La r\u00e9gression logistique : quand les maths disent oui ou non \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 --violet: #4a1a6b;\n --violet-clair: #9b59b6;\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 background: #1a0a2e;\n color: var(--creme);\n padding: 80px 40px 70px;\n position: relative;\n overflow: hidden;\n }\n .hero::before {\n content: \"\u03c3 P() log e\";\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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Une banque qui d\u00e9cide d’accorder un cr\u00e9dit. Un algorithme qui filtre les spams. Tous utilisent le m\u00eame outil math\u00e9matique : la r\u00e9gression logistique \u2014 l’art de transformer des chiffres en d\u00e9cisions.<\/p>\n <div class=\"meta\">Par Odilon AKOWANOU \u00b7 MathsVivantes \u00b7 Lecture : 12 min<\/div>\n <\/div>\n<\/div>\n\n<article class=\"article-body\">\n\n <p>\n Dans l’article pr\u00e9c\u00e9dent, nous avons vu que la r\u00e9gression lin\u00e9aire pr\u00e9dit une <em>valeur continue<\/em> : combien de kilogrammes de ma\u00efs, quelle temp\u00e9rature, quel prix. Mais parfois, la question n’est pas \u00ab\u00a0combien ?\u00a0\u00bb \u2014 elle est \u00ab\u00a0<strong>oui ou non ?<\/strong>\u00a0\u00bb\n <\/p>\n\n <p>\n Est-ce que ce patient a le paludisme ? <strong>Oui ou non.<\/strong>\n Est-ce que cet email est un spam ? <strong>Oui ou non.<\/strong>\n Est-ce que cet \u00e9tudiant va r\u00e9ussir son examen ? <strong>Oui ou non.<\/strong>\n <\/p>\n\n <p>\n Pour r\u00e9pondre \u00e0 ces questions binaires, nous avons besoin d’un outil diff\u00e9rent : la <strong>r\u00e9gression logistique<\/strong>. Et derri\u00e8re ce nom intimidant se cache une id\u00e9e d’une \u00e9l\u00e9gance remarquable.\n <\/p>\n\n <h2>Pourquoi pas utiliser la r\u00e9gression lin\u00e9aire ?<\/h2>\n\n <p>\n Imaginons qu’on veuille pr\u00e9dire si un \u00e9l\u00e8ve r\u00e9ussit son BEPC (1 = r\u00e9ussite, 0 = \u00e9chec) en fonction de ses heures de r\u00e9vision. On pourrait essayer la r\u00e9gression lin\u00e9aire…\n <\/p>\n\n <div class=\"schema-container\">\n <div class=\"schema-titre\">\u26a0\ufe0f Probl\u00e8me de la r\u00e9gression lin\u00e9aire pour une sortie binaire<\/div>\n <svg width=\"100%\" viewBox=\"0 0 520 280\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\">\n <rect width=\"520\" height=\"280\" fill=\"white\"\/>\n \n <line x1=\"60\" y1=\"230\" x2=\"480\" y2=\"230\" stroke=\"#0f0e0b\" stroke-width=\"2\"\/>\n <line x1=\"60\" y1=\"230\" x2=\"60\" y2=\"30\" stroke=\"#0f0e0b\" stroke-width=\"2\"\/>\n \n <text x=\"200\" y=\"260\" font-size=\"11\" fill=\"#6b6560\">Heures de r\u00e9vision<\/text>\n <text x=\"10\" y=\"230\" font-size=\"11\" fill=\"#6b6560\">0<\/text>\n <text x=\"10\" y=\"80\" font-size=\"11\" fill=\"#6b6560\">1<\/text>\n <text x=\"30\" y=\"155\" font-size=\"10\" fill=\"#9b59b6\">0.5<\/text>\n \n <line x1=\"60\" y1=\"80\" x2=\"480\" y2=\"80\" stroke=\"#e8e2d8\" stroke-width=\"1\" stroke-dasharray=\"4,3\"\/>\n \n <line x1=\"70\" y1=\"210\" x2=\"460\" y2=\"50\" stroke=\"#c8850a\" stroke-width=\"2\" stroke-dasharray=\"6,3\"\/>\n \n <circle cx=\"90\" cy=\"230\" r=\"5\" fill=\"#b03a2e\"\/>\n <circle cx=\"120\" cy=\"230\" r=\"5\" fill=\"#b03a2e\"\/>\n <circle cx=\"100\" cy=\"230\" r=\"5\" fill=\"#b03a2e\"\/>\n <circle cx=\"140\" cy=\"230\" r=\"5\" fill=\"#b03a2e\"\/>\n <circle cx=\"160\" cy=\"230\" r=\"5\" fill=\"#b03a2e\"\/>\n \n <circle cx=\"300\" cy=\"80\" r=\"5\" fill=\"#2e6b4f\"\/>\n <circle cx=\"340\" cy=\"80\" r=\"5\" fill=\"#2e6b4f\"\/>\n <circle cx=\"370\" cy=\"80\" r=\"5\" fill=\"#2e6b4f\"\/>\n <circle cx=\"400\" cy=\"80\" r=\"5\" fill=\"#2e6b4f\"\/>\n <circle cx=\"430\" cy=\"80\" r=\"5\" fill=\"#2e6b4f\"\/>\n \n <text x=\"430\" y=\"45\" font-size=\"10\" fill=\"#b03a2e\">\u2190 Valeurs > 1 !<\/text>\n <text x=\"62\" y=\"245\" font-size=\"10\" fill=\"#b03a2e\">Valeurs < 0 ! \u2192<\/text>\n \n <path d=\"M 70,225 Q 180,220 240,155 Q 300,90 460,72\" stroke=\"#4a1a6b\" stroke-width=\"2.5\" fill=\"none\"\/>\n \n <line x1=\"300\" y1=\"190\" x2=\"330\" y2=\"190\" stroke=\"#c8850a\" stroke-width=\"2\" stroke-dasharray=\"6,3\"\/>\n <text x=\"335\" y=\"194\" font-size=\"10\" fill=\"#6b6560\">R\u00e9gression lin\u00e9aire (probl\u00e8me)<\/text>\n <line x1=\"300\" y1=\"210\" x2=\"330\" y2=\"210\" stroke=\"#4a1a6b\" stroke-width=\"2.5\"\/>\n <text x=\"335\" y=\"214\" font-size=\"10\" fill=\"#6b6560\">Courbe logistique (solution)<\/text>\n <\/svg>\n <\/div>\n\n <p>\n Le probl\u00e8me est clair : la droite lin\u00e9aire peut pr\u00e9dire des valeurs inf\u00e9rieures \u00e0 0 ou sup\u00e9rieures \u00e0 1 \u2014 ce qui n’a aucun sens pour une probabilit\u00e9. On ne peut pas avoir une probabilit\u00e9 de r\u00e9ussite de 1,3 ou de -0,2 !\n <\/p>\n\n <p>\n Il nous faut une fonction qui reste <strong>toujours entre 0 et 1<\/strong>. C’est exactement ce que fait la fonction sigmo\u00efde.\n <\/p>\n\n <h2>La fonction sigmo\u00efde : le c\u0153ur de tout<\/h2>\n\n <div class=\"definition\">\n <strong>Fonction sigmo\u00efde :<\/strong> Une fonction math\u00e9matique en forme de S qui transforme n’importe quel nombre r\u00e9el en une valeur comprise entre 0 et 1. C’est la fonction porte d’entr\u00e9e de la r\u00e9gression logistique \u2014 et de tous les r\u00e9seaux de neurones artificiels.\n <\/div>\n\n <div class=\"formule\">\u03c3(z) = 1 \/ (1 + e\u207b\u1dbb)<\/div>\n\n <p>O\u00f9 <strong>e<\/strong> est la constante d’Euler (\u2248 2,718) et <strong>z<\/strong> = ax + b comme dans la r\u00e9gression lin\u00e9aire.