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  • 1. Arthur CHARPENTIER - Analyse des donn´ees Analyse des donn´ees (1) L’Analyse en Composantes Principales Arthur Charpentier http ://perso.univ-rennes1.fr/arthur.charpentier/ blog.univ-rennes1.fr/arthur.charpentier/ Master 2, Universit´e Rennes 1 1
  • 2. Arthur CHARPENTIER - Analyse des donn´ees Introduction `a l’analyse des donn´ees Dans ce cours, nous verrrons essentiellement deux types de m´ethodes • les m´ethodes factorielles, o`u on cherchera `a r´eduire le nombre de variables en les r´esumant en un petit nombre de composantes synth´etiques ◦ en particulier l’ACP, Analyse en Composantes Principales si les variables sont quantitatives ◦ en particulier l’AC, Analyse des Correspondances si les variables sont qualitatives, o`u on cherchera les liens entre les modalit´es, avec l’ACF Analyse des Correspondances Factorielles (simples) dans le cas o`u on dispose de 2 variables, et l’ACM Analyse des Correspondances Multiples dans le cas o`u on dispose de plus de 2 variables 2
  • 3. Arthur CHARPENTIER - Analyse des donn´ees Introduction `a l’analyse des donn´ees • les m´ethodes de classification, o`u on cherchera `a r´eduire la taille de l’ensemble des individus en les regroupant en un petit nombre de groupes homog`enes ◦ en particulier la CAH, Classification Ascendante Hi´erarchique ... ◦ en particulier l’Analyse Discriminante ... Remarque Ce cours est davantage un cours d’alg`ebre lin´eaire qu’un cours de probabilit´e ou de statistique. Mais une interpr´etation sera parfois possible en terme de moyenne ou de variance (voire de covariance). 3
  • 4. Arthur CHARPENTIER - Analyse des donn´ees Exemple, ville et (in)s´ecurit´e “Le palmar`es des d´epartements : o`u vit-on en s´ecurit´e ?, dans L’Express (no 2589, 15 f´evrier 2001) • infra Nombre d’infractions totale pour 1000 habitants (2000) • vvi Nombre de vols avec violance pour 1000 habitants (2000) • auto Nombre de vols d’automobiles pour 1000 habitants (2000) > add=read.table("http://perso.univ-rennes1.fr/arthur.charpentier/securite.txt",header=TRU > base=add[,2:ncol(add)] > rownames(base)=add$dep > base=base[,c(1,6,9)] > head(base) infra vvi auto D1 44.11 0.27 4.47 D2 45.97 0.55 4.39 D3 38.83 0.41 2.39 D4 49.68 0.21 4.17 D5 47.67 0.33 2.35 D6 109.21 4.10 8.83 4
  • 5. Arthur CHARPENTIER - Analyse des donn´ees Exemple, ville et (in)s´ecurit´e q q q qq q q q q q q q q q q q qqq q q q q q qqq q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q qq q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q 20 40 60 80 100 120 140 0246810 infractions volsavecviolence 20 40 60 80 100 120 140 02468101214 0 2 4 6 8 10 infractions volsavecviolence volsautomobile qqq qq q q q q q qq q q q q qqq qqq q q qqqq qq q q q qq q q q q q q q qq q qq qq q qq qqq qq qq q qq qq q q q q qq q qq q q qq q q q q qq qqqq qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 5
  • 6. Arthur CHARPENTIER - Analyse des donn´ees Exemple, ville et (in)s´ecurit´e qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q 20 40 60 80 100 120 140 02468101214 infractions volsautomobile 20 40 60 80 100 120 140 02468101214 0 2 4 6 8 10 infractions volsavecviolence volsautomobile qq q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 6
  • 7. Arthur CHARPENTIER - Analyse des donn´ees Exemple, ville et (in)s´ecurit´e q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q 0 2 4 6 8 10 02468101214 vols avec violence volsautomobile 20 40 60 80 100 120 140 02468101214 0 2 4 6 8 10 infractions volsavecviolence volsautomobile qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 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q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 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q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 7
  • 8. Arthur CHARPENTIER - Analyse des donn´ees Exemple, ville et (in)s´ecurit´e Les variables semblent plutˆot corr´el´ees positivement, > cor(base) infra vvi auto infra 1.0000000 0.8583172 0.7808855 vvi 0.8583172 1.0000000 0.5032206 auto 0.7808855 0.5032206 1.0000000 Supposons que l’on cherche `a regrouper les villes “proches”. =⇒ Comme on a du mal `a voir dans R3 , on va essayer de projeter le nuage. • projection sur un axe (droite) • projection sur un plan 8
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Arthur CHARPENTIER - Analyse des donn´ees Exemple, ville et (in)s´ecurit´e 20 40 60 80 100 120 140 02468101214 0 2 4 6 8 10 infractions volsavecviolence volsautomobile q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq q qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq q qqqqqqqqqqqqqqqqqqqq q qqqqqqqqqqqqqqqqqqqqqqqqqqqqqq q 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Arthur CHARPENTIER - Analyse des donn´ees Exemple, ville et (in)s´ecurit´e 20 40 60 80 100 120 140 02468101214 0 2 4 6 8 10 infractions volsavecviolence volsautomobile q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 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  • 11. Arthur CHARPENTIER - Analyse des donn´ees Exemple, ville et (in)s´ecurit´e 20 40 60 80 100 120 140 02468101214 0 2 4 6 8 10 infractions volsavecviolence volsautomobile qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq 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q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 12
  • 13. Arthur CHARPENTIER - Analyse des donn´ees Exemple, ville et (in)s´ecurit´e Peut-ˆetre faut-il normer les axes pour les rendre comparable ? −2 −1 0 1 2 3 4 5 −2−10123 −1 0 1 2 3 4 5 6 7 infractions volsavecviolence volsautomobile q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q −2 −1 0 1 2 3 4 5 −2−10123 −1 0 1 2 3 4 5 6 7 infractions volsavecviolence volsautomobile q q qq q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 13
  • 14. Arthur CHARPENTIER - Analyse des donn´ees Analyse de la “meilleur” projection d = 2 D1 D2 D3 D4 D5 D6 D7 D8D9 D10 D11 D12 D13 D14 D15 D16 D17 D18 D19 D21 D22 D23 D24 D25 D26 D27 D28D29 D30 D31 D32 D33 D34 D35 D36 D37 D38 D39D40 D41 D42 D43 D44 D45 D46 D47 D48 D49 D50 D51 D52 D53 D54 D55 D56 D57D58 D59 D60 D61 D62D63D64 D65 D66 D67D68 D69 D70 D71 D72 D73D74 D75 D76 D77 D78 D79 D80 D81 D82 D83 D84 D85 D86 D87 D88 D89 D90 D91 D92 D93 D94 D95 14
  • 15. Arthur CHARPENTIER - Analyse des donn´ees Analyse de la “meilleur” projection d = 2 40] 60] 80] 100] 120] D1 D2 D3 D4 D5 D6 D7 D8D9 D10 D11 D12 D13 D14 D15 D16 D17 D18 D19 D21 D22 D23 D24 D25 D26 D27 D28D29 D30 D31 D32 D33 D34 D35 D36 D37 D38 D39 D40 D41 D42 D43 D44 D45 D46 D47 D48 D49 D50 D51 D52 D53 D54 D55 D56 D57D58 D59 D60 D61 D62 D63 D64 D65 D66 D67D68 D69 D70 D71 D72 D73 D74 D75 D76 D77 D78 D79 D80 D81 D82 D83 D84 D85 D86 D87 D88 D89 D90 D91 D92 D93 D94 D95 Infractions (total) d = 2 40] 60] 80] 100] 120] D1 D2 D3 D4 D5 D6 D7 D8D9 D10 D11 D12 D13 D14 D15 D16 D17 D18 D19 D21 D22 D23 D24 D25 D26 D27 D28D29 D30 D31 D32 D33 D34 D35 D36 D37 D38 D39 D40 D41 D42 D43 D44 D45 D46 D47 D48 D49 D50 D51 D52 D53 D54 D55 D56 D57D58 D59 D60 D61 D62 D63 D64 D65 D66 D67D68 D69 D70 D71 D72 D73 D74 D75 D76 D77 D78 D79 D80 D81 D82 D83 D84 D85 D86 D87 D88 D89 D90 D91 D92 D93 D94 D95 q Infractions (total) 15
  • 16. Arthur CHARPENTIER - Analyse des donn´ees Analyse de la “meilleur” projection d = 2 2] 4] 6] 8] D1 D2 D3 D4 D5 D6 D7 D8D9 D10 D11 D12 D13 D14 D15 D16 D17 D18 D19 D21 D22 D23 D24 D25 D26 D27 D28D29 D30 D31 D32 D33 D34 D35 D36 D37 D38 D39 D40 D41 D42 D43 D44 D45 D46 D47 D48 D49 D50 D51 D52 D53 D54 D55 D56 D57D58 D59 D60 D61 D62 D63 D64 D65 D66 D67D68 D69 D70 D71 D72 D73 D74 D75 D76 D77 D78 D79 D80 D81 D82 D83 D84 D85 D86 D87 D88 D89 D90 D91 D92 D93 D94 D95 Vols avec violence d = 2 2] 4] 6] 8] D1 D2 D3 D4 D5 D6 D7 D8D9 D10 D11 D12 D13 D14 D15 D16 D17 D18 D19 D21 D22 D23 D24 D25 D26 D27 D28D29 D30 D31 D32 D33 D34 D35 D36 D37 D38 D39 D40 D41 D42 D43 D44 D45 D46 D47 D48 D49 D50 D51 D52 D53 D54 D55 D56 D57D58 D59 D60 D61 D62 D63 D64 D65 D66 D67D68 D69 D70 D71 D72 D73 D74 D75 D76 D77 D78 D79 D80 D81 D82 D83 D84 D85 D86 D87 D88 D89 D90 D91 D92 D93 D94 D95 q Vols avec violence 16
