title: Functional Parallel Factor Analysis for Functions of One- and Two-dimensional Arguments.
abstract: Parallel factor analysis (PARAFAC) is a useful multivariate method for decomposing three-way data that consist of three different types of entities simultaneously. This method estimates trilinear components, each of which is a low-dimensional representation of a set of entities, often called a mode, to explain the maximum variance of the data. Functional PARAFAC permits the entities in different modes to be smooth functions or curves, varying over a continuum, rather than a collection of unconnected responses. The existing functional PARAFAC methods handle functions of a one-dimensional argument (e.g., time) only. In this paper, we propose a new extension of functional PARAFAC for handling three-way data whose responses are sequenced along both a two-dimensional domain (e.g., a plane with x- and y-axis coordinates) and a one-dimensional argument. Technically, the proposed method combines PARAFAC with basis function expansion approximations, using a set of piecewise quadratic finite element basis functions for estimating two-dimensional smooth functions and a set of one-dimensional basis functions for estimating one-dimensional smooth functions. In a simulation study, the proposed method appeared to outperform the conventional PARAFAC. We apply the method to EEG data to demonstrate its empirical usefulness.
authors.affiliation: Department of Psychology, McGill University, 1205 Dr. Penfield Avenue, Montreal, QC, H3A 1B1 , Canada. ji.yeh.choi@mail.mcgill.ca.
authors.forename: Ji Yeh
authors.initials: JY
authors.lastname: Choi
authors.affiliation: Department of Psychology, McGill University, 1205 Dr. Penfield Avenue, Montreal, QC, H3A 1B1 , Canada.
authors.forename: Heungsun
authors.initials: H
authors.lastname: Hwang
authors.affiliation: University of Groningen, Groningen, The Netherlands.
authors.forename: Marieke E
authors.initials: ME
authors.lastname: Timmerman
mesh_terms: D000465:Algorithms; D001921:Brain / Q000502:physiology; D003198:Computer Simulation; D003627:Data Interpretation, Statistical; D004569:Electroencephalography; D005163:Factor Analysis, Statistical; D006801:Humans; D016018:Least-Squares Analysis; D010364:Pattern Recognition, Visual / Q000502:physiology; D021641:Recognition, Psychology / Q000502:physiology; D012984:Software
keywords: functional data analysis; parallel factor analysis; spatial and temporal variation; three-way data