La r\u00e9gression lin\u00e9aire : pr\u00e9dire l’avenir avec une droite \u2014 MathsVivantes<\/title>\n<link href=\"https:\/\/fonts.googleapis.com\/css2?family=Playfair+Display:ital,wght@0,400;0,700;1,400&family=Source+Serif+4:ital,wght@0,300;0,400;0,600;1,300;1,400&display=swap\" rel=\"stylesheet\">\n<style>\n :root {\n --noir: #0f0e0b;\n --creme: #f5f0e8;\n --bleu: #1a3a5c;\n --bleu-clair: #4a90d9;\n --ocre: #c8850a;\n --ocre-clair: #f0c060;\n --rouge: #b03a2e;\n --gris: #6b6560;\n --gris-clair: #e8e2d8;\n }\n\n * { margin: 0; padding: 0; box-sizing: border-box; }\n\n body {\n background-color: var(--creme);\n color: var(--noir);\n font-family: 'Source Serif 4', Georgia, serif;\n font-weight: 300;\n line-height: 1.8;\n }\n\n \/* HERO *\/\n .hero {\n background: #0d2137;\n color: var(--creme);\n padding: 80px 40px 70px;\n position: relative;\n overflow: hidden;\n }\n .hero::before {\n content: \"\u0177 \u03b2 \u03c3 \u2211\";\n position: absolute;\n top: 20px; right: 40px;\n font-size: 5rem;\n opacity: 0.05;\n font-family: 'Playfair Display', serif;\n letter-spacing: 0.3em;\n }\n .hero-inner { max-width: 760px; 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}\n .article-body { padding: 40px 20px 60px; }\n .exercice { padding: 24px 20px; }\n .code-block { padding: 16px; font-size: 0.78rem; }\n }\n<\/style>\n<\/head>\n<body>\n\n\n<div class=\"hero\">\n <div class=\"hero-inner\">\n <div class=\"article-rubrique\">Data & Maths \u00b7 Article #3<\/div>\n <h1>La r\u00e9gression lin\u00e9aire :<br><em>pr\u00e9dire l’avenir avec une droite<\/em><\/h1>\n <p class=\"hero-intro\">Une vendeuse de tomates au march\u00e9 de Dantokpa le fait intuitivement chaque matin. Un data scientist le fait avec Python. Et Galton l’a d\u00e9couvert au XIX<sup>e<\/sup> si\u00e8cle en \u00e9tudiant la taille des p\u00e8res et de leurs fils. C’est la r\u00e9gression lin\u00e9aire \u2014 l’un des outils les plus puissants et les plus \u00e9l\u00e9gants des math\u00e9matiques appliqu\u00e9es.<\/p>\n <div class=\"meta\">Par Odilon AKOWANOU \u00b7 MathsVivantes \u00b7 Lecture : 11 min<\/div>\n <\/div>\n<\/div>\n\n\n<article class=\"article-body\">\n\n <p>\n Chaque matin au march\u00e9 de Dantokpa \u00e0 Cotonou, Mama Adjoavi regarde ses stocks de tomates, observe les prix des jours pr\u00e9c\u00e9dents, et d\u00e9cide combien vendre aujourd’hui pour maximiser ses b\u00e9n\u00e9fices. Elle ne le sait pas, mais elle fait de la r\u00e9gression lin\u00e9aire \u2014 instinctivement, dans sa t\u00eate.\n <\/p>\n\n <p>\n La r\u00e9gression lin\u00e9aire, c’est l’art de <strong>trouver la droite qui passe au plus pr\u00e8s d’un nuage de points<\/strong>. C’est simple \u00e0 comprendre, puissant \u00e0 utiliser, et c’est la porte d’entr\u00e9e de toute la data science moderne.\n <\/p>\n\n <h2>L’histoire : Galton et les fils qui ressemblent \u00e0 leurs p\u00e8res<\/h2>\n\n <div class=\"encadre-histoire encadre\">\n <span class=\"encadre-titre\">\ud83d\udcdc Naissance d’une id\u00e9e<\/span>\n <p>\n En 1886, le statisticien britannique <strong>Francis Galton<\/strong> \u00e9tudie un ph\u00e9nom\u00e8ne curieux : les fils de p\u00e8res tr\u00e8s grands sont souvent grands, mais rarement aussi grands que leurs p\u00e8res. Et les fils de p\u00e8res tr\u00e8s petits sont souvent petits, mais rarement aussi petits.\n <\/p>\n <p>\n Il appelle ce ph\u00e9nom\u00e8ne la <em>\u00ab\u00a0r\u00e9gression vers la moyenne\u00a0\u00bb<\/em> \u2014 les valeurs extr\u00eames tendent \u00e0 revenir vers la moyenne de la population. Pour mod\u00e9liser cette relation entre la taille du p\u00e8re et celle du fils, il invente une m\u00e9thode graphique : tracer la <strong>droite qui r\u00e9sume le mieux la relation<\/strong> entre deux variables.\n <\/p>\n <p>\n N\u00e9 d’une observation biologique sur des familles anglaises du XIX<sup>e<\/sup> si\u00e8cle, cet outil voyage aujourd’hui dans les algorithmes de Netflix, les mod\u00e8les m\u00e9t\u00e9o de l’ASECNA, et les pr\u00e9dictions de r\u00e9colte des agriculteurs b\u00e9ninois.\n <\/p>\n <\/div>\n\n <h2>L’intuition : une droite dans un nuage<\/h2>\n\n <p>\n Imagine que tu collectes des donn\u00e9es dans un village agricole du Borgou : la quantit\u00e9 de pluie (en mm) et la production de ma\u00efs (en kg) pour 6 saisons cons\u00e9cutives.\n <\/p>\n\n <div class=\"table-container\">\n <table>\n <tr>\n <th>Saison<\/th>\n <th>Pluie (mm) \u2014 x<\/th>\n <th>Ma\u00efs (kg) \u2014 y<\/th>\n <\/tr>\n <tr><td>1<\/td><td>100<\/td><td>200<\/td><\/tr>\n <tr><td>2<\/td><td>150<\/td><td>280<\/td><\/tr>\n <tr><td>3<\/td><td>200<\/td><td>350<\/td><\/tr>\n <tr><td>4<\/td><td>120<\/td><td>230<\/td><\/tr>\n <tr><td>5<\/td><td>180<\/td><td>320<\/td><\/tr>\n <tr><td>6<\/td><td>250<\/td><td>420<\/td><\/tr>\n <\/table>\n <\/div>\n\n <p>\n Si tu traces ces points sur un graphe, tu verras qu’ils forment un nuage avec une tendance claire : <strong>plus il pleut, plus la production est \u00e9lev\u00e9e<\/strong>. La r\u00e9gression lin\u00e9aire consiste \u00e0 trouver la droite qui r\u00e9sume le mieux cette tendance.