  • 17. Arthur CHARPENTIER - Analyse des donn´ees Analyse de la “meilleur” projection d = 2 2] 4] 6] 8] 10] 12] D1 D2 D3 D4 D5 D6 D7 D8D9 D10 D11 D12 D13 D14 D15 D16 D17 D18 D19 D21 D22 D23 D24 D25 D26 D27 D28D29 D30 D31 D32 D33 D34 D35 D36 D37 D38 D39 D40 D41 D42 D43 D44 D45 D46 D47 D48 D49 D50 D51 D52 D53 D54 D55 D56 D57D58 D59 D60 D61 D62 D63 D64 D65 D66 D67D68 D69 D70 D71 D72 D73 D74 D75 D76 D77 D78 D79 D80 D81 D82 D83 D84 D85 D86 D87 D88 D89 D90 D91 D92 D93 D94 D95 Vols d'automobiles d = 2 2] 4] 6] 8] 10] 12] D1 D2 D3 D4 D5 D6 D7 D8D9 D10 D11 D12 D13 D14 D15 D16 D17 D18 D19 D21 D22 D23 D24 D25 D26 D27 D28D29 D30 D31 D32 D33 D34 D35 D36 D37 D38 D39 D40 D41 D42 D43 D44 D45 D46 D47 D48 D49 D50 D51 D52 D53 D54 D55 D56 D57D58 D59 D60 D61 D62 D63 D64 D65 D66 D67D68 D69 D70 D71 D72 D73 D74 D75 D76 D77 D78 D79 D80 D81 D82 D83 D84 D85 D86 D87 D88 D89 D90 D91 D92 D93 D94 D95 q Vols d'automobiles 17
  • 18. Arthur CHARPENTIER - Analyse des donn´ees Un peu de g´eom´etrie euclidienne On observe n individus, et q variables (quantitatives, sur R). Les nuages de points peuvent se d´ecomposer de deux mani`eres, – l’espace des individus, i.e. Rq – l’espace des variables, i.e. Rn On note xij l’observation de la j`eme variable sur le i`eme individu. variables 1 · · · j · · · q individus 1 x11 · · · x1j · · · x1q ... ... ... ... i xi1 · · · xij · · · xiq ... ... ... ... n xn1 · · · xnj · · · xnq 18
  • 19. Arthur CHARPENTIER - Analyse des donn´ees Un peu de g´eom´etrie euclidienne Chaque individu est charact´eris´e par Li = (xi1, · · · , xiq)t , appartenant `a Rq , exprim´e dans la base canonique {e1, · · · , eq}. Definition 1. Les points individus dans l’espace vectoriel Rq , muni´e de {e1, · · · , eq} est appel´e espace des individus. =⇒ comment mesurer la distance entre deux individus ? 19
  • 20. Arthur CHARPENTIER - Analyse des donn´ees Distance entre individus Definition 2. Soit D une matrice diagonale q × q, dont les ´el´ements diagonaux sont strictement positifs (dii > 0 pour i = 1, · · · , q). Alors la fonction ϕ : Rq × Rq → R d´efinie par (u, v) → ut Dv = q j=1 djjujvj est un produit scalaire, not´e < ·, · >D. Definition 3. Soit D une telle matrice diagonale q × q, et < ·, · >D le produit scalaire associ´e. On note alors · D la norme associ´ee, u D = √ < u, u >D = q j=1 djjujuj et dD(·, ·) la distance associ´ee, dD(u, v) = u − v D. 20
  • 21. Arthur CHARPENTIER - Analyse des donn´ees Exemples de produits scalaires • D = Id correspond au produit scalaire canonique, < u, v >Id= q j=1 ujvj • Consid´erons le produit scalaire associ´e `a D =   3/4 0 0 1/4   Les points `a ´egale distance de l’origine 0 sont les points M = (x, y) ∈ R2 tels que 0M D = α > 0, i.e. 3 4 x2 + 1 4 y2 = α, c’est `a dire une ellipse dans R2 . 21
  • 22. Arthur CHARPENTIER - Analyse des donn´ees D´eformation de l’espace −2 −1 0 1 2 −2−1012 Produit scalaire canonique, Id q −2 −1 0 1 2 −2−1012 Produit scalaire associé à la matrice D q 22
  • 23. Arthur CHARPENTIER - Analyse des donn´ees Les m´etriques usuelles Il y a fondamentalement trois types de m´etriques `a retenir, • la m´etrique usuelle i.e. M = I, la matrice identi´e Dans ce cas, la distance d´epend de l’unit´e de mesure, et de la dispersion des variables. • la m´etrique r´eduite i.e. M = diag(s−2 1 , · · · , s−2 q ), la matrice diagonale des inverses des variances empiriques Rappelons que pour une s´erie d’observations {x1, · · · , xq}, la moyenne (empirique) est mx = x = 1 n n i=1 xi et que la variance (empirique) est s2 x = 1 n n i=1 (xi − x)2 = 1 n n i=1 x2 i − x2 . 23
  • 24. Arthur CHARPENTIER - Analyse des donn´ees Enfin, rappelons que la covariance entre x et y est sxy = 1 n n i=1 (xi − x)(yi − y) = 1 n n i=1 xiyi − xy. On appele corr´elation (au sens de Pearson) la grandeur rxy = sxy sxsy = n i=1(xi − x)(yi − y) n i=1(xi − x)2 · n i=1(yi − y)2 . • la m´etrique transform´ee i.e. M = T T, Cela est ´equivalent `a travailler avec la m´etrique classique I sur le tableau transform´ee XT . Notons que pour toute matrice symm´etrique positive M, il existe une telle matrice T, appel´e racine carr´ee de M 24
  • 25. Arthur CHARPENTIER - Analyse des donn´ees D´eformation de l’espace Proposition 4. Munir l’espace de la m´etrique issue de D q × q, diagonale, est ´equivalent `a attribuer des poids { √ d11, · · · , dqq} aux q variables et d’utiliser la m´etrique canonique. D´emonstration. Pour tout u, v ∈ Rq , < u, v >D= ut Dv = q j=1 djjujvj = q j=1 djjuj djjvj soit < u, v >D=< ˜u, ˜v >Id o`u ˜u = (˜u1, · · · , ˜uq), ˜uj = djjuj. 25
  • 26. Arthur CHARPENTIER - Analyse des donn´ees Les variables, cas de la dimension 2 On cherche ici `a mesurer une distance, ou une proximit´e, entre des variables. Intuitivement, cette notion doit ˆetre proche de la notion de corr´elation. Soient deux variables X1 et X2 continues. Remarque La r´egression propose d’´etudier le lien entre deux variables, dans l’optique d’en utiliser une pour pr´evoir l’autre. 26
  • 27. Arthur CHARPENTIER - Analyse des donn´ees q q q q qq q q q q q q q q q q q q q q q q qqq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q qq q q q −2 −1 0 1 2 −2−1012 q q q q qq q q q q q q q q q q q q q q q q qqq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q qq q q q −2 −1 0 1 2 −2−1012 Ici, on s’int´eresse davantage `a des projections (orthogonales). On parlera alors de direction principal du nuage. 27
  • 28. Arthur CHARPENTIER - Analyse des donn´ees q q q q qq q q q q q q q q q q q q q q q q qqq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q qq q q q −2 −1 0 1 2 −2−1012 q q q q qq q q q q q q q q q q q q q q q q qqq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q qq q q q −2 −1 0 1 2 −2−1012 q q q On peut montrer que cet axe passe par le centre de gravit´e du nuage (comme les deux autres r´egressions). Changeons les coordonn´ees pour simplifier, Y1 = X1 − X1 et Y2 = X2 − X2. On notera O ce barycentre, X les points d’origine et P les projections 28
  • 29. Arthur CHARPENTIER - Analyse des donn´ees orthongonales. On cherche `a minimiser I = n i=1 XiPi 2 = n i=1 OiXi 2 − OiPi 2 (qu’on appelera inertie), par des propri´et´es d’orthogonalit´e. Les points O et X ´etant fixer, si u est le vecteur directeur de l’axe, u = (a, b), suppos´e unitaire, minimiser I devient `a maximiser I2 = n i=1 OiPi 2 = (Y u) Y Y uu (Y Y )uu (nΣ)u o`u Σ correspond `a la matrice de variance-covariance de Y (et donc de X). Σ est symm´etrique, elle poss`ede toujours deux valeurs propres, et deux vecteurs propres, et Σ = UΛU =   u1,1 u1,2 u2,1 u2,2     λ1 0 0 λ2     u1,1 u1,2 u2,1 u2,2   29
  • 30. Arthur CHARPENTIER - Analyse des donn´ees o`u U est une matrice othonorm´ee. Aussi, I2 = λ1α2 + λ2β2 ≤ max{λ1, λ2} [α2 + β2 ] =1 , o`u (α, β) sont les nouvelles coordon´ees de u. L’inertie ne peut donc d´epasser la plus grande valeur propre (on supposera que c’est λ1), et elle atteint cette valeur lorsque u est le premier vecteur propre. =⇒ l’axe principal d’un nuage de points bivari´e est le vecteur propre associ´e `a la plus grande valeur propre de la matrice de variance-covariance des deux variables. Ce r´esultat va se g´en´eraliser en plus grande dimension. 30
  • 31. Arthur CHARPENTIER - Analyse des donn´ees L’espace des variables De la mˆeme mani`ere, chaque variable est charact´eris´e par Cj = (x1j, · · · , xnj)t , appartenant `a Rn , exprim´e dans la base canonique {f1, · · · , fn}. G´en´eralement, dans l’espace des variables, un poids identique sera donn´e `a chaque individu. 31
  • 32. Arthur CHARPENTIER - Analyse des donn´ees Projeter un nuage de points 32
  • 33. Arthur CHARPENTIER - Analyse des donn´ees Sous R, on peut utiliser le code suivant > library(mnormt);library(rgl) > mu <- c(0,0,0) > Sigma <- matrix(c(1,0.5,0.4,0.5,1,-0.5,0.4,-0.5,1), 3, 3) > Z <- rmnorm(80, mu, Sigma) > plot3d(Z,type="s",col="blue") > plot3d(ellipse3d(cor(Z)),col="light green",alpha=0.5,add=TRUE) =⇒ la recherche d’axes principaux est li´e `a la recherche des axes de l’ellipse. 33
  • 34. Arthur CHARPENTIER - Analyse des donn´ees Projeter un nuage de points Attention des points proches dans Rk ont des projections proches, mais deux points dont les projections sont proches ne sont pas n´ecessairement proches. 34
