Mixture density

Densité mélange

En statistiques, on appelle densité mélange, ou loi mélange une fonction de densité qui est issue d'une combinaison linéaire de plusieurs fonctions de densité.

En prenant une fonction $f (x,θ)$ , densité d'une variable aléatoire $x$ paramétrée par $θ$ . Par exemple, si f est une loi normale, alors $θ$ est constitué de la moyenne et de la variance. Si on appelle $\theta_1,\dots,\theta_g$ une famille de paramètres et $\pi_1,\dots,\pi_g$ une famille de scalaires tels que

$\sum_{k=1}^g\pi_k=1$ ,

alors, la fonction $g$ définie par

$g(x,\theta_1,\dots,\theta_g)=\sum_{k=1}^g\pi_kf(x,\theta_k)$

est une fonction la fonction de densité d'une loi mélange à $g$ composantes.

On peut également étendre cette définition au cas où le nombre des composantes est infini. En considérant un ensemble $Ω$ de paramètres, si on a

$\int_\Omega \pi\left(\theta\right)d\theta=1,$

alors, la fonction

$g\left(x,\Omega\right)=\int_\Omega\pi(\theta)f(x,\theta)d\theta$

est une densité mélange.

Par exemple, l'image suivante représente la fonction de densité d'un mélange gaussien à deux composantes de dimension 2

Autres articles

Modèle de mélanges gaussiens

Ce document provient de « Densit%C3%A9 m%C3%A9lange ».

Catégorie : Loi de probabilité

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Densité mélange

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Densité mélange

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