Robust Principal Components by Casewise and Cellwise Weighting

Principal component analysis (PCA) is a fundamental tool for analyzing multivariate data. Here the focus is on dimension reduction to the principal subspace, characterized by its projection matrix. The classical principal subspace can be strongly affected by the presence of outliers. Traditional robust approaches consider casewise outliers, that is, cases generated by an unspecified outlier distribution that differs from that of the clean cases. But there may also be cellwise outliers, which are suspicious entries that can occur anywhere in the data matrix. Another common issue is that some cells may be missing. This paper proposes a new robust PCA method, called cellPCA, that can simultaneously deal with casewise outliers, cellwise outliers, and missing cells. Its single objective function combines two robust loss functions, that together mitigate the effect of casewise and cellwise outliers. The objective function is minimized by an iteratively reweighted least squares (IRLS) algorithm. Residual cellmaps and enhanced outlier maps are proposed for outlier detection. The casewise and cellwise influence functions of the principal subspace are derived, and its asymptotic distribution is obtained. Extensive simulations and two real data examples illustrate the performance of cellPCA.主成分分析（Principal Component Analysis，PCA）是分析多元数据的基础性工具。本文的研究重点为将数据降维至主成分子空间，该子空间以其投影矩阵为核心特征。经典主成分子空间易受异常值的显著影响。传统鲁棒方法多关注逐样本异常值，即由与干净样本分布不同的未知异常值分布所生成的样本。但数据矩阵中也可能存在逐单元格异常值，即可出现在数据矩阵任意位置的可疑数值项。另一常见问题是部分单元格存在缺失值。本文提出一种新型鲁棒主成分分析方法cellPCA，可同时处理逐样本异常值、逐单元格异常值与缺失单元格。该方法的单一目标函数结合了两种鲁棒损失函数，二者协同削弱逐样本与逐单元格异常值的影响。目标函数通过迭代重加权最小二乘（Iteratively Reweighted Least Squares，IRLS）算法实现最小化求解。本文还提出残差单元格图与增强型异常值图用于异常值检测。推导得到主成分子空间的逐样本与逐单元格影响函数，并获取其渐近分布。大量仿真实验与两个真实数据案例验证了cellPCA的性能表现。