  • 35. Arthur CHARPENTIER - Analyse des donn´ees Projeter des points, la notion d’inertie Consid´erons le tableau de donn´ees X = (xij)1≤i≤n,1≤j≤q = {L1, · · · , Ln}. L’espace individus (de Rq ) est muni de la m´etrique issue D. Definition 5. On appelle inertie du nuage des points {L1, · · · , Ln} la quantit´e I(X, D) = n i=1 di Li 2 D = n i=1 q j=1 diDjjx2 ij =⇒ on cherche des axes ou des plans de projections telle que l’intertie soit maximale. 35
  • 36. Arthur CHARPENTIER - Analyse des donn´ees Projection sur un plan Plan de projection, en dimension 3 0 1 2 3 4 5 6 0123456 0 1 2 3 4 5 6 q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q qq q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q qq q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q qq qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q qq q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 0 1 2 3 4 5 6 0123456 Projection sur le plan q 36
  • 37. Arthur CHARPENTIER - Analyse des donn´ees Projection sur un plan Plan de projection, en dimension 3 0 1 2 3 4 5 6 0123456 0 1 2 3 4 5 6 q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q qq q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q qq q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q qq q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 0 1 2 3 4 5 6 0123456 Projection sur le plan q 37
  • 38. Arthur CHARPENTIER - Analyse des donn´ees Projection sur un plan Plan de projection, en dimension 3 0 1 2 3 4 5 6 0123456 0 1 2 3 4 5 6 q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q qq q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 0 1 2 3 4 5 6 0123456 Projection sur le plan q 38
  • 39. Arthur CHARPENTIER - Analyse des donn´ees Projection sur un plan Plan de projection, en dimension 3 0 1 2 3 4 5 6 0123456 0 1 2 3 4 5 6 q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q qq q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q 0 1 2 3 4 5 6 0123456 Projection sur le plan q 39
  • 40. Arthur CHARPENTIER - Analyse des donn´ees Projection sur un plan Plan de projection, en dimension 3 0 1 2 3 4 5 6 0123456 0 1 2 3 4 5 6 q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q qq q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q 0 1 2 3 4 5 6 0123456 Projection sur le plan q 40
  • 41. Arthur CHARPENTIER - Analyse des donn´ees Projection sur un plan Plan de projection, en dimension 3 0 1 2 3 4 5 6 0123456 0 1 2 3 4 5 6 q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q qq q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q 0 1 2 3 4 5 6 0123456 Projection sur le plan q 41
  • 42. Arthur CHARPENTIER - Analyse des donn´ees Projection sur un plan Plan de projection, en dimension 3 0 1 2 3 4 5 6 0123456 0 1 2 3 4 5 6 q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q qq q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q qq q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q qq q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 0 1 2 3 4 5 6 0123456 Projection sur le plan q 42
  • 43. Arthur CHARPENTIER - Analyse des donn´ees L’inertie expliqu´ee par un axe Consid´erons le tableau de donn´ees X = (xij)1≤i≤n,1≤j≤q = {L1, · · · , Ln}. L’espace individus (de Rq ) est muni de la m´etrique issue D. Definition 6. Soit u ∈ Rq . On appelle inertie du nuage des points {L1, · · · , Ln} expliqu´ee par l’axe u la quantit´e I(X, u, D) correspondant `a l’intertie du nuage project´e orthogonalement sur u (pour < ·, · >D). D’apr`es le th´eor`eme de Pytaghore inertie totale ≥ inertie expliqu´ee par l’axe u. Consid´erons le cas de la projection de R2 sur un axe u. 43
  • 44. Arthur CHARPENTIER - Analyse des donn´ees Le (premier) axe principal Consid´erons le tableau de donn´ees X = (xij)1≤i≤n,1≤j≤q = {L1, · · · , Ln}. L’espace individus (de Rq ) est muni de la m´etrique issue D. Definition 7. L’axe principal, ou premier axe principal, pour un nuage d’individus {L1, · · · , Ln} est un vecteur unitaire u ∈ Rq qui maximise l’inertie I(X, u, D) (pour < ·, · >D). On cherche alors u = argmax{u DX XDu}, avec u D = 1. Ce probl`eme est ´equivalent `a chercher v = D1/2 u qui maximize v = argmax{v D1/2 X XD1/2 v}, avec v = 1. (1) la derni`ere norme ´etant la norme euclidienne. Proposition 8. Le vecteur unitaire v ∈ Rq solution de ?? est le vecteur propre associ´e `a la plus grande valeur propre de la matrice (XD1/2 ) (XD1/2 ). 44
  • 45. Arthur CHARPENTIER - Analyse des donn´ees D´emonstration. v est n´ecessaire un vecteur propre car (utilisation du Lagrangien pour d´eterminer l’optimum) (XD1/2 ) (XD1/2 )v − λv = 0. Rappelons que (XD1/2 ) (XD1/2 ) est diagonalisable dans une base orthonom´ee (car symm´etrique r´eelle). Soient λ1 > · · · > λk toutes les valeurs propres, i.e. (XD1/2 ) (XD1/2 )vk = λkvk. Comme on cherche `a maximiser v D1/2 X XD1/2 v, c’est que v = v1. Corollaire 9. Le vecteur D-unitaire u = u1 ∈ Rq maximisant I(X, u, D) est d´efini de mani‘ere unique (au signe pr`es) par u = D−1/2 v1 o`u v1 est le vecteur propre associ´e `a la plus grande valeur propre de la matrice (XD1/2 ) (XD1/2 ). Et l’inertie expliqu´ee par cet axe vaut alors λ1. 45
  • 46. Arthur CHARPENTIER - Analyse des donn´ees Un r´esultat d’alg`ebre lin´eaire Proposition 10. On a ´equivalence entre les r´esultats suivants • Si Ek est le sous-espace de dimension k portant l’inertie principale, alors Ek+1 = Ek ⊕ uk+1 o`u uk+1 est l’axe (espace de dimension 1) D-orthogonal `a Ek portant l’inertie maximale. • Ek est engendr´e par les k vecteurs propres de (XD1/2 ) (XD1/2 ) associ´es aux k plus grandes valeurs propres. Aussi, l’ACP sur k + 1 variables est obtenue par ajout d’une composante d’inertie maximale `a l’ACP sur k variable. C’est un m´echanisme it´eratif, il est inutile de refaire tourner des algorithmes. 46
  • 47. Arthur CHARPENTIER - Analyse des donn´ees Les autres axes principaux Le 2`eme axe principal est • un axe orthogonal `a u1 pour < ·, · >D • maximisant l’inertie En fait, u2 = D−1/2 v2 o`u v2 est le vecteur propre associ´e `a la plus seconde grande valeur propre de la matrice (XD1/2 ) (XD1/2 ). Et l’inertie expliqu´ee par cet axe vaut alors λ2. Rappelons que < u1, u2 >D=< v1, v2 >D= 0. 47
  • 48. Arthur CHARPENTIER - Analyse des donn´ees Les autres axes principaux De mani`ere plus g´en´erale, le k`eme axe principal est • un axe orthogonal `a u1, · · · , uk−1 pour < ·, · >D • maximisant l’inertie En fait, uk = D−1/2 vk o`u vk est le vecteur propre associ´e `a la plus k`eme grande valeur propre de la matrice (XD1/2 ) (XD1/2 ). Et l’inertie expliqu´ee par cet axe vaut alors λk. Rappelons que < uj, uk >D=< vj, vk >D= 0 pour j = 1, 2, · · · , k − 1. 48
  • 49. Arthur CHARPENTIER - Analyse des donn´ees Le (premier) axe principal Consid´erons le tableau de donn´ees X = (xij)1≤i≤n,1≤j≤q = {L1, · · · , Ln}. L’espace individus (de Rq ) est muni de la m´etrique issue D. Definition 11. Le plan principal, ou premier plan principal, pour un nuage d’individus {L1, · · · , Ln} est le plan engendr´e par u1, u2. 49
  • 50. Arthur CHARPENTIER - Analyse des donn´ees Rappel de la m´ethodologie Consid´erons le tableau de donn´ees X = (xij)1≤i≤n,1≤j≤q = {L1, · · · , Ln}. L’espace des individus (de Rq ) est muni de la m´etrique issue D. • on diagonalise (XD1/2 ) (XD1/2 ). • soient λ1 ≥ λ2 ≥ λ3 ≥ · · · ≥ λq les valeurs propres, et vj les vecteurs propres • les axes principaux sont les uj = D−1/2 vj. Consid´erons le tableau de donn´ees X = (xij)1≤i≤n,1≤j≤q = {L1, · · · , Ln}. L’espace des variables (de Rn ) est muni de la m´etrique issue ∆. • on diagonalise (∆1/2 X) (∆1/2 X). • soient λ1 ≥ λ2 ≥ λ3 ≥ · · · ≥ λq les valeurs propres, et νi les vecteurs propres • les axes principaux sont les µi = ∆−1/2 νi. 50
  • 51. Arthur CHARPENTIER - Analyse des donn´ees Combien d’axes principaux doit-on retenir ? Rappelons que l’on cherche `a r´esumer l’information apport´ee par les variables par un “petit” nombre de facteurs, en tenant compte des corr´elation existant entre les variables. =⇒ on veut garder peu d’axes principaux, avec • un soucis d’interpr´etation : on ne garde que des axes que l’on puisse interpr´eter, • des axes qui expliquent suffisement d’inertie. Pour cela, on a deux m´ethodes ◦ la m´ethode du coude, correspondant `a un d´ecrochage au niveau des valeurs propres ◦ la r`egle de Kaiser, pour les variables centr´ees r´eduites : on ne garde que les valeurs propres sup´erieures `a 1. (ce seuil de 1 correspond `a la moyenne des valeurs propres). 51
