3e:null 3f:["$","title",null,{"children":"Illustration de la méthode ACP : un exemple avec des pains | Laurent Jégou | Observable"}] 45:T4f8,stats = [ { nom: "poids", moyenne: maths.mean(poids), médiane: maths.median(poids), ecart_type: maths.std(poids), minimum: maths.min(poids), maximum: maths.max(poids) }, { nom: "longueur", moyenne: maths.mean(longueurs), médiane: maths.median(longueurs), ecart_type: maths.std(longueurs), minimum: maths.min(longueurs), maximum: maths.max(longueurs) }, { nom: "largeur", moyenne: maths.mean(largeurs), médiane: maths.median(largeurs), ecart_type: maths.std(largeurs), minimum: maths.min(largeurs), maximum: maths.max(largeurs) }, { nom: "épaisseur", moyenne: maths.mean(épaisseurs), médiane: maths.median(épaisseurs), ecart_type: maths.std(épaisseurs), minimum: maths.min(épaisseurs), maximum: maths.max(épaisseurs) }, { nom: "prix", moyenne: maths.mean(prix), médiane: maths.median(prix), ecart_type: maths.std(prix), minimum: maths.min(prix), maximum: maths.max(prix) } ];46:T4af,matrice_r = [ {nom: "poids", poids: 1, longueurs: coef_correlation(poids, longueurs), largeurs: coef_correlation(poids, largeurs), épaisseurs: coef_correlation(poids, épaisseurs), prix: coef_correlation(poids, prix), }, {nom: "longueurs", poids: coef_correlation(longueurs, poids), longueurs: 1, largeurs: coef_correlation(longueurs, largeurs), épaisseurs: coef_correlation(longueurs, épaisseurs), prix: coef_correlation(longueurs, prix), }, {nom: "largeurs", poids: coef_correlation(largeurs, poids), longueurs: coef_correlation(largeurs, longueurs), largeurs: 1, épaisseurs: coef_correlation(largeurs, épaisseurs), prix: coef_correlation(largeurs, prix), }, {nom: "épaisseurs", poids: coef_correlation(épaisseurs, poids), longueurs: coef_correlation(épaisseurs, longueurs), largeurs: coef_correlation(épaisseurs, largeurs), épaisseurs: 1, prix: coef_correlation(épaisseurs, prix), }, {nom: "prix", poids: coef_correlation(prix, poids), longueurs: coef_correlation(prix, longueurs), largeurs: coef_correlation(prix, largeurs), épaisseurs: coef_correlation(prix, épaisseurs), prix: 1, } ]47:T581,matrice_varcovar = [ {nom: "poids", poids: maths.variance(poids_cr, 'uncorrected'), longueurs: covariance(poids_cr, longueurs_cr), largeurs: covariance(poids_cr, largeurs_cr), épaisseurs: covariance(poids_cr, épaisseurs_cr), prix: covariance(poids_cr, prix_cr), }, {nom: "longueurs", poids: covariance(longueurs_cr, poids_cr), longueurs: maths.variance(longueurs_cr, 'uncorrected'), largeurs: covariance(longueurs_cr, largeurs_cr), épaisseurs: covariance(longueurs_cr, épaisseurs_cr), prix: covariance(longueurs_cr, prix_cr), }, {nom: "largeurs", poids: covariance(largeurs_cr, poids_cr), longueurs: covariance(largeurs_cr, longueurs_cr), largeurs: maths.variance(largeurs_cr, 'uncorrected'), épaisseurs: covariance(largeurs_cr, épaisseurs_cr), prix: covariance(largeurs_cr, prix_cr), }, {nom: "épaisseurs", poids: covariance(épaisseurs_cr, poids_cr), longueurs: covariance(épaisseurs_cr, longueurs_cr), largeurs: covariance(épaisseurs_cr, largeurs_cr), épaisseurs: maths.variance(épaisseurs_cr, 'uncorrected'), prix: covariance(épaisseurs_cr, prix_cr), }, {nom: "prix", poids: covariance(prix_cr, poids_cr), longueurs: covariance(prix_cr, longueurs_cr), largeurs: covariance(prix_cr, largeurs_cr), épaisseurs: covariance(prix_cr, épaisseurs_cr), prix: maths.variance(prix_cr, 'uncorrected'), } ]48:T43d,On représente traditionnellement les variables en projection sur les deux premières composantes, les plus explicatives, par des vecteurs partant de l'origine. Plus le vecteur est long (proche de 1), plus la qualité de la décomposition est bonne. On remarque nettement la proximité des trois variables repérées précédemment dans le tableau des corrélations : longueur, poids et prix, qui se situent à l'extrémité positive de l'axe horiznotal, de la première composante. À l'inverse, la largeur est sur l'extrémité négative de cet axe horizontal. *L'axe horizontal, la première composante, permet donc de distinguer les pains longs, lourds et chers des autres, nos trois variables très corrélées sont regroupées.