<![CDATA[Gaussian Mixture Model (GMM) is a probabilistic clustering method capable of modeling heterogeneous data through a mixture of Gaussian distributions with flexible covariance structures. Parameter estimation in GMM is commonly performed using the Expectation–Maximization (EM) algorithm; however, the algorithm is sensitive to the choice of initial parameter values and may converge to local optima. To address this issue, this study presents a systematic mathematical derivation of GMM parameter estimation using the EM algorithm and employs Ward–K-Means initialization to obtain representative initial partitions that improve convergence stability and estimation quality. The proposed framework consists of initialization using Ward–K-Means, iterative parameter estimation through the EM algorithm, selection of the optimal number of Gaussian components based on the Bayesian Information Criterion (BIC), and cluster assignment using the Maximum A Posteriori (MAP) rule. The derived update equations provide maximum likelihood estimates for the mixture weights, mean vectors, and covariance matrices of the Gaussian components. The proposed approach is applied to flood vulnerability characteristic data from 44 districts in DKI Jakarta in 2025. Based on the minimum BIC value of 319,180746, the optimal model consists of eight Gaussian components, and the MAP rule assigns each district to the component with the highest posterior probability. The resulting clusters reveal distinct flood vulnerability characteristics across districts, demonstrating that the proposed GMM-EM framework with Ward–K-Means initialization provides an interpretable approach for probabilistic clustering of heterogeneous data.]]>
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