Variable Selection in Nonlinear Principal Component Analysis

Abstract

Principal components analysis (PCA) is a popular dimension reduction method and is applied to analyze quantitative data. For PCA to qualitative data, nonlinear PCA can be applied, where the data are quantified by using optimal scaling that nonlinearly transforms qualitative data into quantitative data. Then nonlinear PCA reveals nonlinear relationships among variables with different measurement levels. Using this quantification, we can consider variable selection in the context of PCA for qualitative data. In PCA for quantitative data, modified PCA (M.PCA) of Tanaka and Mori derives principal components which are computed as a linear combination of a subset of variables but can reproduce all the variables very well. This means that M.PCA can select a reasonable subset of variables with different measurement levels if it is extended so as to deal with qualitative data by using the idea of nonlinear PCA. A nonlinear M.PCA is therefore proposed for variable selection in nonlinear PCA. The method, in this chapter, is based on the idea in “Nonlinear Principal Component Analysis and its Applications” by Mori et al. (Springer). The performance of the method is evaluated in a numerical example.