Treating fuzziness in subjective evaluation data

Abstract

This paper proposes a technique to deal with fuzziness in subjective evaluation data, and applies it to principal component analysis and correspondence analysis. In the existing method, or techniques developed directly from it, fuzzy sets are defined from some standpoint on a data space, and the fuzzy parameters of the statistical model are identified with linear programming or the method of least squares. In this paper, we try to map the variation in evaluation data into the parameter space while preserving information as much as possible, and thereby define fuzzy sets in the parameter space. Clearly, it is possible to use the obtained fuzzy model to derive things like the principal component scores from the extension principle. However, with a fuzzy model which uses the extension principle, the possibility distribution spreads out as the explanatory variable values increase. This does not necessarily make sense for subjective evaluations, such as a 5-level evaluation, for instance. Instead of doing so, we propose a method for explicitly expressing the vagueness of evaluation, using certain quantities related to the eigenvalues of a matrix which specifies the fuzzy parameter spread. As a numerical example, we present an analysis of subjective evaluation data on local environments.