Cours add-r1-part1

Arthur CHARPENTIER - Analyse des données
Analyse des données (1)
L’Analyse en Composantes Principales
Arthur Charpentier
http ://perso.univ-rennes1.fr/arthur.charpentier/
blog.univ-rennes1.fr/arthur.charpentier/
Master 2, Université Rennes 1
1

Introduction à l’analyse des données
Dans ce cours, nous verrrons essentiellement deux types de méthodes
• les méthodes factorielles, où on cherchera à réduire le nombre de variables en
les résumant en un petit nombre de composantes synthétiques
◦ 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, où on cherchera les liens entre les modalités, avec l’ACF
Analyse des Correspondances Factorielles (simples) dans le cas où on dispose
de 2 variables, et l’ACM Analyse des Correspondances Multiples dans le cas
où on dispose de plus de 2 variables
2

Introduction à l’analyse des données
• les méthodes de classification, où on cherchera à réduire la taille de l’ensemble
des individus en les regroupant en un petit nombre de groupes homogènes
◦ en particulier la CAH, Classification Ascendante Hiérarchique ...
◦ en particulier l’Analyse Discriminante ...
Remarque Ce cours est davantage un cours d’algèbre linéaire qu’un cours de
probabilité ou de statistique. Mais une interprétation sera parfois possible en
terme de moyenne ou de variance (voire de covariance).
3

Exemple, ville et (in)sécurité
“Le palmarès des départements : où vit-on en sécurité ?, dans L’Express (no
2589, 15 février 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

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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
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5

qq
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qq
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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
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vols avec violence
volsautomobile
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infractions
volsavecviolence
volsautomobile
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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
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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
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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
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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
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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
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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 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q
q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q
q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 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

Les variables semblent plutôt corrélées 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 à regrouper les villes “proches”.
=⇒ Comme on a du mal à voir dans R3
, on va essayer de projeter le nuage.
• projection sur un axe (droite)
• projection sur un plan
8

20 40 60 80 100 120 140
02468101214
0
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4
6
8
10
infractions
volsavecviolence
volsautomobile
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q
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9

20 40 60 80 100 120 140
02468101214
0
2
4
6
8
10
infractions
volsavecviolence
volsautomobile
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qqqqqqqqqqqqqqqqqqqqqqq
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qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
q
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
q
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
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qqqqq
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qqq
qqqqqqqqqqqqqq
qqq
10

20 40 60 80 100 120 140
02468101214
0
2
4
6
8
10
infractions
volsavecviolence
volsautomobile
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
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qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
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=⇒ recherche de la projection “la plus représentative”, cf. idée des moindres
carrés, qui minimise l’erreur de projection comise
11

Pourquoi pas projecter sur un plan ?
20 40 60 80 100 120 140
02468101214
0
2
4
6
8
10
infractions
volsavecviolence
volsautomobile
q
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12

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
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−2 −1 0 1 2 3 4 5
−2−10123
−1
0
1
2
3
4
5
6
7
infractions
volsavecviolence
volsautomobile
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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
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q
q
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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
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q
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q
q
q
q
q
q
q
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q
q
q
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q
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q
q
q
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q
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q
q
q
q
q
q
q
q
q
q
q
q
q
13

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

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

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

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

Un peu de géométrie euclidienne
On observe n individus, et q variables (quantitatives, sur R).
Les nuages de points peuvent se décomposer de deux manières,
– l’espace des individus, i.e. Rq
– l’espace des variables, i.e. Rn
On note xij l’observation de la jème variable sur le ième individu.
variables
1 · · · j · · · q
individus 1 x11 · · · x1j · · · x1q
...
...
...
...
i xi1 · · · xij · · · xiq
...
...
...
...
n xn1 · · · xnj · · · xnq
18

Un peu de géométrie euclidienne
Chaque individu est charactérisé par Li = (xi1, · · · , xiq)t
, appartenant à Rq
,
exprimé dans la base canonique {e1, · · · , eq}.
Definition 1. Les points individus dans l’espace vectoriel Rq
, munié de
{e1, · · · , eq} est appelé espace des individus.
=⇒ comment mesurer la distance entre deux individus ?
19

Distance entre individus
Definition 2. Soit D une matrice diagonale q × q, dont les éléments diagonaux
sont strictement positifs (dii > 0 pour i = 1, · · · , q). Alors la fonction
ϕ : Rq
× Rq
→ R définie par
(u, v) → ut
Dv =
q
j=1
djjujvj
est un produit scalaire, noté < ·, · >D.
Definition 3. Soit D une telle matrice diagonale q × q, et < ·, · >D le produit
scalaire associé. On note alors · D la norme associée,
u D =
√
< u, u >D =
q
j=1
djjujuj
et dD(·, ·) la distance associée,
dD(u, v) = u − v D.
20

