Sparse principal component regression for generalized linear models

Abstract

Principal component regression (PCR) is a widely used two-stage procedure: principal component analysis (PCA), followed by regression in which the selected principal components are regarded as new explanatory variables in the model. Note that PCA is based only on the explanatory variables, so the principal components are not selected using the information on the response variable. In this paper, we propose a one-stage procedure for PCR in the framework of generalized linear models. The basic loss function is based on a combination of the regression loss and PCA loss. An estimate of the regression parameter is obtained as the minimizer of the basic loss function with a sparse penalty. We call the proposed method sparse principal component regression for generalized linear models (SPCR-glm). Taking the two loss function into consideration simultaneously, SPCR-glm enables us to obtain sparse principal component loadings that are related to a response variable. However, a combination of loss functions may cause a parameter identification problem, but this potential problem is avoided by virtue of the sparse penalty. Thus, the sparse penalty plays two roles in this method. The parameter estimation procedure is proposed using various update algorithms with the coordinate descent algorithm. We apply SPCR-glm to two real datasets, doctor visits data and mouse consomic strain data. SPCR-glm provides more easily interpretable principal component (PC) scores and clearer classification on PC plots than the usual PCA.