  • 52. Arthur CHARPENTIER - Analyse des donn´ees Les composantes principales prendre x pour les individus, y pour les individu centr´es, et z pour les individus centr´es r´eduits Les coordonn´ees d’un individu centr´e yi sur un axe principal ∆k sont obtenues par D-projection ci,j =< yi, uk >D= yiDuk Definition 12. On appelera composantes principales les variables ck, dans RI , d´efinies par ck = Y Duk Il s’agit des coordonn´ees des projections D-orthongales sur les axes principaux. 52
  • 53. Arthur CHARPENTIER - Analyse des donn´ees Les composantes principales Definition 13. La repr´esentation graphique du nuage des individus dans le plan principal est alors le nuage des points c1, c2. On notera que, par construction, ck = 0 car les colonnes de y sont centr´ees. De plus, V ar(ck) = λk et Cov(ck1 , ck2 ) = 0, i.e. les composantes principales sont orthogonales 53
  • 54. Arthur CHARPENTIER - Analyse des donn´ees Les donn´ees centr´ees r´eduites Il peut parfois ˆetre pertinant de travailler avec la m´etrique D1/s2 , car les distances entre variables sont tr`es sensibles aux unit´ees (et donc `a la dispersion). Rappelons que travailler avec la matrice D1/s2 sur le nuage y est ´equivalent `a travailler avecla m´etrique usuelle I sur le nuage de points centr´es r´eduits. Definition 14. On appelera nuage centr´e r´eduit le tableau Z contenant les zi,j = xi,j − xj sj i.e. z = (x − x)D1/s = yD1/s. 54
  • 55. Arthur CHARPENTIER - Analyse des donn´ees Le “cercle des corr´elations” On suppose que l’espace des variables est muni d’une m´etrique D. On prendra la m´etrique des poids. Alors s2 x = V ar(x) = x 2 D et sxy = cov(x, y) =< x, y >D . De plus, r(x, y) = < x, y >D x D y D . Si les variables sont suppos´ees centr´ees et r´eduite, la corr´elation entre une composante principale ck et une variable zj , o`u z = (x − x)D1/s est r(zj, ck) = cov(zj , ck) √ ck = xj Dck √ λk , donc le vecteur des corr´elations du facteur ck avec toutes les variables z est r(z, ck) = z Dck √ λk , 55
  • 56. Arthur CHARPENTIER - Analyse des donn´ees or comme z Dck = z Dck = λkuk, on en d´duit simplement que r(z, ck) = λkuk. De cette expression, notons que p k=1 r(zj, ck)2 = zj 2 D = 1 et donc, en particulier, r(zj, c1)2 + r(zj, c2)2 ≤ 1. Definition 15. On appelera cercle des corr´elations (e.g. dans le plan principal) le nuage de points (r(zj, c1), r(zj, c2)) pour k = 1, · · · , ????, o`u sont projet´ees les variables. La notion de “cercle” vient de la premi`ere propri´et´e. Mais l’interpr´etation de la proximit´e des points n’est possible qu’au bord du cercle. 56
  • 57. Arthur CHARPENTIER - Analyse des donn´ees 57
  • 58. Arthur CHARPENTIER - Analyse des donn´ees Ls contributions des individus Nous avions not´e que λk = 1 n n i=1 c2 i,k. Definition 16. On appelera contribution d’un individu i `a un axe k la quantit´e c2 i,k nλk . La contribution sera importante si elle exc`ede le poids de l’individu 1/n, i.e. |ci,k| > √ λk. 58
  • 59. Arthur CHARPENTIER - Analyse des donn´ees Enlever/rajouter des variables/individus Il est possible de faire une analyse en enlevant certaines variables et/ou individus, quite `a les rajouter par la suite, • certains individus vont ˆetre sur-repr´esent´es, et risqueront de tirer le nuage dans une direction. On peut les exclure de la r´egression, quite `a les rajouter par la suite • certains individus vont ˆetre sur-repr´esent´es, et risqueront de tirer le nuage des individus dans une direction. On peut les exclure de la r´egression, quite `a les rajouter par la suite • certaines variables peuvent, par un comportement assez diff´erent, d´eformer le nuage des variables. Consid´erons ici la base ACPsup.csv) t´el´echargeables sur ma page internet, dont l’ACP brute donne 59
  • 60. Arthur CHARPENTIER - Analyse des donn´ees Enlever/rajouter des variables/individus −80 −60 −40 −20 −505 Les individus cl1 cl2 1 2 3 4 5 6 7 8 9 10 11 121314 15 16 17 18 19 20 2122 23 24 25 26 2728 29 30 31 3233 34 35 36 37 38 39 40 4142 43 44 4546 47 48 49 50 51 52 53 54 55 56 57 58 59 6061 62 63 64 65 66 67 68 69 70 71 72 7374 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 9394 95 96 97 98 99 100 −14 −12 −10 −8 −6 −4 −2 −4−20246 Les variables Comp1Comp2 A B CD E 60
  • 61. Arthur CHARPENTIER - Analyse des donn´ees Enlever/rajouter des variables/individus −15 −10 −5 0 −4−3−2−1012 Les individus cl1 cl2 12 3 4 5 6 7 8 9 10 11 12 13 14 15 161718 19 20 21 22 23 2425 26 2728 29 30 31 32 3334 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 −1 0 1 2 −3−2−1012 Les individus cl1[1:99] cl2[1:99] 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 1718 19 20 21 22 23 24 25 26 2728 29 30 31 32 3334 35 36 37 38 3940 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 −1.4 −1.2 −1.0 −0.8 −0.6 −0.4 −0.2 0.0 −0.8−0.6−0.4−0.20.00.2 Les variables Comp1 Comp2 A B C D E 61
  • 62. Arthur CHARPENTIER - Analyse des donn´ees Les variables suppl´ementaires Pour les individus suppl´ementaires, on peut calculer la corr´elation entr la variable et les composantes principales, plus placer ce point dans le cercle des corr´elations. Si ˜z est la variable centr´ee r´eduite suppl´ementaire, on calcule r(˜z, ck) = ˜z Dck √ λk = 1 n √ λk n i=1 ˜zici,k. Notons qu’il est possible de tester la significativit´e de la corr´elation. z<- dudi.pca(don, center = T, scale = T, scannf = F) ligsup<-suprow(z,donsup) 62
  • 63. Arthur CHARPENTIER - Analyse des donn´ees Les individus suppl´ementaires De mˆeme ici, si Si ˜z est l’individu centr´ee r´eduite suppl´ementaire, on calcule pour chaque axe principal k ˜ck =< ˜z, uk) = p j=1 ˜zjuk,j. 63
  • 64. Arthur CHARPENTIER - Analyse des donn´ees Exemple sur donn´ees simul´ees 0.6 0.8 1.0 1.2 −0.2−0.10.00.10.20.30.4 Les variables Comp1 Comp2 A B C D 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4−0.6−0.4−0.20.00.20.4 Les variables Comp1 Comp2 A B C D E 64
  • 65. Arthur CHARPENTIER - Analyse des donn´ees −4 −2 0 2 4 6 −2−101 Les individus cl1 cl2 1 2 3 4 5 6 7 8 9 10 11 12 1314 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 7172 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 −25 −20 −15 −10 −5 0 5 0510 Les individus cl1 cl2 1 2 3 45 6 7 8 9 10 1112 1314 15 16 17 18 19 20 2122 2324 25 26 27 28 29 30 3132 3334 35 36 37 38 39 4041 42 43 44 45 46 4748 49 50 51 52 53 54 55 5657 58 59 6061 62 63 64 65 66 67 68 69 70 7172 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 9091 929394 95 96 97 98 99 100 65
  • 66. Arthur CHARPENTIER - Analyse des donn´ees Un cas d’´ecole Consid´erons les r´esultats de l’´election pr´esidentielle de 1995, au premier tour (base election95.csv). Notons que la personne pour laquelle on vote peut ˆetre vue comme une variable qualitative (cf cours 3 sur l’ACM). Les variables principales sont les variables suivantes • VOY95 Pourcentage de vote de Mme Voynet • HUE95 Pourcentage de vote de M. Hue • JOS95 Pourcentage de vote de M. Jospin • LAG95 Pourcentage de vote de Mme Laguiller • VIL95 Pourcentage de vote de M. de Villiers • CHEM95 Pourcentage de vote de M. Cheminade • CHI95 Pourcentage de vote de M. Chirac • BAL95 Pourcentage de vote de M. Balladur • LEP95 Pourcentage de vote de M. Le Pen • inscrits 95 Nombre d’inscrits sur les listes ´electorales en mai 1995 • exprimes 95 Nombre de suffrages exprim´es au premier tour de l’election 66
  • 67. Arthur CHARPENTIER - Analyse des donn´ees pr´esidentielle de 1995 On obtient les graphiques suivants q −0.5 0.0 0.5 1.0 −0.50.00.51.0 CA factor map Dim 1 (71.57%) Dim2(12.4%) q q q q q q q q q q q q q q q q q q Agriculteurs Artisans Commercants ChefsEntreprise ProfLiberales CadresPublic CadresEntreprProfIntPublic ProfIntEntrepr Techniciens Contremaitres EmployesPublic EmployesEntrepr EmployesCommerc PersonnelsServ OuvriersQualif OuvriersNonQual OuvriersAgricol Espagnol Italien Portugais AutresUE Algerien Marocain Tunisien Turc Autres −1.5 −1.0 −0.5 0.0 0.5 1.0 −0.50.00.51.0 Axe 1 Axe2 Agriculteurs Artisans Commercants ChefsEntreprise ProfLiberales CadresPublic CadresEntreprProfIntPublic ProfIntEntrepr Techniciens Contremaitres EmployesPublicEmployesEntreprEmployesCommerc PersonnelsServ OuvriersQualif OuvriersNonQual OuvriersAgricol Espagnol Italien Portugais AutresUE Algerien Marocain Tunisien Turc Autres • il y a plusieurs variables suppl´ementaire, li´es `a la r´epartition par CSP dans un d´epartement, le niveau de diplˆome, la nationalit´e. 67