\n <\/p>\n\n <div class=\"schema-container\">\n <div class=\"schema-titre\">\ud83d\udcc8 Nuage de points et droite de r\u00e9gression<\/div>\n <svg width=\"100%\" viewBox=\"0 0 520 320\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\">\n \n <rect width=\"520\" height=\"320\" fill=\"white\"\/>\n\n \n <line x1=\"60\" y1=\"270\" x2=\"480\" y2=\"270\" stroke=\"#0f0e0b\" stroke-width=\"2\"\/>\n <line x1=\"60\" y1=\"270\" x2=\"60\" y2=\"30\" stroke=\"#0f0e0b\" stroke-width=\"2\"\/>\n\n \n <polygon points=\"480,265 490,270 480,275\" fill=\"#0f0e0b\"\/>\n <polygon points=\"55,30 60,20 65,30\" fill=\"#0f0e0b\"\/>\n\n \n <text x=\"490\" y=\"274\" font-size=\"12\" fill=\"#6b6560\" font-family=\"serif\">x<\/text>\n <text x=\"40\" y=\"25\" font-size=\"12\" fill=\"#6b6560\" font-family=\"serif\">y<\/text>\n <text x=\"220\" y=\"300\" font-size=\"11\" fill=\"#6b6560\" font-family=\"serif\">Pluie (mm)<\/text>\n <text x=\"10\" y=\"160\" font-size=\"11\" fill=\"#6b6560\" font-family=\"serif\" transform=\"rotate(-90,15,160)\">Ma\u00efs (kg)<\/text>\n\n \n <text x=\"88\" y=\"285\" font-size=\"10\" fill=\"#6b6560\">100<\/text>\n <text x=\"148\" y=\"285\" font-size=\"10\" fill=\"#6b6560\">150<\/text>\n <text x=\"208\" y=\"285\" font-size=\"10\" fill=\"#6b6560\">200<\/text>\n <text x=\"268\" y=\"285\" font-size=\"10\" fill=\"#6b6560\">250<\/text>\n\n \n <text x=\"30\" y=\"235\" font-size=\"10\" fill=\"#6b6560\">200<\/text>\n <text x=\"30\" y=\"185\" font-size=\"10\" fill=\"#6b6560\">280<\/text>\n <text x=\"30\" y=\"140\" font-size=\"10\" fill=\"#6b6560\">350<\/text>\n <text x=\"30\" y=\"75\" font-size=\"10\" fill=\"#6b6560\">420<\/text>\n\n \n <line x1=\"80\" y1=\"248\" x2=\"440\" y2=\"58\" stroke=\"#4a90d9\" stroke-width=\"2.5\" stroke-dasharray=\"0\"\/>\n\n \n \n <circle cx=\"100\" cy=\"238\" r=\"6\" fill=\"#1a3a5c\"\/>\n \n <circle cx=\"150\" cy=\"192\" r=\"6\" fill=\"#1a3a5c\"\/>\n \n <circle cx=\"210\" cy=\"138\" r=\"6\" fill=\"#1a3a5c\"\/>\n \n <circle cx=\"122\" cy=\"226\" r=\"6\" fill=\"#1a3a5c\"\/>\n \n <circle cx=\"190\" cy=\"158\" r=\"6\" fill=\"#1a3a5c\"\/>\n \n <circle cx=\"268\" cy=\"72\" r=\"6\" fill=\"#1a3a5c\"\/>\n\n \n <circle cx=\"330\" cy=\"50\" r=\"6\" fill=\"#1a3a5c\"\/>\n <text x=\"342\" y=\"54\" font-size=\"11\" fill=\"#6b6560\">Donn\u00e9es r\u00e9elles<\/text>\n <line x1=\"325\" y1=\"70\" x2=\"355\" y2=\"70\" stroke=\"#4a90d9\" stroke-width=\"2.5\"\/>\n <text x=\"362\" y=\"74\" font-size=\"11\" fill=\"#6b6560\">Droite de r\u00e9gression<\/text>\n\n \n <text x=\"290\" y=\"110\" font-size=\"12\" fill=\"#4a90d9\" font-style=\"italic\">\u0177 = ax + b<\/text>\n <\/svg>\n <\/div>\n\n <h2>La formule : d’o\u00f9 vient cette droite ?<\/h2>\n\n <p>\n La droite de r\u00e9gression s’\u00e9crit :\n <\/p>\n\n <div class=\"formule\">\u0177 = ax + b<\/div>\n\n <p>O\u00f9 :<\/p>\n\n <div class=\"encadre\">\n <span class=\"encadre-titre\">\ud83d\udccc Signification des param\u00e8tres<\/span>\n <p><strong>\u0177<\/strong> (y chapeau) = la valeur <em>pr\u00e9dite<\/em> par le mod\u00e8le<\/p>\n <p><strong>x<\/strong> = la variable d’entr\u00e9e (ici, la quantit\u00e9 de pluie)<\/p>\n <p><strong>a<\/strong> = la pente de la droite (combien y augmente quand x augmente de 1)<\/p>\n <p><strong>b<\/strong> = l’ordonn\u00e9e \u00e0 l’origine (valeur de y quand x = 0)<\/p>\n <\/div>\n\n <p>\n Pour trouver les valeurs de <strong>a<\/strong> et <strong>b<\/strong> qui donnent la <em>meilleure<\/em> droite possible, on utilise la m\u00e9thode des <strong>moindres carr\u00e9s<\/strong> : on minimise la somme des carr\u00e9s des \u00e9carts entre les valeurs r\u00e9elles et les valeurs pr\u00e9dites.\n <\/p>\n\n <div class=\"definition\">\n <strong>M\u00e9thode des moindres carr\u00e9s :<\/strong> Trouver a et b tels que la somme \u2211(y\u1d62 – \u0177\u1d62)\u00b2 soit minimale. Autrement dit, la droite doit \u00eatre aussi proche que possible de tous les points \u00e0 la fois.