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Cours add-r1-part1

  1. 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. 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. 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. 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. 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. 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. 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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  8. 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. 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 12
  13. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 32. Arthur CHARPENTIER - Analyse des donn´ees Projeter un nuage de points 32
  33. 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. 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. 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. 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. 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. 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. 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

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