre pass\u00e9.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Mod\u00e8les auto-r\u00e9gressifs\u00a0: informations d\u00e9taill\u00e9es<\/h2>\n\n\n\n<p>Les mod\u00e8les auto-r\u00e9gressifs (AR) sont des outils essentiels dans l&#039;analyse de s\u00e9ries chronologiques, utilis\u00e9s pour pr\u00e9voir les valeurs futures sur la base de donn\u00e9es historiques. Ces mod\u00e8les supposent que les valeurs pass\u00e9es influencent les valeurs actuelles et futures de mani\u00e8re lin\u00e9aire. Ils sont largement utilis\u00e9s en \u00e9conomie, en finance, en pr\u00e9visions m\u00e9t\u00e9orologiques et dans divers autres domaines o\u00f9 les donn\u00e9es de s\u00e9ries chronologiques sont r\u00e9pandues.<\/p><h3>Repr\u00e9sentation math\u00e9matique<\/h3><p>Un mod\u00e8le d\u2019ordre auto-r\u00e9gressif <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>p<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">p<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.625em; vertical-align: -0.1944em;\"><\/span><span class=\"mord mathnormal\">p<\/span><\/span><\/span><\/span> (AR(p)) est math\u00e9matiquement exprim\u00e9 comme suit\u00a0:\n<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>Oui<\/mi><mi>t<\/mi><\/msub><mo>=<\/mo><msub><mi>\u03d5<\/mi><mn>1<\/mn><\/msub><msub><mi>Oui<\/mi><mrow><mi>t<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msub><mo>+<\/mo><msub><mi>\u03d5<\/mi><mn>2<\/mn><\/msub><msub><mi>Oui<\/mi><mrow><mi>t<\/mi><mo>\u2212<\/mo><mn>2<\/mn><\/mrow><\/msub><mo>+<\/mo><mo>\u22ef<\/mo><mo>+<\/mo><msub><mi>\u03d5<\/mi><mi>p<\/mi><\/msub><msub><mi>Oui<\/mi><mrow><mi>t<\/mi><mo>\u2212<\/mo><mi>p<\/mi><\/mrow><\/msub><mo>+<\/mo><msub><mi>\u03f5<\/mi><mi>t<\/mi><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">Y_t = \\phi_1 Y_{t-1} + \\phi_2 Y_{t-2} + \\cdots + \\phi_p Y_{tp} + \\epsilon_t<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.8333em; vertical-align: -0.15em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.22222em;\">Oui<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.2806em;\"><span style=\"top: -2.55em; margin-left: -0.2222em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">t<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2778em;\"><\/span><span class=\"mrel\">=<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.9028em; vertical-align: -0.2083em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">\u03d5<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3011em;\"><span style=\"top: -2.55em; margin-left: 0em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.22222em;\">Oui<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3011em;\"><span style=\"top: -2.55em; margin-left: -0.2222em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">t<\/span><span class=\"mbin mtight\">\u2212<\/span><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.2083em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2222em;\"><\/span><span class=\"mbin\">+<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.9028em; vertical-align: -0.2083em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">\u03d5<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3011em;\"><span style=\"top: -2.55em; margin-left: 0em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.22222em;\">Oui<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3011em;\"><span style=\"top: -2.55em; margin-left: -0.2222em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">t<\/span><span class=\"mbin mtight\">\u2212<\/span><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.2083em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2222em;\"><\/span><span class=\"mbin\">+<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.6667em; vertical-align: -0.0833em;\"><\/span><span class=\"minner\">\u22ef<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em;\"><\/span><span class=\"mbin\">+<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.9805em; vertical-align: -0.2861em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">\u03d5<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.1514em;\"><span style=\"top: -2.55em; margin-left: 0em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">p<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.2861em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.22222em;\">Oui<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.2806em;\"><span style=\"top: -2.55em; margin-left: -0.2222em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">t<\/span><span class=\"mbin mtight\">\u2212<\/span><span class=\"mord mathnormal mtight\">p<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.2861em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2222em;\"><\/span><span class=\"mbin\">+<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.5806em; vertical-align: -0.15em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">\u03f5<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.2806em;\"><span style=\"top: -2.55em; margin-left: 