\n <\/div>\n\n <p>Les formules math\u00e9matiques pour calculer a et b sont :<\/p>\n\n <div class=\"etape\"><span class=\"etape-numero\">\u2460<\/span> a = \u2211(x\u1d62 – x\u0304)(y\u1d62 – \u0233) \/ \u2211(x\u1d62 – x\u0304)\u00b2<\/div>\n <div class=\"etape\"><span class=\"etape-numero\">\u2461<\/span> b = \u0233 – a \u00d7 x\u0304<\/div>\n <div class=\"etape\"><span class=\"etape-numero\">\u2462<\/span> o\u00f9 x\u0304 = moyenne de x et \u0233 = moyenne de y<\/div>\n\n <h2>Application : les donn\u00e9es du Borgou<\/h2>\n\n <p>Reprenons nos donn\u00e9es agricoles et calculons a et b :<\/p>\n\n <div class=\"table-container\">\n <table>\n <tr>\n <th>x<\/th><th>y<\/th><th>x – x\u0304<\/th><th>y – \u0233<\/th><th>(x-x\u0304)(y-\u0233)<\/th><th>(x-x\u0304)\u00b2<\/th>\n <\/tr>\n <tr><td>100<\/td><td>200<\/td><td>-66,7<\/td><td>-100<\/td><td>6670<\/td><td>4448,9<\/td><\/tr>\n <tr><td>150<\/td><td>280<\/td><td>-16,7<\/td><td>-20<\/td><td>334<\/td><td>278,9<\/td><\/tr>\n <tr><td>200<\/td><td>350<\/td><td>33,3<\/td><td>50<\/td><td>1665<\/td><td>1108,9<\/td><\/tr>\n <tr><td>120<\/td><td>230<\/td><td>-46,7<\/td><td>-70<\/td><td>3269<\/td><td>2180,9<\/td><\/tr>\n <tr><td>180<\/td><td>320<\/td><td>13,3<\/td><td>20<\/td><td>266<\/td><td>176,9<\/td><\/tr>\n <tr><td>250<\/td><td>420<\/td><td>83,3<\/td><td>120<\/td><td>9996<\/td><td>6938,9<\/td><\/tr>\n <tr><td><strong>x\u0304=166,7<\/strong><\/td><td><strong>\u0233=300<\/strong><\/td><td>\u2014<\/td><td>\u2014<\/td><td><strong>\u2211=22200<\/strong><\/td><td><strong>\u2211=15133<\/strong><\/td><\/tr>\n <\/table>\n <\/div>\n\n <div class=\"etape\"><span class=\"etape-numero\">\u2460<\/span> a = 22200 \/ 15133 \u2248 1,47<\/div>\n <div class=\"etape\"><span class=\"etape-numero\">\u2461<\/span> b = 300 – 1,47 \u00d7 166,7 \u2248 54,9<\/div>\n\n <div class=\"formule\">\u0177 = 1,47x + 54,9<\/div>\n\n <p>\n Interpr\u00e9tation concr\u00e8te : <strong>chaque millim\u00e8tre de pluie suppl\u00e9mentaire apporte environ 1,47 kg de ma\u00efs en plus<\/strong>. Si la m\u00e9t\u00e9o annonce 220 mm de pluie pour la prochaine saison, on pr\u00e9dit : \u0177 = 1,47 \u00d7 220 + 54,9 \u2248 <strong>378 kg de ma\u00efs<\/strong>.\n <\/p>\n\n <h2>Et maintenant, avec Python<\/h2>\n\n <p>\n Faire ces calculs \u00e0 la main, c’est bien pour comprendre. Mais en pratique, on utilise Python. Voici comment faire la m\u00eame chose en quelques lignes :\n <\/p>\n\n <div class=\"code-block\">\n <span class=\"code-label\">Python<\/span>\n<span class=\"code-comment\"># R\u00e9gression lin\u00e9aire – Production de ma\u00efs au Borgou<\/span>\n<span class=\"code-keyword\">import<\/span> numpy <span class=\"code-keyword\">as<\/span> np\n<span class=\"code-keyword\">from<\/span> sklearn.linear_model <span class=\"code-keyword\">import<\/span> LinearRegression\n<span class=\"code-keyword\">import<\/span> matplotlib.pyplot <span class=\"code-keyword\">as<\/span> plt\n\n<span class=\"code-comment\"># Donn\u00e9es : pluie (mm) et production ma\u00efs (kg)<\/span>\nx = np.array([<span class=\"code-number\">100<\/span>, <span class=\"code-number\">150<\/span>, <span class=\"code-number\">200<\/span>, <span class=\"code-number\">120<\/span>, <span class=\"code-number\">180<\/span>, <span class=\"code-number\">250<\/span>]).reshape(-<span class=\"code-number\">1<\/span>, <span class=\"code-number\">1<\/span>)\ny = np.array([<span class=\"code-number\">200<\/span>, <span class=\"code-number\">280<\/span>, <span class=\"code-number\">350<\/span>, <span class=\"code-number\">230<\/span>, <span class=\"code-number\">320<\/span>, <span class=\"code-number\">420<\/span>])\n\n<span class=\"code-comment\"># Cr\u00e9er et entra\u00eener le mod\u00e8le<\/span>\nmodele = <span class=\"code-function\">LinearRegression<\/span>()\nmodele.<span class=\"code-function\">fit<\/span>(x, y)\n\n<span class=\"code-comment\"># Afficher les param\u00e8tres<\/span>\n<span class=\"code-function\">print<\/span>(<span class=\"code-string\">f\u00a0\u00bbPente a = {modele.coef_[0]:.2f}\u00a0\u00bb<\/span>)\n<span class=\"code-function\">print<\/span>(<span class=\"code-string\">f\u00a0\u00bbOrdonn\u00e9e b = {modele.intercept_:.2f}\u00a0\u00bb<\/span>)\n\n<span class=\"code-comment\"># Pr\u00e9dire pour 220 mm de pluie<\/span>\nprediction = modele.<span class=\"code-function\">predict<\/span>([[<span class=\"code-number\">220<\/span>]])\n<span class=\"code-function\">print<\/span>(<span class=\"code-string\">f\u00a0\u00bbPr\u00e9diction pour 220mm : {prediction[0]:.0f} kg\u00a0\u00bb<\/span>)\n\n<span class=\"code-comment\"># Tracer le graphique<\/span>\nplt.<span class=\"code-function\">scatter<\/span>(x, y, color=<span class=\"code-string\">‘#1a3a5c’<\/span>, label=<span class=\"code-string\">‘Donn\u00e9es r\u00e9elles’<\/span>)\nplt.<span class=\"code-function\">plot<\/span>(x, modele.<span class=\"code-function\">predict<\/span>(x), color=<span class=\"code-string\">‘#4a90d9’<\/span>, label=<span class=\"code-string\">‘Droite de r\u00e9gression’<\/span>)\nplt.<span class=\"code-function\">xlabel<\/span>(<span class=\"code-string\">‘Pluie (mm)’<\/span>)\nplt.<span class=\"code-function\">ylabel<\/span>(<span class=\"code-string\">‘Production ma\u00efs (kg)’<\/span>)\nplt.<span class=\"code-function\">title<\/span>(<span class=\"code-string\">‘R\u00e9gression lin\u00e9aire – Borgou, B\u00e9nin’<\/span>)\nplt.<span class=\"code-function\">legend<\/span>()\nplt.