Statistica Sinica 24 (2014), 463-492

DIMENSION FOLDING PCA AND PFC FOR MATRIX-

VALUED PREDICTORS

Shanshan Ding and R. Dennis Cook

University of Minnesota

Abstract: Conventional dimension reduction methods deal mainly with simple data structure and are inappropriate for data with matrix-valued predictors. Li, Kim and Altman (2010) proposed dimension folding methods that effectively improve major moment-based dimension reduction techniques for the more complex data structure. Their methods, however, are moment-based and rely on slicing the responses to gain information about the conditional distribution of $X\vert Y$. This can be inadequate when the number of slices is not chosen properly. We propose model-based dimension folding methods that can be treated as extensions of conventional principal components analysis (PCA) and principal fitted components (PFC). We refer to them as dimension folding PCA and dimension folding PFC. The proposed methods can simultaneously reduce a predictor's multiple dimensions and inherit asymptotic properties from maximum likelihood estimation. Dimension folding PFC gains further efficiency by effective use of the response information. Both methods can provide robust estimation and are computationally efficient. We demonstrated their advantages by both simulation and data analysis.

Key words and phrases: Central dimension folding subspace, central subspace, inverse regression, matrix normal distribution, sufficient dimension reduction.