Modelo de Mistura Gaussiana (GMM)
Uma Gaussiana Modelo de Mistura (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 aprendizado de máquina for tasks such as clustering, estimação de densidade, and classification.
Cada distribuição Gaussiana 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.
Matematicamente, a probability função de densidade de um GMM pode ser expressa como:
P(x) = Σ (πk * N(x | μk, Σk))
Aqui, πk represents the weight do componente Gaussiano k-ésimo, e N(x | μk, Σk) denotes the probability density of the data point x under the k-th Gaussian with mean μk e covariância 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 dados observados.
GMMs are particularly useful in scenarios where the data exhibits cluster-like structures and can be applied in various fields, including finance, processamento de imagens, and bioinformatics.