* *Sur l'autre axe, vertical, correspondant à la deuxième composante, on voir que l'épaisseur s'oppose à la largeur*. Les pains larges sont souvent peu épais. Pour en avoir une illustration, on peut maintenant projeter les individus sur ces axes et repérer leurs groupes, donc potentiellement des types de pains aux caractéristiques proches.49:[{"id":"db424aba23d1092e48a919acc1402d0df9e971b65300d5c6851be4be4b9b577a0ff5f9a553fad51c50843d0eb0f790888886e6727dd13d177ffa8fb02e055a58","href":"https://static.observableusercontent.com/files/db424aba23d1092e48a919acc1402d0df9e971b65300d5c6851be4be4b9b577a0ff5f9a553fad51c50843d0eb0f790888886e6727dd13d177ffa8fb02e055a58","name":"pains@2.csv","type":"text/csv","lastModified":1676734168763.814,"size":1121}] 43:{"id":"e2b88c7ac3d1a73e","slug":"acp-des-pains","title":"Illustration de la méthode ACP : un exemple avec des pains","thumbnail":"a3740c6074e58f7d9a5d4c87c633dfca351e5c4e920a0627be9f3e20cf982dae","access_level":"public","create_time":"$D2023-02-18T14:33:06.522Z","update_time":"$D2024-11-19T10:00:39.162Z","view_time":null,"first_public_time":"$D2023-02-21T16:13:19.109Z","has_unpublished_edits":null,"fork":false,"stars":16,"starred":false,"owner":{"type":"team","id":"1936870012656af6","login":"ljegou","name":"Laurent Jégou","avatar_url":"https://avatars-next.observableusercontent.com/447aa44a872789c3c13d2505fdb5a34daaaa3a44590422f4f782b6ea9a4bd3ac","tier":"starter_2024","bio":"Cartography associate professor (MCF) at Toulouse-Jean Jaurès university."},"version":1096,"publish_version":null,"role":null,"authored":false,"authors":[{"type":"individual","id":"dc56e6af58493346","login":"ljegou","name":"Laurent Jégou","avatar_url":"https://avatars-next.observableusercontent.com/447aa44a872789c3c13d2505fdb5a34daaaa3a44590422f4f782b6ea9a4bd3ac","tier":null,"bio":"Cartography associate professor (MCF) at Toulouse-Jean Jaurès university."}],"members":[{"type":"individual","id":"dc56e6af58493346","login":"ljegou","name":"Laurent Jégou","avatar_url":"https://avatars-next.observableusercontent.com/447aa44a872789c3c13d2505fdb5a34daaaa3a44590422f4f782b6ea9a4bd3ac","tier":null,"role":"editor"}],"comments":[],"custom_thumbnail":true,"trashed":false,"lapsed":null,"subscribed":null,"forks":1,"collection":null,"type":"notebook","body":{"titling":"auto","title":"Illustration de la méthode ACP : un exemple avec des pains","cells":[{"id":0,"value":"# Illustration de la méthode ACP : un exemple avec des pains","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":979,"value":"À partir du catalogue d'un fournisseur de boulangerie, nous avons extrait 5 caractéristiques *quantitatives* des types de pains disponibles : le poids, la longueur, la largeur, l'épaisseur et le prix.\n\nNous cherchons à savoir si l'on peut trouver des groupes, des types de pains, à partir de ces variables. 5 variables c'est un peu trop, il faudrait concentrer cette information en deux ou trois variables synthétiques seulement.\n\nLa méthode de l'**Analyse en Composantes Principales** va nous y aider, en déterminant quelles sont les variables qui décrivent le mieux les différences entre les pains et en les regroupant par composantes, de nouvelles variables synthétiques.\n\nCommençons par mettre à plat notre tableau de valeurs par individu.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":986,"value":"## Tableau des individus / variables","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":10,"value":"table = Inputs.table(pains);","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":989,"value":"Certains types d'analyse statistiques peuvent nous aider à explorer les données, par exemple les fréquences pour avoir une vision synthétique de la répartition des individus selon une variable, ici les prix.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":961,"value":"## Graphique des fréquences de la variable prix","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":13,"value":"Plot.rectY(pains, Plot.binX({y: \"count\"}, {x: \"prix\", fill: \"orange\"})).plot()","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":992,"value":"On remarque que les valeurs des prix sont assez dispersées, avec un mode assez net (la barre la plus haute) et des valeurs élevées bien présentes.\n\nCalculer les indicateurs statistiques classiques, de *centralité* et de *dispersion* nous aidera à y voir plus clair.