<\/p>\n\n <div class=\"schema-container\">\n <div class=\"schema-titre\">\ud83d\udcc8 La courbe sigmo\u00efde \u2014 toujours entre 0 et 1<\/div>\n <svg width=\"100%\" viewBox=\"0 0 520 280\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\">\n <rect width=\"520\" height=\"280\" fill=\"white\"\/>\n \n <line x1=\"60\" y1=\"140\" x2=\"480\" y2=\"140\" stroke=\"#0f0e0b\" stroke-width=\"1.5\"\/>\n <line x1=\"260\" y1=\"20\" x2=\"260\" y2=\"260\" stroke=\"#0f0e0b\" stroke-width=\"1.5\"\/>\n \n <text x=\"485\" y=\"144\" font-size=\"11\" fill=\"#6b6560\">z<\/text>\n <text x=\"263\" y=\"18\" font-size=\"11\" fill=\"#6b6560\">\u03c3<\/text>\n <text x=\"30\" y=\"55\" font-size=\"10\" fill=\"#9b59b6\">1.0<\/text>\n <text x=\"30\" y=\"144\" font-size=\"10\" fill=\"#9b59b6\">0.5<\/text>\n <text x=\"30\" y=\"235\" font-size=\"10\" fill=\"#9b59b6\">0.0<\/text>\n \n <line x1=\"60\" y1=\"50\" x2=\"480\" y2=\"50\" stroke=\"#e8e2d8\" stroke-width=\"1\" stroke-dasharray=\"4,3\"\/>\n <line x1=\"60\" y1=\"230\" x2=\"480\" y2=\"230\" stroke=\"#e8e2d8\" stroke-width=\"1\" stroke-dasharray=\"4,3\"\/>\n \n <path d=\"M 60,228 C 100,226 140,222 180,210 C 200,200 220,180 240,160 C 255,143 265,120 280,105 C 300,85 320,65 360,55 C 390,50 430,49 480,48\"\n stroke=\"#4a1a6b\" stroke-width=\"3\" fill=\"none\"\/>\n \n <circle cx=\"260\" cy=\"140\" r=\"5\" fill=\"#9b59b6\"\/>\n <text x=\"268\" y=\"136\" font-size=\"10\" fill=\"#9b59b6\">\u03c3(0) = 0.5<\/text>\n \n <rect x=\"260\" y=\"50\" width=\"220\" height=\"90\" fill=\"rgba(46,107,79,0.08)\"\/>\n <text x=\"330\" y=\"100\" font-size=\"10\" fill=\"#2e6b4f\">Pr\u00e9dit : OUI (\u22650.5)<\/text>\n <rect x=\"60\" y=\"140\" width=\"200\" height=\"90\" fill=\"rgba(176,58,46,0.08)\"\/>\n <text x=\"100\" y=\"185\" font-size=\"10\" fill=\"#b03a2e\">Pr\u00e9dit : NON (<0.5)<\/text>\n <\/svg>\n <\/div>\n\n <p>\n La r\u00e8gle de d\u00e9cision est simple : si \u03c3(z) \u2265 0,5 \u2192 on pr\u00e9dit <strong>OUI (1)<\/strong>. Si \u03c3(z) < 0,5 \u2192 on pr\u00e9dit <strong>NON (0)<\/strong>. Le seuil de 0,5 peut \u00eatre ajust\u00e9 selon le contexte.\n <\/p>\n\n <h2>Lin\u00e9aire vs Logistique : le tableau comparatif<\/h2>\n\n <div class=\"comparatif\">\n <div class=\"comparatif-bloc\">\n <div class=\"comparatif-titre\" style=\"border-color:#c8850a\">\ud83d\udccf R\u00e9gression Lin\u00e9aire<\/div>\n <p>\u2192 Pr\u00e9dit une <strong>valeur continue<\/strong><\/p>\n <p>\u2192 Sortie : n’importe quel nombre r\u00e9el<\/p>\n <p>\u2192 Question : <em>\u00ab\u00a0Combien ?\u00a0\u00bb<\/em><\/p>\n <p>\u2192 Ex : pr\u00e9voir le prix du ma\u00efs<\/p>\n <p>\u2192 Fonction : \u0177 = ax + b<\/p>\n <\/div>\n <div class=\"comparatif-bloc\">\n <div class=\"comparatif-titre\" style=\"border-color:#9b59b6\">\ud83d\udd00 R\u00e9gression Logistique<\/div>\n <p>\u2192 Pr\u00e9dit une <strong>probabilit\u00e9<\/strong><\/p>\n <p>\u2192 Sortie : entre 0 et 1<\/p>\n <p>\u2192 Question : <em>\u00ab\u00a0Oui ou non ?\u00a0\u00bb<\/em><\/p>\n <p>\u2192 Ex : pr\u00e9voir si r\u00e9ussite au BEPC<\/p>\n <p>\u2192 Fonction : \u03c3(ax + b)<\/p>\n <\/div>\n <\/div>\n\n <h2>Application : pr\u00e9dire la r\u00e9ussite au BEPC<\/h2>\n\n <p>\n Voici des donn\u00e9es fictives mais r\u00e9alistes sur 8 \u00e9l\u00e8ves b\u00e9ninois, leurs heures de r\u00e9vision hebdomadaires et leur r\u00e9sultat au BEPC :\n <\/p>\n\n <div class=\"table-container\">\n <table>\n <tr>\n <th>\u00c9l\u00e8ve<\/th>\n <th>Heures\/semaine (x)<\/th>\n <th>R\u00e9sultat (y)<\/th>\n <th>P(r\u00e9ussite) pr\u00e9dite<\/th>\n <\/tr>\n <tr><td>Kofi<\/td><td>2<\/td><td>0 (\u00c9chec)<\/td><td>0,12<\/td><\/tr>\n <tr><td>Amina<\/td><td>4<\/td><td>0 (\u00c9chec)<\/td><td>0,25<\/td><\/tr>\n <tr><td>S\u00e9kou<\/td><td>6<\/td><td>1 (R\u00e9ussite)<\/td><td>0,45<\/td><\/tr>\n <tr><td>Fatou<\/td><td>8<\/td><td>1 (R\u00e9ussite)<\/td><td>0,66<\/td><\/tr>\n <tr><td>Brice<\/td><td>10<\/td><td>1 (R\u00e9ussite)<\/td><td>0,82<\/td><\/tr>\n <tr><td>Adjovi<\/td><td>12<\/td><td>1 (R\u00e9ussite)<\/td><td>0,92<\/td><\/tr>\n <tr><td>Moussa<\/td><td>3<\/td><td>0 (\u00c9chec)<\/td><td>0,18<\/td><\/tr>\n <tr><td>C\u00e9leste<\/td><td>9<\/td><td>1 (R\u00e9ussite)<\/td><td>0,75<\/td><\/tr>\n <\/table>\n <\/div>\n\n <p>\n On voit clairement la tendance : plus un \u00e9l\u00e8ve r\u00e9vise, plus sa probabilit\u00e9 de r\u00e9ussite augmente. Avec 6 heures par semaine, la probabilit\u00e9 est de 45% \u2014 trop risqu\u00e9. Avec 10 heures, elle monte \u00e0 82%. Le mod\u00e8le dit clairement : <strong>r\u00e9vise plus !<\/strong>\n <\/p>\n\n <h2>Le code Python<\/h2>\n\n <div class=\"code-block\">\n <span class=\"code-label\">Python<\/span>\n<span class=\"code-comment\"># R\u00e9gression logistique – R\u00e9ussite au BEPC<\/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> LogisticRegression\n<span class=\"code-keyword\">from<\/span> sklearn.metrics <span class=\"code-keyword\">import<\/span> accuracy_score\n<span class=\"code-keyword\">import<\/span> matplotlib.pyplot <span class=\"code-keyword\">as<\/span> plt\n\n<span class=\"code-comment\"># Donn\u00e9es : heures de r\u00e9vision et r\u00e9sultats<\/span>\nx = np.array([<span class=\"code-number\">2<\/span>, <span class=\"code-number\">4<\/span>, <span class=\"code-number\">6<\/span>, <span class=\"code-number\">8<\/span>, <span class=\"code-number\">10<\/span>, <span class=\"code-number\">12<\/span>, <span class=\"code-number\">3<\/span>, <span class=\"code-number\">9<\/span>]).reshape(-<span class=\"code-number\">1<\/span>, <span class=\"code-number\">1<\/span>)\ny = np.array([<span class=\"code-number\">0<\/span>, <span class=\"code-number\">0<\/span>, <span class=\"code-number\">1<\/span>, <span class=\"code-number\">1<\/span>, <span class=\"code-number\">1<\/span>, <span class=\"code-number\">1<\/span>, <span class=\"code-number\">0<\/span>, <span class=\"code-number\">1<\/span>])\n\n<span class=\"code-comment\"># Cr\u00e9er et entra\u00eener le mod\u00e8le<\/span>\nmodele = <span class=\"code-function\">LogisticRegression<\/span>()\nmodele.