A dimension reduction technique for two-mode non-convex fuzzy data

Fuzzy modeling and fuzzy statistics provide useful tools for handling empirical situations affected by vagueness and imprecision in the data. Several fuzzy statistical models and methods (e.g., fuzzy regression, fuzzy principal component analysis, fuzzy clustering) have been developed over the years. Generally the standard LR-fuzzy data representation has been used in these methods. However, several empirical contexts, such as human ratings and decision making, may show more complex fuzzy structures which cannot be successfully modeled by the LR representation. In all these cases another type of fuzzy data representation, the so-called LHIR representation, should be preferred instead. In particular, this novel representation allows to handle with fuzzy data which are characterized by non-convex membership functions. In this paper, we address the problem of summarizing large datasets characterized by two-mode non-convex fuzzy data. We introduce a novel dimension reduction technique (NCFCA) based on the framework of Component Analysis and Least squares programming. Finally, to better highlight some important characteristics of the proposed model, we apply NCFCA to three empirical datasets concerning behavioral and socio-economic issues.