0em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">t<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p><p>O\u00f9:<\/p><ul><li><span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>Oui<\/mi><mi>t<\/mi><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">Yt<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.8333em; vertical-align: -0.15em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.22222em;\">Oui<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.2806em;\"><span style=\"top: -2.55em; margin-left: -0.2222em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">t<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> est la valeur de la s\u00e9rie \u00e0 l&#039;instant <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>t<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">t<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6151em;\"><\/span><span class=\"mord mathnormal\">t<\/span><\/span><\/span><\/span>.<\/li><li><span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>\u03d5<\/mi><mn>1<\/mn><\/msub><mo separator=\"true\">,<\/mo><msub><mi>\u03d5<\/mi><mn>2<\/mn><\/msub><mo separator=\"true\">,<\/mo><mo>\u2026<\/mo><mo separator=\"true\">,<\/mo><msub><mi>\u03d5<\/mi><mi>p<\/mi><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">\\phi_1, \\phi_2, \\ldots, \\phi_p<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.9805em; vertical-align: -0.2861em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">\u03d5<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3011em;\"><span style=\"top: -2.55em; margin-left: 0em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">\u03d5<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3011em;\"><span style=\"top: -2.55em; margin-left: 0em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em;\"><\/span><span class=\"minner\">\u2026<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em;\"><\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">\u03d5<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.1514em;\"><span style=\"top: -2.55em; margin-left: 0em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">p<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.2861em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> sont les coefficients du mod\u00e8le.<\/li><li><span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>Oui<\/mi><mrow><mi>t<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msub><mo separator=\"true\">,<\/mo><msub><mi>Oui<\/mi><mrow><mi>t<\/mi><mo>\u2212<\/mo><mn>2<\/mn><\/mrow><\/msub><mo separator=\"true\">,<\/mo><mo>\u2026<\/mo><mo separator=\"true\">,<\/mo><msub><mi>Oui<\/mi><mrow><mi>t<\/mi><mo>\u2212<\/mo><mi>p<\/mi><\/mrow><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">Y_{t-1}, Y_{t-2}, \\ldots, Y_{tp}<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.9694em; vertical-align: -0.2861em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.22222em;\">Oui<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3011em;\"><span style=\"top: -2.55em; margin-left: -0.2222em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">t<\/span><span class=\"mbin mtight\">\u2212<\/span><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.2083em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.22222em;\">Oui<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3011em;\"><span style=\"top: -2.55em; margin-left: -0.2222em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">t<\/span><span class=\"mbin mtight\">\u2212<\/span><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.2083em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em;\"><\/span><span class=\"minner\">\u2026<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em;\"><\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.22222em;\">Oui<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.2806em;\"><span style=\"top: -2.55em; margin-left: -0.2222em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">t<\/span><span class=\"mbin mtight\">\u2212<\/span><span class=\"mord mathnormal mtight\">p<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.2861em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> sont les valeurs pass\u00e9es de la s\u00e9rie.<\/li><li><span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>\u03f5<\/mi><mi>t<\/mi><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">\\epsilon_t<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.5806em; vertical-align: -0.15em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">\u03f5<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.2806em;\"><span style=\"top: -2.55em; margin-left: 0em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">t<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> est le terme d&#039;erreur \u00e0 l&#039;instant <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>t<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">t<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6151em;\"><\/span><span class=\"mord mathnormal\">t<\/span><\/span><\/span><\/span>, g\u00e9n\u00e9ralement suppos\u00e9 \u00eatre du bruit blanc avec une moyenne nulle et une variance constante.