Fuzzy clustering and dimensionality reduction of a three-way data matrix

A three-way three-mode data array X, where modes are units, variables, and occasions, is the data structure that can comprehensively and statistically analyze a collective phenomenon. When X has a large dimension, it is important to synthesize its information by identifying classes of similar occasions where units are described by a reduced set of LVs. In this paper, a simultaneous reduction of the occasions and variables of X is proposed. A fuzzy clustering of the occasions allows the identification of <i>K</i> clusters of multivariate data matrices that are within-cluster perceived similar. For each cluster, a consensus matrix with respect to the units is identified. Variables in the cluster are correlated and maintain their covariance structure that can be synthesized for each consensus matrix by applying a Second-Order Disjoint Factor Analysis. The proposal allows therefore to softly cluster occasions into <i>K</i> clusters and, for each consensus matrix, firstly identify a set of <i>Q</i> first-order factors and secondly identify a unique general factor, which can be considered as the most synthetic indicator summarizing the original <i>J</i> variables. The performance of the methodology is tested through a detailed simulation study. Finally, it is also applied to a real dataset, where its strength and usefulness are revealed.