Data Clustering Analysis Using Gaussian Mixture Models (GMM)

The Gaussian Mixture Model (GMM) is a probabilistic model-based clustering approach suitable for datasets with complex distributions. Unlike K-means, GMM assumes that data is generated from a mixture of multiple Gaussian distributions, where each Gaussian component corresponds to one cluster. This allows GMM to more flexibly capture clusters of varying shapes.

In MATLAB, the built-in `fitgmdist` function enables Gaussian Mixture Model fitting and clustering. This function employs the Expectation-Maximization (EM) algorithm to iteratively optimize model parameters, including means, covariance matrices, and mixture weights. The EM algorithm alternates between estimating the posterior probabilities of data points belonging to each component (E-step) and updating the model parameters to maximize the expected log-likelihood (M-step).

The implementation workflow can be summarized as follows: Data Preprocessing: Ensure data is standardized or normalized to prevent scale differences from affecting model performance. Model Initialization: Specify the number of Gaussian components (clusters), which can be determined using the elbow method or information criteria such as AIC and BIC. Parameter Training: Call the `fitgmdist` function to fit the model to the data, automatically calculating parameters for each Gaussian component. MATLAB's implementation includes regularization options to handle ill-conditioned covariance matrices. Cluster Assignment: Use the `cluster` function to assign data points to the most likely Gaussian distribution based on posterior probabilities, enabling soft clustering where each point has membership probabilities across all clusters. Result Evaluation: Assess clustering quality through metrics like silhouette coefficients or log-likelihood values, adjusting model complexity when necessary.

GMM's key advantage lies in its ability to quantify the probability of data points belonging to each cluster (soft clustering) and its adaptability to elliptical cluster distributions. Limitations include sensitivity to initialization, higher computational complexity compared to distance-based methods, and the requirement to pre-specify the number of components.