<span class=\"code-function\">fit<\/span>(x, y)\n\n<span class=\"code-comment\"># Pr\u00e9cision du mod\u00e8le<\/span>\npredictions = modele.<span class=\"code-function\">predict<\/span>(x)\nprecision = <span class=\"code-function\">accuracy_score<\/span>(y, predictions)\n<span class=\"code-function\">print<\/span>(<span class=\"code-string\">f\u00a0\u00bbPr\u00e9cision du mod\u00e8le : {precision*100:.0f}%\u00a0\u00bb<\/span>)\n\n<span class=\"code-comment\"># Pr\u00e9dire pour un nouvel \u00e9l\u00e8ve (7h de r\u00e9vision)<\/span>\nnouvel_eleve = np.array([[<span class=\"code-number\">7<\/span>]])\nproba = modele.<span class=\"code-function\">predict_proba<\/span>(nouvel_eleve)[<span class=\"code-number\">0<\/span>][<span class=\"code-number\">1<\/span>]\ndecision = modele.<span class=\"code-function\">predict<\/span>(nouvel_eleve)[<span class=\"code-number\">0<\/span>]\n<span class=\"code-function\">print<\/span>(<span class=\"code-string\">f\u00a0\u00bbProbabilit\u00e9 de r\u00e9ussite (7h) : {proba:.0%}\u00a0\u00bb<\/span>)\n<span class=\"code-function\">print<\/span>(<span class=\"code-string\">f\u00a0\u00bbD\u00e9cision : {‘R\u00e9ussite \u2713’ if decision == 1 else ‘\u00c9chec \u2717’}\u00a0\u00bb<\/span>)\n\n<span class=\"code-comment\"># Tracer la courbe sigmo\u00efde<\/span>\nx_courbe = np.linspace(<span class=\"code-number\">0<\/span>, <span class=\"code-number\">15<\/span>, <span class=\"code-number\">200<\/span>).reshape(-<span class=\"code-number\">1<\/span>, <span class=\"code-number\">1<\/span>)\nprobas_courbe = modele.<span class=\"code-function\">predict_proba<\/span>(x_courbe)[:, <span class=\"code-number\">1<\/span>]\n\nplt.<span class=\"code-function\">figure<\/span>(figsize=(<span class=\"code-number\">8<\/span>, <span class=\"code-number\">5<\/span>))\nplt.<span class=\"code-function\">plot<\/span>(x_courbe, probas_courbe, color=<span class=\"code-string\">‘#4a1a6b’<\/span>, linewidth=<span class=\"code-number\">2.5<\/span>)\nplt.<span class=\"code-function\">scatter<\/span>(x, y, color=<span class=\"code-string\">‘#9b59b6’<\/span>, s=<span class=\"code-number\">80<\/span>, zorder=<span class=\"code-number\">5<\/span>)\nplt.<span class=\"code-function\">axhline<\/span>(y=<span class=\"code-number\">0.5<\/span>, color=<span class=\"code-string\">‘#c8850a’<\/span>, linestyle=<span class=\"code-string\">‘–‘<\/span>, label=<span class=\"code-string\">‘Seuil 0.5’<\/span>)\nplt.<span class=\"code-function\">xlabel<\/span>(<span class=\"code-string\">‘Heures de r\u00e9vision \/ semaine’<\/span>)\nplt.<span class=\"code-function\">ylabel<\/span>(<span class=\"code-string\">‘Probabilit\u00e9 de r\u00e9ussite au BEPC’<\/span>)\nplt.<span class=\"code-function\">title<\/span>(<span class=\"code-string\">‘R\u00e9gression Logistique – Pr\u00e9diction BEPC’<\/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>Pr\u00e9cision du mod\u00e8le : 100%<\/strong> (sur ces 8 exemples d’entra\u00eenement)<\/p>\n <p><strong>Probabilit\u00e9 de r\u00e9ussite pour 7h : 57%<\/strong><\/p>\n <p><strong>D\u00e9cision : R\u00e9ussite \u2713<\/strong> (juste au-dessus du seuil de 0,5)<\/p>\n <\/div>\n\n <h2>Les applications r\u00e9elles en Afrique<\/h2>\n\n <div class=\"encadre-histoire encadre\">\n <span class=\"encadre-titre\">\ud83c\udf0d Cas d’usage africains<\/span>\n <p>\n <strong>1. Sant\u00e9 :<\/strong> Des h\u00f4pitaux \u00e0 Cotonou, Lagos et Dakar utilisent des mod\u00e8les similaires pour pr\u00e9dire le risque de paludisme s\u00e9v\u00e8re chez un patient, en fonction de ses sympt\u00f4mes et de son \u00e2ge.\n <\/p>\n <p>\n <strong>2. Finance :<\/strong> Les microfinances b\u00e9ninoises (comme CLCAM) commencent \u00e0 explorer ces outils pour d\u00e9cider d’accorder ou non un pr\u00eat \u00e0 un agriculteur, selon ses ant\u00e9c\u00e9dents et sa r\u00e9gion.\n <\/p>\n <p>\n <strong>3. Agriculture :<\/strong> Pr\u00e9dire si une r\u00e9colte sera bonne ou mauvaise selon les donn\u00e9es m\u00e9t\u00e9o, le type de sol et les intrants utilis\u00e9s.\n <\/p>\n <p>\n <strong>4. \u00c9ducation :<\/strong> Identifier les \u00e9l\u00e8ves \u00e0 risque de d\u00e9crochage scolaire pour intervenir \u00e0 temps \u2014 exactement le genre d’outil qu’un enseignant-data scientist peut construire.\n <\/p>\n <\/div>\n\n <blockquote>\n Les donn\u00e9es sont la nouvelle mati\u00e8re premi\u00e8re. Mais sans les math\u00e9matiques pour les raffiner, elles ne valent rien.\n <cite>\u2014 Inspiration de la data science africaine<\/cite>\n <\/blockquote>\n\n \n <div class=\"exercice\">\n <h3>\u270f\ufe0f Exercice corrig\u00e9<\/h3>\n <p>\n Un agent de sant\u00e9 \u00e0 Porto-Novo a recueilli des donn\u00e9es sur 6 patients suspects de paludisme. Il dispose du nombre de sympt\u00f4mes pr\u00e9sent\u00e9s (x) et du diagnostic final (y : 1 = paludisme confirm\u00e9, 0 = n\u00e9gatif).