The Analysis of Two-Way Functional Data Using Two-Way Regularized Singular Value Decompositions

Two-way functional data consist of a data matrix whose row and column domains are both structured, for example, temporally or spatially, as when the data are time series collected at different locations in space. We extend one-way functional principal component analysis (PCA) to two-way functional data by introducing regularization of both left and right singular vectors in the singular value decomposition (SVD) of the data matrix. We focus on a penalization approach and solve the non-trivial problem of constructing proper two-way penalties from one-way regression penalties. We introduce conditional crossvalidated smoothing parameter selection whereby left-singular vectors are crossvalidated conditional on right-singular vectors, and vice versa. The concept can be realized as part of an alternating optimization algorithm. In addition to the penalization approach, we briefly consider two-way regularization with basis expansion. The proposed methods are illustrated with one simulated and two Jianhua Z. Huang is Professor (Email: [email protected]), Department of Statistics, Texas A&M University, College Station, TX 77843. Haipeng Shen (Email: [email protected]) is Assistant Professor, Department of Statistics and Operations Research, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599. Andreas Buja (Email: [email protected]) is Liem Sioe Liong/First Pacific Company Professor, Department of Statistics, The Wharton School, University of Pennsylvania, Philadelphia, PA 19104. Jianhua Z. Huang’s work was partially supported by NSF grant DMS-0606580, NCI grant CA57030, and Award Number KUS-CI-016-04, made by King Abdullah University of Science and Technology (KAUST). Haipeng Shen’s work was partially supported by NSF grant DMS-0606577, CMMI-0800575, and UNC-CH R. J. Reynolds Fund Award for Junior Faculty Development.