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.

以个体（units）、变量（variables）与测量场合（occasions）为模态的三维三模态数据阵列X，是可对集体现象开展全面统计分析的数据结构。当X维度较高时，通过识别相似场合的类别以综合其信息至关重要，此类场合下个体可通过精简后的潜变量（Latent Variables, LVs）集合进行描述。本文提出了一种可同时对X的测量场合与变量进行降维的方法。通过对测量场合实施模糊聚类，可识别出K个簇，每个簇内的多元数据矩阵均具备类内相似性。针对每个簇，可识别出关于个体的共识矩阵。该簇内的变量相互关联且保持协方差结构，可通过二阶不相交因子分析（Second-Order Disjoint Factor Analysis）为每个共识矩阵综合该结构。因此，本方法可将测量场合软聚类为K个簇；针对每个共识矩阵，首先识别出Q个一阶因子，其次确定唯一的通用因子，该通用因子可作为概括原始J个变量的最具综合性的指标。本文通过详尽的仿真研究验证了该方法的性能。最后，本方法还被应用于真实数据集，其优势与应用价值得以验证。