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 $\mathcal{K}$ -BM, for iterative numerical computation of the minimum $\mathcal{K}$ -divergence estimator (M $\mathcal{K}$ DE). This estimator leverages Parzen's non-parametric $\mathcal{K}$ ernel density estimate to down-weight low density regions associated with outlying measurements. Akin to the conventional EM, the $\mathcal{K}$ -BM involves successive M aximizations of lower B ounds on the objective function of the M $\mathcal{K}$ DE. However, differently from EM, these bounds are not exclusively reliant on conditional expectations. The $\mathcal{K}$ -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 $\mathcal{K}$ DE's objective function. The proposed criterion, called $\mathcal{K}$ -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 $\mathcal{K}$ -BM, the $\mathcal{K}$ -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 $\mathcal{K}$ -divergence based framework over other robust approaches are illustrated in simulation studies involving synthetic and real data.

中文翻译：

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鲁棒高斯混合建模：基于$\mathcal{K}$-散度的方法


本文解决了存在异常值的情况下鲁棒高斯混合建模的问题。我们首先引入一个类似于期望最大化（EM）的通用方案，称为$\mathcal{K}$ -BM，用于最小值的迭代数值计算$\mathcal{K}$ -散度估计器（M $\mathcal{K}$德）。该估计器利用 Parzen 的非参数$\mathcal{K}$内核密度估计，以减轻与外围测量相关的低密度区域的权重。与传统的 EM 类似， $\mathcal{K}$ -BM 涉及 M $\mathcal{K}$ DE 目标函数下界的连续 M 最大化。然而，与新兴市场不同的是，这些界限并不完全依赖于条件预期。 $\mathcal{K}$ -BM 算法应用于有限阶多元高斯混合模型 (GMM) 的鲁棒参数估计。我们继续引入贝叶斯信息准则（BIC）的一个新的鲁棒变体，它惩罚 M $\mathcal{K}$ DE 的目标函数。所提出的标准称为 $\mathcal{K}$ -BIC，可方便地应用于稳健的 GMM 阶次选择。在本文中，我们还建立了一个数据驱动的程序来选择内核的带宽参数。该过程通过最小化基础密度和估计 GMM 密度之间的均方积分误差 (MISE) 的经验渐近近似来进行操作。 最后，将 $\mathcal{K}$ -BM、$\mathcal{K}$ -BIC 和基于 MISE 的核带宽选择组合成一个统一的框架，用于 GMM 的联合阶数选择和参数估计。涉及合成数据和真实数据的模拟研究说明了基于 $\mathcal{K}$ 散度的框架相对于其他稳健方法的优势。