A Data-Adaptive Principal Component Analysis: Use of Composite Asymmetric Huber Function

This article considers a new type of principal component analysis (PCA) that adaptively reflects the information of data. The ordinary PCA is useful for dimension reduction and identifying important features of multivariate data. However, it uses the second moment of data only, and consequently, it is not efficient for analyzing real observations in the case that these are skewed or asymmetric data. To extend the scope of PCA to non-Gaussian distributed data that cannot be well represented by the second moment, a new approach for PCA is proposed. The core of the methodology is to use a composite asymmetric Huber function defined as a weighted linear combination of modified Huber loss functions, which replaces the conventional square loss function. A practical algorithm to implement the data-adaptive PCA is discussed. Results from numerical studies including simulation study and real data analysis demonstrate the promising empirical properties of the proposed approach. Supplementary materials for this article are available online.