\n <\/p>\n <p>\n x (sympt\u00f4mes) : 1, 2, 3, 4, 5, 6<br>\n y (diagnostic) : 0, 0, 0, 1, 1, 1\n <\/p>\n <p><strong>a)<\/strong> Quel type de r\u00e9gression est adapt\u00e9 ici, et pourquoi ?<\/p>\n <p><strong>b)<\/strong> Si on a trouv\u00e9 z = 1,5x – 5, calcule \u03c3(z) pour x = 3 et x = 5.<\/p>\n <p><strong>c)<\/strong> Pour x = 3, le mod\u00e8le diagnostiquerait-il un paludisme ?<\/p>\n\n <div class=\"correction\">\n <div class=\"correction-titre\">\u25b8 Correction<\/div>\n <p><strong>a)<\/strong> La r\u00e9gression <strong>logistique<\/strong> est adapt\u00e9e car la sortie est binaire (0 ou 1 \u2014 n\u00e9gatif ou positif). La r\u00e9gression lin\u00e9aire donnerait des valeurs hors de [0,1], ce qui n’aurait pas de sens pour une probabilit\u00e9.<\/p>\n <p><strong>b)<\/strong> Pour x = 3 : z = 1,5(3) – 5 = 4,5 – 5 = <strong>-0,5<\/strong><\/p>\n <p>\u03c3(-0,5) = 1 \/ (1 + e^0,5) = 1 \/ (1 + 1,649) \u2248 <strong>0,38<\/strong> soit 38%<\/p>\n <p>Pour x = 5 : z = 1,5(5) – 5 = 7,5 – 5 = <strong>2,5<\/strong><\/p>\n <p>\u03c3(2,5) = 1 \/ (1 + e^(-2,5)) = 1 \/ (1 + 0,082) \u2248 <strong>0,92<\/strong> soit 92%<\/p>\n <p><strong>c)<\/strong> Pour x = 3, \u03c3 = 0,38 < 0,5 \u2192 le mod\u00e8le pr\u00e9dit <strong>NON (0)<\/strong> : pas de paludisme confirm\u00e9. \u2705<\/p>\n <\/div>\n <\/div>\n\n <div class=\"conclusion\">\n <h2>Ce qu’on retient<\/h2>\n <p>\n La r\u00e9gression logistique, c’est le passage des math\u00e9matiques \u00e0 la <strong>prise de d\u00e9cision<\/strong>. L\u00e0 o\u00f9 la r\u00e9gression lin\u00e9aire mesure, la r\u00e9gression logistique tranche. Ensemble, ces deux outils forment le socle de la data science \u2014 et des intelligences artificielles qui transforment aujourd’hui la m\u00e9decine, la finance et l’\u00e9ducation en Afrique.\n <\/p>\n <p>\n Maintenant que tu comprends ces deux r\u00e9gressions, tu n’es plus seulement un observateur de l’IA. Tu en comprends les fondations.\n <\/p>\n <p>\n Dans le prochain article, nous sortons des chiffres et rentrons dans la nature pour poser une question fascinante : <strong>pourquoi les abeilles construisent-elles des hexagones ?<\/strong> La r\u00e9ponse cache l’un des probl\u00e8mes d’optimisation les plus \u00e9l\u00e9gants de toute la math\u00e9matique.\n <\/p>\n <\/div>\n\n<\/article>\n\n<\/body>\n<\/html>\n","protected":false},"excerpt":{"rendered":"<p>La r\u00e9gression logistique : quand les maths disent oui ou non \u2014 MathsVivantes Data & Maths \u00b7 Article #4 La […]<\/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],"tags":[],"class_list":["post-74","post","type-post","status-publish","format-standard","hentry","category-data-et-maths"],"yoast_head":"\n<title>La r\u00e9gression logistique - AfrikIntelligent<\/title>\n<meta name=\"description\" content=\"La r\u00e9gression logistique expliqu\u00e9e simplement . 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De la fonction sigmo\u00cfde aux applications m\u00e9dicales en Afrique - avec code python .","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/vmi3217629.contaboserver.net\/?p=74","og_locale":"fr_FR","og_type":"article","og_title":"La r\u00e9gression logistique - AfrikIntelligent","og_description":"La r\u00e9gression logistique expliqu\u00e9e simplement . 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De la fonction sigmo\u00cfde aux applications m\u00e9dicales en Afrique - avec code python .","breadcrumb":{"@id":"https:\/\/vmi3217629.contaboserver.net\/?p=74#breadcrumb"},"inLanguage":"fr-FR","potentialAction":[{"@type":"ReadAction","target":["https:\/\/vmi3217629.contaboserver.net\/?p=74"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/vmi3217629.contaboserver.net\/?p=74#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Accueil","item":"https:\/\/vmi3217629.contaboserver.net\/"},{"@type":"ListItem","position":2,"name":"La r\u00e9gression logistique"}]},{"@type":"WebSite","@id":"https:\/\/vmi3217629.contaboserver.net\/#website","url":"https:\/\/vmi3217629.contaboserver.net\/","name":"AfrikIntelligent","description":"Append autrement","potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/vmi3217629.contaboserver.net\/?s={search_term_string}"},"query-input":{"@type":"PropertyValueSpecification","valueRequired":true,"valueName":"search_term_string"}}],"inLanguage":"fr-FR"},{"@type":"Person","@id":"https:\/\/vmi3217629.contaboserver.net\/#\/schema\/person\/a3da39574d729b6b3ff7514ccc055923","name":"Odilon","image":{"@type":"ImageObject","inLanguage":"fr-FR","@id":"https:\/\/secure.gravatar.com\/avatar\/57bcd6a76fa395ae5e994943b66ccd3dd7e71cd0b1b77b351986e88c27801523?s=96&d=mm&r=g","url":"https:\/\/secure.gravatar.com\/avatar\/57bcd6a76fa395ae5e994943b66ccd3dd7e71cd0b1b77b351986e88c27801523?s=96&d=mm&r=g","contentUrl":"https:\/\/secure.gravatar.com\/avatar\/57bcd6a76fa395ae5e994943b66ccd3dd7e71cd0b1b77b351986e88c27801523?s=96&d=mm&r=g","caption":"Odilon"},"sameAs":["https:\/\/vmi3217629.contaboserver.net"],"url":"https:\/\/vmi3217629.contaboserver.net\/?author=1"}]}},"uagb_featured_image_src":{"full":false,"thumbnail":false,"medium":false,"medium_large":false,"large":false,"1536x1536":false,"2048x2048":false},"uagb_author_info":{"display_name":"Odilon","author_link":"https:\/\/vmi3217629.contaboserver.net\/?author=1"},"uagb_comment_info":0,"uagb_excerpt":"La r\u00e9gression logistique : quand les maths disent oui ou non \u2014 MathsVivantes Data & Maths \u00b7 Article #4 La […]","_links":{"self":[{"href":"https:\/\/vmi3217629.contaboserver.net\/index.php?rest_route=\/wp\/v2\/posts\/74","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/vmi3217629.contaboserver.net\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/vmi3217629.contaboserver.net\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/vmi3217629.contaboserver.net\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/vmi3217629.contaboserver.net\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=74"}],"version-history":[{"count":1,"href":"https:\/\/vmi3217629.contaboserver.net\/index.php?rest_route=\/wp\/v2\/posts\/74\/revisions"}],"predecessor-version":[{"id":75,"href":"https:\/\/vmi3217629.contaboserver.net\/index.php?rest_route=\/wp\/v2\/posts\/74\/revisions\/75"}],"wp:attachment":[{"href":"https:\/\/vmi3217629.contaboserver.net\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=74"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/vmi3217629.contaboserver.net\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=74"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/vmi3217629.contaboserver.net\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=74"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}

{"id":74,"date":"2026-04-17T15:48:09","date_gmt":"2026-04-17T14:48:09","guid":{"rendered":"https:\/\/vmi3217629.contaboserver.net\/?p=74"},"modified":"2026-05-02T08:20:19","modified_gmt":"2026-05-02T07:20:19","slug":"la-regression-logistique-classification","status":"publish","type":"post","link":"https:\/\/vmi3217629.contaboserver.net\/?p=74","title":{"rendered":"La r\u00e9gression logistique"},"content":{"rendered":"\n\n\n\n\n\nLa r\u00e9gression logistique : quand les maths disent oui ou non \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 --violet: #4a1a6b;\n --violet-clair: #9b59b6;\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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top: -10px;\n font-size: 3.5rem;\n color: var(--violet-clair);\n font-family: 'Playfair Display', serif;\n line-height: 1;\n }\n blockquote cite { display: block; font-size: 0.82rem; font-style: normal; letter-spacing: 0.1em; text-transform: uppercase; color: var(--gris); margin-top: 8px; }\n\n .conclusion { border-top: 2px solid var(--noir); margin-top: 3em; padding-top: 2em; }\n\n @media (max-width: 600px) {\n .hero { padding: 50px 20px 40px; }\n .article-body { padding: 40px 20px 60px; }\n .exercice { padding: 24px 20px; }\n .code-block { padding: 16px; font-size: 0.78rem; }\n .comparatif { grid-template-columns: 1fr; }\n }\n<\/style>\n<\/head>\n<body>\n\n<div class=\"hero\">\n <div class=\"hero-inner\">\n <div class=\"article-rubrique\">Data & Maths \u00b7 Article #4<\/div>\n <h1>La r\u00e9gression logistique :<br><em>quand les maths disent oui ou non<\/em><\/h1>\n <p class=\"hero-intro\">Un m\u00e9decin de Cotonou qui pr\u00e9dit si un patient a le paludisme. Une banque qui d\u00e9cide d’accorder un cr\u00e9dit. Un algorithme qui filtre les spams. Tous utilisent le m\u00eame outil math\u00e9matique : la r\u00e9gression logistique \u2014 l’art de transformer des chiffres en d\u00e9cisions.<\/p>\n <div class=\"meta\">Par Odilon AKOWANOU \u00b7 MathsVivantes \u00b7 Lecture : 12 min<\/div>\n <\/div>\n<\/div>\n\n<article class=\"article-body\">\n\n <p>\n Dans l’article pr\u00e9c\u00e9dent, nous avons vu que la r\u00e9gression lin\u00e9aire pr\u00e9dit une <em>valeur continue<\/em> : combien de kilogrammes de ma\u00efs, quelle temp\u00e9rature, quel prix. Mais parfois, la question n’est pas \u00ab\u00a0combien ?\u00a0\u00bb \u2014 elle est \u00ab\u00a0<strong>oui ou non ?<\/strong>\u00a0\u00bb\n <\/p>\n\n <p>\n Est-ce que ce patient a le paludisme ? <strong>Oui ou non.<\/strong>\n Est-ce que cet email est un spam ? <strong>Oui ou non.<\/strong>\n Est-ce que cet \u00e9tudiant va r\u00e9ussir son examen ? <strong>Oui ou non.<\/strong>\n <\/p>\n\n <p>\n Pour r\u00e9pondre \u00e0 ces questions binaires, nous avons besoin d’un outil diff\u00e9rent : la <strong>r\u00e9gression logistique<\/strong>. Et derri\u00e8re ce nom intimidant se cache une id\u00e9e d’une \u00e9l\u00e9gance remarquable.\n <\/p>\n\n <h2>Pourquoi pas utiliser la r\u00e9gression lin\u00e9aire ?<\/h2>\n\n <p>\n Imaginons qu’on veuille pr\u00e9dire si un \u00e9l\u00e8ve r\u00e9ussit son BEPC (1 = r\u00e9ussite, 0 = \u00e9chec) en fonction de ses heures de r\u00e9vision. On pourrait essayer la r\u00e9gression lin\u00e9aire…\n <\/p>\n\n <div class=\"schema-container\">\n <div class=\"schema-titre\">\u26a0\ufe0f Probl\u00e8me de la r\u00e9gression lin\u00e9aire pour une sortie binaire<\/div>\n <svg width=\"100%\" viewBox=\"0 0 520 280\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\">\n <rect width=\"520\" height=\"280\" fill=\"white\"\/>\n \n <line x1=\"60\" y1=\"230\" x2=\"480\" y2=\"230\" stroke=\"#0f0e0b\" stroke-width=\"2\"\/>\n <line x1=\"60\" y1=\"230\" x2=\"60\" y2=\"30\" stroke=\"#0f0e0b\" stroke-width=\"2\"\/>\n \n <text x=\"200\" y=\"260\" font-size=\"11\" fill=\"#6b6560\">Heures de r\u00e9vision<\/text>\n <text x=\"10\" y=\"230\" font-size=\"11\" fill=\"#6b6560\">0<\/text>\n <text x=\"10\" y=\"80\" font-size=\"11\" fill=\"#6b6560\">1<\/text>\n <text x=\"30\" y=\"155\" font-size=\"10\" fill=\"#9b59b6\">0.5<\/text>\n \n <line x1=\"60\" y1=\"80\" x2=\"480\" y2=\"80\" stroke=\"#e8e2d8\" stroke-width=\"1\" stroke-dasharray=\"4,3\"\/>\n \n <line x1=\"70\" y1=\"210\" x2=\"460\" y2=\"50\" stroke=\"#c8850a\" stroke-width=\"2\" stroke-dasharray=\"6,3\"\/>\n \n <circle cx=\"90\" cy=\"230\" r=\"5\" fill=\"#b03a2e\"\/>\n <circle cx=\"120\" cy=\"230\" r=\"5\" fill=\"#b03a2e\"\/>\n <circle cx=\"100\" cy=\"230\" r=\"5\" fill=\"#b03a2e\"\/>\n <circle cx=\"140\" cy=\"230\" r=\"5\" fill=\"#b03a2e\"\/>\n <circle cx=\"160\" cy=\"230\" r=\"5\" fill=\"#b03a2e\"\/>\n \n <circle cx=\"300\" cy=\"80\" r=\"5\" fill=\"#2e6b4f\"\/>\n <circle cx=\"340\" cy=\"80\" r=\"5\" fill=\"#2e6b4f\"\/>\n <circle cx=\"370\" cy=\"80\" r=\"5\" fill=\"#2e6b4f\"\/>\n <circle cx=\"400\" cy=\"80\" r=\"5\" fill=\"#2e6b4f\"\/>\n <circle cx=\"430\" cy=\"80\" r=\"5\" fill=\"#2e6b4f\"\/>\n \n <text x=\"430\" y=\"45\" font-size=\"10\" fill=\"#b03a2e\">\u2190 Valeurs > 1 !