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Finite mixtures of multivariate skew t-distributions: some recent and new results

机译：多元偏斜t分布的有限混合：一些新近的结果

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Finite mixtures of multivariate skew t (MST) distributions have proven to be useful in modelling heterogeneous data with asymmetric and heavy tail behaviour. Recently, they have been exploited as an effective tool for modelling flow cytometric data. A number of algorithms for the computation of the maximum likelihood (ML) estimates for the model parameters of mixtures of MST distributions have been put forward in recent years. These implementations use various characterizations of the MST distribution, which are similar but not identical. While exact implementation of the expectation-maximization (EM) algorithm can be achieved for 'restricted' characterizations of the component skew t-distributions, Monte Carlo (MC) methods have been used to fit the 'unrestricted' models. In this paper, we review several recent fitting algorithms for finite mixtures of multivariate skew t-distributions, at the same time clarifying some of the connections between the various existing proposals. In particular, recent results have shown that the EM algorithm can be implemented exactly for faster computation of ML estimates for mixtures with unrestricted MST components. The gain in computational time is effected by noting that the semi-infinite integrals on the E-step of the EM algorithm can be put in the form of moments of the truncated multivariate non-central t-distribution, similar to the restricted case, which subsequently can be expressed in terms of the non-truncated form of the central t-distribution function for which fast algorithms are available. We present comparisons to illustrate the relative performance of the restricted and unrestricted models, and demonstrate the usefulness of the recently proposed methodology for the unrestricted MST mixture, by some applications to three real datasets.
机译：事实证明，多元偏斜t（MST）分布的有限混合可用于对具有不对称和重尾行为的异构数据进行建模。最近，它们已被用作建模流式细胞仪数据的有效工具。近年来，已经提出了许多用于计算MST分布的混合模型参数的最大似然（ML）估计的算法。这些实现使用MST分布的各种特征，这些特征相似但不相同。虽然可以对组件偏斜t分布的“受限”特征实现期望最大化（EM）算法的精确实现，但蒙特卡罗（MC）方法已用于拟合“不受限制”模型。在本文中，我们回顾了几种针对多元偏斜t分布的有限混合的最近拟合算法，同时阐明了各种现有建议之间的一些联系。特别是，最近的结果表明，EM算法可以准确实现，以更快地计算出具有不受限制的MST成分的混合物的ML估计值。计算时间的增加是通过注意到EM算法的E步上的半无限积分可以采用截断的多元非中心t分布的矩的形式来进行的，类似于受限情况，即随后可以用中央t分布函数的非截断形式表示，为此可以使用快速算法。我们目前进行比较，以说明受限和非受限模型的相对性能，并通过对三个真实数据集的一些应用，证明了最近提出的用于非受限MST混合物的方法的有用性。

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