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":175,"value":"## Statistiques descriptives des variables","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":110,"value":"tableStats = Inputs.table(stats, {format: {\n moyenne: sparkbar(260),\n médiane: sparkbar(260),\n ecart_type: sparkbar(140),\n minimum: sparkbar(60),\n maximum: sparkbar(500)\n }\n});","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":58,"value":"$45","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1008,"value":"## Corrélations","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":1010,"value":"Pour repérer si des dimensions sont corrélées, on peut utiliser l'outil de la corrélation linéaire simple, en comparant les variables 2 à 2.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":177,"value":"## Table/matrice des corrélations (sans C-R)","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":166,"value":"tableMatriceR = Inputs.table(matrice_r, {\n format: {\n poids: sparkbar(1),\n longueurs: sparkbar(1),\n largeurs: sparkbar(1),\n épaisseurs: sparkbar(1),\n prix: sparkbar(1)\n }\n});","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":158,"value":"$46","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1013,"value":"Étant donné que les unités de nos variables leur donnent des valeurs d'étendue très variable (grammes, centimètres, euros...), il est préférable de normaliser les valeurs en utilisant un *centrage* (soustraction de la moyenne) et une *réduction* (division par l'écart-type.\n\nOn obtient ainsi des valeurs qui ne représentent que les éloignements proportionnels des individus à la moyenne et sont donc comparables entre elles. Ces valeurs peuvent être comparées 2 à 2 par l'outil statistique de la covariance, qui mesure la différence des valeurs des individus sur deux variables selon la mesure de la variance (carré des écarts moyens à la moyenne).\n\nSur la diagonale de la matrice, on indique la variance simple de la variable, qui est ici toujours égale à 1 car les valeurs sont centrées-réduites.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":202,"value":"## Table/Matrice des variances-covariances (avec C-R)","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":199,"value":"tableMatriceV = Inputs.table(matrice_varcovar, {\n format: {\n poids: sparkbar(1),\n longueurs: sparkbar(1),\n largeurs: sparkbar(1),\n épaisseurs: sparkbar(1),\n prix: sparkbar(1)\n } \n});","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":193,"value":"$47","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1018,"value":"Ici, on remarque nettement de fortes corrélations :\n- poids / longueur\n- poids / prix\n- longueur / prix\n\nCes trois variables sont corrélées, c'est à dire que si un pain a une valeur particulière (haute, basse...) sur l'une d'elle, il aura des valeurs de même types sur les autres variables. *Un pain long et lourd sera plus cher*.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":1016,"value":"Ces corrélations linéaires peuvent être visualisées par un diagramme par points X,Y. Si les points suivent une ligne croissante, c'est que les deux variables sont corrélées positivement, la valeur de l'une d'elle sera proche de la valeur sur l'autre variable, avec un décalage (a) et un coefficient (b) pour l'équation de la droite de régression ax+b.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":969,"value":"## Une forte corrélation entre longueur et poids / prix","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":929,"value":"Plot.plot({\n grid: true,\n color: {\n type: \"linear\",\n scheme: \"magma\",\n reverse: true,\n legend: true\n }, \n marks: [\n Plot.dot(pains,\n {\n x: \"longueur\", \n y: \"poids_g\",\n fill: \"prix\",\n r:\"épaisseur\"\n }),\n Plot.linearRegressionY(pains, \n {\n x: \"longueur\",\n y: \"poids_g\",\n ci: 0.95\n }),\n Plot.ruleY([0])\n ]\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":965,"value":"## Une corrélation faiblement négative entre longueur et largeur","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":19,"value":"Plot.plot({\n grid: true,\n x: {domain: [5, 58]},\n y: {domain: [4, 14]},\n color: {\n type: \"linear\",\n scheme: \"magma\",\n