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Functional Parallel Factor Analysis for Functions of One- and Two-dimensional Arguments

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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.

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Authors and Affiliations

  1. Department of Psychology, McGill University, 1205 Dr. Penfield Avenue, Montreal, QC, H3A 1B1 , Canada

    Ji Yeh Choi & Heungsun Hwang

  2. University of Groningen, Groningen, The Netherlands

    Marieke E. Timmerman

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  1. Ji Yeh Choi
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  2. Heungsun Hwang
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Correspondence to Ji Yeh Choi.

Additional information

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.

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Choi, J.Y., Hwang, H. & Timmerman, M.E. Functional Parallel Factor Analysis for Functions of One- and Two-dimensional Arguments. Psychometrika 83, 1–20 (2018). https://doi.org/10.1007/s11336-017-9558-9

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