  • 68. Arthur CHARPENTIER - Analyse des donn´ees • on notera que des d´epartements ont un comportement “singulier”, il serait peut-ˆetre judicieux de les traiter comme individus suppl´ementaires Le diplˆome est trait´e comme variable “normale” `a gauche, mais comme variable suppl ´mentaire `a droite. Les modalit´es sont les suivantes DIPL0 Personne g´ee de moins de 15 ans, DIPL1 Aucun diplme, DIPL2 Certificat d’´etudes primaires, DIPL3 BEPC, brevet ´el´ementaire, brevet des coll`eges, DIPL4 CAP, DIPL5 BEP, DIPL6 Baccalaur´eat g´en´eral, DIPL7 Baccalaur´eat technologique ou professionnel, DIPL8 Diplme universitaire de 1er cycle, DIPL9 Diplme universitaire de 2e ou 3e cycle. 68
  • 69. Arthur CHARPENTIER - Analyse des donn´ees −1.0 −0.5 0.0 0.5 1.0 −1.0−0.50.00.51.0 Les variables Comp1 Comp2 VOY95 HUE95 JOS95 LAG95 VIL95 CHEM95 CHI95 BAL95 LEP95 DIPLOME0 DIPLOME1 DIPLOME2 DIPLOME3 DIPLOME4 DIPLOME5 DIPLOME6 DIPLOME7 DIPLOME8 DIPLOME9 −0.5 0.0 0.5 −0.6−0.4−0.20.00.20.40.60.8 Les variables Comp1 Comp2 VOY95 HUE95 JOS95 LAG95 VIL95 CHEM95 CHI95 BAL95 LEP95 DIPLOME0DIPLOME1DIPLOME2DIPLOME3DIPLOME4DIPLOME5DIPLOME6DIPLOME7DIPLOME8DIPLOME9 69
  • 70. Arthur CHARPENTIER - Analyse des donn´ees −1.0 −0.5 0.0 0.5 1.0 −1.0−0.50.00.51.0 Les variables Comp1 Comp2 VOY95 HUE95 JOS95 LAG95 VIL95 CHEM95 CHI95 BAL95 LEP95 CHOMEURS ETUDIANTS MILITAIRES −1.0 −0.5 0.0 0.5 −0.50.00.51.0 Les variables Comp1 Comp2 VOY95 HUE95 JOS95 LAG95 VIL95 CHEM95 CHI95 BAL95 LEP95 CHOMEURS ETUDIANTSMILITAIRES Pour les CSP, on notera CS1· Agriculteurs exploitants, CS2· Artisans, commerants et chefs d’entreprises, CS3· Cadres et professions intellectuelles sup´erieures, CS4· Professions interm´ediaires (dont CS44 pour le clerg´e), CS5· Employ´es, CS6· Ouvriers, CS7· Retrait´es (dont CS72 Anciens artisans, 70
  • 71. Arthur CHARPENTIER - Analyse des donn´ees commerants, chefs d’entreprise), CS8· Autres personnes inactives (dont CS81 Chmeurs n’ayant jamais travaill´e). −1.0 −0.5 0.0 0.5 1.0 −1.0−0.50.00.51.0 Les variables Comp1 Comp2 VOY95 HUE95 JOS95 LAG95 VIL95CHEM95 CHI95 BAL95 LEP95 CS11 CS12 CS13 CS21 CS22 CS23 CS31 CS33 CS34 CS35 CS37 CS38 CS42 CS43 CS44 CS45 CS46 CS47 CS48 CS52 CS53 CS54CS55 CS56 CS62 CS63 CS64 CS65 CS67 CS68 CS69 CS71 CS72 CS74 CS75 CS77 CS78 CS81 CS83 CS84 CS85 CS86 −0.5 0.0 0.5 −0.6−0.4−0.20.00.20.40.60.8 Les variables Comp1 Comp2 VOY95 HUE95 JOS95 LAG95 VIL95 CHEM95 CHI95 BAL95 LEP95 CS11CS12CS13 CS21 CS22 CS23 CS31 CS33 CS34 CS35CS37 CS38 CS42 CS43 CS44 CS45 CS46 CS47CS48 CS52 CS53 CS54CS55 CS56 CS62 CS63 CS64 CS65 CS67 CS68 CS69 CS71 CS72 CS74CS75 CS77 CS78 CS81 CS83 CS84 CS85 CS86 Pour les d´epartements, on peut commencer par ´ecarter la corr`eze 71
  • 72. Arthur CHARPENTIER - Analyse des donn´ees −1.0 −0.5 0.0 0.5 1.0 −1.0−0.50.00.51.0 Les variables Comp1 Comp2 VOY95 HUE95 JOS95 LAG95 VIL95 CHEM95 CHI95 BAL95 LEP95 −6 −4 −2 0 2 4 −5−4−3−2−1012 Les individus cl1cl2 AIN AISNE ALLIER ALPES−DE−HAUTE−PROVENCE HAUTES−ALPESALPES−MARITIMES ARDECHE ARDENNES ARIEGE AUBE AUDE AVEYRON BOUCHES−DU−RHONE CALVADOS CANTAL CHARENTE CHARENTE−MARITIME CHER CORSE−DU−SUD HAUTE−CORSE COTE−D−OR COTES−D−ARMOR CREUSE DORDOGNE DOUBS DROME EURE EURE−ET−LOIRFINISTERE GARD HAUTE−GARONNE GERS GIRONDE HERAULT ILLE−ET−VILAINE INDRE INDRE−ET−LOIRE ISERE JURA LANDES LOIR−ET−CHER LOIRE HAUTE−LOIRE LOIRE−ATLANTIQUE LOIRET LOT LOT−ET−GARONNE LOZERE MAINE−ET−LOIREMANCHE MARNE HAUTE−MARNE MAYENNE MEURTHE−ET−MOSELLE MEUSE MORBIHAN MOSELLE NIEVRE NORD OISE ORNE PAS−DE−CALAIS PUY−DE−DOME PYRENEES−ATLANTIQUES HAUTES−PYRENEES PYRENEES−ORIENTALES BAS−RHIN HAUT−RHIN RHONE HAUTE−SAONE SAONE−ET−LOIRE SARTHE SAVOIE HAUTE−SAVOIE PARIS SEINE−MARITIME SEINE−ET−MARNE YVELINES DEUX−SEVRES SOMME TARN TARN−ET−GARONNE VAR VAUCLUSE VENDEE VIENNEHAUTE−VIENNE VOSGES YONNE TERRITOIRE−DE−BELFORT ESSONNE HAUTS−DE−SEINE SEINE−SAINT−DENIS CORREZE On peut aussi ´etudier l’impact de la Vend´ee 72
  • 73. Arthur CHARPENTIER - Analyse des donn´ees −1.0 −0.5 0.0 0.5 1.0 −1.0−0.50.00.51.0 Les variables Comp1 Comp2 VOY95HUE95 JOS95 LAG95 VIL95CHEM95 CHI95 BAL95 LEP95 −8 −6 −4 −2 0 2 4 6 −3−2−101234 Les individus cl1cl2 AIN AISNE ALLIER ALPES−DE−HAUTE−PROVENCE HAUTES−ALPES ALPES−MARITIMES ARDECHE ARDENNES ARIEGE AUBE AUDE AVEYRON BOUCHES−DU−RHONE CALVADOS CANTAL CHARENTE CHARENTE−MARITIME CHER CORREZE CORSE−DU−SUD HAUTE−CORSE COTE−D−OR COTES−D−ARMOR CREUSE DORDOGNE DOUBS DROME EURE EURE−ET−LOIR FINISTERE GARD HAUTE−GARONNE GERS GIRONDE HERAULT ILLE−ET−VILAINE INDREINDRE−ET−LOIRE ISERE JURA LANDES LOIR−ET−CHER LOIRE HAUTE−LOIRE LOIRE−ATLANTIQUE LOIRET LOT LOT−ET−GARONNE LOZERE MAINE−ET−LOIRE MANCHEMARNEHAUTE−MARNE MAYENNE MEURTHE−ET−MOSELLE MEUSE MORBIHAN MOSELLE NIEVRE NORD OISE ORNE PAS−DE−CALAIS PUY−DE−DOME PYRENEES−ATLANTIQUES HAUTES−PYRENEES PYRENEES−ORIENTALES BAS−RHIN HAUT−RHIN RHONE HAUTE−SAONESAONE−ET−LOIRE SARTHE SAVOIE HAUTE−SAVOIEPARIS SEINE−MARITIME SEINE−ET−MARNE YVELINES DEUX−SEVRES SOMME TARN TARN−ET−GARONNE VAR VAUCLUSE VIENNE HAUTE−VIENNE VOSGES YONNE TERRITOIRE−DE−BELFORT ESSONNE HAUTS−DE−SEINE SEINE−SAINT−DENIS VENDEE Et enfin l’impact de l’Alsace (Bas et Haut Rhin) 73
  • 74. Arthur CHARPENTIER - Analyse des donn´ees −1.0 −0.5 0.0 0.5 1.0 −1.0−0.50.00.51.0 Les variables Comp1 Comp2 VOY95 HUE95 JOS95 LAG95 VIL95CHEM95 CHI95 BAL95 LEP95 −8 −6 −4 −2 0 2 4 −4−3−2−1012 Les individus cl1 cl2 AIN AISNE ALLIERALPES−DE−HAUTE−PROVENCE HAUTES−ALPESALPES−MARITIMES ARDECHE ARDENNES ARIEGE AUBE AUDE AVEYRON BOUCHES−DU−RHONE CALVADOS CANTAL CHARENTE CHARENTE−MARITIME CHER CORREZE CORSE−DU−SUD HAUTE−CORSE COTE−D−OR COTES−D−ARMOR CREUSE DORDOGNE DOUBS DROMEEURE EURE−ET−LOIR FINISTERE GARD HAUTE−GARONNE GERS GIRONDE HERAULT ILLE−ET−VILAINE INDREINDRE−ET−LOIRE ISERE JURA LANDES LOIR−ET−CHER LOIRE HAUTE−LOIRE LOIRE−ATLANTIQUE LOIRET LOT LOT−ET−GARONNE LOZERE MAINE−ET−LOIRE MANCHE MARNE HAUTE−MARNE MAYENNE MEURTHE−ET−MOSELLE MEUSEMORBIHAN MOSELLE NIEVRE NORD OISE ORNE PAS−DE−CALAIS PUY−DE−DOME PYRENEES−ATLANTIQUES HAUTES−PYRENEES PYRENEES−ORIENTALES RHONE HAUTE−SAONE SAONE−ET−LOIRESARTHE SAVOIE HAUTE−SAVOIE PARIS SEINE−MARITIME SEINE−ET−MARNE YVELINES DEUX−SEVRES SOMME TARN TARN−ET−GARONNE VAR VAUCLUSE VENDEE VIENNE HAUTE−VIENNE VOSGES YONNE TERRITOIRE−DE−BELFORT ESSONNE HAUTS−DE−SEINE SEINE−SAINT−DENIS BAS−RHIN HAUT−RHIN 74
  • 75. Arthur CHARPENTIER - Analyse des donn´ees Mise en oeuvre pratique 75
  • 76. Arthur CHARPENTIER - Analyse des donn´ees Les donn´ees, en ACP “Le palmar`es des d´epartements. O`u vit-on en s´ecurit´e ?, dans L’Express (no 2589, 15 f´evrier 2001). • infra Nombre d’infractions totale pour 1000 habitants (2000) • vols Nombre total de vols pour 1000 habitants (2000) • eco Nombre d’infractions ´economiques et finaci`eres pour 1000 habitants (2000) • crim Nombre de crimes et d´elits contre les personnes pour 1000 habitants (2000) • vma Nombre de vols `a main arm´ee pour 1000 habitants (2000) • vvi Nombre de vols avec violance pour 1000 habitants (2000) • camb Nombre de cambriolages pour 1000 habitants (2000) • roul Nombre de vols `a la roulotte pour 1000 habitants (2000) • auto Nombre de vols d’automobiles pour 1000 habitants (2000) 76
  • 77. Arthur CHARPENTIER - Analyse des donn´ees Les donn´ees, en ACP robuste Dans les ACP robuste, on ne s’int´eresse plus aux niveaux mais aux rangs 77
