Principal component analysis in an asymmetric norm

Abstract

Principal component analysis (PCA) is a widely used dimension reduction tool in high-dimensional data analysis. In risk quantification in finance, climatology and many other applications, however, the interest lies in capturing the tail variations rather than variation around the mean. To this end, 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 fluctuations in the τ-expectile level of the data are approximated by a low-dimensional subspace. We provide algorithms based on iterative least squares, derive bounds on their convergence time, and compare their performance through simulations. We apply the algorithms to a Chinese weather dataset and fMRI data from an investment decision study.