Fuzzy Clusterwise Functional Extended Redundancy Analysis

Abstract

Functional data refer to data that are assumed to be generated from an underlying smooth function varying over a continuum such as time or space. Functional linear models (FLMs) and functional extended redundancy analysis (FERA) are major regression analysis methods for investigating directional associations between predictor and dependent variables that can be functional In practice, functional data may often arise from heterogeneous subgroups of the population, which involve distinctive directional relationships between predictor and dependent variables. When such cluster-level heterogeneity is present, ignoring this heterogeneity would likely lead to biased statistical inferences. FLMs have been extended to capture cluster-level heterogeneity. Conversely, there has been no attempt to take into account cluster-level heterogeneity in FERA. In this paper, we propose to extend FERA to accommodate cluster-level heterogeneity by combining the method with fuzzy clusterwise regression (FCR) into a unified framework. The proposed method, called fuzzy clusterwise functional extended redundancy analysis (FCFERA), aims to estimate fuzzy memberships of individuals and clusterwise regression coefficient functions at the same time. A penalized least squares criterion is minimized to estimate these parameters by adopting an alternating least squares algorithm in combination with basis function expansions. We conduct simulation studies to investigate the performance of the proposed method. We also apply this method to real data to demonstrate its empirical usefulness.