Weitere ähnliche Inhalte Ähnlich wie Cours add-r1-part1 (16) Mehr von Arthur Charpentier (20) Cours add-r1-part11. 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).
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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
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5
6. Arthur CHARPENTIER - Analyse des donn´ees
Exemple, ville et (in)s´ecurit´e
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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
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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
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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
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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
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7
8. Arthur CHARPENTIER - Analyse des donn´ees
Exemple, ville et (in)s´ecurit´e
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8
9. Arthur CHARPENTIER - Analyse des donn´ees
Exemple, ville et (in)s´ecurit´e
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9
10. Arthur CHARPENTIER - Analyse des donn´ees
Exemple, ville et (in)s´ecurit´e
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qqq
10
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
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qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
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qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
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qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
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qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
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=⇒ recherche de la projection “la plus repr´esentative”, cf. id´ee des moindres
carr´es, qui minimise l’erreur de projection comise
11
12. Arthur CHARPENTIER - Analyse des donn´ees
Exemple, ville et (in)s´ecurit´e
Pourquoi pas projecter sur un plan ?
20 40 60 80 100 120 140
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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
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q
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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
q
q
qq
q
q
q
q
q
q
q
q
q
q
q
q
q
qq
q
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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
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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
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
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q q
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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
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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
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qqq
q
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q q
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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
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
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q q
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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
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q q
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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
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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
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0 1 2 3 4 5 6
0123456
Projection sur le plan
q
39