reverse: true,\n legend: true\n }, \n marks: [\n Plot.dot(pains,\n {\n x: \"longueur\", \n y: \"largeur\",\n fill: \"épaisseur\",\n r:\"poids_g\"\n }),\n Plot.linearRegressionY(pains, \n {\n x: \"longueur\",\n y: \"largeur\",\n ci: 0.95\n }),\n Plot.ruleY([0])\n ]\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1021,"value":"## Regrouper les variables en axes synthétiques : l'ACP","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":1023,"value":"Pour aller plus loin, comme on a remarqué que plusieurs variables étaient corrélées, on peut utiliser la technique de l'**Analyse en Composantes Principales** pour simplifier notre tableau de variables en quelques *composantes* (ou variables synthétiques), qui pourront devenir des *axes* de représentations sur lesquels nos pains seraient projetés et que l'on pourrait alors regrouper en *types* de pains.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":1026,"value":"La méthode de l'ACP utilise un outil matriciel appelé la *décomposition en valeurs singulières* (avec l'acroynme SVD en anglais). Cet outil va produire automatiquement des composantes, sur lesquels nos individus (les pains) et nos variables (les caractéristiques) pourront être projetés.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":519,"value":"svd = new Matrix.SingularValueDecomposition(matrice_cr, {autoTranspose: true});","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":260,"value":"## Table des composantes principales","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":262,"value":"table_facteurs = Inputs.table(tableau_composantes, {\n format: {\n Pct_var_exp: sparkbar_pct(1),\n Pct_var_exp_cumul: sparkbar_pct(1)\n }\n});","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1035,"value":"La *SVD* trouve **5** *composantes* pour expliquer 99% de la variance entre individus, elles sont ici classées dans l'ordre décroissant de leur pouvoir d'explication de la variance totale (la part de leur valeur propre dans le total des valeurs propres de composantes). Avec les deux premières on explique plus de 80% des différences de valeurs entre les individus, ces composantes (ou variables synthétiques) sont donc de bons résumés de l'ensemble du tableau. En ajoutant la troisième composante, on obtient 95% d'explication.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":803,"value":"## Contributions des composantes à l'explication des variances","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":369,"value":"Plot.plot({\n x: {\n type: \"point\",\n label: \"Composante\",\n grid: true\n },\n y: {\n label: \"↑ Valeur propre\",\n grid: true\n },\n marks: [\n Plot.line(valpplot),\n ]\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1039,"value":"On représente les valeurs propres de composantes par un graphique en courbes pour repérer à quel moment le gain en pouvoir explicatif diminue, à partir de quelle composante. Ici, après la deuxième composante, la courbe marque un \"*coude*\" net (*elbow* en anglais).","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":1041,"value":"Une fois la SVD calculée, on peut utiliser ses résultats pour projeter les individus et les variables sur les axes que forment les composantes (par une multiplication matricielle). On peut ainsi visualiser les variables qui \"*expliquent*\" les composantes et voir la façon dont les individus, les pains, se regroupent sur ces axes.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":809,"value":"## Position des variables et des individus sur les deux premières composantes","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":656,"value":"\nplots_vars12 = Plot.plot({\n height: 500,\n x: {\n label: \"Composante 1\",\n domain: [-1.2, 1.8],\n grid: true\n },\n y: {\n label: \"Composante 2\",\n domain: [-1, 1],\n grid: true\n },\n marks: [\n Plot.arrow(coords_vars12, {\n x1: \"ox\",\n y1: \"oy\",\n x2: \"dx\",\n y2: \"dy\",\n stroke: \"red\",\n strokeOpacity: 0.8\n }),\n ,\n Plot.text(coords_vars12, {\n x: \"dx\",\n y: d => d.dy + d.pos,\n text: \"nom\",\n fill: \"red\",\n stroke: \"white\",\n dy: 1\n }),\n Plot.ruleX([0]),\n Plot.ruleY([0])\n ]\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1044,"value":"$48","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":417,"value":"plots_individus12 = \nPlot.plot({\n height: 500,\n x: {\n label: \"Composante 1\",\n domain: [-1.2, 1.8],\n grid: true\n },\n y: {\n label: \"Composante 2\",\n domain: [-1, 1],\n grid: true\n },\n marks: [\n Plot.dot(coords_inds12,\n { x: \"x\", y: \"y\",\n }),\n Plot.text(coords_inds12, {\n x: \"x\",\n y: \"y\",\n text: \"nom\",\n fill: \"black\",\n stroke: \"white\",\n dy: -12\n }),\n Plot.arrow(coords_vars12, {\n x1: \"ox\",\n y1: \"oy\",\n x2: d => d.dx,\n y2: d => d.dy,\n stroke: \"red\",\n strokeOpacity: 0.8\n }),\n ,\n Plot.text(coords_vars12, {\n x: d => d.dx,\n y: d => d.dy + d.pos,\n text: \"nom\",\n fill: \"red\",\n stroke: \"white\",\n dy: 1\n }),\n Plot.ruleX([0]),\n Plot.ruleY([0])\n ]\n});","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1048,"value":"Les pains sont bien répartis sur l'axe horizontal, un peu moins sur l'axe vertical, qui est moins discriminant. On retrouve des exemples de pains aux caractéristiques particulières en fonction de leur position par rapport aux axes et aux variables : la pita est large mais fine, la flute rustique longue, lourde et chère, la baguette gastronomique pile au milieu de notre échantillonnage boulanger.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":815,"value":"## Position des variables et des individus sur les composantes 2 et 3","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":778,"value":"\nplots_vars23 = Plot.plot({\n height: 500,\n x: {\n label: \"Composante 2\",\n domain: [-0.8, 1],\n grid: true\n },\n y: {\n label: \"Composante 3\",\n domain: [-0.4, 0.6],\n grid: true\n },\n marks: [\n Plot.arrow(coords_vars23, {\n x1: \"ox\",\n y1: \"oy\",\n x2: \"dx\",\n y2: \"dy\",\n stroke: \"red\",\n strokeOpacity: 0.8\n }),\n ,\n Plot.text(coords_vars23, {\n x: \"dx\",\n y: d => d.dy + d.pos,\n text: \"nom\",\n fill: \"red\",\n stroke: \"white\",\n dy: 0\n }),\n Plot.ruleX([0]),\n Plot.ruleY([0])\n ]\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":786,"value":"plots_individus23 = \nPlot.plot({\n height: 500,\n x: {\n label: \"Composante 2\",\n domain: [-0.8, 1],\n grid: true\n },\n y: {\n label: \"Composante 3\",\n domain: [-0.4, 0.6],\n grid: true\n },\n marks: [\n Plot.dot(coords_inds23,\n { x: \"x\", y: \"y\",\n }),\n Plot.text(coords_inds23, {\n x: \"x\",\n y: \"y\",\n text: \"nom\",\n fill: \"black\",\n stroke: \"white\",\n dy: -12\n }),\n Plot.arrow(coords_vars23, {\n x1: \"ox\",\n y1: \"oy\",\n x2: d => d.dx,\n y2: d => d.dy,\n stroke: \"red\",\n strokeOpacity: 0.8\n }),\n ,\n Plot.text(coords_vars23, {\n x: d => d.dx,\n y: d => d.dy + d.pos,\n text: \"nom\",\n fill: \"red\",\n stroke: \"white\",\n dy: 0\n }),\n Plot.ruleX([0]),\n Plot.ruleY([0])\n ]\n});","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1050,"value":"L'observation des projections sur la composante suivante, la troisième, montre une répartition moins nette. Elle oppose les pains larges et épais (focaccia, marguerite) aux pains plutôt longs (baguettes).","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":1076,"value":"## Conclusion","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":1078,"value":"Au final, la méthode, appliquée ici sur un jeu de donnée évocateur mais possédant volontairement un nombre d'individus limité et des variables parfois fortement corrélées, nous permet de produire des dimensions synthétiques qui restent représentatives des caractéristiques des individus, ce qui nous permet de les regrouper en types : longs, lourds et chers, larges et fins...\n\nPour aller plus loin, on peut tenter d'utiliser des techniques de regroupement automatique des individus (*clustering* en anglais), basées sur ces composantes principales, ce sera l'objet d'un prochain carnet Observable !","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":124,"value":"## Données, calculs et indicateurs","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":5,"value":"pains = (FileAttachment(\"pains@2.csv\").csv().then(\n function(data) { //Conversion des types numériques\n data.forEach(function(d) {\n d.poids_g = +d.poids_g;\n d.longueur = +d.longueur;\n d.largeur = +d.largeur;\n d.épaisseur = +d.