MATLAB Source Code for Gaussian Mixture EM Algorithm with Three-Parameter Estimation

Gaussian Mixture Model and EM Algorithm Principles The Gaussian Mixture Model describes complex data distributions through linear combinations of multiple Gaussian distributions. Its core parameters include: weight coefficients for each Gaussian component, mean vectors, and covariance matrices. The EM algorithm optimizes these parameters through iterative E-step (Expectation calculation) and M-step (Maximization) procedures.

MATLAB Implementation Key Steps Initialization phase: Randomly set weights, means, and covariances for each Gaussian component, ensuring weights sum to 1. This can be implemented using rand() and normalization functions. E-step computation: Calculate posterior probabilities (responsibility values) for each data point belonging to each component based on current parameters. This involves Gaussian probability density calculations using mvnpdf() function. M-step update: Re-estimate weights, means, and covariances using responsibility values. Weight updates become means of responsibility values, mean updates become weighted averages, and covariance updates become weighted scatter matrices. These can be vectorized using MATLAB's matrix operations. Convergence check: Terminate iteration when the log-likelihood function change falls below a threshold. Implement with while-loop and likelihood difference calculation.

Parameter Estimation Considerations Covariance matrices require regularization terms to avoid singularity issues. Add small diagonal values using eye() function. Multiple random initializations can avoid local optima. Implement with for-loop over different initial conditions. Component number K is typically selected using BIC/AIC criteria through model comparison.

Extended Application Scenarios This algorithm suits image segmentation, speech feature modeling tasks. MATLAB's matrix computation capabilities enable efficient vectorized implementation. For high-dimensional data, consider diagonal covariance matrices to simplify calculations using diag() function.