psychometrika—vol. 83, no. 1, 1–20
FUNCTIONAL PARALLEL FACTOR ANALYSIS FOR FUNCTIONS OF ONE- AND
Ji Yeh Choi and Heungsun Hwang
Marieke E. Timmerman
UNIVERSITY OF GRONINGEN
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-andy-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
Key words: three-way data, parallel factor analysis, functional data analysis, spatial and temporal variation.
Three-way data consist of three different types of entities simultaneously (e.g., subjects,
variables, and occasions), each of which is also called a mode. A few examples of three-way data
include neuroimaging data arranged as an array of brain locations by subjects by time points (e.g.,
Cox, 1996; Germond, Dojat, Taylor, & Garbay, 2000; Thirion & Faugeras, 2003), multivariate
longitudinal data as an array of subjects by variables by occasions (e.g., Kroonenberg, 1987; Kuze
et al., 1985;Oort,2001), and gene expression data as an array of genes by subjects by time points
(Holter, Maritan, Cieplak, Fedoroff, & Banavar, 2001; Xiao, Wang, & Khodursky, 2011).
Parallel factor analysis (PARAFAC) (Harshman, 1970), or equivalently canonical decompo-
sition (CANDECOMP) (Carroll & Chang, 1970), is a well-known method for explaining vari-
ations in three-way data. PARAFAC aims to extract trilinear components, each of which is a
low-dimensional representation of a mode, in such a way that they retain the maximum variance
of the data. The number of the extracted components is the same across all modes. It generally
provides unique solutions up to permutation, reﬂection and scaling of the components, under
some mild conditions (e.g., Bro, 1997; Kruskal, 1977; Stegeman & Sidiropoulos, 2007) such that
there exist no alternative ways to decompose the data that yield the same model ﬁt. Thus, any
post-processing technique (i.e., rotation) is unnecessary for the PARAFAC solution.
The authors wish to thank Jelena Ristic for her constructive comments on the analysis of EEG data.
The MATLAB code used in this paper is available upon request from the author Ji Yeh Choi.
Correspondence should be made to Ji Yeh Choi and Heungsun Hwang, Department of Psychology, McGill University,
1205 Dr. Penﬁeld Avenue, Montreal, QC H3A 1B1 Canada. Email: firstname.lastname@example.org
© 2017 The Psychometric Society