Robust Gaussian Mixture Modeling: A <inline-formula><tex-math notation="LaTeX">$\mathcal{K}$</tex-math></inline-formula>-Divergence Based Approach

This paper addresses the problem of robust Gaussian mixture modeling in the presence of outliers. We commence by introducing a general expectation-maximization (EM)-like scheme, called <inline-formula><tex-math notation="LaTeX">$\mathcal{K}$</tex-math></inline-formula>-BM, for iterative numerical computation of the minimum <inline-formula><tex-math notation="LaTeX">$\mathcal{K}$</tex-math></inline-formula>-divergence estimator (M<inline-formula><tex-math notation="LaTeX">$\mathcal{K}$</tex-math></inline-formula>DE). This estimator leverages Parzen's non-parametric <inline-formula><tex-math notation="LaTeX">$\mathcal{K}$</tex-math></inline-formula>ernel density estimate to down-weight low density regions associated with outlying measurements. Akin to the conventional EM, the <inline-formula><tex-math notation="LaTeX">$\mathcal{K}$</tex-math></inline-formula>-BM involves successive <underline>M</underline>aximizations of lower <underline>B</underline>ounds on the objective function of the M<inline-formula><tex-math notation="LaTeX">$\mathcal{K}$</tex-math></inline-formula>DE. However, differently from EM, these bounds are not exclusively reliant on conditional expectations. The <inline-formula><tex-math notation="LaTeX">$\mathcal{K}$</tex-math></inline-formula>-BM algorithm is applied to robust parameter estimation of a finite-order multivariate Gaussian mixture model (GMM). We proceed by introducing a new robust variant of the Bayesian information criterion (BIC) that penalizes the M<inline-formula><tex-math notation="LaTeX">$\mathcal{K}$</tex-math></inline-formula>DE's objective function. The proposed criterion, called <inline-formula><tex-math notation="LaTeX">$\mathcal{K}$</tex-math></inline-formula>-BIC, is conveniently applied for robust GMM order selection. In the paper, we also establish a data-driven procedure for selection of the kernel's bandwidth parameter. This procedure operates by minimizing an empirical asymptotic approximation of the mean-integrated-squared-error (MISE) between the underlying density and the estimated GMM density. Lastly, the <inline-formula><tex-math notation="LaTeX">$\mathcal{K}$</tex-math></inline-formula>-BM, the <inline-formula><tex-math notation="LaTeX">$\mathcal{K}$</tex-math></inline-formula>-BIC, and the MISE based selection of the kernel's bandwidth are combined into a unified framework for joint order selection and parameter estimation of a GMM. The advantages of the <inline-formula><tex-math notation="LaTeX">$\mathcal{K}$</tex-math></inline-formula>-divergence based framework over other robust approaches are illustrated in simulation studies involving synthetic and real data.