12. Analyse de corrélation
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−3 −2 −1 0 1 2 3
−3
−1
1 2 3
ρ ρ=−1
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−3 −2 −1 0 1 2 3
−3
−1
1 2 3
ρ ρ=−0.5
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−3 −2 −1 0 1 2 3
−3
−1
1 2 3
ρ ρ=0
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−3 −2 −1 0 1 2 3
−3
−1
1 2 3
ρ ρ=0.1
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−3 −2 −1 0 1 2 3
−3
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1 2 3
ρ ρ=0.3
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−3 −2 −1 0 1 2 3
−3
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1 2 3
ρ ρ=0.5
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−3 −2 −1 0 1 2 3
−3
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1 2 3
ρ ρ=0.7
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−3 −2 −1 0 1 2 3
−3
−1
1 2 3
ρ ρ=0.9
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−3 −2 −1 0 1 2 3
−3
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1 2 3
ρ ρ=1
Figure 12.8 – Exemples de données provenant d’une variable normale bivariée.
On dira que la distribution de (X, Y ) est normale bivariée ou binormale si sa
densité de probabilité est définie par :
f (x, y) =
exp
−
1
2(1−ρ 2 )
(x−μ X )
2
σ 2
X
−
2ρ(x−μ X )(y−μ Y )
σ X σ Y
+
(y−μ Y )
2
σ 2
Y
2πσ X σ Y
1 − ρ 2
où μ X et μ Y dénotent les moyennes et σ X et σ Y les écarts types des composantes X et Y de la variable bivariée (X, Y ), et où ρ dénote leur coefficient de
corrélation. La densité de probabilité d’une variable bivariée nous informe sur
la probabilité d’obtenir des observations aux alentours des différentes valeurs
possible (x, y) de (X, Y ) (et qui définit ainsi sa distribution). La famille des
distributions normales bivariées est caractérisée par cinq paramètres : μ X , μ Y ,
σ X , σ Y et ρ. Alors que les deux premiers sont des mesures de position et les
deux suivants des mesures de dispersion de X et de Y , le cinquième est un
paramètre d’association entre X et Y . Ce cinquième paramètre ρ caractérise
l’association entre X et Y lorsque la distribution de (X, Y ) est normale bivariée.