<\/li><\/ul><h3>D\u00e9termination de la commande (p)<\/h3><p>L&#039;ordre <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>p<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">p<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.625em; vertical-align: -0.1944em;\"><\/span><span class=\"mord mathnormal\">p<\/span><\/span><\/span><\/span> d&#039;un mod\u00e8le AR est crucial car il d\u00e9termine le nombre d&#039;observations pass\u00e9es \u00e0 inclure dans le mod\u00e8le. Le choix de <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>p<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">p<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.625em; vertical-align: -0.1944em;\"><\/span><span class=\"mord mathnormal\">p<\/span><\/span><\/span><\/span> implique un compromis\u00a0:<\/p><ul><li><strong>Ordre inf\u00e9rieur<\/strong> mod\u00e8les (petits <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>p<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">p<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.625em; vertical-align: -0.1944em;\"><\/span><span class=\"mord mathnormal\">p<\/span><\/span><\/span><\/span>) peut ne pas parvenir \u00e0 capturer tous les mod\u00e8les pertinents dans les donn\u00e9es, conduisant \u00e0 un sous-apprentissage.<\/li><li><strong>Ordre sup\u00e9rieur<\/strong> mod\u00e8les (grands <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>p<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">p<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.625em; vertical-align: -0.1944em;\"><\/span><span class=\"mord mathnormal\">p<\/span><\/span><\/span><\/span>) peut capturer des mod\u00e8les plus complexes mais risque un surajustement, o\u00f9 le mod\u00e8le d\u00e9crit un bruit al\u00e9atoire au lieu du processus sous-jacent.<\/li><\/ul><p>M\u00e9thodes courantes pour d\u00e9terminer l\u2019ordre optimal <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>p<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">p<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.625em; vertical-align: -0.1944em;\"><\/span><span class=\"mord mathnormal\">p<\/span><\/span><\/span><\/span> inclure:<\/p><ul><li><strong>Fonction d&#039;autocorr\u00e9lation partielle (PACF)<\/strong>: Identifie les d\u00e9calages importants qui doivent \u00eatre inclus.<\/li><li><strong>Crit\u00e8res d&#039;information<\/strong>: Des crit\u00e8res tels que l&#039;ad\u00e9quation et la complexit\u00e9 du mod\u00e8le d&#039;\u00e9quilibre du crit\u00e8re d&#039;information d&#039;Akaike (AIC) et du crit\u00e8re d&#039;information bay\u00e9sien (BIC) permettent de choisir un mod\u00e8le appropri\u00e9. <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>p<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">p<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.625em; vertical-align: -0.1944em;\"><\/span><span class=\"mord mathnormal\">p<\/span><\/span><\/span><\/span>.<\/li><\/ul><h3>Estimation du mod\u00e8le<\/h3><p>Estimation des param\u00e8tres <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>\u03d5<\/mi><mn>1<\/mn><\/msub><mo separator=\"true\">,<\/mo><msub><mi>\u03d5<\/mi><mn>2<\/mn><\/msub><mo separator=\"true\">,<\/mo><mo>\u2026<\/mo><mo separator=\"true\">,<\/mo><msub><mi>\u03d5<\/mi><mi>p<\/mi><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">\\phi_1, \\phi_2, \\ldots, \\phi_p<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.9805em; vertical-align: -0.2861em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">\u03d5<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3011em;\"><span style=\"top: -2.55em; margin-left: 0em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">\u03d5<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3011em;\"><span style=\"top: -2.55em; margin-left: 0em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em;\"><\/span><span class=\"minner\">\u2026<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em;\"><\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">\u03d5<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.1514em;\"><span style=\"top: -2.55em; margin-left: 0em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">p<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.2861em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> implique d\u2019ajuster le mod\u00e8le aux donn\u00e9es historiques. Cela peut \u00eatre fait en utilisant des techniques telles que\u00a0:<\/p><ul><li><strong>Estimation des moindres carr\u00e9s<\/strong>: minimise la somme des erreurs quadratiques entre les valeurs observ\u00e9es et pr\u00e9dites.<\/li><li><strong>Estimation de vraisemblance maximale<\/strong>: Trouve les param\u00e8tres qui maximisent la probabilit\u00e9 d\u2019observer les donn\u00e9es donn\u00e9es.<\/li><\/ul><h3>Diagnostic du mod\u00e8le<\/h3><p>Apr\u00e8s avoir ajust\u00e9 un mod\u00e8le AR, il est essentiel d\u2019\u00e9valuer son ad\u00e9quation. Les principaux contr\u00f4les de diagnostic comprennent\u00a0:<\/p><ul><li><strong>Analyse r\u00e9siduelle<\/strong>: garantit que les r\u00e9sidus (erreurs) ressemblent \u00e0 du bruit blanc, indiquant qu&#039;aucun motif n&#039;est laiss\u00e9 inexpliqu\u00e9 par le mod\u00e8le.<\/li><li><strong>Test de Ljung-Box<\/strong>: \u00e9value si l&#039;une des autocorr\u00e9lations des r\u00e9sidus est significativement diff\u00e9rente de z\u00e9ro.