  • 78. Arthur CHARPENTIER - Analyse des donn´ees Base de donn´ees pour les 25 villes compar´ees Angers 14 19 12 12 11 19 19 7 6 14 21 Bordeaux 20 7 18 18 9 3 7 19 19 23 13 Caen 8 17 16 6 24 13 15 12 5 13 18 Clermont-Ferrand 14 25 8 16 7 20 5 5 9 1 24 Dijon 17 20 18 14 13 24 16 11 11 10 16 Douai-Lens 1 23 3 5 23 17 21 3 2 5 19 Grenoble 22 11 16 21 7 4 8 23 14 7 6 Lille 10 6 8 5 20 6 24 21 16 3 8 Lyon 10 8 23 17 5 2 13 22 23 24 4 Marseille-Aix-en-Provence 24 3 24 25 3 5 6 5 24 19 3 Metz 4 12 3 2 13 14 19 9 10 12 22 Montpellier 25 2 14 22 4 12 4 18 20 9 10 Nancy 24 16 12 10 16 18 19 24 5 21 17 Nantes 6 10 12 12 17 9 12 17 18 22 11 Nice 20 1 23 23 2 8 1 6 22 16 2 Orl ?ns 4 13 8 13 15 15 24 17 3 11 14 Paris 12 4 25 8 19 1 25 25 25 25 1 Rennes 12 14 8 7 21 11 11 15 12 15 7 Rouen 6 22 20 1 22 25 22 13 7 8 20 Saint-Etienne 15 24 12 19 8 21 9 1 13 2 25 Strasbourg 20 9 21 9 14 16 10 15 17 20 15 Toulon 8 18 4 24 1 10 3 2 15 4 5 Toulouse 22 5 20 20 10 7 2 20 21 17 12 Tours 17 15 14 15 18 22 20 8 8 6 9 Valenciennes 2 21 3 5 25 23 14 10 1 18 23 78
  • 79. Arthur CHARPENTIER - Analyse des donn´ees Nombre de m´edecins (pour 1000 habitants) Angers 14 19 12 12 11 19 19 7 6 14 21 Bordeaux 20 7 18 18 9 3 7 19 19 23 13 Caen 8 17 16 6 24 13 15 12 5 13 18 Clermont-Ferrand 14 25 8 16 7 20 5 5 9 1 24 Dijon 17 20 18 14 13 24 16 11 11 10 16 Douai-Lens 1 23 3 5 23 17 21 3 2 5 19 Grenoble 22 11 16 21 7 4 8 23 14 7 6 Lille 10 6 8 5 20 6 24 21 16 3 8 Lyon 10 8 23 17 5 2 13 22 23 24 4 Marseille-Aix-en-Provence 24 3 24 25 3 5 6 5 24 19 3 Metz 4 12 3 2 13 14 19 9 10 12 22 Montpellier 25 2 14 22 4 12 4 18 20 9 10 Nancy 24 16 12 10 16 18 19 24 5 21 17 Nantes 6 10 12 12 17 9 12 17 18 22 11 Nice 20 1 23 23 2 8 1 6 22 16 2 Orl ?ns 4 13 8 13 15 15 24 17 3 11 14 Paris 12 4 25 8 19 1 25 25 25 25 1 Rennes 12 14 8 7 21 11 11 15 12 15 7 Rouen 6 22 20 1 22 25 22 13 7 8 20 Saint-Etienne 15 24 12 19 8 21 9 1 13 2 25 Strasbourg 20 9 21 9 14 16 10 15 17 20 15 Toulon 8 18 4 24 1 10 3 2 15 4 5 Toulouse 22 5 20 20 10 7 2 20 21 17 12 Tours 17 15 14 15 18 22 20 8 8 6 9 Valenciennes 2 21 3 5 25 23 14 10 1 18 23 79
  • 80. Arthur CHARPENTIER - Analyse des donn´ees Nombre de crimes et d´elits (pour 1000 habitants) Angers 14 19 12 12 11 19 19 7 6 14 21 Bordeaux 20 7 18 18 9 3 7 19 19 23 13 Caen 8 17 16 6 24 13 15 12 5 13 18 Clermont-Ferrand 14 25 8 16 7 20 5 5 9 1 24 Dijon 17 20 18 14 13 24 16 11 11 10 16 Douai-Lens 1 23 3 5 23 17 21 3 2 5 19 Grenoble 22 11 16 21 7 4 8 23 14 7 6 Lille 10 6 8 5 20 6 24 21 16 3 8 Lyon 10 8 23 17 5 2 13 22 23 24 4 Marseille-Aix-en-Provence 24 3 24 25 3 5 6 5 24 19 3 Metz 4 12 3 2 13 14 19 9 10 12 22 Montpellier 25 2 14 22 4 12 4 18 20 9 10 Nancy 24 16 12 10 16 18 19 24 5 21 17 Nantes 6 10 12 12 17 9 12 17 18 22 11 Nice 20 1 23 23 2 8 1 6 22 16 2 Orl ?ns 4 13 8 13 15 15 24 17 3 11 14 Paris 12 4 25 8 19 1 25 25 25 25 1 Rennes 12 14 8 7 21 11 11 15 12 15 7 Rouen 6 22 20 1 22 25 22 13 7 8 20 Saint-Etienne 15 24 12 19 8 21 9 1 13 2 25 Strasbourg 20 9 21 9 14 16 10 15 17 20 15 Toulon 8 18 4 24 1 10 3 2 15 4 5 Toulouse 22 5 20 20 10 7 2 20 21 17 12 Tours 17 15 14 15 18 22 20 8 8 6 9 Valenciennes 2 21 3 5 25 23 14 10 1 18 23 80
  • 81. Arthur CHARPENTIER - Analyse des donn´ees Ensoleillement moyen, entre 1991 et 2000 Angers 14 19 12 12 11 19 19 7 6 14 21 Bordeaux 20 7 18 18 9 3 7 19 19 23 13 Caen 8 17 16 6 24 13 15 12 5 13 18 Clermont-Ferrand 14 25 8 16 7 20 5 5 9 1 24 Dijon 17 20 18 14 13 24 16 11 11 10 16 Douai-Lens 1 23 3 5 23 17 21 3 2 5 19 Grenoble 22 11 16 21 7 4 8 23 14 7 6 Lille 10 6 8 5 20 6 24 21 16 3 8 Lyon 10 8 23 17 5 2 13 22 23 24 4 Marseille-Aix-en-Provence 24 3 24 25 3 5 6 5 24 19 3 Metz 4 12 3 2 13 14 19 9 10 12 22 Montpellier 25 2 14 22 4 12 4 18 20 9 10 Nancy 24 16 12 10 16 18 19 24 5 21 17 Nantes 6 10 12 12 17 9 12 17 18 22 11 Nice 20 1 23 23 2 8 1 6 22 16 2 Orl ?ns 4 13 8 13 15 15 24 17 3 11 14 Paris 12 4 25 8 19 1 25 25 25 25 1 Rennes 12 14 8 7 21 11 11 15 12 15 7 Rouen 6 22 20 1 22 25 22 13 7 8 20 Saint-Etienne 15 24 12 19 8 21 9 1 13 2 25 Strasbourg 20 9 21 9 14 16 10 15 17 20 15 Toulon 8 18 4 24 1 10 3 2 15 4 5 Toulouse 22 5 20 20 10 7 2 20 21 17 12 Tours 17 15 14 15 18 22 20 8 8 6 9 Valenciennes 2 21 3 5 25 23 14 10 1 18 23 81
  • 82. Arthur CHARPENTIER - Analyse des donn´ees Cumul des emboutillages Angers 14 19 12 12 11 19 19 7 6 14 21 Bordeaux 20 7 18 18 9 3 7 19 19 23 13 Caen 8 17 16 6 24 13 15 12 5 13 18 Clermont-Ferrand 14 25 8 16 7 20 5 5 9 1 24 Dijon 17 20 18 14 13 24 16 11 11 10 16 Douai-Lens 1 23 3 5 23 17 21 3 2 5 19 Grenoble 22 11 16 21 7 4 8 23 14 7 6 Lille 10 6 8 5 20 6 24 21 16 3 8 Lyon 10 8 23 17 5 2 13 22 23 24 4 Marseille-Aix-en-Provence 24 3 24 25 3 5 6 5 24 19 3 Metz 4 12 3 2 13 14 19 9 10 12 22 Montpellier 25 2 14 22 4 12 4 18 20 9 10 Nancy 24 16 12 10 16 18 19 24 5 21 17 Nantes 6 10 12 12 17 9 12 17 18 22 11 Nice 20 1 23 23 2 8 1 6 22 16 2 Orl ?ns 4 13 8 13 15 15 24 17 3 11 14 Paris 12 4 25 8 19 1 25 25 25 25 1 Rennes 12 14 8 7 21 11 11 15 12 15 7 Rouen 6 22 20 1 22 25 22 13 7 8 20 Saint-Etienne 15 24 12 19 8 21 9 1 13 2 25 Strasbourg 20 9 21 9 14 16 10 15 17 20 15 Toulon 8 18 4 24 1 10 3 2 15 4 5 Toulouse 22 5 20 20 10 7 2 20 21 17 12 Tours 17 15 14 15 18 22 20 8 8 6 9 Valenciennes 2 21 3 5 25 23 14 10 1 18 23 82
  • 83. Arthur CHARPENTIER - Analyse des donn´ees > add=read.table("http://perso.univ-rennes1.fr/arthur.charpentier/ADD-ex-villes.txt",heade > base=add[,2:ncol(add)] > rownames(base)=add$Agglo Consid´erons comme matrice D la matrice 1 n I pour l’espace des individus, et la matrice identit´e pour l’espace des variables, ∆ = I. On diagonale alors 1 n X X, et on note v1, · · · , vq les vecteurs propres associ´es aux valeurs propres λ1 > · · · > λq. On obtient alors les vecteurs uk engendrant les axes principaux, qui expliquent chacun 100 × λk k j=1 λj % de l’inertie totale. > X <- as.matrix(base) > n <- nrow(base) > eigen(1/n * t(X) %*% X) > eigen(1/n * t(X) %*% X)$vectors [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11] [1,] -0.31 -0.29 -0.2684 -0.254 -0.493 -0.331 -0.096 -0.097 -0.242 0.3880 0.318 [2,] -0.29 0.38 -0.2735 0.016 0.190 0.272 -0.083 -0.122 0.527 0.2165 0.492 [3,] -0.32 -0.27 0.0340 0.099 0.236 -0.590 0.156 0.108 0.539 0.0492 -0.277 [4,] -0.29 -0.29 -0.4430 -0.157 0.208 0.393 -0.380 -0.080 -0.062 -0.0011 -0.506 83
  • 84. Arthur CHARPENTIER - Analyse des donn´ees [5,] -0.30 0.30 0.3507 -0.013 0.039 -0.054 0.104 -0.742 -0.161 0.1818 -0.270 [6,] -0.30 0.35 -0.2859 0.022 -0.051 -0.337 -0.153 -0.042 -0.174 -0.7280 0.073 [7,] -0.30 0.26 0.3767 -0.412 0.344 -0.078 -0.236 0.493 -0.262 0.1879 0.034 [8,] -0.31 -0.16 0.4032 -0.329 -0.485 0.346 0.075 0.063 0.349 -0.3510 -0.022 [9,] -0.29 -0.39 -0.0098 0.013 0.409 0.190 0.560 -0.059 -0.304 -0.1865 0.341 [10,] -0.30 -0.19 0.3090 0.734 -0.073 0.097 -0.423 0.081 -0.113 0.0173 0.152 [11,] -0.30 0.35 -0.2256 0.288 -0.301 0.161 0.476 0.384 -0.131 0.2088 -0.325 > eigen(1/n * t(X) %*% X)$values [1] 1940.974149 275.990875 123.372428 31.594830 28.553309 23.945216 [7] 14.283394 12.950240 10.841242 6.205078 4.849239 Pour mieux comprendre quelle part est expliqu´ee par les premiers axes propres, on utilise > valp <- eigen(1/n * t(X) %*% X)$values > 100 * valp/sum(valp) [1] 78.4688525 11.1576382 4.9876465 1.2773020 1.1543407 0.9680467 [7] 0.5774428 0.5235466 0.4382850 0.2508562 0.1960429 i.e. le premier axe explique 78.5% de l’inertie, et le second 11% de l’inertie (soit pr`es de 90% pour le plan principal). 84
  • 85. Arthur CHARPENTIER - Analyse des donn´ees Une autre possibilit´e est d’utiliser dudi.pca de library(ade4). > acp <- dudi.pca(base, scale = F, center = F,scannf = F, nf = ncol(base)) > acp$c1 CS1 CS2 CS3 CS4 CS5 CS6 CS7 CS8 CS9 CS10 CS11 Medecins -0.31 -0.29 -0.2684 -0.254 -0.493 -0.331 -0.096 -0.097 -0.242 0.3880 0.318 Crimin -0.29 0.38 -0.2735 0.016 0.190 0.272 -0.083 -0.122 0.527 0.2165 0.492 Musees -0.32 -0.27 0.0340 0.099 0.236 -0.590 0.156 0.108 0.539 0.0492 -0.277 Soleil -0.29 -0.29 -0.4430 -0.157 0.208 0.393 -0.380 -0.080 -0.062 -0.0011 -0.506 Polution -0.30 0.30 0.3507 -0.013 0.039 -0.054 0.104 -0.742 -0.161 0.1818 -0.270 Embout -0.30 0.35 -0.2859 0.022 -0.051 -0.337 -0.153 -0.042 -0.174 -0.7280 0.073 LienParis -0.30 0.26 0.3767 -0.412 0.344 -0.078 -0.236 0.493 -0.262 0.1879 0.034 Cadres -0.31 -0.16 0.4032 -0.329 -0.485 0.346 0.075 0.063 0.349 -0.3510 -0.022 CreatEntrp -0.29 -0.39 -0.0098 0.013 0.409 0.190 0.560 -0.059 -0.304 -0.1865 0.341 Revenu -0.30 -0.19 0.3090 0.734 -0.073 0.097 -0.423 0.081 -0.113 0.0173 0.152 PrixImmob -0.30 0.35 -0.2256 0.288 -0.301 0.161 0.476 0.384 -0.131 0.2088 -0.325 Les valeurs propres sont elles > acp$eig [1] 1940.974149 275.990875 123.372428 31.594830 28.553309 23.945216 14.283394 [8] 12.950240 10.841242 6.205078 4.849239 85