<\/text>\n <text x=\"62\" y=\"245\" font-size=\"10\" fill=\"#b03a2e\">Valeurs < 0 ! \u2192<\/text>\n \n <path d=\"M 70,225 Q 180,220 240,155 Q 300,90 460,72\" stroke=\"#4a1a6b\" stroke-width=\"2.5\" fill=\"none\"\/>\n \n <line x1=\"300\" y1=\"190\" x2=\"330\" y2=\"190\" stroke=\"#c8850a\" stroke-width=\"2\" stroke-dasharray=\"6,3\"\/>\n <text x=\"335\" y=\"194\" font-size=\"10\" fill=\"#6b6560\">R\u00e9gression lin\u00e9aire (probl\u00e8me)<\/text>\n <line x1=\"300\" y1=\"210\" x2=\"330\" y2=\"210\" stroke=\"#4a1a6b\" stroke-width=\"2.5\"\/>\n <text x=\"335\" y=\"214\" font-size=\"10\" fill=\"#6b6560\">Courbe logistique (solution)<\/text>\n <\/svg>\n <\/div>\n\n <p>\n Le probl\u00e8me est clair : la droite lin\u00e9aire peut pr\u00e9dire des valeurs inf\u00e9rieures \u00e0 0 ou sup\u00e9rieures \u00e0 1 \u2014 ce qui n’a aucun sens pour une probabilit\u00e9. On ne peut pas avoir une probabilit\u00e9 de r\u00e9ussite de 1,3 ou de -0,2 !\n <\/p>\n\n <p>\n Il nous faut une fonction qui reste <strong>toujours entre 0 et 1<\/strong>. C’est exactement ce que fait la fonction sigmo\u00efde.\n <\/p>\n\n <h2>La fonction sigmo\u00efde : le c\u0153ur de tout<\/h2>\n\n <div class=\"definition\">\n <strong>Fonction sigmo\u00efde :<\/strong> Une fonction math\u00e9matique en forme de S qui transforme n’importe quel nombre r\u00e9el en une valeur comprise entre 0 et 1. C’est la fonction porte d’entr\u00e9e de la r\u00e9gression logistique \u2014 et de tous les r\u00e9seaux de neurones artificiels.\n <\/div>\n\n <div class=\"formule\">\u03c3(z) = 1 \/ (1 + e\u207b\u1dbb)<\/div>\n\n <p>O\u00f9 <strong>e<\/strong> est la constante d’Euler (\u2248 2,718) et <strong>z<\/strong> = ax + b comme dans la r\u00e9gression lin\u00e9aire.<\/p>\n\n <div class=\"schema-container\">\n <div class=\"schema-titre\">\ud83d\udcc8 La courbe sigmo\u00efde \u2014 toujours entre 0 et 1<\/div>\n <svg width=\"100%\" viewBox=\"0 0 520 280\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\">\n <rect width=\"520\" height=\"280\" fill=\"white\"\/>\n \n <line x1=\"60\" y1=\"140\" x2=\"480\" y2=\"140\" stroke=\"#0f0e0b\" stroke-width=\"1.5\"\/>\n <line x1=\"260\" y1=\"20\" x2=\"260\" y2=\"260\" stroke=\"#0f0e0b\" stroke-width=\"1.5\"\/>\n \n <text x=\"485\" y=\"144\" font-size=\"11\" fill=\"#6b6560\">z<\/text>\n <text x=\"263\" y=\"18\" font-size=\"11\" fill=\"#6b6560\">\u03c3<\/text>\n <text x=\"30\" y=\"55\" font-size=\"10\" fill=\"#9b59b6\">1.0<\/text>\n <text x=\"30\" y=\"144\" font-size=\"10\" fill=\"#9b59b6\">0.5<\/text>\n <text x=\"30\" y=\"235\" font-size=\"10\" fill=\"#9b59b6\">0.0<\/text>\n \n <line x1=\"60\" y1=\"50\" x2=\"480\" y2=\"50\" stroke=\"#e8e2d8\" stroke-width=\"1\" stroke-dasharray=\"4,3\"\/>\n <line x1=\"60\" y1=\"230\" x2=\"480\" y2=\"230\" stroke=\"#e8e2d8\" stroke-width=\"1\" stroke-dasharray=\"4,3\"\/>\n \n <path d=\"M 60,228 C 100,226 140,222 180,210 C 200,200 220,180 240,160 C 255,143 265,120 280,105 C 300,85 320,65 360,55 C 390,50 430,49 480,48\"\n stroke=\"#4a1a6b\" stroke-width=\"3\" fill=\"none\"\/>\n \n <circle cx=\"260\" cy=\"140\" r=\"5\" fill=\"#9b59b6\"\/>\n <text x=\"268\" y=\"136\" font-size=\"10\" fill=\"#9b59b6\">\u03c3(0) = 0.5<\/text>\n \n <rect x=\"260\" y=\"50\" width=\"220\" height=\"90\" fill=\"rgba(46,107,79,0.08)\"\/>\n <text x=\"330\" y=\"100\" font-size=\"10\" fill=\"#2e6b4f\">Pr\u00e9dit : OUI (\u22650.5)<\/text>\n <rect x=\"60\" y=\"140\" width=\"200\" height=\"90\" fill=\"rgba(176,58,46,0.08)\"\/>\n <text x=\"100\" y=\"185\" font-size=\"10\" fill=\"#b03a2e\">Pr\u00e9dit : NON (<0.5)<\/text>\n <\/svg>\n <\/div>\n\n <p>\n La r\u00e8gle de d\u00e9cision est simple : si \u03c3(z) \u2265 0,5 \u2192 on pr\u00e9dit <strong>OUI (1)<\/strong>. Si \u03c3(z) < 0,5 \u2192 on pr\u00e9dit <strong>NON (0)<\/strong>. Le seuil de 0,5 peut \u00eatre ajust\u00e9 selon le contexte.\n <\/p>\n\n <h2>Lin\u00e9aire vs Logistique : le tableau comparatif<\/h2>\n\n <div class=\"comparatif\">\n <div class=\"comparatif-bloc\">\n <div class=\"comparatif-titre\" style=\"border-color:#c8850a\">\ud83d\udccf R\u00e9gression Lin\u00e9aire<\/div>\n <p>\u2192 Pr\u00e9dit une <strong>valeur continue<\/strong><\/p>\n <p>\u2192 Sortie : n’importe quel nombre r\u00e9el<\/p>\n <p>\u2192 Question : <em>\u00ab\u00a0Combien ?\u00a0\u00bb<\/em><\/p>\n <p>\u2192 Ex : pr\u00e9voir le prix du ma\u00efs<\/p>\n <p>\u2192 Fonction : \u0177 = ax + b<\/p>\n <\/div>\n <div class=\"comparatif-bloc\">\n <div class=\"comparatif-titre\" style=\"border-color:#9b59b6\">\ud83d\udd00 R\u00e9gression Logistique<\/div>\n <p>\u2192 Pr\u00e9dit une <strong>probabilit\u00e9<\/strong><\/p>\n <p>\u2192 Sortie : entre 0 et 1<\/p>\n <p>\u2192 Question : <em>\u00ab\u00a0Oui ou non ?