Principal Components Regression by using Generalized Principal Components Analysis

Abstract

Principal components analysis (PCA) is one method for reducing the dimension of the explanatory variables, although the principal components are derived by using all the explanatory variables. Several authors have proposed a modified PCA (MPCA), which is based on using only selected explanatory variables in order to obtain the principal components (see e.g., Jolliffie, 1972, 1986; Robert and Escoufier, 1976; Tanaka and Mori, 1997). However, MPCA uses all of the selected explanatory variables to obtain the principal components. There may, therefore, be extra variables for some of the principal components. Hence, in the present paper, we propose a generalized PCA (GPCA) by extending the partitioning of the explanatory variables. In this paper, we estimate the unknown vector in the linear regression model based on the result of a GPCA. We also propose some improvements in the method to reduce the computational cost.