gmmTrain: Parameter Training for Gaussian Mixture Model (GMM)

The gmmTrain function implements the core parameter training process for Gaussian Mixture Models (GMM), which are powerful probabilistic models widely applied in data science, speech recognition, and image processing applications. The function accepts input data vectors and employs the Expectation-Maximization (EM) algorithm to learn optimal parameters that best fit a GMM to the observed data. This training procedure is critical as it directly impacts the model's predictive accuracy and generalization capabilities. During execution, the function iteratively estimates three key parameter sets: the mean vectors and covariance matrices defining each Gaussian component's distribution characteristics, along with the mixture weights representing each component's proportional contribution to the overall model. The implementation typically includes convergence checking based on log-likelihood improvements and may incorporate regularization techniques to prevent covariance matrix singularity issues. By accurately determining these parameters through maximum likelihood estimation, the trained GMM can effectively perform density estimation, clustering, and classification tasks on new datasets. The gmmTrain function serves as a fundamental building block for GMM-based systems and is essential for researchers and practitioners working with mixture models in statistical modeling and machine learning applications.