Gaussian Mixture Model Parameter Estimation Using Expectation-Maximization Algorithm

A Gaussian Mixture Model (GMM) is a widely used probabilistic model that assumes data is generated from a mixture of multiple Gaussian distributions. To estimate the parameters of these Gaussian components (means, covariance matrices, and mixture weights), the Expectation-Maximization (EM) algorithm serves as a classical solution.

In the MATLAB environment, implementing the EM algorithm for GMM estimation typically involves the following steps with corresponding code considerations:

Initialization: Begin by specifying the number of Gaussian components (K) and randomly initializing the mean vectors, covariance matrices, and mixture weights for each component. In MATLAB, this can be efficiently done using functions like randn for means and eye for initial covariance matrices, often with k-means clustering for better starting points.

E-step (Expectation Step): Calculate the posterior probabilities (responsibilities) for each data point belonging to each Gaussian component. This step utilizes Bayes' theorem with current parameter estimates to determine the probability that each point was generated by each distribution. MATLAB implementation typically involves computing multivariate Gaussian probabilities using mvnpdf function and normalizing the results across components.

M-step (Maximization Step): Re-estimate the means, covariances, and mixture weights based on the responsibilities computed in the E-step. The new parameters are calculated using weighted averages that maximize the likelihood function. Code implementation involves efficient matrix operations using MATLAB's vectorized functions like sum and mean with responsibility weights.

Iterative Optimization: Repeatedly alternate between E-steps and M-steps until the log-likelihood function converges or the maximum iteration count is reached. MATLAB implementations typically include convergence checks using tolerance thresholds on likelihood changes and iteration counters to prevent infinite loops.

In MATLAB, the EM algorithm can be efficiently implemented using matrix operations to avoid explicit loops, significantly improving computational performance. Common optimizations include preventing covariance matrix singularity (e.g., by adding small regularization terms) and using log-probabilities to avoid numerical underflow. The final EM algorithm automatically discovers the clustering structure of data, making it suitable for unsupervised learning tasks such as data clustering, density estimation, and pattern recognition.