<\/li><\/ul><h3>Applications<\/h3><p>Les mod\u00e8les AR sont polyvalents et trouvent des applications dans divers domaines\u00a0:<\/p><ul><li><strong>\u00c9conomie et Finance<\/strong>: Pr\u00e9visions des cours boursiers, des taux d&#039;int\u00e9r\u00eat et des indicateurs \u00e9conomiques.<\/li><li><strong>Pr\u00e9vision m\u00e9t\u00e9o<\/strong>: Pr\u00e9dire les mod\u00e8les de temp\u00e9rature et de pr\u00e9cipitations.<\/li><li><strong>Ing\u00e9nierie<\/strong>: Syst\u00e8mes de traitement et de contr\u00f4le du signal.<\/li><li><strong>Biostatistique<\/strong>: Mod\u00e9lisation de donn\u00e9es de s\u00e9ries chronologiques biologiques.<\/li><\/ul><h3>Avantages et limites<\/h3><p><strong>Avantages :<\/strong><\/p><ul><li>Simplicit\u00e9 et facilit\u00e9 de mise en \u0153uvre.<\/li><li>Interpr\u00e9tation claire des param\u00e8tres.<\/li><li>Efficace pour les pr\u00e9visions \u00e0 court terme.<\/li><\/ul><p><strong>Limites:<\/strong><\/p><ul><li>Suppose des relations lin\u00e9aires.<\/li><li>Peut \u00eatre inad\u00e9quat pour les donn\u00e9es pr\u00e9sentant une forte saisonnalit\u00e9 ou des mod\u00e8les non lin\u00e9aires.<\/li><li>Sensible au choix de la commande <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>p<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">p<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.625em; vertical-align: -0.1944em;\"><\/span><span class=\"mord mathnormal\">p<\/span><\/span><\/span><\/span>.<\/li><\/ul><h3>Exemple<\/h3><p>Consid\u00e9rons un mod\u00e8le AR(2) (ordre 2) pour les donn\u00e9es de s\u00e9ries chronologiques\u00a0:\n<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>Oui<\/mi><mi>t<\/mi><\/msub><mo>=<\/mo><mn>0.5<\/mn><msub><mi>Oui<\/mi><mrow><mi>t<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msub><mo>+<\/mo><mn>0.2<\/mn><msub><mi>Oui<\/mi><mrow><mi>t<\/mi><mo>\u2212<\/mo><mn>2<\/mn><\/mrow><\/msub><mo>+<\/mo><msub><mi>\u03f5<\/mi><mi>t<\/mi><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">Y_t = 0,5 Y_{t-1} + 0,2 Y_{t-2} + \\epsilon_t<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.8333em; vertical-align: -0.15em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.22222em;\">Oui<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.2806em;\"><span style=\"top: -2.55em; margin-left: -0.2222em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">t<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2778em;\"><\/span><span class=\"mrel\">=<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.8917em; vertical-align: -0.2083em;\"><\/span><span class=\"mord\">0.5<\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.22222em;\">Oui<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3011em;\"><span style=\"top: -2.55em; margin-left: -0.2222em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">t<\/span><span class=\"mbin mtight\">\u2212<\/span><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.2083em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2222em;\"><\/span><span class=\"mbin\">+<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.8917em; vertical-align: -0.2083em;\"><\/span><span class=\"mord\">0.2<\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.22222em;\">Oui<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3011em;\"><span style=\"top: -2.55em; margin-left: -0.2222em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">t<\/span><span class=\"mbin mtight\">\u2212<\/span><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.2083em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2222em;\"><\/span><span class=\"mbin\">+<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.5806em; vertical-align: -0.15em;\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">\u03f5<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.2806em;\"><span style=\"top: -2.55em; margin-left: 0em; margin-right: 0.05em;\"><span class=\"pstrut\" style=\"height: 2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">t<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em;\"><span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\nIci, la valeur \u00e0 l&#039;heure <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>t<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">t<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6151em;\"><\/span><span class=\"mord mathnormal\">t<\/span><\/span><\/span><\/span> d\u00e9pend des valeurs aux deux instants pr\u00e9c\u00e9dents, avec des coefficients respectivement de 0,5 et 0,2.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Analyse des principales caract\u00e9ristiques des mod\u00e8les auto-r\u00e9gressifs<\/h2>\n\n\n\n<p>Les mod\u00e8les auto-r\u00e9gressifs offrent plusieurs fonctionnalit\u00e9s cl\u00e9s qui les rendent utiles pour diverses applications\u00a0:<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>Pr\u00e9diction de s\u00e9quence<\/strong>: Les mod\u00e8les auto-r\u00e9gressifs excellent dans la pr\u00e9vision des valeurs futures dans une s\u00e9quence chronologique, ce qui les rend id\u00e9aux pour la pr\u00e9vision de s\u00e9ries chronologiques.