<span class=\"code-function\">show<\/span>()\n <\/div>\n\n <div class=\"encadre\">\n <span class=\"encadre-titre\">\ud83d\udcbb R\u00e9sultat attendu<\/span>\n <p><strong>Pente a = 1.47<\/strong> \u2192 identique \u00e0 notre calcul manuel \u2705<\/p>\n <p><strong>Ordonn\u00e9e b = 54.90<\/strong> \u2192 identique aussi \u2705<\/p>\n <p><strong>Pr\u00e9diction pour 220mm : 378 kg<\/strong><\/p>\n <\/div>\n\n <h2>Les limites \u00e0 conna\u00eetre<\/h2>\n\n <p>\n La r\u00e9gression lin\u00e9aire est puissante, mais elle n’est pas magique. Voici ce qu’il faut garder en t\u00eate :\n <\/p>\n\n <div class=\"encadre\">\n <span class=\"encadre-titre\">\u26a0\ufe0f Attention !<\/span>\n <p><strong>1. Elle suppose une relation lin\u00e9aire.<\/strong> Si la vraie relation est courbe, la droite sera une mauvaise approximation.<\/p>\n <p><strong>2. Elle est sensible aux valeurs aberrantes.<\/strong> Un point tr\u00e8s \u00e9loign\u00e9 des autres peut tirer la droite vers lui.<\/p>\n <p><strong>3. Corr\u00e9lation \u2260 causalit\u00e9.<\/strong> Ce que la pluie et le ma\u00efs ont en commun ne prouve pas que l’un cause l’autre \u2014 m\u00eame si ici c’est logique !<\/p>\n <p><strong>4. Ne pas extrapoler trop loin.<\/strong> Pr\u00e9dire pour 2000 mm de pluie avec un mod\u00e8le calibr\u00e9 entre 100 et 250 mm serait hasardeux.<\/p>\n <\/div>\n\n <blockquote>\n Tous les mod\u00e8les sont faux, mais certains sont utiles.\n <cite>\u2014 George Box, statisticien britannique<\/cite>\n <\/blockquote>\n\n \n <div class=\"exercice\">\n <h3>\u270f\ufe0f Exercice corrig\u00e9<\/h3>\n <p>\n Un vendeur de sachets d’eau \u00e0 Cotonou note ses ventes sur 5 jours de forte chaleur :\n <\/p>\n <p>\n Temp\u00e9rature (\u00b0C) : 32, 34, 36, 35, 38<br>\n Sachets vendus : 120, 145, 170, 158, 200\n <\/p>\n <p><strong>a)<\/strong> Calcule la moyenne de x (temp\u00e9rature) et de y (sachets).<\/p>\n <p><strong>b)<\/strong> Calcule la pente a de la droite de r\u00e9gression.<\/p>\n <p><strong>c)<\/strong> Si demain la temp\u00e9rature atteint 40\u00b0C, combien de sachets faut-il pr\u00e9voir ?<\/p>\n\n <div class=\"correction\">\n <div class=\"correction-titre\">\u25b8 Correction<\/div>\n <p><strong>a)<\/strong> x\u0304 = (32+34+36+35+38)\/5 = 175\/5 = <strong>35\u00b0C<\/strong><\/p>\n <p>\u0233 = (120+145+170+158+200)\/5 = 793\/5 = <strong>158,6 sachets<\/strong><\/p>\n <p><strong>b)<\/strong> Calcul de a :<\/p>\n <p>\u2211(x\u1d62-x\u0304)(y\u1d62-\u0233) = (-3)(-38,6)+(-1)(-13,6)+(1)(11,4)+(0)(-0,6)+(3)(41,4)<\/p>\n <p>= 115,8 + 13,6 + 11,4 + 0 + 124,2 = <strong>265<\/strong><\/p>\n <p>\u2211(x\u1d62-x\u0304)\u00b2 = 9+1+1+0+9 = <strong>20<\/strong><\/p>\n <p>a = 265\/20 = <strong>13,25<\/strong><\/p>\n <p>b = 158,6 – 13,25 \u00d7 35 = 158,6 – 463,75 = <strong>-305,15<\/strong><\/p>\n <p><strong>c)<\/strong> \u0177 = 13,25 \u00d7 40 – 305,15 = 530 – 305,15 \u2248 <strong>225 sachets<\/strong> \u00e0 pr\u00e9voir \u2705<\/p>\n <\/div>\n <\/div>\n\n \n <div class=\"conclusion\">\n <h2>Ce qu’on retient<\/h2>\n <p>\n La r\u00e9gression lin\u00e9aire, c’est le pont entre les math\u00e9matiques pures et la data science. Elle transforme un nuage de points chaotique en une droite qui pr\u00e9dit, qui explique, qui d\u00e9cide. De Galton aux march\u00e9s de Cotonou, en passant par les champs du Borgou, c’est le m\u00eame outil \u2014 \u00e9l\u00e9gant, simple, universel.\n <\/p>\n <p>\n Et le plus beau ? Tu viens de comprendre le fondement math\u00e9matique d’algorithmes qui font tourner des entreprises valant des milliards. Pas mal pour une droite.\n <\/p>\n <p>\n Dans le prochain article, nous d\u00e9couvrirons pourquoi les <strong>abeilles construisent des hexagones<\/strong> \u2014 et ce que \u00e7a nous apprend sur l’optimisation math\u00e9matique. Un probl\u00e8me que les abeilles ont r\u00e9solu bien avant les ing\u00e9nieurs.\n <\/p>\n <\/div>\n\n<\/article>\n\n<\/body>\n<\/html>\n\n\n\n<p><\/p>\n","protected":false},"excerpt":{"rendered":"<p>La r\u00e9gression lin\u00e9aire : pr\u00e9dire l’avenir avec une droite \u2014 MathsVivantes Data & Maths \u00b7 Article #3 La r\u00e9gression lin\u00e9aire […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_uag_custom_page_level_css":"","site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-4)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-4)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-4)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[10,5],"tags":[],"class_list":["post-72","post","type-post","status-publish","format-standard","hentry","category-data-et-maths","category-data-science-pratique"],"yoast_head":"\n<title>La r\u00e9gression lin\u00e9aire - AfrikIntelligent<\/title>\n<meta name=\"description\" content=\"Comprendre la r\u00e9gression lin\u00e9aire avec des exemples concrets b\u00e9ninois . 