EM Algorithm for Estimating k-Dimensional Gaussian Mixture Models with Implementation Details

This paper presents an implementation of the Expectation-Maximization (EM) algorithm designed for estimating parameters of k-dimensional Gaussian mixture models. The input data consists of matrix X(n,d) where n represents the number of observations and d denotes the dimensionality of each variable. Additional required parameters include: maximum number of Gaussian components k, likelihood convergence threshold ltol (optional, default empty), maximum iteration count maxiter (optional, default empty), plotting flag pflag for 1D/2D visualization (optional, default empty), and initial parameter structure containing weights W, means M, and covariance matrices V (optional, default empty). The algorithm implementation estimates the Gaussian mixture model parameters including: mixture weights W(1,k), mean vectors M(d,k), covariance matrices V(d,d,k), and the log-likelihood value L. The EM algorithm operates through iterative expectation and maximization steps, where the E-step computes posterior probabilities using current parameter estimates, while the M-step updates parameters by maximizing the expected complete-data log-likelihood. To facilitate understanding of this methodology, we provide mathematical explanations of key concepts including likelihood computation, parameter initialization strategies, and convergence criteria. Practical implementation considerations are discussed, such as handling covariance matrix singularity through regularization techniques. The paper includes numerical examples demonstrating application scenarios and performance characteristics, with special attention to computational efficiency and numerical stability in high-dimensional settings. Code implementation details cover efficient matrix operations for parameter updates and strategies for avoiding local optima through multiple random initializations.