<\/li>\n\n\n\n<li><strong>Capacit\u00e9s g\u00e9n\u00e9ratives<\/strong>: Ces mod\u00e8les peuvent g\u00e9n\u00e9rer de nouveaux \u00e9chantillons de donn\u00e9es qui ressemblent aux donn\u00e9es d&#039;entra\u00eenement, ce qui les rend utiles pour l&#039;augmentation des donn\u00e9es et les t\u00e2ches cr\u00e9atives telles que la g\u00e9n\u00e9ration de texte et d&#039;images.<\/li>\n\n\n\n<li><strong>La flexibilit\u00e9<\/strong>: Les mod\u00e8les auto-r\u00e9gressifs peuvent s&#039;adapter \u00e0 diff\u00e9rents types de donn\u00e9es et ne sont pas limit\u00e9s \u00e0 un domaine sp\u00e9cifique, permettant leur application dans divers domaines.<\/li>\n\n\n\n<li><strong>Interpr\u00e9tabilit\u00e9<\/strong>: La simplicit\u00e9 de la structure du mod\u00e8le permet une interpr\u00e9tation facile de ses param\u00e8tres et pr\u00e9dictions.<\/li>\n\n\n\n<li><strong>Adaptabilit\u00e9<\/strong>: Les mod\u00e8les auto-r\u00e9gressifs peuvent s&#039;adapter \u00e0 l&#039;\u00e9volution des mod\u00e8les de donn\u00e9es et incorporer de nouvelles informations au fil du temps.<\/li>\n<\/ol>\n\n\n\n<h2 class=\"wp-block-heading\">Types de mod\u00e8les auto-r\u00e9gressifs<\/h2>\n\n\n\n<p>Les mod\u00e8les auto-r\u00e9gressifs se pr\u00e9sentent sous diverses formes, chacune ayant ses propres caract\u00e9ristiques sp\u00e9cifiques. Les principaux types de mod\u00e8les auto-r\u00e9gressifs comprennent\u00a0:<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>Mod\u00e8les auto-r\u00e9gressifs \u00e0 moyenne mobile (ARMA)<\/strong>: Combine les composants d&#039;auto-r\u00e9gression et de moyenne mobile pour tenir compte \u00e0 la fois des erreurs pr\u00e9sentes et pass\u00e9es.<\/li>\n\n\n\n<li><strong>Mod\u00e8les de moyenne mobile int\u00e9gr\u00e9e auto-r\u00e9gressive (ARIMA)<\/strong>: \u00e9tend ARMA en incorporant la diff\u00e9renciation pour obtenir la stationnarit\u00e9 des donn\u00e9es de s\u00e9ries chronologiques non stationnaires.<\/li>\n\n\n\n<li><strong>Mod\u00e8les saisonniers de moyenne mobile int\u00e9gr\u00e9e auto-r\u00e9gressive (SARIMA)<\/strong>: Une version saisonni\u00e8re d&#039;ARIMA, adapt\u00e9e aux donn\u00e9es de s\u00e9ries chronologiques avec des mod\u00e8les saisonniers.<\/li>\n\n\n\n<li><strong>Mod\u00e8les vectoriels auto-r\u00e9gressifs (VAR)<\/strong>: Une extension multivari\u00e9e des mod\u00e8les auto-r\u00e9gressifs, utilis\u00e9e lorsque plusieurs variables s&#039;influencent mutuellement.<\/li>\n\n\n\n<li><strong>R\u00e9seaux de m\u00e9moire \u00e0 long terme et \u00e0 court terme (LSTM)<\/strong>: Type de r\u00e9seau neuronal r\u00e9current capable de capturer des d\u00e9pendances \u00e0 longue port\u00e9e dans des donn\u00e9es s\u00e9quentielles, souvent utilis\u00e9 dans les t\u00e2ches de traitement du langage naturel et de reconnaissance vocale.<\/li>\n\n\n\n<li><strong>Mod\u00e8les de transformateurs<\/strong>: Un type d&#039;architecture de r\u00e9seau neuronal qui utilise des m\u00e9canismes d&#039;attention pour traiter des donn\u00e9es s\u00e9quentielles, connu pour son succ\u00e8s dans la traduction linguistique et la g\u00e9n\u00e9ration de texte.<\/li>\n<\/ol>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"1061\" height=\"440\" src=\"https:\/\/oneproxy.pro\/wp-content\/uploads\/2024\/06\/auto-egressive-model.png\" alt=\"Mod\u00e8les autor\u00e9gressifs pour le traitement du langage naturel\" class=\"wp-image-505503\" title=\"Mod\u00e8les autor\u00e9gressifs pour le traitement du langage naturel\" srcset=\"https:\/\/oneproxy.pro\/wp-content\/uploads\/2024\/06\/auto-egressive-model.png 1061w, https:\/\/oneproxy.pro\/wp-content\/uploads\/2024\/06\/auto-egressive-model-150x62.png 150w, https:\/\/oneproxy.pro\/wp-content\/uploads\/2024\/06\/auto-egressive-model-768x318.png 768w, https:\/\/oneproxy.pro\/wp-content\/uploads\/2024\/06\/auto-egressive-model-18x7.png 18w\" sizes=\"auto, (max-width: 1061px) 100vw, 1061px\" \/><figcaption class=\"wp-element-caption\">Mod\u00e8les autor\u00e9gressifs pour le traitement du langage naturel<\/figcaption><\/figure>\n\n\n\n<p>Voici un tableau comparatif r\u00e9sumant les principales caract\u00e9ristiques de ces mod\u00e8les auto-r\u00e9gressifs :<\/p>\n\n\n\n<figure class=\"wp-block-table\"><table><thead><tr><th>Mod\u00e8le<\/th><th>Principales caract\u00e9ristiques<\/th><th>Application<\/th><\/tr><\/thead><tbody><tr><td>ARM\u00c9E<\/td><td>R\u00e9gression automatique, moyenne mobile<\/td><td>Pr\u00e9visions de s\u00e9ries chronologiques<\/td><\/tr><tr><td>ARIMA<\/td><td>Auto-r\u00e9gression, int\u00e9gr\u00e9e, moyenne mobile<\/td><td>Donn\u00e9es financi\u00e8res, tendances \u00e9conomiques<\/td><\/tr><tr><td>SARIMA<\/td><td>Auto-r\u00e9gression saisonni\u00e8re, int\u00e9gr\u00e9e, moyenne mobile<\/td><td>Donn\u00e9es climatiques, tendances saisonni\u00e8res<\/td><\/tr><tr><td>VAR<\/td><td>Multivari\u00e9, auto-r\u00e9gression<\/td><td>Mod\u00e9lisation macro\u00e9conomique<\/td><\/tr><tr><td>LSTM<\/td><td>R\u00e9seau neuronal r\u00e9current<\/td><td>Traitement du langage naturel<\/td><\/tr><tr><td>Transformateur<\/td><td>M\u00e9canisme d&#039;attention, traitement parall\u00e8le<\/td><td>G\u00e9n\u00e9ration de texte, traduction<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<h2 class=\"wp-block-heading\">Fa\u00e7ons d&#039;utiliser les mod\u00e8les auto-r\u00e9gressifs, probl\u00e8mes et leurs solutions li\u00e9s \u00e0 l&#039;utilisation<\/h2>\n\n\n\n<p>Les mod\u00e8les auto-r\u00e9gressifs trouvent des applications dans un large \u00e9ventail de domaines\u00a0:<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>Pr\u00e9visions de s\u00e9ries chronologiques<\/strong>: Pr\u00e9dire les cours des actions, les conditions m\u00e9t\u00e9orologiques ou le trafic sur un site Web.