Correspondence Analysis for Fuzzy Data (CAFD): The practical application; Representative example with defuzzified-geometrical display.

Abstract

This paper is the continuation of our previous works (Theodorou and Alevizos, 2006; Theodorou et al., 2007) where Correspondence Analysis (CA) for fuzzy data was introduced, and the mathematical foundation for this fuzzy extension was investigated. Since algebraically CA is mainly an eigendecomposition problem, a two-step method was developed there for the theoretical treatment of the fuzzy eigenvalue problem, when CA is applied on a contingency table with fuzzy data (in short, CAFD).

In this work, we complete these theoretical results with the practical approach of implementing CAFD, i.e., we present the geometrical results of CA on a fuzzy contingency table, where its entries-data are fuzzy numbers. A real problem is considered and the employed computational processes are explained in detail through a simple numerical but representative example. For the critical geometrical point of view of CAFD, and in order to project the fuzzy profiles on a low dimensional space, a new defuzzification-step is developed. As crisp outputs-projections of CAFD, we indicate certain variants of some usual defuzzification techniques. Finally, on a first simultaneous geometrical display, we are presenting and illustrating the defuzzified-factorial projections of CAFD in comparison with those of standard CA.

The main objective of this paper is to show that CAFD can work geometrically and computationally (and how) in practice. Thus, the last step that remains for the end use of CAFD is the creation of suitable software.