Simultaneous estimation and clustering with finite mixture of nonparanormal graphical models

Abstract

Graphical mixture models provide a powerful tool to visually depict conditional independencies or dependencies between heterogeneous high-dimensional data. When the random variables corresponding to the vertices are continuous, mixture component densities are assumed to be multivariate normal with different covariance matrices, leading to introduction of the Gaussian graphical mixture model (GGMM). The nonparanormal graphical mixture model (NGMM) replaces the restrictive normal assumption with a semiparametric Gaussian copula, which extends the nonparanormal graphical model and mixture models. Such an extension allows us to simultaneously estimate cluster assignments and cluster-specific graphical model structure. To enable such analyses, we propose a regularized estimation scheme with two forms of penalty function (conventional and unconventional) via the expectation-maximization algorithm to learn a finite mixture of nonparanormal graphical models. We illustrate the performance of our method through a simulation study under both ideal and noisy settings. We also apply the proposed methodology to a breast cancer data set to diagnose malignant or benign tumors in patients. The results showed that the combination of NGMM together with unconventional penalty, which was named as NGMM1 during this study, provides the most efficient approach in clustering and estimating the graphical model structure.