  • 86. Arthur CHARPENTIER - Analyse des donn´ees Les projections sur les deux premiers axes sont donn´ees par acp$c1[,1 :2]. Toutes les variables contribuent `a l’axe 1 (sens n´egatif). On utilise s.label(acp$li) et s.label(acp$co) pour projeter lignes et colonnes respectivement d = 10 Angers Bordeaux CaenClermontFerrand Dijon Douai Grenoble Lille Lyon Marseille Metz Montpellier Nancy Nantes Nice Orléans Paris Rennes Rouen SaintEtienne Strasbourg Toulon Toulouse Tours Valenciennes d = 5 Medecins Crimin Musees Soleil Polution Embout LienParis Cadres CreatEntrp Revenu PrixImmob 86
  • 87. Arthur CHARPENTIER - Analyse des donn´ees ACP centr´ee ou pas Parmi les transformations usuelles des variables, on peut les centrer. La nouvelle origine G a pour coordonn´ees (C1, · · · , Cq), correspondant au centre de gravit´e du nuage de points. On note ˜Cj les colonnes (centr´ees) de ˜X, i.e. ˜Cj = Cj − Cj . Alors la norme de ˜Cj correspond `a l’´ecart-type de Cj, puisque ˜Cj 2 = 1 n i=1 n(xi,j − Cj)2 = V ar(Cj). 87
  • 88. Arthur CHARPENTIER - Analyse des donn´ees d = 10 Angers Bordeaux Caen ClermontFerrand Dijon Douai Grenoble Lille Lyon Marseille Metz Montpellier Nancy Nantes Nice Orléans Paris RennesRouen SaintEtienne Strasbourg Toulon Toulouse Tours Valenciennes d = 2d = 2 Medecins Crimin Musees Soleil Polution Embout LienParis Cadres CreatEntrp Revenu PrixImmob 88
  • 89. Arthur CHARPENTIER - Analyse des donn´ees ACP norm´ee ou pas Parmi les transformations usuelles des variables, on peut les normer. Ceci permet de r´e´equilibrer des variables qui peuvent ˆetre exprim´ees dans des unit´ees diff´erentes. d = 2 Angers Bordeaux Caen ClermontFerrand Dijon Douai Grenoble Lille Lyon Marseille Metz Montpellier Nancy Nantes Nice Orléans Paris RennesRouen SaintEtienne Strasbourg Toulon Toulouse Tours Valenciennes d = 0.5d = 0.5 Medecins Crimin Musees Soleil Polution Embout LienParis Cadres CreatEntrp Revenu PrixImmob Attention On a seulement normalis´e les variables. 89
  • 90. Arthur CHARPENTIER - Analyse des donn´ees Angers Bordeaux Caen ClermontFerrand Dijon Douai Grenoble Lille Lyon Marseille Metz Montpellier Nancy Nantes Nice Orléans Paris Rennes Rouen SaintEtienne Strasbourg Toulon Toulouse Tours Valenciennes Medecins Crimin Musees Soleil Polution Embout LienParis Cadres CreatEntrp Revenu PrixImmob L’´etude de ces inerties peut se faire `a l’aide de plot(princomp(base)) sous R. biplot(princomp(base)) permet de projeter les individus sur le premier plan principal. 90
  • 91. Arthur CHARPENTIER - Analyse des donn´ees Comp.1 Comp.3 Comp.5 Comp.7 Comp.9 Variances 050100150200250 −0.4 −0.2 0.0 0.2 0.4 −0.4−0.20.00.20.4 Comp.1 Comp.2 Angers Bordeaux Caen ClermontFerrand Dijon Douai Grenoble Lille Lyon Marseille Metz Montpellier Nancy Nantes Nice Orléans Paris RennesRouen SaintEtienne Strasbourg Toulon Toulouse Tours Valenciennes −30 −20 −10 0 10 20 30 40 −30−20−10010203040 Medecins Crimin Musees Soleil Polution Embout LienParis Cadres CreatEntrp Revenu PrixImmob Attention le signe peut changer d’un logiciel `a l’autre. Par exemple, le calcul complet `a partir de la diagonalisation donne x=as.matrix(base) n <- nrow(x); p <- ncol(x) centre <- apply(x, 2, mean) 91
  • 92. Arthur CHARPENTIER - Analyse des donn´ees x <- x - matrix(centre, nr=n, nc=p, byrow=T) e1 <- eigen( t(x) %*% x, symmetric=T ) e2 <- eigen( x %*% t(x), symmetric=T ) variables <- t(e2$vectors) %*% x individus <- t(e1$vectors) %*% t(x) variables <- t(variables) individus <- t(individus) valeurs.propres <- e1$values plot( individus[,1:2], xlim=c( min(c(individus[,1],-individus[,1])), max(c(individus[,1],-individus[,1])) ), ylim=c( min(c(individus[,2],-individus[,2])), max(c(individus[,2],-individus[,2])) ), xlab=’’, ylab=’’, frame.plot=F ) par(new=T) plot( variables[,1:2], col=’red’, xlim=c( min(c(variables[,1],-variables[,1])), max(c(variables[,1],-variables[,1])) ), ylim=c( min(c(variables[,2],-variables[,2])), max(c(variables[,2],-variables[,2])) ), axes=F, xlab=’’, ylab=’’, pch=’.’) 92
  • 93. Arthur CHARPENTIER - Analyse des donn´ees axis(3, col=’red’) axis(4, col=’red’) arrows(0,0,variables[,1],variables[,2],col=’red’) q q q q q q q q q q q q q q q q q qq q q q q q q −30 −20 −10 0 10 20 30 −20−1001020 −30 −20 −10 0 10 20 30 −20−1001020 −0.4 −0.2 0.0 0.2 0.4−0.4−0.20.00.20.4 Comp.1 Comp.2 Angers Bordeaux Caen ClermontFerrand Dijon Douai Grenoble Lille Lyon Marseille Metz Montpellier Nancy Nantes Nice Orléans Paris RennesRouen SaintEtienne Strasbourg Toulon Toulouse Tours Valenciennes −30 −20 −10 0 10 20 30 40 −30−20−10010203040 Medecins Crimin Musees Soleil Polution Embout LienParis Cadres CreatEntrp Revenu PrixImmob 93
  • 94. Arthur CHARPENTIER - Analyse des donn´ees Explication des axes Pour interpr´eter le premier axe, rappelons que > names(base) [1] "Medecins" "Crimin" "Musees" "Soleil" "Polution" [6] "Embout" "LienParis" "Cadres" "CreatEntrp" "Revenu" [11] "PrixImmob" > acp$co[, 1] [1] 0.6847162 -0.8499417 0.7013389 0.7029398 -0.6656761 [6] -0.7872159 -0.5619327 0.3887672 0.9139772 0.4715448 [11] -0.7837233 94
  • 95. Arthur CHARPENTIER - Analyse des donn´ees variable axe 1 axe 2 Medecins 0.6847162 -0.27456842 Criminalit´e -0.8499417 -0.31990547 Musees 0.7013389 0.20111087 Soleil 0.7029398 -0.63249825 Polution -0.6656761 0.63938379 Embouteillages -0.7872159 -0.30661893 Lien Paris -0.5619327 0.65398028 Cadres 0.3887672 0.72323380 Creation Entreprises 0.9139772 0.04395415 Revenu 0.4715448 0.58525352 Prix Immobilier -0.7837233 -0.22212682 95
  • 96. Arthur CHARPENTIER - Analyse des donn´ees Recherche des points affluents L’´etude de la projection de l’espace des individus permet d’associer ou de dissocier des individus au comportement proche, ou radicalement diff´erent : deux points tr`es ´eloign´es sur le premier axe sont tr`es ´eloign´es dans le nuage initial. L’intertie du nuage s’´ecrit I(X, D, u) = n i=1 < Li, u >2 D . On peut ainsi chercher les points qui contribuent le plus au positionnement de l’axe, i.e. i pour lequels < Li, u >2 D est grand. Definition 17. On appelle contribution (absolue) du point Li `a la position de l’axe uk la quantit´e CTk(Li) = < Li, uk >2 D λk . 96
  • 97. Arthur CHARPENTIER - Analyse des donn´ees Notons que n i=1 CTk(Li) = I(X, D, uk) λk = 1. 97
  • 98. Arthur CHARPENTIER - Analyse des donn´ees Qualit´e d’une projection Definition 18. On appelle qualit´e de la repr´esentation du point Li sur l’axe uk la quantit´e QRk(Li) = < Li, uk >2 D Li, D . Notons que q k=1 CTk(Li) = 1. On parle aussi de contribution relative > intacp <- inertia.dudi(acp, col.inertia = T,row.inertia = T) > intacp$row.rel[, 1] Angers Bordeaux Caen ClermontFerrand -5500 7565 -5479 -2464 Dijon Douai Grenoble Lille -2131 -8886 5132 -144 Lyon Marseille Metz Montpellier 6621 8075 -5311 6134 Nancy Nantes Nice Orleans -455 572 7660 -4276 Paris Rennes Rouen SaintEtienne 98
  • 99. Arthur CHARPENTIER - Analyse des donn´ees 2726 -559 -6401 -1760 Strasbourg Toulon Toulouse Tours 1865 150 8115 -2283 Valenciennes -7911 > intacp$row.abs[, 1] Angers Bordeaux Caen ClermontFerrand 210 483 251 243 Dijon Douai Grenoble Lille 72 1212 347 12 Lyon Marseille Metz Montpellier 678 1168 374 523 Nancy Nantes Nice Orleans 30 19 1024 226 Paris Rennes Rouen SaintEtienne 444 16 681 172 Strasbourg Toulon Toulouse Tours 60 17 605 94 Valenciennes 1040 Le signe indique le signe de la coordonn´ee sur l’axe 1. 99
  • 100. Arthur CHARPENTIER - Analyse des donn´ees acp$co fournit l’interpr´etation des axes µk dans l’espace des variables , acp$c1 fournit l’interpr´etation des axes uk dans l’espace des individus Les valeurs sont identiques `a une constante de normalisation pr`es. =⇒ Les axes uk et µk ont la mˆeme interpr´etation par rapport aux variables initiales. On peut alors envisager une rep´esentation simultan´ee des espaces individus ou variables. On peut regarder la projection des villes, en fonction de diff´erentes variables explicatives, s.value(acpli, scale(base$Medecins))s.value(acpli, scale(base$Crimin )) 100
  • 101. Arthur CHARPENTIER - Analyse des donn´ees d = 2 −1.5 −0.5 0.5 1.5 d = 2 −1.5 −0.5 0.5 1.5 d = 2 −1.5 −0.5 0.5 1.5 d = 2 −1.5 −0.5 0.5 1.5 d = 2 −1.5 −0.5 0.5 1.5 d = 2 −1.5 −0.5 0.5 1.5 101