<\/li>\n\n\n\n<li><strong>Traitement du langage naturel<\/strong>: G\u00e9n\u00e9ration de texte, traduction linguistique, analyse des sentiments.<\/li>\n\n\n\n<li><strong>G\u00e9n\u00e9ration d&#039;images<\/strong>: Cr\u00e9ation d&#039;images r\u00e9alistes \u00e0 l&#039;aide de r\u00e9seaux contradictoires g\u00e9n\u00e9ratifs (GAN).<\/li>\n\n\n\n<li><strong>Composition musicale<\/strong>: G\u00e9n\u00e9rer de nouvelles s\u00e9quences et compositions musicales.<\/li>\n\n\n\n<li><strong>D\u00e9tection d&#039;une anomalie<\/strong>: Identification des valeurs aberrantes dans les donn\u00e9es de s\u00e9ries chronologiques.<\/li>\n<\/ol>\n\n\n\n<p>Malgr\u00e9 leurs atouts, les mod\u00e8les auto-r\u00e9gressifs pr\u00e9sentent certaines limites\u00a0:<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>M\u00e9moire \u00e0 court terme<\/strong>: Ils peuvent avoir du mal \u00e0 capturer les d\u00e9pendances \u00e0 longue port\u00e9e dans les donn\u00e9es.<\/li>\n\n\n\n<li><strong>Surapprentissage<\/strong>: Les mod\u00e8les auto-r\u00e9gressifs d&#039;ordre \u00e9lev\u00e9 peuvent surajuster le bruit dans les donn\u00e9es.<\/li>\n\n\n\n<li><strong>Stationnarit\u00e9 des donn\u00e9es<\/strong>: Les mod\u00e8les de type ARIMA n\u00e9cessitent des donn\u00e9es stationnaires, ce qui peut \u00eatre difficile \u00e0 r\u00e9aliser en pratique.<\/li>\n<\/ol>\n\n\n\n<p>Pour relever ces d\u00e9fis, les chercheurs ont propos\u00e9 diverses solutions :<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>R\u00e9seaux de neurones r\u00e9currents (RNN)<\/strong>: Ils offrent de meilleures capacit\u00e9s de m\u00e9moire \u00e0 long terme.<\/li>\n\n\n\n<li><strong>Techniques de r\u00e9gularisation<\/strong>: Utilis\u00e9 pour \u00e9viter le surajustement dans les mod\u00e8les d&#039;ordre \u00e9lev\u00e9.<\/li>\n\n\n\n<li><strong>Diff\u00e9rence saisonni\u00e8re<\/strong>: Pour obtenir la stationnarit\u00e9 des donn\u00e9es dans les donn\u00e9es saisonni\u00e8res.<\/li>\n\n\n\n<li><strong>M\u00e9canismes d&#039;attention<\/strong>: Am\u00e9liorez la gestion des d\u00e9pendances \u00e0 longue port\u00e9e dans les mod\u00e8les Transformer.<\/li>\n<\/ol>\n\n\n\n<h2 class=\"wp-block-heading\">Principales caract\u00e9ristiques et autres comparaisons avec des termes similaires<\/h2>\n\n\n\n<p>Les mod\u00e8les auto-r\u00e9gressifs sont souvent compar\u00e9s \u00e0 d\u2019autres mod\u00e8les de s\u00e9ries chronologiques, tels que\u00a0:<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>Mod\u00e8les de moyenne mobile (MA)<\/strong>: Concentrez-vous uniquement sur la relation entre la valeur actuelle et les erreurs pass\u00e9es, alors que les mod\u00e8les auto-r\u00e9gressifs consid\u00e8rent les valeurs pass\u00e9es de la variable.<\/li>\n\n\n\n<li><strong>Mod\u00e8les de moyenne mobile auto-r\u00e9gressive (ARMA)<\/strong>: Combinez les composants auto-r\u00e9gressifs et de moyenne mobile, offrant une approche plus compl\u00e8te de la mod\u00e9lisation des donn\u00e9es de s\u00e9ries chronologiques.<\/li>\n\n\n\n<li><strong>Mod\u00e8les de moyenne mobile int\u00e9gr\u00e9e auto-r\u00e9gressive (ARIMA)<\/strong>: Incorporer la diff\u00e9renciation pour obtenir la stationnarit\u00e9 des donn\u00e9es de s\u00e9ries chronologiques non stationnaires.<\/li>\n<\/ol>\n\n\n\n<p>Voici un tableau comparatif mettant en \u00e9vidence les principales diff\u00e9rences entre ces mod\u00e8les de s\u00e9ries chronologiques\u00a0:<\/p>\n\n\n\n<figure class=\"wp-block-table\"><table><thead><tr><th>Mod\u00e8le<\/th><th>Principales caract\u00e9ristiques<\/th><th>Application<\/th><\/tr><\/thead><tbody><tr><td>Auto-r\u00e9gressif (AR)<\/td><td>R\u00e9gression par rapport aux valeurs pass\u00e9es<\/td><td>Pr\u00e9visions de s\u00e9ries chronologiques<\/td><\/tr><tr><td>Moyenne mobile (MA)<\/td><td>R\u00e9gression contre les erreurs pass\u00e9es<\/td><td>Filtrage du bruit<\/td><\/tr><tr><td>Moyenne mobile auto-r\u00e9gressive (ARMA)<\/td><td>Combinaison de composants AR et MA<\/td><td>Pr\u00e9vision de s\u00e9ries chronologiques, filtrage du bruit<\/td><\/tr><tr><td>Moyenne mobile int\u00e9gr\u00e9e auto-r\u00e9gressive (ARIMA)<\/td><td>Diff\u00e9rence pour la stationnarit\u00e9<\/td><td>Donn\u00e9es financi\u00e8res, tendances \u00e9conomiques<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<h2 class=\"wp-block-heading\">Perspectives et technologies du futur li\u00e9es aux mod\u00e8les auto-r\u00e9gressifs<\/h2>\n\n\n\n<p>Les mod\u00e8les auto-r\u00e9gressifs continuent d&#039;\u00e9voluer, stimul\u00e9s par les progr\u00e8s de l&#039;apprentissage profond et du traitement du langage naturel. L\u2019avenir des mod\u00e8les auto-r\u00e9gressifs impliquera probablement\u00a0:<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>Architectures plus complexes<\/strong>: Les chercheurs exploreront des structures de r\u00e9seau plus complexes et des combinaisons de mod\u00e8les auto-r\u00e9gressifs avec d&#039;autres architectures telles que les transformateurs et les LSTM.