Three‐way clustering around latent variables approach with constraints on the configurations to facilitate interpretation

Abstract

The set‐up of comprehensive studies in life sciences involving a longitudinal dimension—as appears in time‐scale metabolomics—calls for the use of dimension reduction techniques for three‐way data structures (e.g., samples by variables by time points). For this purpose, a clustering around latent variables for three‐way data approach, CLV3W, has been proposed. CLV3W aims at both partitioning the variables into nonoverlapping clusters and estimating within each cluster a rank‐one Parafac model consisting of a latent component (resp. a weighting system) associated with the first mode (resp. third mode) and a vector of loadings reflecting the degree of closeness of each variable of the second mode to its cluster. In this paper, two constrained CLV3W models are discussed. First, a nonnegativity constraint is defined implying that clusters are composed of positively correlated variables. Second, it is proposed to constrain the weighting system to be the same for all clusters. These two constraints aim at providing more parsimonious models with configurations that are easier to interpret. The appropriateness of both constraints is evaluated in a simulation study and illustrated on two case studies pertaining to sensory evaluation and metabolomics data. Regarding the first case study, CLV3W yields the identification of two consumer segments together with one common emotional pleasantness dimension associated with coffee aromas. CLV3W analysis of human preterm breast milk metabolomics data provided three clusters of lipid species that are responsible for specific functions (i.e., milk fat globules membrane‐constituents, fatty acid oxidation‐products, lipid mediators as eicosanoids and endocannabinoids).

Least-squares bilinear clustering of three-way data

“…Whereas these rely on maximum likelihood estimation, another research stream have focused on nonparametric data approximation methods mainly via least-squares estimation. The papers of Vichi (1999), Rocci and Vichi (2005), Vichi et al (2007), Papalexakis et al (2013), Wilderjans and Ceulemans (2013), Llobell et al (2019), Llobell et al (2020) and Cariou et al (2021) belong to this stream, as does our paper. An approach often used in this stream is to generalize a three-way data decomposition, such as Tucker's three-mode factor analysis (Tucker3; Tucker 1966) or the Candecomp/Parafac decomposition (CP; Carroll and Chang 1970;Harshman 1970;Hitchcock 1927), by adding clustering over one or more of the modes.…”

“…In the social sciences, a marketing research survey asking multiple individuals to rate several products on various characteristics using a Likert scale generates a three-way array of rating scores (e.g., DeSarbo et al 1982). Similar research designs are commonly used in sensometrics (e.g., Cariou et al 2021). Other applications include: high-throughput molecular data in bioinformatics (e.g., Lonsdale et al 2013); spectroscopic data in chemometrics (e.g., Faber et al 2003;Bro 2006); and neuroimaging data, collected using electroencephalography (EEG) or functional magnetic resonance imaging (fMRI), for example (e.g., Genevsky and Knutson 2015).…”