épaisseur;\n d.prix = +d.prix;\n });\n return data;\n }\n));","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":315,"value":"pains_cr = {\n const nbval = pains.length;\n var pains_cr = [];\n var moy_poids = maths.mean(poids);\n var ect_poids = maths.std(poids, 'uncorrected');\n var moy_lng = maths.mean(longueurs);\n var ect_lng = maths.std(longueurs, 'uncorrected');\n var moy_lrg = maths.mean(largeurs);\n var ect_lrg = maths.std(largeurs, 'uncorrected');\n var moy_ép = maths.mean(épaisseurs);\n var ect_ép = maths.std(épaisseurs, 'uncorrected');\n var moy_prix = maths.mean(prix);\n var ect_prix = maths.std(prix, 'uncorrected');\n for (var i = 0; i {return el.longueur});\n","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":326,"value":"longueurs_cr = pains_cr.map(el => {return el.longueur});","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":80,"value":"largeurs = pains.map(el => {return el.largeur})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":332,"value":"largeurs_cr = pains_cr.map(el => {return el.largeur})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":83,"value":"épaisseurs = pains.map(el => {return el.épaisseur})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":335,"value":"épaisseurs_cr = pains_cr.map(el => {return el.épaisseur})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":86,"value":"prix = pains.map(el => {return el.prix})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":338,"value":"prix_cr = pains_cr.map(el => {return el.prix})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":118,"value":"poids = pains.map(el => {return el.poids_g})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":341,"value":"poids_cr = pains_cr.map(el => {return el.poids_g})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":181,"value":"matrice_valeursvcv = matrice_varcovar.map(el => {return [el.poids, el.longueurs, el.largeurs, el.épaisseurs, el.prix]});","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":126,"value":"coef_correlation = function (vara, varb) {\n const nbval = pains.length;\n var moya = maths.mean(vara);\n var moyb = maths.mean(varb);\n var numerateur = 0.0;\n var diviseur = 0.0;\n for (var i = 1; i < nbval; i++) {\n numerateur += (vara[i]-moya)*(varb[i]-moyb);\n diviseur += maths.sqrt(maths.square(vara[i]-moya)*maths.square(varb[i]-moyb));\n }\n return numerateur/diviseur;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":184,"value":"covariance = function (vara, varb) {\n const nbval = pains.length;\n var moya = maths.mean(vara);\n var moyb = maths.mean(varb);\n var numerateur = 0;\n for (var i = 1; i < nbval; i++) {\n numerateur += (vara[i]-moya)*(varb[i]-moyb);\n }\n return numerateur/(nbval - 1.0);\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":493,"value":"matrice_cr = new Matrix.Matrix(pains_cr.map(el => {return [el.poids_g, el.longueur, el.largeur, el.épaisseur, el.prix]})); ","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":238,"value":"val_propres = (svd.s).map(e => {return Math.pow((e/5),2)});","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":640,"value":"valpplot = {\n var vp = Array.from(val_propres);\n return vp.map((e, i) => {return [i+1, e]});\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":304,"value":"vect_propres = svd.V.data;","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":250,"value":"sommes_vp = {\n var s = 0;\n for (var i=0;i {return el * 100 / sommes_vp});","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":272,"value":"tableau_composantes = {\n const nbval = val_propres.length;\n var facteurs = [];\n var pct_cumul = 0;\n for (var i = 0; i < nbval; i++) {\n pct_cumul += pct_vp[i];\n facteurs.push({Composante: i+1, Valeur_propre: val_propres[i], Pct_var_exp: pct_vp[i].toFixed(2)+\" %\", Pct_var_exp_cumul: pct_cumul.toFixed(2)+\" %\"});\n }\n return facteurs;\n}\n","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":430,"value":"coords_inds = {\n var ci = Array();\n var U = svd.U.data;\n for (var i=0; i htl.html`

${x.toLocaleString(\"en\")}`\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1083,"value":"Source de l'imagette : Scott Bauer, U.S. Department of Agriculture, PIXNIO, licence CC0","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"}],"files":"$W49","stdlib":"1"}} 44:"$43:body"