ガウス混合モデル（GMM）
ガウス分布 混合モデル (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 機械学習 for tasks such as clustering, 密度推定, and classification.
各 ガウス分布 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.
数学的には、 probability GMMの確率密度関数は次のように表される：
P(x) = Σ (πk * N(x | μk, Σk))
ここで、πk represents the weight k番目のガウス成分のものであり、N(x | μk, Σk) denotes the probability density of the data point x under the k-th Gaussian with mean μk と共分散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 観測データ.
GMMs are particularly useful in scenarios where the data exhibits cluster-like structures and can be applied in various fields, including finance, 画像処理, and bioinformatics.