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{"id":72,"date":"2026-04-17T01:19:18","date_gmt":"2026-04-17T00:19:18","guid":{"rendered":"https:\/\/vmi3217629.contaboserver.net\/?p=72"},"modified":"2026-05-02T08:19:23","modified_gmt":"2026-05-02T07:19:23","slug":"la-regression-lineaire-explication","status":"publish","type":"post","link":"https:\/\/vmi3217629.contaboserver.net\/?p=72","title":{"rendered":"La r\u00e9gression lin\u00e9aire"},"content":{"rendered":"\n\n\n\n\n\nLa r\u00e9gression lin\u00e9aire : pr\u00e9dire l’avenir avec une droite \u2014 MathsVivantes<\/title>\n<link href=\"https:\/\/fonts.googleapis.com\/css2?family=Playfair+Display:ital,wght@0,400;0,700;1,400&family=Source+Serif+4:ital,wght@0,300;0,400;0,600;1,300;1,400&display=swap\" rel=\"stylesheet\">\n<style>\n :root {\n --noir: #0f0e0b;\n --creme: #f5f0e8;\n --bleu: #1a3a5c;\n --bleu-clair: #4a90d9;\n --ocre: #c8850a;\n --ocre-clair: #f0c060;\n --rouge: #b03a2e;\n --gris: #6b6560;\n --gris-clair: #e8e2d8;\n }\n\n * { margin: 0; padding: 0; box-sizing: border-box; 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}\n .article-body { padding: 40px 20px 60px; }\n .exercice { padding: 24px 20px; }\n .code-block { padding: 16px; font-size: 0.78rem; }\n }\n<\/style>\n<\/head>\n<body>\n\n\n<div class=\"hero\">\n <div class=\"hero-inner\">\n <div class=\"article-rubrique\">Data & Maths \u00b7 Article #3<\/div>\n <h1>La r\u00e9gression lin\u00e9aire :<br><em>pr\u00e9dire l’avenir avec une droite<\/em><\/h1>\n <p class=\"hero-intro\">Une vendeuse de tomates au march\u00e9 de Dantokpa le fait intuitivement chaque matin. Un data scientist le fait avec Python. Et Galton l’a d\u00e9couvert au XIX<sup>e<\/sup> si\u00e8cle en \u00e9tudiant la taille des p\u00e8res et de leurs fils. C’est la r\u00e9gression lin\u00e9aire \u2014 l’un des outils les plus puissants et les plus \u00e9l\u00e9gants des math\u00e9matiques appliqu\u00e9es.<\/p>\n <div class=\"meta\">Par Odilon AKOWANOU \u00b7 MathsVivantes \u00b7 Lecture : 11 min<\/div>\n <\/div>\n<\/div>\n\n\n<article class=\"article-body\">\n\n <p>\n Chaque matin au march\u00e9 de Dantokpa \u00e0 Cotonou, Mama Adjoavi regarde ses stocks de tomates, observe les prix des jours pr\u00e9c\u00e9dents, et d\u00e9cide combien vendre aujourd’hui pour maximiser ses b\u00e9n\u00e9fices. Elle ne le sait pas, mais elle fait de la r\u00e9gression lin\u00e9aire \u2014 instinctivement, dans sa t\u00eate.\n <\/p>\n\n <p>\n La r\u00e9gression lin\u00e9aire, c’est l’art de <strong>trouver la droite qui passe au plus pr\u00e8s d’un nuage de points<\/strong>. C’est simple \u00e0 comprendre, puissant \u00e0 utiliser, et c’est la porte d’entr\u00e9e de toute la data science moderne.\n <\/p>\n\n <h2>L’histoire : Galton et les fils qui ressemblent \u00e0 leurs p\u00e8res<\/h2>\n\n <div class=\"encadre-histoire encadre\">\n <span class=\"encadre-titre\">\ud83d\udcdc Naissance d’une id\u00e9e<\/span>\n <p>\n En 1886, le statisticien britannique <strong>Francis Galton<\/strong> \u00e9tudie un ph\u00e9nom\u00e8ne curieux : les fils de p\u00e8res tr\u00e8s grands sont souvent grands, mais rarement aussi grands que leurs p\u00e8res. Et les fils de p\u00e8res tr\u00e8s petits sont souvent petits, mais rarement aussi petits.\n <\/p>\n <p>\n Il appelle ce ph\u00e9nom\u00e8ne la <em>\u00ab\u00a0r\u00e9gression vers la moyenne\u00a0\u00bb<\/em> \u2014 les valeurs extr\u00eames tendent \u00e0 revenir vers la moyenne de la population. Pour mod\u00e9liser cette relation entre la taille du p\u00e8re et celle du fils, il invente une m\u00e9thode graphique : tracer la <strong>droite qui r\u00e9sume le mieux la relation<\/strong> entre deux variables.\n <\/p>\n <p>\n N\u00e9 d’une observation biologique sur des familles anglaises du XIX<sup>e<\/sup> si\u00e8cle, cet outil voyage aujourd’hui dans les algorithmes de Netflix, les mod\u00e8les m\u00e9t\u00e9o de l’ASECNA, et les pr\u00e9dictions de r\u00e9colte des agriculteurs b\u00e9ninois.\n <\/p>\n <\/div>\n\n <h2>L’intuition : une droite dans un nuage<\/h2>\n\n <p>\n Imagine que tu collectes des donn\u00e9es dans un village agricole du Borgou : la quantit\u00e9 de pluie (en mm) et la production de ma\u00efs (en kg) pour 6 saisons cons\u00e9cutives.\n <\/p>\n\n <div class=\"table-container\">\n <table>\n <tr>\n <th>Saison<\/th>\n <th>Pluie (mm) \u2014 x<\/th>\n <th>Ma\u00efs (kg) \u2014 y<\/th>\n <\/tr>\n <tr><td>1<\/td><td>100<\/td><td>200<\/td><\/tr>\n <tr><td>2<\/td><td>150<\/td><td>280<\/td><\/tr>\n <tr><td>3<\/td><td>200<\/td><td>350<\/td><\/tr>\n <tr><td>4<\/td><td>120<\/td><td>230<\/td><\/tr>\n <tr><td>5<\/td><td>180<\/td><td>320<\/td><\/tr>\n <tr><td>6<\/td><td>250<\/td><td>420<\/td><\/tr>\n <\/table>\n <\/div>\n\n <p>\n Si tu traces ces points sur un graphe, tu verras qu’ils forment un nuage avec une tendance claire : <strong>plus il pleut, plus la production est \u00e9lev\u00e9e<\/strong>. La r\u00e9gression lin\u00e9aire consiste \u00e0 trouver la droite qui r\u00e9sume le mieux cette tendance.