Candecomp/Parafac with zero constraints at arbitrary positions in a loading matrix

Henk A.L Kiers,Paolo Giordani

doi:10.1016/j.chemolab.2020.104145

Abstract

When one interprets Candecomp/Parafac (CP) solutions for analyzing three-way data, small loadings are often ignored, that is, considered to be zero. Rather than just considering them zero, it seems better to actually model such values as zero. This can be done by successive modeling approaches as well as by a simultaneous modeling approach. This paper offers algorithms for three such approaches, and compares them on the basis of empirical data and a simulation study. The conclusion of the latter was that, under realistic circumstances, all approaches recovered the underlying structure well, when the number of values to constrain to zero was given. Whereas the simultaneous modeling approach seemed to perform slightly better, differences were very small and not substantial. Given that the simultaneous approach is far more time consuming than the successive approaches, the present study suggests that for practical purposes successive approaches for modeling zeros in the CP model seem to be indicated.

Highlights

For analyses using the correct number of zero loadings, all individual results for the recovery of A-mode loadings and the binary weights, as well as the fit percentages and numbers of local optima were plotted on one screen for each of the 80 conditions for quick visual inspection
This indicated that fit percentages were as expected, numbers of local optima for CP_Simult varied much, and recovery results often showed a big majority of fairly equal results and a small set of clearly poorer results
The explanation is that, in case of degeneracy, loadings on two components tend to become high in absolute sense, and as a consequence the loadings on the third may often all be lower than the other ones

Summary

Introduction

The idea is to find a number (R) of components that jointly represent the data well. Let X denote the (preprocessed) three-way data, with elements xijk for i 1⁄4 1, ...,I, j 1⁄4 1,..,J and k 1⁄4 1, ...,K, where I, J and K denote the number of units for the three modes, labelled modes A, B and C, respectively. R1⁄41 where air, bjr, and ckr denote loadings for units from each of the three modes, on component r, r 1⁄4 1, ...,R, and eijk denotes the residual or error terms. The idea of fitting the model to data is that the R components jointly give a simple summary of assessed by means of the the data. In sensory analysis, the data may represent scores of I experts (mode A), on J attributes (mode B) of K products

Objectives

Results

Conclusion