EM Algorithm for Gaussian Mixture Model (GMM)

The Gaussian Mixture Model (GMM) is a widely used probabilistic model that assumes data is generated from a mixture of multiple Gaussian distributions. The Expectation-Maximization (EM) algorithm serves as the classical method for estimating GMM parameters.

### Algorithm Core Concept Initialization: Randomly assign initial values for each Gaussian component's mean vector, covariance matrix, and mixing coefficient. E-step (Expectation step): Compute the posterior probability (responsibility) of each data point belonging to each Gaussian component using Bayes' theorem. M-step (Maximization step): Update the Gaussian parameters (means, covariances, mixing coefficients) by maximizing the expected complete-data log-likelihood based on E-step results. Iterative convergence: Alternate between E-step and M-step until parameter changes fall below a threshold or maximum iterations are reached.

### MATLAB Implementation Key Points - Calculate probability densities using multivariate Gaussian distribution formula: pdf = (2*pi)^(-k/2)*det(Σ)^(-1/2)*exp(-0.5*(x-μ)'Σ^(-1)(x-μ)) - Efficiently update parameters through matrix operations: means = (responsibility' * data) / sum(responsibility) - Implement log-probability computations to prevent numerical underflow in likelihood calculations - Use MATLAB's gmdistribution.fit or implement custom EM with regularization for covariance matrices

### Application Scenarios GMM with EM algorithm is extensively applied in clustering, image segmentation, and anomaly detection. Its strength lies in modeling complex data distributions, though it's sensitive to initial conditions and may converge to local optima.

For improved performance, consider K-means initialization for better starting points or cross-validation to determine optimal component numbers through criteria like AIC or BIC.