<\/li>\n\n\n\n<li><strong>M\u00e9canismes d&#039;attention<\/strong>: Les m\u00e9canismes d&#039;attention seront affin\u00e9s pour am\u00e9liorer les d\u00e9pendances \u00e0 longue port\u00e9e dans les donn\u00e9es s\u00e9quentielles.<\/li>\n\n\n\n<li><strong>Formation efficace<\/strong>: Des efforts seront faits pour r\u00e9duire les exigences de calcul pour la formation de mod\u00e8les auto-r\u00e9gressifs \u00e0 grande \u00e9chelle.<\/li>\n\n\n\n<li><strong>Apprentissage non supervis\u00e9<\/strong>: Des mod\u00e8les auto-r\u00e9gressifs seront utilis\u00e9s pour des t\u00e2ches d&#039;apprentissage non supervis\u00e9es, telles que la d\u00e9tection d&#039;anomalies et l&#039;apprentissage de repr\u00e9sentations.<\/li>\n<\/ol>\n\n\n\n<h2 class=\"wp-block-heading\">Comment les serveurs proxy peuvent \u00eatre utilis\u00e9s ou associ\u00e9s \u00e0 des mod\u00e8les auto-r\u00e9gressifs<\/h2>\n\n\n\n<p>Les serveurs proxy peuvent jouer un r\u00f4le important dans l\u2019am\u00e9lioration des performances des mod\u00e8les auto-r\u00e9gressifs, notamment dans certaines applications :<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>Collecte de donn\u00e9es<\/strong>: Lors de la collecte de donn\u00e9es de formation pour des mod\u00e8les auto-r\u00e9gressifs, des serveurs proxy peuvent \u00eatre utilis\u00e9s pour anonymiser et diversifier les sources de donn\u00e9es, garantissant ainsi une repr\u00e9sentation plus compl\u00e8te de la distribution des donn\u00e9es.<\/li>\n\n\n\n<li><strong>Augmentation des donn\u00e9es<\/strong>: Les serveurs proxy permettent de g\u00e9n\u00e9rer des points de donn\u00e9es suppl\u00e9mentaires en acc\u00e9dant \u00e0 diff\u00e9rentes sources en ligne et en simulant diverses interactions des utilisateurs, ce qui contribue \u00e0 am\u00e9liorer la g\u00e9n\u00e9ralisation du mod\u00e8le.<\/li>\n\n\n\n<li><strong>L&#039;\u00e9quilibrage de charge<\/strong>: Dans les applications \u00e0 grande \u00e9chelle, les serveurs proxy peuvent r\u00e9partir la charge d&#039;inf\u00e9rence sur plusieurs serveurs, garantissant ainsi un d\u00e9ploiement efficace et \u00e9volutif de mod\u00e8les auto-r\u00e9gressifs.<\/li>\n\n\n\n<li><strong>Confidentialit\u00e9 et s\u00e9curit\u00e9<\/strong>: Les serveurs proxy agissent comme interm\u00e9diaires entre les clients et les serveurs, fournissant une couche suppl\u00e9mentaire de s\u00e9curit\u00e9 et de confidentialit\u00e9 pour les applications sensibles utilisant des mod\u00e8les auto-r\u00e9gressifs.<\/li>\n<\/ol>\n\n\n\n<h2 class=\"wp-block-heading\">Liens connexes<\/h2>\n\n\n\n<p>Pour plus d&#039;informations sur les mod\u00e8les auto-r\u00e9gressifs, vous pouvez explorer les ressources suivantes\u00a0:<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li><a href=\"https:\/\/www.wiley.com\/en-us\/Time+Series+Analysis%3A+Forecasting+and+Control%2C+5th+Edition-p-9781118675021\" target=\"_new\" rel=\"noopener nofollow\">Analyse des s\u00e9ries chronologiques\u00a0: pr\u00e9vision et contr\u00f4le par George Box et Gwilym Jenkins<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/www.mitpressjournals.org\/doi\/pdf\/10.1162\/neco.1997.9.8.1735\" target=\"_new\" rel=\"noopener nofollow\">R\u00e9seaux de m\u00e9moire \u00e0 long terme (LSTM)<\/a><\/li>\n\n\n\n<li><a href=\"http:\/\/jalammar.github.io\/illustrated-transformer\/\" target=\"_new\" rel=\"noopener nofollow\">Le transformateur illustr\u00e9 par Jay Alammar<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/towardsdatascience.com\/an-introduction-to-time-series-analysis-and-forecasting-in-python-af7eeb238a64\" target=\"_new\" rel=\"noopener nofollow\">Une introduction \u00e0 l&#039;analyse et \u00e0 la pr\u00e9vision des s\u00e9ries chronologiques en Python<\/a><\/li>\n<\/ol>\n\n\n\n<p>Les mod\u00e8les auto-r\u00e9gressifs sont devenus un outil fondamental pour diverses t\u00e2ches li\u00e9es aux donn\u00e9es, permettant des pr\u00e9dictions pr\u00e9cises et une g\u00e9n\u00e9ration de donn\u00e9es r\u00e9alistes. \u00c0 mesure que la recherche progresse dans ce domaine, nous pouvons nous attendre \u00e0 l\u2019\u00e9mergence de mod\u00e8les encore plus avanc\u00e9s et efficaces, r\u00e9volutionnant ainsi la fa\u00e7on dont nous traitons les donn\u00e9es s\u00e9quentielles \u00e0 l\u2019avenir.