  • 102. Arthur CHARPENTIER - Analyse des donn´ees d = 2 −1.5 −0.5 0.5 1.5 d = 2 −1.5 −0.5 0.5 1.5 d = 2 −1.5 −0.5 0.5 1.5 d = 2 −1.5 −0.5 0.5 1.5 d = 2 −1.5 −0.5 0.5 1.5 102
  • 103. Arthur CHARPENTIER - Analyse des donn´ees Retour sur la m´thodologie de l’ACP Pour r´esumer, on part d’un nuage de n individus dont on connaˆıt p variables quantitatives, not´ees x. On g´en´eral, on centre et on r´duit les variables pour obtenir une matrice z. La matrice p × p de corr´elation de z poss`ede des valeurs propres que l’on ordonne λ1 ≥ λ2 ≥ · · · ≥ λk ≥ 0. Les facteurs principaux uk sont les vecteurs propres orthonorm´es de la matrice de corr´elation, associ´es aux valeurs propres λk. uk,j est le poids de la variable j dans la composante k. Les composantes principales ont les vecteurs ck = zuk, de taille n. ck,i est la valeur de la composante k pour l’individu i. Notons que la variance de ck vaut λk. Le cercle des corr´elation permet de visualiser les corr´elations entre les variables avec les axes principaux. Seules les variables au bord du cercle sont interpr´etables (car bien repr´esent´es par les deux axes). 103
  • 104. Arthur CHARPENTIER - Analyse des donn´ees Petite digression : mod´eliser des taux Consid´erons le jeu de donn´ees suivant, load(url("http://pbil.univ-lyon1.fr/R/donnees/pps066.rda")) Trois tableaux croisent 20 pays et 39 ann´ees pour la consommation individuelle de bi`ere, vin et spiritueux. Le but est d´etudier la r´epartition entre ces trois types d’alcool (et non pas les niveaux d’alcool consomm´es). Pour cela, comme on est en dimension 3 (3 alcools possibles), on peut utiliser une repr´esentation dite triangulaire. 104
  • 105. Arthur CHARPENTIER - Analyse des donn´ees 0 1 Vin 1 0 Bière 1 0 Spiriteux qq q q q q q q q q q q q q q q q qq q AllAut Bel Chy Dan Esp Fin Fra Gre Hon Irl Ita Lux PBa Pol Por Rep RUSlq Sue 0 1 Vin 0.9 0 Bière 0.9 0.1 Spiriteux q q q q q q q q q q qq q q q q q q q q All Aut Bel Chy Dan Esp Fin Fra Gre Hon IrlIta Lux PBa Pol Por Rep RU Slq Sue 105
  • 106. Arthur CHARPENTIER - Analyse des donn´ees Retour sur la m´ethodologie de l’ACP Sous R, plusieurs fonctions permettent de faire des ACP • dans library(base), la fonction princomp, • dans library(ade4), la fonction dudi.pca, qui permet simplement de centrer et r´eduire les variables. • dans library(FactoMineR), la fonction PCA 106
  • 107. Arthur CHARPENTIER - Analyse des donn´ees L’ACP avec dudi.pca Cette partie sera inspir´ee de Dufour & Lobry (2008), tdr601.pdf. Consid´erons les donn´ees survey de library(MASS). On retiendra 4 variables, • survey$Wr.Hnd correspondant `a l’empan de la main d’´ecriture • survey$NW.Hnd correspondant `a l’empan de la main qui n’´ecrit pas • survey$Height correspondant `a la taille de la personne • survey$sex correspondant au sexe de la personne. 107
  • 108. Arthur CHARPENTIER - Analyse des donn´ees L’ACP avec dudi.pca 108
  • 109. Arthur CHARPENTIER - Analyse des donn´ees L’ACP avec dudi.pca 109
  • 110. Arthur CHARPENTIER - Analyse des donn´ees L’ACP avec dudi.pca survey.cc <- survey[complete.cases(survey), ] mesures <- survey.cc[, c("Wr.Hnd", "NW.Hnd", "Height")] La premi`ere commande permet de ne garder que les individus ne pr´esentant pas de valeurs manquantes. L’ACP se fait en utilisant simplement acp <- dudi.pca(mesures, scann = FALSE, nf = 3). Pour r´ecup´erer toutes les informations, on peut utiliser la fonction suivantes > eval(acp$call) Duality diagramm class: pca dudi $call: dudi.pca(df = mesures, scannf = FALSE, nf = 3) $nf: 3 axis-components saved $rank: 3 eigen values: 2.509 0.4568 0.03445 vector length mode content 1 $cw 3 numeric column weights 2 $lw 168 numeric row weights 110
  • 111. Arthur CHARPENTIER - Analyse des donn´ees 3 $eig 3 numeric eigen values data.frame nrow ncol content 1 $tab 168 3 modified array 2 $li 168 3 row coordinates 3 $l1 168 3 row normed scores 4 $co 3 3 column coordinates 5 $c1 3 3 column normed scores acp$tab est la matrice z obtenue en centrant puis en r´eduisant la table initiale x. acp$cw contient des points attibu´es `a chaque variable (colonne), i.e. ici 1 partout. acp$lw contient des points attibu´es `a chaque individu (ligne), i.e. ici 1/n partout. acp$eig contient le vecteur des valeurs propres. acp$c1 donne les coordon´ees des variables sur les 3 permiers axes principaux. Ces vecteurs sont de norme 1. acp$co donne les coordon´ees des variables sur les 3 permiers axes principaux. Ces vecteurs sont de norme √ λ. > acp$c1 111
  • 112. Arthur CHARPENTIER - Analyse des donn´ees CS1 CS2 CS3 Empan1 0.6084890 -0.3420962 0.71603859 Empan2 0.6040404 -0.3855223 -0.69750107 Taille 0.5146613 0.8569380 -0.02794614 > acp$co Comp1 Comp2 Comp3 Empan1 0.9637816 -0.2312213 0.132897100 Empan2 0.9567355 -0.2605728 -0.129456527 Taille 0.8151685 0.5792006 -0.005186817 > t(t(acp$c1)*sqrt(acp$eig)) CS1 CS2 CS3 Empan1 0.9637816 -0.2312213 0.132897100 Empan2 0.9567355 -0.2605728 -0.129456527 Taille 0.8151685 0.5792006 -0.005186817 acp$l1 donne les coordon´ees des individus sur les 3 permiers axes principaux, ces vecteurs ´etant unitaires acp$li donne les coordon´ees des individus sur les 3 permiers axes principaux, ces vecteurs ´etant unitaires > head(acp$l1) 112
  • 113. Arthur CHARPENTIER - Analyse des donn´ees RS1 RS2 RS3 1 -0.18511289 0.35854289 0.7771789 2 0.65642982 -0.01654562 -2.0459458 5 0.24122137 -1.63843055 0.1095513 6 -0.35270516 0.54186131 0.3453116 7 -0.08047126 1.91845520 -0.4136080 8 -1.14527378 -1.08502402 -0.6698669 > head(acp$li) Axis1 Axis2 Axis3 1 -0.2931990 0.24233756 0.14424478 2 1.0397147 -0.01118311 -0.37972849 5 0.3820689 -1.10740798 0.02033277 6 -0.5586473 0.36624167 0.06409000 7 -0.1274579 1.29667540 -0.07676584 8 -1.8139913 -0.73336294 -0.12432761 > head(t(t(acp$l1) * sqrt(acp$eig))) RS1 RS2 RS3 1 -0.2931990 0.24233756 0.14424478 2 1.0397147 -0.01118311 -0.37972849 5 0.3820689 -1.10740798 0.02033277 6 -0.5586473 0.36624167 0.06409000 113
  • 114. Arthur CHARPENTIER - Analyse des donn´ees 7 -0.1274579 1.29667540 -0.07676584 8 -1.8139913 -0.73336294 -0.12432761 Enfin, pour faire quelques graphiques, on utilise s.label ou s.class pour visualiser les individus > s.label(acp$li, xax = 1, yax = 2) > s.class(acp$li, fac=sexe,col=c("red","blue"),xax = 1, yax = 2) 114
  • 115. Arthur CHARPENTIER - Analyse des donn´ees d = 2 1 2 5 6 7 8 9 10 11 14 17 18 20 21 22 23 24 272830 32 33 34 36 38 39 42 44 47 48 49 50 51 52 53 54 55 57 59 6162 63 65 71 73 74 75 76 77 79 82 85 86 87 88 89 91 93 95 97 98 100 102 104 105 106 109 110111 112 113 114115 116117 118 119 120 122 123 124 125 127 128 129 130 131132 134 135 136 138 140 141 143 144 145 146 147 148 149 150 151 152 153 154 155 156158 160 161 163 164 166 167 168 170 172 174 175 176 177 178180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 196 197 198 199 200 201 202 204 205 206 207 208 209 211 212 214 215 218 220 222 223 227 228 229 230 231233 234 236 237 d = 2 q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q qq q q q q q Female Male Pour visualiser les variables, on utilise s.corcircle ou pour tout repr´esenter ensemble, la fonction scatter > s.corcircle(acp$co, xax = 1, yax = 2) > scatter(acp) 115
  • 116. Arthur CHARPENTIER - Analyse des donn´ees Empan1Empan2 Taille d = 2 1 2 5 6 7 8 9 10 11 14 17 18 20 21 22 23 24 27 28 30 32 33 34 36 38 39 42 44 47 48 49 50 51 52 53 54 55 57 59 6162 63 65 71 73 74 75 76 77 79 82 85 86 87 88 89 91 93 95 97 98 100 102 104 105 106 109 110111 112 113 114115 116117 118 119 120 122 123 124 125 127 128 129 130 131132 134 135 136 138 140 141 143 144 145 146 147 148 149 150 151 152 153 154 155 156158 160 161 163 164 166 167 168 170 172 174 175 176 177 178180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 196 197 198 199 200 201 202 204 205 206 207 208 209 211 212 214 215 218 220 222 223 227 228 229 230 231233 234 236 237 Empan1 Empan2 TailleEigenvalues 116
  • 117. Arthur CHARPENTIER - Analyse des donn´ees Travaux dirig´es Le TD portera sur la base de donn´ees departement.xls (dont une codification est donn´e dans le fichier code-departement.xls) t´el´echargeables sur ma page internet. 117