Multifactor Analysis Based on Factor-Dependent Geometry

Abstract

This paper proposes a novel method that preserves the geometrical structure created by variation of multiple factors in analysis of multiple factor models, i.e., multifactor analysis. We use factor-dependent submanifolds as constituent elements of the factor-dependent geometry in a multiple factor framework. In this paper, a submanifold is defined as some subset of a manifold in the data space, and factor-dependent submanifolds are defined as the submani-folds created for each factor by varying only this factor. In this paper, we show that MPCA is formulated using factor-dependent submanifolds, as is our proposed method. We show, however, that MPCA loses the original shapes of these submanifolds because MPCA's parameterization is based on averaging the shapes of factor-dependent subman-ifolds for each factor. On the other hand, our proposed multifactor analysis preserves the shapes of individual factor-dependent submanifolds in low-dimensional spaces. Because the parameters obtained by our method do not lose their structures, our method, unlike MPCA, sufficiently covers original factor-dependent submanifolds. As a result of sufficient coverage, our method is appropriate for accurate classification of each sample.