使用多元斜正态分布的尺度混合对随机缺失值数据进行聚类
Clustering data with values missing at random using scale mixtures of multivariate skew-normal distributions
AI总结：
针对含偏态、厚尾特征的随机缺失数据聚类难题，扩展FMSMSN族并提出适配的增广EM类算法，经仿真验证性能后应用于全球CO₂排放数据，兼具灵活性与实用性。
摘要：
在基于模型的聚类中，处理缺失数据是一项重大挑战，尤其是当数据呈现偏态和厚尾特征时。我们通过扩展有限混合多元斜正态尺度混合（FMSMSN，finite mixture of scale mixtures of multivariate skew-normal）族，以适配随机缺失（MAR，missing at random）机制下的不完整数据，从而解决这一问题。\n以往的研究仅局限于FMSMSN族的某一特殊情况，而我们的方法为整个族提供了一套聚类分析方法，能够在存在缺失值的数据中兼顾偏态和超额峰度的影响。由\textit{azzalini1996}和\textit{arnoldbeaver}参数化的多元斜正态分布将正态分布作为特殊情况包含在内，这确保了我们的方法能够灵活适配正态性假设下现有的对称型基于模型的聚类技术。\n我们推导了数据中缺失分量的分布特性，并提出了一种针对不完整观测的增广EM类算法。改进后的E步给出了缺失值条件期望的闭式表达式。仿真实验表明，在不同缺失值比例下，FMSMSN族在聚类性能和参数恢复方面均展现出灵活性，同时实验还纳入了样本量和簇间邻近度的影响。最后，我们将FMSMSN族的特殊情况应用于全球CO₂排放数据，验证了所提方法的实际效用。
英文摘要：
Handling missing data is a major challenge in model-based clustering, especially when the data exhibit skewness and heavy tails. We address this by extending the finite mixture of scale mixtures of multivariate skew-normal (FMSMSN) family to accommodate incomplete data under a missing at random (MAR) mechanism. Unlike previous work that is limited to one of the special cases of the FMSMSN family, our method offers a cluster analysis methodology for the entire family that accounts for skewness and excess kurtosis amidst data with missing values. The multivariate skew-normal distribution, as parameterised by \cite{azzalini1996} and \cite{arnoldbeaver} includes the normal distribution as a special case, which ensures that our method is flexible toward existing symmetric model-based clustering techniques under a normality assumption. We derive the distributional properties of the missing components of the data and propose an augmented EM-type algorithm tailored for incomplete observations. The modified E-step yields closed-form expressions for the conditional expectations of the missing values. The simulation experiments showcase the flexibility of the FMSMSN family in both clustering performance and parameter recovery for varying percentages of missing values, while incorporating the effects of sample size and cluster proximity. Finally, we illustrate the practical utility of the proposed method by applying special cases of the FMSMSN family to global CO2 emissions data.
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