Gaußsches Mischmodell (GMM)
Eine Gaußsche Verteilung Mischmodell (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 maschinellem Lernen for tasks such as clustering, Dichteschätzung, and classification.
Jedes Gaußsche Verteilung 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.
Mathematisch ist das probability Die Dichtefunktion eines GMM kann ausgedrückt werden als:
P(x) = Σ (πk * N(x | μk, Σk))
Hier stellt πk represents the weight der k-ten Gaußschen Komponente, und N(x | μk, Σk) denotes the probability density of the data point x under the k-th Gaussian with mean μk sowie Kovarianz 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 beobachtete Daten.
GMMs are particularly useful in scenarios where the data exhibits cluster-like structures and can be applied in various fields, including finance, der Bildverarbeitung, and bioinformatics.