\u00a0\u00bb<\/em><\/p>\n <p>\u2192 Ex : pr\u00e9voir si r\u00e9ussite au BEPC<\/p>\n <p>\u2192 Fonction : \u03c3(ax + b)<\/p>\n <\/div>\n <\/div>\n\n <h2>Application : pr\u00e9dire la r\u00e9ussite au BEPC<\/h2>\n\n <p>\n Voici des donn\u00e9es fictives mais r\u00e9alistes sur 8 \u00e9l\u00e8ves b\u00e9ninois, leurs heures de r\u00e9vision hebdomadaires et leur r\u00e9sultat au BEPC :\n <\/p>\n\n <div class=\"table-container\">\n <table>\n <tr>\n <th>\u00c9l\u00e8ve<\/th>\n <th>Heures\/semaine (x)<\/th>\n <th>R\u00e9sultat (y)<\/th>\n <th>P(r\u00e9ussite) pr\u00e9dite<\/th>\n <\/tr>\n <tr><td>Kofi<\/td><td>2<\/td><td>0 (\u00c9chec)<\/td><td>0,12<\/td><\/tr>\n <tr><td>Amina<\/td><td>4<\/td><td>0 (\u00c9chec)<\/td><td>0,25<\/td><\/tr>\n <tr><td>S\u00e9kou<\/td><td>6<\/td><td>1 (R\u00e9ussite)<\/td><td>0,45<\/td><\/tr>\n <tr><td>Fatou<\/td><td>8<\/td><td>1 (R\u00e9ussite)<\/td><td>0,66<\/td><\/tr>\n <tr><td>Brice<\/td><td>10<\/td><td>1 (R\u00e9ussite)<\/td><td>0,82<\/td><\/tr>\n <tr><td>Adjovi<\/td><td>12<\/td><td>1 (R\u00e9ussite)<\/td><td>0,92<\/td><\/tr>\n <tr><td>Moussa<\/td><td>3<\/td><td>0 (\u00c9chec)<\/td><td>0,18<\/td><\/tr>\n <tr><td>C\u00e9leste<\/td><td>9<\/td><td>1 (R\u00e9ussite)<\/td><td>0,75<\/td><\/tr>\n <\/table>\n <\/div>\n\n <p>\n On voit clairement la tendance : plus un \u00e9l\u00e8ve r\u00e9vise, plus sa probabilit\u00e9 de r\u00e9ussite augmente. Avec 6 heures par semaine, la probabilit\u00e9 est de 45% \u2014 trop risqu\u00e9. Avec 10 heures, elle monte \u00e0 82%. Le mod\u00e8le dit clairement : <strong>r\u00e9vise plus !<\/strong>\n <\/p>\n\n <h2>Le code Python<\/h2>\n\n <div class=\"code-block\">\n <span class=\"code-label\">Python<\/span>\n<span class=\"code-comment\"># R\u00e9gression logistique – R\u00e9ussite au BEPC<\/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> LogisticRegression\n<span class=\"code-keyword\">from<\/span> sklearn.metrics <span class=\"code-keyword\">import<\/span> accuracy_score\n<span class=\"code-keyword\">import<\/span> matplotlib.pyplot <span class=\"code-keyword\">as<\/span> plt\n\n<span class=\"code-comment\"># Donn\u00e9es : heures de r\u00e9vision et r\u00e9sultats<\/span>\nx = np.array([<span class=\"code-number\">2<\/span>, <span class=\"code-number\">4<\/span>, <span class=\"code-number\">6<\/span>, <span class=\"code-number\">8<\/span>, <span class=\"code-number\">10<\/span>, <span class=\"code-number\">12<\/span>, <span class=\"code-number\">3<\/span>, <span class=\"code-number\">9<\/span>]).reshape(-<span class=\"code-number\">1<\/span>, <span class=\"code-number\">1<\/span>)\ny = np.array([<span class=\"code-number\">0<\/span>, <span class=\"code-number\">0<\/span>, <span class=\"code-number\">1<\/span>, <span class=\"code-number\">1<\/span>, <span class=\"code-number\">1<\/span>, <span class=\"code-number\">1<\/span>, <span class=\"code-number\">0<\/span>, <span class=\"code-number\">1<\/span>])\n\n<span class=\"code-comment\"># Cr\u00e9er et entra\u00eener le mod\u00e8le<\/span>\nmodele = <span class=\"code-function\">LogisticRegression<\/span>()\nmodele.<span class=\"code-function\">fit<\/span>(x, y)\n\n<span class=\"code-comment\"># Pr\u00e9cision du mod\u00e8le<\/span>\npredictions = modele.<span class=\"code-function\">predict<\/span>(x)\nprecision = <span class=\"code-function\">accuracy_score<\/span>(y, predictions)\n<span class=\"code-function\">print<\/span>(<span class=\"code-string\">f\u00a0\u00bbPr\u00e9cision du mod\u00e8le : {precision*100:.0f}%\u00a0\u00bb<\/span>)\n\n<span class=\"code-comment\"># Pr\u00e9dire pour un nouvel \u00e9l\u00e8ve (7h de r\u00e9vision)<\/span>\nnouvel_eleve = np.array([[<span class=\"code-number\">7<\/span>]])\nproba = modele.<span class=\"code-function\">predict_proba<\/span>(nouvel_eleve)[<span class=\"code-number\">0<\/span>][<span class=\"code-number\">1<\/span>]\ndecision = modele.<span class=\"code-function\">predict<\/span>(nouvel_eleve)[<span class=\"code-number\">0<\/span>]\n<span class=\"code-function\">print<\/span>(<span class=\"code-string\">f\u00a0\u00bbProbabilit\u00e9 de r\u00e9ussite (7h) : {proba:.0%}\u00a0\u00bb<\/span>)\n<span class=\"code-function\">print<\/span>(<span class=\"code-string\">f\u00a0\u00bbD\u00e9cision : {‘R\u00e9ussite \u2713’ if decision == 1 else ‘\u00c9chec \u2717’}\u00a0\u00bb<\/span>)\n\n<span class=\"code-comment\"># Tracer la courbe sigmo\u00efde<\/span>\nx_courbe = np.linspace(<span class=\"code-number\">0<\/span>, <span class=\"code-number\">15<\/span>, <span class=\"code-number\">200<\/span>).reshape(-<span class=\"code-number\">1<\/span>, <span class=\"code-number\">1<\/span>)\nprobas_courbe = modele.<span class=\"code-function\">predict_proba<\/span>(x_courbe)[:, <span class=\"code-number\">1<\/span>]\n\nplt.<span class=\"code-function\">figure<\/span>(figsize=(<span class=\"code-number\">8<\/span>, <span class=\"code-number\">5<\/span>))\nplt.<span class=\"code-function\">plot<\/span>(x_courbe, probas_courbe, color=<span class=\"code-string\">‘#4a1a6b’<\/span>, linewidth=<span class=\"code-number\">2.5<\/span>)\nplt.<span class=\"code-function\">scatter<\/span>(x, y, color=<span class=\"code-string\">‘#9b59b6’<\/span>, s=<span class=\"code-number\">80<\/span>, zorder=<span class=\"code-number\">5<\/span>)\nplt.<span class=\"code-function\">axhline<\/span>(y=<span class=\"code-number\">0.5<\/span>, color=<span class=\"code-string\">‘#c8850a’<\/span>, linestyle=<span class=\"code-string\">‘–‘<\/span>, label=<span class=\"code-string\">‘Seuil 0.5’<\/span>)\nplt.