\n <\/p>\n\n <div class=\"schema-container\">\n <div class=\"schema-titre\">\ud83d\udcc8 Nuage de points et droite de r\u00e9gression<\/div>\n <svg width=\"100%\" viewBox=\"0 0 520 320\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\">\n \n <rect width=\"520\" height=\"320\" fill=\"white\"\/>\n\n \n <line x1=\"60\" y1=\"270\" x2=\"480\" y2=\"270\" stroke=\"#0f0e0b\" stroke-width=\"2\"\/>\n <line x1=\"60\" y1=\"270\" x2=\"60\" y2=\"30\" stroke=\"#0f0e0b\" stroke-width=\"2\"\/>\n\n \n <polygon points=\"480,265 490,270 480,275\" fill=\"#0f0e0b\"\/>\n <polygon points=\"55,30 60,20 65,30\" fill=\"#0f0e0b\"\/>\n\n \n <text x=\"490\" y=\"274\" font-size=\"12\" fill=\"#6b6560\" font-family=\"serif\">x<\/text>\n <text x=\"40\" y=\"25\" font-size=\"12\" fill=\"#6b6560\" font-family=\"serif\">y<\/text>\n <text x=\"220\" y=\"300\" font-size=\"11\" fill=\"#6b6560\" font-family=\"serif\">Pluie (mm)<\/text>\n <text x=\"10\" y=\"160\" font-size=\"11\" fill=\"#6b6560\" font-family=\"serif\" transform=\"rotate(-90,15,160)\">Ma\u00efs (kg)<\/text>\n\n \n <text x=\"88\" y=\"285\" font-size=\"10\" fill=\"#6b6560\">100<\/text>\n <text x=\"148\" y=\"285\" font-size=\"10\" fill=\"#6b6560\">150<\/text>\n <text x=\"208\" y=\"285\" font-size=\"10\" fill=\"#6b6560\">200<\/text>\n <text x=\"268\" y=\"285\" font-size=\"10\" fill=\"#6b6560\">250<\/text>\n\n \n <text x=\"30\" y=\"235\" font-size=\"10\" fill=\"#6b6560\">200<\/text>\n <text x=\"30\" y=\"185\" font-size=\"10\" fill=\"#6b6560\">280<\/text>\n <text x=\"30\" y=\"140\" font-size=\"10\" fill=\"#6b6560\">350<\/text>\n <text x=\"30\" y=\"75\" font-size=\"10\" fill=\"#6b6560\">420<\/text>\n\n \n <line x1=\"80\" y1=\"248\" x2=\"440\" y2=\"58\" stroke=\"#4a90d9\" stroke-width=\"2.5\" stroke-dasharray=\"0\"\/>\n\n \n \n <circle cx=\"100\" cy=\"238\" r=\"6\" fill=\"#1a3a5c\"\/>\n \n <circle cx=\"150\" cy=\"192\" r=\"6\" fill=\"#1a3a5c\"\/>\n \n <circle cx=\"210\" cy=\"138\" r=\"6\" fill=\"#1a3a5c\"\/>\n \n <circle cx=\"122\" cy=\"226\" r=\"6\" fill=\"#1a3a5c\"\/>\n \n <circle cx=\"190\" cy=\"158\" r=\"6\" fill=\"#1a3a5c\"\/>\n \n <circle cx=\"268\" cy=\"72\" r=\"6\" fill=\"#1a3a5c\"\/>\n\n \n <circle cx=\"330\" cy=\"50\" r=\"6\" fill=\"#1a3a5c\"\/>\n <text x=\"342\" y=\"54\" font-size=\"11\" fill=\"#6b6560\">Donn\u00e9es r\u00e9elles<\/text>\n <line x1=\"325\" y1=\"70\" x2=\"355\" y2=\"70\" stroke=\"#4a90d9\" stroke-width=\"2.5\"\/>\n <text x=\"362\" y=\"74\" font-size=\"11\" fill=\"#6b6560\">Droite de r\u00e9gression<\/text>\n\n \n <text x=\"290\" y=\"110\" font-size=\"12\" fill=\"#4a90d9\" font-style=\"italic\">\u0177 = ax + b<\/text>\n <\/svg>\n <\/div>\n\n <h2>La formule : d’o\u00f9 vient cette droite ?<\/h2>\n\n <p>\n La droite de r\u00e9gression s’\u00e9crit :\n <\/p>\n\n <div class=\"formule\">\u0177 = ax + b<\/div>\n\n <p>O\u00f9 :<\/p>\n\n <div class=\"encadre\">\n <span class=\"encadre-titre\">\ud83d\udccc Signification des param\u00e8tres<\/span>\n <p><strong>\u0177<\/strong> (y chapeau) = la valeur <em>pr\u00e9dite<\/em> par le mod\u00e8le<\/p>\n <p><strong>x<\/strong> = la variable d’entr\u00e9e (ici, la quantit\u00e9 de pluie)<\/p>\n <p><strong>a<\/strong> = la pente de la droite (combien y augmente quand x augmente de 1)<\/p>\n <p><strong>b<\/strong> = l’ordonn\u00e9e \u00e0 l’origine (valeur de y quand x = 0)<\/p>\n <\/div>\n\n <p>\n Pour trouver les valeurs de <strong>a<\/strong> et <strong>b<\/strong> qui donnent la <em>meilleure<\/em> droite possible, on utilise la m\u00e9thode des <strong>moindres carr\u00e9s<\/strong> : on minimise la somme des carr\u00e9s des \u00e9carts entre les valeurs r\u00e9elles et les valeurs pr\u00e9dites.\n <\/p>\n\n <div class=\"definition\">\n <strong>M\u00e9thode des moindres carr\u00e9s :<\/strong> Trouver a et b tels que la somme \u2211(y\u1d62 – \u0177\u1d62)\u00b2 soit minimale. Autrement dit, la droite doit \u00eatre aussi proche que possible de tous les points \u00e0 la fois.