  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.
