Abstract
Purpose The paper is dedicated to the analysis of fuzzy similarity measures in uncertainty analysis in general, and in economic decision-making in particular. The purpose of this paper is to explain how a similarity measure can be chosen to quantify a qualitative description of similarities provided by experts of a given domain, in the case where the objects to compare are described through imprecise or linguistic attribute values represented by fuzzy sets. The case of qualitative dissimilarities is also addressed and the particular case of their representation by distances is presented. Design/methodology/approach The approach is based on measurement theory, following Tversky’s well-known paradigm. Findings A list of axioms which may or may not be satisfied by a qualitative comparative similarity between fuzzy objects is proposed, as extensions of axioms satisfied by similarities between crisp objects. They enable to express necessary and sufficient conditions for a numerical similarity measure to represent a comparative similarity between fuzzy objects. The representation of comparative dissimilarities is also addressed by means of specific functions depending on the distance between attribute values. Originality/value Examples of functions satisfying certain axioms to represent comparative similarities are given. They are based on the choice of operators to compute intersection, union and difference of fuzzy sets. A simple application of this methodology to economy is given, to show how a measure of similarity can be chosen to represent intuitive similarities expressed by an economist by means of a quantitative measure easily calculable. More detailed and formal results are given in Coletti and Bouchon-Meunier (2020) for similarities and Coletti et al. (2020) for dissimilarities.
Highlights
The purpose of this paper is to revisit results introduced in Coletti and Bouchon-Meunier (2019a) and developed in Coletti and Bouchon-Meunier (2020) in a more intuitive approach, enabling the reader to grasp the meaning of the formal results based on measurement theory with the aim of making a conscious choice between several possible measures
We identify the requirements underlying the choice of a fuzzy similarity measure on the basis of measurement theory, in the spirit of the formalisation of classic similarities given in Tversky (1977)
We consider that a dissimilarity measure D:Y2 ! R represents the comparative dissimilarity 40 if and only if, for all (X, Y), (X0, Y0) [ Y2, it holds that ðX; Y Þ40ðX0; Y 0Þ ) DðX; YÞ # DðX0; Y 0Þ; ðX; Y Þ 00 ðX0; Y0Þ ) DðX; Y Þ < DðX0; Y 0Þ: With the purpose of characterising dissimilarity measures able to represent a given qualitative comparative dissimilarity, we propose a list of axioms that may or may not be satisfied by a dissimilarity measure, and that the expert providing the comparative dissimilarity can consider natural or not
Summary
The purpose of this paper is to revisit results introduced in Coletti and Bouchon-Meunier (2019a) and developed in Coletti and Bouchon-Meunier (2020) in a more intuitive approach, enabling the reader to grasp the meaning of the formal results based on measurement theory with the aim of making a conscious choice between several possible measures. We limit our research of an appropriate similarity measure to the general class introduced by Tversky (1977) in the framework of measurement theory In this setting, the similarity S(X, Y) between two objects X and Y depends on the evaluation of elements common to X and Y, and elements pertaining to only one of them. Axiom FS4G (partial attribute stability) claims that the comparative degree of similarity of a pair of objects does not change if we slightly alter the values of membership of two attributes, the alteration being limited according to the difference between their values. Axiom FS72 (weak monotonicity) considers that the greater the measure of the intersection and the smaller the measure of the attributes present in only one of the objects, the higher their comparative similarity. It can be noted that S2, S3 and S4 are examples of similarity measure S2
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