Exemples de produits scalaires
• D = Id correspond au produit scalaire canonique, < u, v >Id=
q
j=1
ujvj
• Considérons le produit scalaire associé à D =


3/4 0
0 1/4


Les points à égale 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 à dire une ellipse dans R2
.
21

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

Les métriques usuelles
Il y a fondamentalement trois types de métriques à retenir,
• la métrique usuelle i.e. M = I, la matrice identié
Dans ce cas, la distance dépend de l’unité de mesure, et de la dispersion des
variables.
• la métrique réduite i.e. M = diag(s−2
1 , · · · , s−2
q ), la matrice diagonale des
inverses des variances empiriques
Rappelons que pour une série 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

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 corrélation (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 métrique transformée i.e. M = T T,
Cela est équivalent à travailler avec la métrique classique I sur le tableau
transformée XT .
Notons que pour toute matrice symmétrique positive M, il existe une telle
matrice T, appelé racine carrée de M
24

Déformation de l’espace
Proposition 4. Munir l’espace de la métrique issue de D q × q, diagonale, est
équivalent à attribuer des poids {
√
d11, · · · , dqq} aux q variables et d’utiliser la
métrique canonique.
Démonstration. 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 où ˜u = (˜u1, · · · , ˜uq), ˜uj = djjuj.
25

Les variables, cas de la dimension 2
On cherche ici à mesurer une distance, ou une proximité, entre des variables.
Intuitivement, cette notion doit être proche de la notion de corrélation.
Soient deux variables X1 et X2 continues.
Remarque La régression propose d’étudier le lien entre deux variables, dans
l’optique d’en utiliser une pour prévoir l’autre.
26

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

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 gravité du nuage (comme les
deux autres régressions).
Changeons les coordonnées pour simplifier, Y1 = X1 − X1 et Y2 = X2 − X2. On
notera O ce barycentre, X les points d’origine et P les projections
28

orthongonales. On cherche à minimiser
I =
n
i=1
XiPi
2
=
n
i=1
OiXi
2
− OiPi
2
(qu’on appelera inertie),
par des propriétés d’orthogonalité. Les points O et X étant fixer, si u est le
vecteur directeur de l’axe, u = (a, b), supposé unitaire, minimiser I devient à
maximiser
I2 =
n
i=1
OiPi
2
= (Y u) Y Y uu (Y Y )uu (nΣ)u
où Σ correspond à la matrice de variance-covariance de Y (et donc de X).
Σ est symmétrique, elle possède 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

où U est une matrice othonormée. Aussi,
I2 = λ1α2
+ λ2β2
≤ max{λ1, λ2} [α2
+ β2
]
=1
,
où (α, β) sont les nouvelles coordonées de u.
L’inertie ne peut donc dépasser 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 bivarié est le vecteur propre associé à la
plus grande valeur propre de la matrice de variance-covariance des deux variables.
Ce résultat va se généraliser en plus grande dimension.
30

L’espace des variables
De la même manière, chaque variable est charactérisé par Cj = (x1j, · · · , xnj)t
,
appartenant à Rn
, exprimé dans la base canonique {f1, · · · , fn}.
Généralement, dans l’espace des variables, un poids identique sera donné à
chaque individu.
31

Projeter un nuage de points
32

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

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

Projeter des points, la notion d’inertie
Considérons le tableau de données X = (xij)1≤i≤n,1≤j≤q = {L1, · · · , Ln}.
L’espace individus (de Rq
) est muni de la métrique issue D.
Definition 5. On appelle inertie du nuage des points {L1, · · · , Ln} la quantité
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

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
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q
q
q
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q
q q
q
q
q
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q
q q
q
q
q
q
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q
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q
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q q
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q q
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q q
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qq
q q
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q q
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q
q
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qq
q
q
q
q
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q
qq q
q
q
q
q
q
q
qq
q
q
q
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0 1 2 3 4 5 6
0123456
Projection sur le plan
q
36

0 1 2 3 4 5 6
0123456
0
1
2
3
4
5
6
q
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0 1 2 3 4 5 6
0123456
q
37

0 1 2 3 4 5 6
0123456
0
1
2
3
4
5
6
q
q
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0 1 2 3 4 5 6
0123456
q
38

0 1 2 3 4 5 6
0123456
0
1
2
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5
6
q
q
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0 1 2 3 4 5 6
0123456
q
39

Cours add-r1-part1

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