Modèle de mélange gaussien (GMM)
Une Gaussienne Modèle de mélange (GMM) is a probabilistic model that assumes that the data is generated from a mixture of several Gaussian distributions, each representing a different cluster or group within the data. GMMs are widely used in statistics and apprentissage automatique for tasks such as clustering, estimation de densité, and classification.
Chaque distribution gaussienne in a GMM is defined by its mean (the center of the distribution) and covariance (which describes the shape and orientation of the distribution). The overall model is a weighted sum of these Gaussian components, where the weights indicate the proportion of the data that belongs to each cluster.
Mathématiquement, le probability la fonction de densité d'un GMM peut être exprimée comme :
P(x) = Σ (πk * N(x | μk, Σk))
Ici, πk represents the weight de la k-ième composante gaussienne, et N(x | μk, Σk) denotes the probability density of the data point x under the k-th Gaussian with mean μk et covariance Σk.
To fit a GMM to data, algorithms such as the Expectation-Maximization (EM) algorithm are commonly used. The EM algorithm iteratively updates the parameters of the Gaussian components to maximize the likelihood of the données observées.
GMMs are particularly useful in scenarios where the data exhibits cluster-like structures and can be applied in various fields, including finance, traitement d'image, and bioinformatics.