Principal Component Analysis in an Asymmetric Norm

Abstract

Principal component analysis (PCA) is a widely used dimension reduction tool in the analysis of high-dimensional data. However, in many applications such as risk quantification in finance or climatology, one is interested in capturing the tail variations rather than variation around the mean. In this paper, we develop Principal Expectile Analysis (PEC), which generalizes PCA for expectiles. It can be seen as a dimension reduction tool for extreme value theory, where one approximates fluctuations in the τ-expectile level of the data by a low dimensional subspace. We provide algorithms based on iterative least squares, prove upper bounds on their convergence times, and compare their performances in a simulation study. We apply the algorithms to a Chinese weather dataset and fMRI data from an investment decision study.