<\/p>","protected":false},"featured_media":497623,"menu_order":0,"template":"","meta":{"_acf_changed":false,"content-type":"","inline_featured_image":false,"footnotes":""},"class_list":["post-475955","wiki","type-wiki","status-publish","has-post-thumbnail","hentry"],"acf":{"faq_title":"Frequently Asked Questions about <mark>Auto-regressive models: A Comprehensive Overview<\/mark>","faq_items":[{"question":"Question 1: What are Auto-regressive models?","answer":"Answer 1: Auto-regressive models are statistical models used to predict future values based on past observations. They are particularly effective for tasks involving sequential data, such as time-series analysis, natural language processing, and image generation. These models regress a variable against its own lagged values to capture dependencies and patterns in the data."},{"question":"Question 2: How did Auto-regressive models originate?","answer":"Answer 2: The concept of auto-regression dates back to the early 20th century, with contributions from statisticians such as Yule and economist Ragnar Frisch. The term \"auto-regressive\" was first introduced by Norbert Wiener in the 1940s, who laid the foundation for modern auto-regressive models through his work on stochastic processes and prediction."},{"question":"Question 3: How do Auto-regressive models work?","answer":"Answer 3: Auto-regressive models use past values of a variable to predict its current value. The model is trained using the method of least squares to estimate its parameters. Once trained, it can generate future values by recursively predicting based on its own past predictions."},{"question":"Question 4: What are the key features of Auto-regressive models?","answer":"Answer 4: Auto-regressive models offer sequence prediction, generative capabilities, flexibility, interpretability, and adaptability. They excel at forecasting future values in a time-ordered sequence and can generate new data samples resembling the training data. Their simplicity allows for easy interpretation, making them valuable in various applications."},{"question":"Question 5: What types of Auto-regressive models exist?","answer":"Answer 5: There are various types of Auto-regressive models, including Moving Average Auto-regressive (ARMA), Auto-regressive Integrated Moving Average (ARIMA), Seasonal Auto-regressive Integrated Moving Average (SARIMA), Vector Auto-regressive (VAR), Long Short-Term Memory (LSTM) networks, and Transformer models. Each type has specific characteristics suitable for different applications."},{"question":"Question 6: How can Auto-regressive models be used, and what challenges do they face?","answer":"Answer 6: Auto-regressive models are used in time-series forecasting, natural language processing, image generation, music composition, and anomaly detection. However, they may struggle with long-term memory, overfitting, and the need for data stationarity in ARIMA-type models. Solutions include using RNNs for better long-term memory and regularization techniques to prevent overfitting."},{"question":"Question 7: How do Auto-regressive models compare to other time-series models?","answer":"Answer 7: Auto-regressive models are compared with Moving Average (MA) models, Auto-regressive Moving Average (ARMA) models, and Auto-regressive Integrated Moving Average (ARIMA) models. Each model has distinct characteristics, with ARIMA incorporating differencing for stationarity in non-stationary time-series data."},{"question":"Question 8: What are the perspectives and future technologies related to Auto-regressive models?","answer":"Answer 8: The future of Auto-regressive models involves more complex architectures, improved attention mechanisms for better long-range dependencies, and efforts to reduce training computational requirements. They will likely find applications in unsupervised learning, anomaly detection, and representation learning."},{"question":"Question 9: How are proxy servers associated with Auto-regressive models?","answer":"Answer 9: Proxy servers can enhance the performance of Auto-regressive models by anonymizing and diversifying data sources during data collection. They enable data augmentation, load balancing, and add an extra layer of privacy and security for sensitive applications using Auto-regressive models."},{"question":"Question 10: Where can I find more information about Auto-regressive models?","answer":"Answer 10: For further information, you can explore the book \"Time Series Analysis: Forecasting and Control\" by George Box and Gwilym Jenkins, or learn more about Long Short-Term Memory (LSTM) networks from the article \"The Illustrated Transformer\" by Jay Alammar. Additionally, you can find resources on time series analysis and forecasting in Python for practical insights."}]},"_links":{"self":[{"href":"https:\/\/oneproxy.pro\/fr\/wp-json\/wp\/v2\/wiki\/475955","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/oneproxy.pro\/fr\/wp-json\/wp\/v2\/wiki"}],"about":[{"href":"https:\/\/oneproxy.pro\/fr\/wp-json\/wp\/v2\/types\/wiki"}],"version-history":[{"count":4,"href":"https:\/\/oneproxy.pro\/fr\/wp-json\/wp\/v2\/wiki\/475955\/revisions"}],"predecessor-version":[{"id":505505,"href":"https:\/\/oneproxy.pro\/fr\/wp-json\/wp\/v2\/wiki\/475955\/revisions\/505505"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/oneproxy.pro\/fr\/wp-json\/wp\/v2\/media\/497623"}],"wp:attachment":[{"href":"https:\/\/oneproxy.pro\/fr\/wp-json\/wp\/v2\/media?parent=475955"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}