\n <\/div>\n\n <p>Les formules math\u00e9matiques pour calculer a et b sont :<\/p>\n\n <div class=\"etape\"><span class=\"etape-numero\">\u2460<\/span> a = \u2211(x\u1d62 – x\u0304)(y\u1d62 – \u0233) \/ \u2211(x\u1d62 – x\u0304)\u00b2<\/div>\n <div class=\"etape\"><span class=\"etape-numero\">\u2461<\/span> b = \u0233 – a \u00d7 x\u0304<\/div>\n <div class=\"etape\"><span class=\"etape-numero\">\u2462<\/span> o\u00f9 x\u0304 = moyenne de x et \u0233 = moyenne de y<\/div>\n\n <h2>Application : les donn\u00e9es du Borgou<\/h2>\n\n <p>Reprenons nos donn\u00e9es agricoles et calculons a et b :<\/p>\n\n <div class=\"table-container\">\n <table>\n <tr>\n <th>x<\/th><th>y<\/th><th>x – x\u0304<\/th><th>y – \u0233<\/th><th>(x-x\u0304)(y-\u0233)<\/th><th>(x-x\u0304)\u00b2<\/th>\n <\/tr>\n <tr><td>100<\/td><td>200<\/td><td>-66,7<\/td><td>-100<\/td><td>6670<\/td><td>4448,9<\/td><\/tr>\n <tr><td>150<\/td><td>280<\/td><td>-16,7<\/td><td>-20<\/td><td>334<\/td><td>278,9<\/td><\/tr>\n <tr><td>200<\/td><td>350<\/td><td>33,3<\/td><td>50<\/td><td>1665<\/td><td>1108,9<\/td><\/tr>\n <tr><td>120<\/td><td>230<\/td><td>-46,7<\/td><td>-70<\/td><td>3269<\/td><td>2180,9<\/td><\/tr>\n <tr><td>180<\/td><td>320<\/td><td>13,3<\/td><td>20<\/td><td>266<\/td><td>176,9<\/td><\/tr>\n <tr><td>250<\/td><td>420<\/td><td>83,3<\/td><td>120<\/td><td>9996<\/td><td>6938,9<\/td><\/tr>\n <tr><td><strong>x\u0304=166,7<\/strong><\/td><td><strong>\u0233=300<\/strong><\/td><td>\u2014<\/td><td>\u2014<\/td><td><strong>\u2211=22200<\/strong><\/td><td><strong>\u2211=15133<\/strong><\/td><\/tr>\n <\/table>\n <\/div>\n\n <div class=\"etape\"><span class=\"etape-numero\">\u2460<\/span> a = 22200 \/ 15133 \u2248 1,47<\/div>\n <div class=\"etape\"><span class=\"etape-numero\">\u2461<\/span> b = 300 – 1,47 \u00d7 166,7 \u2248 54,9<\/div>\n\n <div class=\"formule\">\u0177 = 1,47x + 54,9<\/div>\n\n <p>\n Interpr\u00e9tation concr\u00e8te : <strong>chaque millim\u00e8tre de pluie suppl\u00e9mentaire apporte environ 1,47 kg de ma\u00efs en plus<\/strong>. Si la m\u00e9t\u00e9o annonce 220 mm de pluie pour la prochaine saison, on pr\u00e9dit : \u0177 = 1,47 \u00d7 220 + 54,9 \u2248 <strong>378 kg de ma\u00efs<\/strong>.\n <\/p>\n\n <h2>Et maintenant, avec Python<\/h2>\n\n <p>\n Faire ces calculs \u00e0 la main, c’est bien pour comprendre. Mais en pratique, on utilise Python. Voici comment faire la m\u00eame chose en quelques lignes :\n <\/p>\n\n <div class=\"code-block\">\n <span class=\"code-label\">Python<\/span>\n<span class=\"code-comment\"># R\u00e9gression lin\u00e9aire – Production de ma\u00efs au Borgou<\/span>\n<span class=\"code-keyword\">import<\/span> numpy <span class=\"code-keyword\">as<\/span> np\n<span class=\"code-keyword\">from<\/span> sklearn.linear_model <span class=\"code-keyword\">import<\/span> LinearRegression\n<span class=\"code-keyword\">import<\/span> matplotlib.pyplot <span class=\"code-keyword\">as<\/span> plt\n\n<span class=\"code-comment\"># Donn\u00e9es : pluie (mm) et production ma\u00efs (kg)<\/span>\nx = np.array([<span class=\"code-number\">100<\/span>, <span class=\"code-number\">150<\/span>, <span class=\"code-number\">200<\/span>, <span class=\"code-number\">120<\/span>, <span class=\"code-number\">180<\/span>, <span class=\"code-number\">250<\/span>]).reshape(-<span class=\"code-number\">1<\/span>, <span class=\"code-number\">1<\/span>)\ny = np.array([<span class=\"code-number\">200<\/span>, <span class=\"code-number\">280<\/span>, <span class=\"code-number\">350<\/span>, <span class=\"code-number\">230<\/span>, <span class=\"code-number\">320<\/span>, <span class=\"code-number\">420<\/span>])\n\n<span class=\"code-comment\"># Cr\u00e9er et entra\u00eener le mod\u00e8le<\/span>\nmodele = <span class=\"code-function\">LinearRegression<\/span>()\nmodele.<span class=\"code-function\">fit<\/span>(x, y)\n\n<span class=\"code-comment\"># Afficher les param\u00e8tres<\/span>\n<span class=\"code-function\">print<\/span>(<span class=\"code-string\">f\u00a0\u00bbPente a = {modele.coef_[0]:.2f}\u00a0\u00bb<\/span>)\n<span class=\"code-function\">print<\/span>(<span class=\"code-string\">f\u00a0\u00bbOrdonn\u00e9e b = {modele.intercept_:.2f}\u00a0\u00bb<\/span>)\n\n<span class=\"code-comment\"># Pr\u00e9dire pour 220 mm de pluie<\/span>\nprediction = modele.<span class=\"code-function\">predict<\/span>([[<span class=\"code-number\">220<\/span>]])\n<span class=\"code-function\">print<\/span>(<span class=\"code-string\">f\u00a0\u00bbPr\u00e9diction pour 220mm : {prediction[0]:.0f} kg\u00a0\u00bb<\/span>)\n\n<span class=\"code-comment\"># Tracer le graphique<\/span>\nplt.<span class=\"code-function\">scatter<\/span>(x, y, color=<span class=\"code-string\">‘#1a3a5c’<\/span>, label=<span class=\"code-string\">‘Donn\u00e9es r\u00e9elles’<\/span>)\nplt.<span class=\"code-function\">plot<\/span>(x, modele.<span class=\"code-function\">predict<\/span>(x), color=<span class=\"code-string\">‘#4a90d9’<\/span>, label=<span class=\"code-string\">‘Droite de r\u00e9gression’<\/span>)\nplt.<span class=\"code-function\">xlabel<\/span>(<span class=\"code-string\">‘Pluie (mm)’<\/span>)\nplt.<span class=\"code-function\">ylabel<\/span>(<span class=\"code-string\">‘Production ma\u00efs (kg)’<\/span>)\nplt.<span class=\"code-function\">title<\/span>(<span class=\"code-string\">‘R\u00e9gression lin\u00e9aire – Borgou, B\u00e9nin’<\/span>)\nplt.<span class=\"code-function\">legend<\/span>()\nplt.