GaussianMixtureModellen

A Gaussian mixture model is a probabilistic model that represents a distribution as a weighted sum of Gaussian densities. It is used for clustering, density estimation, and anomaly detection, offering a flexible alternative to single Gaussian assumptions. By combining multiple Gaussians, the model can capture complex, multimodal data distributions.

Mathematically, the probability density of a Gaussian mixture model is p(x) = sum_{k=1}^K pi_k N(x | mu_k, Sigma_k), where pi_k are nonnegative mixture weights that sum to one, mu_k are the component means, and Sigma_k are the covariance matrices. The model assumes that each data point x is generated by selecting a component with probability pi_k and then sampling from the corresponding Gaussian. The indices are latent, often denoted z, indicating which component produced each observation.

Estimation of the parameters (pi_k, mu_k, Sigma_k) is commonly performed via the Expectation-Maximization algorithm. The E-step computes responsibilities, the posterior probabilities that a given data point was generated by each component. The M-step updates the mixture weights, means, and covariances to maximize the expected complete-data log-likelihood. Convergence is typically assessed by the change in log-likelihood or in parameter values.

Model selection involves choosing the number of components K, often using information criteria such as BIC or AIC, and considering covariance structure (full, diagonal, spherical, or tied). Initialization affects convergence and results, with common strategies including K-means clustering or random initialization.

Applications span clustering, density estimation, and unsupervised pattern recognition across fields such as computer vision, speech processing, and finance. Limitations include sensitivity to initialization, potential identifiability issues, and computational demands for high-dimensional data. Regularization and scalable variants help address these challenges.