<span class=\"code-function\">xlabel<\/span>(<span class=\"code-string\">‘Heures de r\u00e9vision \/ semaine’<\/span>)\nplt.<span class=\"code-function\">ylabel<\/span>(<span class=\"code-string\">‘Probabilit\u00e9 de r\u00e9ussite au BEPC’<\/span>)\nplt.<span class=\"code-function\">title<\/span>(<span class=\"code-string\">‘R\u00e9gression Logistique – Pr\u00e9diction BEPC’<\/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>Pr\u00e9cision du mod\u00e8le : 100%<\/strong> (sur ces 8 exemples d’entra\u00eenement)<\/p>\n <p><strong>Probabilit\u00e9 de r\u00e9ussite pour 7h : 57%<\/strong><\/p>\n <p><strong>D\u00e9cision : R\u00e9ussite \u2713<\/strong> (juste au-dessus du seuil de 0,5)<\/p>\n <\/div>\n\n <h2>Les applications r\u00e9elles en Afrique<\/h2>\n\n <div class=\"encadre-histoire encadre\">\n <span class=\"encadre-titre\">\ud83c\udf0d Cas d’usage africains<\/span>\n <p>\n <strong>1. Sant\u00e9 :<\/strong> Des h\u00f4pitaux \u00e0 Cotonou, Lagos et Dakar utilisent des mod\u00e8les similaires pour pr\u00e9dire le risque de paludisme s\u00e9v\u00e8re chez un patient, en fonction de ses sympt\u00f4mes et de son \u00e2ge.\n <\/p>\n <p>\n <strong>2. Finance :<\/strong> Les microfinances b\u00e9ninoises (comme CLCAM) commencent \u00e0 explorer ces outils pour d\u00e9cider d’accorder ou non un pr\u00eat \u00e0 un agriculteur, selon ses ant\u00e9c\u00e9dents et sa r\u00e9gion.\n <\/p>\n <p>\n <strong>3. Agriculture :<\/strong> Pr\u00e9dire si une r\u00e9colte sera bonne ou mauvaise selon les donn\u00e9es m\u00e9t\u00e9o, le type de sol et les intrants utilis\u00e9s.\n <\/p>\n <p>\n <strong>4. \u00c9ducation :<\/strong> Identifier les \u00e9l\u00e8ves \u00e0 risque de d\u00e9crochage scolaire pour intervenir \u00e0 temps \u2014 exactement le genre d’outil qu’un enseignant-data scientist peut construire.\n <\/p>\n <\/div>\n\n <blockquote>\n Les donn\u00e9es sont la nouvelle mati\u00e8re premi\u00e8re. Mais sans les math\u00e9matiques pour les raffiner, elles ne valent rien.\n <cite>\u2014 Inspiration de la data science africaine<\/cite>\n <\/blockquote>\n\n \n <div class=\"exercice\">\n <h3>\u270f\ufe0f Exercice corrig\u00e9<\/h3>\n <p>\n Un agent de sant\u00e9 \u00e0 Porto-Novo a recueilli des donn\u00e9es sur 6 patients suspects de paludisme. Il dispose du nombre de sympt\u00f4mes pr\u00e9sent\u00e9s (x) et du diagnostic final (y : 1 = paludisme confirm\u00e9, 0 = n\u00e9gatif).\n <\/p>\n <p>\n x (sympt\u00f4mes) : 1, 2, 3, 4, 5, 6<br>\n y (diagnostic) : 0, 0, 0, 1, 1, 1\n <\/p>\n <p><strong>a)<\/strong> Quel type de r\u00e9gression est adapt\u00e9 ici, et pourquoi ?<\/p>\n <p><strong>b)<\/strong> Si on a trouv\u00e9 z = 1,5x – 5, calcule \u03c3(z) pour x = 3 et x = 5.<\/p>\n <p><strong>c)<\/strong> Pour x = 3, le mod\u00e8le diagnostiquerait-il un paludisme ?<\/p>\n\n <div class=\"correction\">\n <div class=\"correction-titre\">\u25b8 Correction<\/div>\n <p><strong>a)<\/strong> La r\u00e9gression <strong>logistique<\/strong> est adapt\u00e9e car la sortie est binaire (0 ou 1 \u2014 n\u00e9gatif ou positif). La r\u00e9gression lin\u00e9aire donnerait des valeurs hors de [0,1], ce qui n’aurait pas de sens pour une probabilit\u00e9.<\/p>\n <p><strong>b)<\/strong> Pour x = 3 : z = 1,5(3) – 5 = 4,5 – 5 = <strong>-0,5<\/strong><\/p>\n <p>\u03c3(-0,5) = 1 \/ (1 + e^0,5) = 1 \/ (1 + 1,649) \u2248 <strong>0,38<\/strong> soit 38%<\/p>\n <p>Pour x = 5 : z = 1,5(5) – 5 = 7,5 – 5 = <strong>2,5<\/strong><\/p>\n <p>\u03c3(2,5) = 1 \/ (1 + e^(-2,5)) = 1 \/ (1 + 0,082) \u2248 <strong>0,92<\/strong> soit 92%<\/p>\n <p><strong>c)<\/strong> Pour x = 3, \u03c3 = 0,38 < 0,5 \u2192 le mod\u00e8le pr\u00e9dit <strong>NON (0)<\/strong> : pas de paludisme confirm\u00e9. \u2705<\/p>\n <\/div>\n <\/div>\n\n <div class=\"conclusion\">\n <h2>Ce qu’on retient<\/h2>\n <p>\n La r\u00e9gression logistique, c’est le passage des math\u00e9matiques \u00e0 la <strong>prise de d\u00e9cision<\/strong>. L\u00e0 o\u00f9 la r\u00e9gression lin\u00e9aire mesure, la r\u00e9gression logistique tranche. Ensemble, ces deux outils forment le socle de la data science \u2014 et des intelligences artificielles qui transforment aujourd’hui la m\u00e9decine, la finance et l’\u00e9ducation en Afrique.\n <\/p>\n <p>\n Maintenant que tu comprends ces deux r\u00e9gressions, tu n’es plus seulement un observateur de l’IA. Tu en comprends les fondations.\n <\/p>\n <p>\n Dans le prochain article, nous sortons des chiffres et rentrons dans la nature pour poser une question fascinante : <strong>pourquoi les abeilles construisent-elles des hexagones ?<\/strong> La r\u00e9ponse cache l’un des probl\u00e8mes d’optimisation les plus \u00e9l\u00e9gants de toute la math\u00e9matique.\n <\/p>\n <\/div>\n\n<\/article>\n\n<\/body>\n<\/html>\n","protected":false},"excerpt":{"rendered":"<p>La r\u00e9gression logistique : quand les maths disent oui ou non \u2014 MathsVivantes Data & Maths \u00b7 Article #4 La […]<\/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],"tags":[],"class_list":["post-74","post","type-post","status-publish","format-standard","hentry","category-data-et-maths"],"yoast_head":"\n<title>La r\u00e9gression logistique - AfrikIntelligent<\/title>\n<meta name=\"description\" content=\"La r\u00e9gression logistique expliqu\u00e9e simplement . 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