<span class=\"code-function\">show<\/span>()\n <\/div>\n\n <div class=\"encadre\">\n <span class=\"encadre-titre\">\ud83d\udcbb R\u00e9sultat attendu<\/span>\n <p><strong>Pente a = 1.47<\/strong> \u2192 identique \u00e0 notre calcul manuel \u2705<\/p>\n <p><strong>Ordonn\u00e9e b = 54.90<\/strong> \u2192 identique aussi \u2705<\/p>\n <p><strong>Pr\u00e9diction pour 220mm : 378 kg<\/strong><\/p>\n <\/div>\n\n <h2>Les limites \u00e0 conna\u00eetre<\/h2>\n\n <p>\n La r\u00e9gression lin\u00e9aire est puissante, mais elle n’est pas magique. Voici ce qu’il faut garder en t\u00eate :\n <\/p>\n\n <div class=\"encadre\">\n <span class=\"encadre-titre\">\u26a0\ufe0f Attention !<\/span>\n <p><strong>1. Elle suppose une relation lin\u00e9aire.<\/strong> Si la vraie relation est courbe, la droite sera une mauvaise approximation.<\/p>\n <p><strong>2. Elle est sensible aux valeurs aberrantes.<\/strong> Un point tr\u00e8s \u00e9loign\u00e9 des autres peut tirer la droite vers lui.<\/p>\n <p><strong>3. Corr\u00e9lation \u2260 causalit\u00e9.<\/strong> Ce que la pluie et le ma\u00efs ont en commun ne prouve pas que l’un cause l’autre \u2014 m\u00eame si ici c’est logique !<\/p>\n <p><strong>4. Ne pas extrapoler trop loin.<\/strong> Pr\u00e9dire pour 2000 mm de pluie avec un mod\u00e8le calibr\u00e9 entre 100 et 250 mm serait hasardeux.<\/p>\n <\/div>\n\n <blockquote>\n Tous les mod\u00e8les sont faux, mais certains sont utiles.\n <cite>\u2014 George Box, statisticien britannique<\/cite>\n <\/blockquote>\n\n \n <div class=\"exercice\">\n <h3>\u270f\ufe0f Exercice corrig\u00e9<\/h3>\n <p>\n Un vendeur de sachets d’eau \u00e0 Cotonou note ses ventes sur 5 jours de forte chaleur :\n <\/p>\n <p>\n Temp\u00e9rature (\u00b0C) : 32, 34, 36, 35, 38<br>\n Sachets vendus : 120, 145, 170, 158, 200\n <\/p>\n <p><strong>a)<\/strong> Calcule la moyenne de x (temp\u00e9rature) et de y (sachets).<\/p>\n <p><strong>b)<\/strong> Calcule la pente a de la droite de r\u00e9gression.<\/p>\n <p><strong>c)<\/strong> Si demain la temp\u00e9rature atteint 40\u00b0C, combien de sachets faut-il pr\u00e9voir ?<\/p>\n\n <div class=\"correction\">\n <div class=\"correction-titre\">\u25b8 Correction<\/div>\n <p><strong>a)<\/strong> x\u0304 = (32+34+36+35+38)\/5 = 175\/5 = <strong>35\u00b0C<\/strong><\/p>\n <p>\u0233 = (120+145+170+158+200)\/5 = 793\/5 = <strong>158,6 sachets<\/strong><\/p>\n <p><strong>b)<\/strong> Calcul de a :<\/p>\n <p>\u2211(x\u1d62-x\u0304)(y\u1d62-\u0233) = (-3)(-38,6)+(-1)(-13,6)+(1)(11,4)+(0)(-0,6)+(3)(41,4)<\/p>\n <p>= 115,8 + 13,6 + 11,4 + 0 + 124,2 = <strong>265<\/strong><\/p>\n <p>\u2211(x\u1d62-x\u0304)\u00b2 = 9+1+1+0+9 = <strong>20<\/strong><\/p>\n <p>a = 265\/20 = <strong>13,25<\/strong><\/p>\n <p>b = 158,6 – 13,25 \u00d7 35 = 158,6 – 463,75 = <strong>-305,15<\/strong><\/p>\n <p><strong>c)<\/strong> \u0177 = 13,25 \u00d7 40 – 305,15 = 530 – 305,15 \u2248 <strong>225 sachets<\/strong> \u00e0 pr\u00e9voir \u2705<\/p>\n <\/div>\n <\/div>\n\n \n <div class=\"conclusion\">\n <h2>Ce qu’on retient<\/h2>\n <p>\n La r\u00e9gression lin\u00e9aire, c’est le pont entre les math\u00e9matiques pures et la data science. Elle transforme un nuage de points chaotique en une droite qui pr\u00e9dit, qui explique, qui d\u00e9cide. De Galton aux march\u00e9s de Cotonou, en passant par les champs du Borgou, c’est le m\u00eame outil \u2014 \u00e9l\u00e9gant, simple, universel.\n <\/p>\n <p>\n Et le plus beau ? Tu viens de comprendre le fondement math\u00e9matique d’algorithmes qui font tourner des entreprises valant des milliards. Pas mal pour une droite.\n <\/p>\n <p>\n Dans le prochain article, nous d\u00e9couvrirons pourquoi les <strong>abeilles construisent des hexagones<\/strong> \u2014 et ce que \u00e7a nous apprend sur l’optimisation math\u00e9matique. Un probl\u00e8me que les abeilles ont r\u00e9solu bien avant les ing\u00e9nieurs.\n <\/p>\n <\/div>\n\n<\/article>\n\n<\/body>\n<\/html>\n\n\n\n<p><\/p>\n","protected":false},"excerpt":{"rendered":"<p>La r\u00e9gression lin\u00e9aire : pr\u00e9dire l’avenir avec une droite \u2014 MathsVivantes Data & Maths \u00b7 Article #3 La r\u00e9gression lin\u00e9aire […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_uag_custom_page_level_css":"","site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-4)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-4)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-4)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[10,5],"tags":[],"class_list":["post-72","post","type-post","status-publish","format-standard","hentry","category-data-et-maths","category-data-science-pratique"],"yoast_head":"\n<title>La r\u00e9gression lin\u00e9aire - AfrikIntelligent<\/title>\n<meta name=\"description\" content=\"Comprendre la r\u00e9gression lin\u00e9aire avec des exemples concrets b\u00e9ninois . 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