Bosteels, Klaas

Kerre, Etienne

English

Ghent University Academic Bibliography

On a reflexivity-preserving family ofcardinality-based fuzzy comparison measuresKlaas Bosteels ∗, Etienne E. KerreFuzziness and Uncertainty Modelling Research GroupDepartment of Applied Mathematics and Computer ScienceGhent University, Krijgslaan 281 (S9), B-9000 Gent, BelgiumAbstractIn this paper, we generalize and extend previous work on a triparametric familyof cardinality-based fuzzy similarity measures. More specifically, we (i) generalizethis family to a broader family of fuzzy comparison measures, (ii) summarize previ-ously obtained results regarding this family and fill up a few gaps by adding somenovel results, (iii) compare our parametric family with an already existing one, and(iv) elaborate on the ranking invariance between certain members of the presentedtriparametric family of cardinality-based fuzzy comparison measures.Key words: Fuzzy sets, Comparison, Compatibility, Inclusion, Resemblance,Similarity, Reflexivity, Ranking invariance1 IntroductionMany applications in information sciences require a way to compare fuzzy sets.In particular, a measure to determine the similarity between two given fuzzysets is a key component of many algorithms in this domain (e.g. [1,13,16,19]).Since different applications have different needs, it makes sense to study sys-tematic ways of constructing and analyzing such measures. Motivated by thisinsight, we presented a systematic way of generating cardinality-based fuzzysimilarity measures in [4]. More precisely, we introduced a general form thatdepends on two parameters, together with constraints on these parameters∗ Corresponding author. Tel.: +32-9-264-9639; fax: +32-9-264-4995.Email addresses: klaas.bosteels@ugent.be (Klaas Bosteels),etienne.kerre@ugent.be (Etienne E. Kerre).Preprint submitted to Elsevier 12 February 2009that ensure various properties of the generated measures. Later on, we gener-alized this biparametric family of cardinality-based fuzzy similarity measuresto a triparametric one, and provided rewritten versions of the previously intro-duced constraints, as well as several additional constraints [6]. In this paper,we further generalize the triparametric family to a broader family of fuzzycomparison measures, and we summarize previously obtained results and con-straints. Moreover, we fill up a few gaps by adding some novel constraints,and we compare the presented triparametric family with the parametric fam-ily of cardinality-based fuzzy comparison measures introduced in [12]. In par-ticular, we provide both theoretical and empirical evidence showing that thereflexivity-preserving nature of our parametric family can be considered anadvantage over the alternative one. In order to explain the somewhat surpris-ing results of the experiment that led to the empirical evidence for this claim,we also take a closer look at the ranking invariance between certain membersof the presented family of fuzzy comparison measures.2 Preliminaries2.1 Basic notationsIn the remainder of this paper, U always denotes a nonempty finite universe.We use the notation F(U) for the class of fuzzy sets in U , i.e., the (crisp)set of all U → [0, 1] mappings. For two fuzzy sets A and B in U , we writeA ⊆ B iff A(u) ≤ B(u) for all u ∈ U , and A = B iff A ⊆ B ∧ B ⊆ A.Furthermore, we let suppA denote the support of a fuzzy set A in U , i.e.,suppA = {u ∈ U | A(u) > 0}, and we define the cardinality |A| of a fuzzyset A in U as |A| = ∑u∈U A(u). Note also that we reuse the name of a crispset for its characteristic mapping. In particular, U denotes both the universeand the mapping that associates 1 with each element from the universe, and∅ denotes the empty set as well as the U → [0, 1] mapping given by ∅(u) = 0,for all u ∈ U .2.2 Conjunctors, copulas, and t-normsA conjunctor is an increasing [0, 1]2 → [0, 1] mapping C that satisfies C(0, 0) =C(0, 1) = C(1, 0) = 0 and C(1, 1) = 1, i.e., that coincides with the Booleanconjunction on {0, 1}2. The following potential properties of a conjunctor Care interesting:C(x, 1) = C(1, x) = x (1 neutral),2Only the first pages of this paper are presented here because of copyright restrictions. If you would like to obtain the full text, please e­mail to Klaas.Bosteels@UGent.be.

On a reflexivity-preserving family of cardinality-based fuzzy comparison measures

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On a reflexivity-preserving family of cardinality-based fuzzy comparison measures

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