Similarity and Compatibility in Fuzzy Set Theory

Similarity is perhaps the most frequently used, most difficult to quantify, and the most universally employed type of compatibility measure. Because of the widespread applications, we begin with an examination of previous efforts to develop mathematical techniques for similarity assessment.

Several properties of compatibility discussed in the previous sections may not be relevant to all the fields of study in which compatibility between objects or concepts is used to analyze information and assist in decision making. An understanding the significance of these properties, however, is necessary for building a general classification framework. In this section we present a brief examination of the measurement and use of compatibility in taxonomy, psychology, and statistics. In these disciplines, the focus has been on the similarity of objects and the assessment has generally been based upon the commonality of crisp sets of attributes possessed by the objects. The analysis of crisp sets has often provided the basis for a generalization to measurement of fuzzy sets and we will focus on crisp techniques that have been subsequently applied to the assessment of the compatibility of objects whose properties are described by fuzzy sets.

In classical set theory, referred to as crisp set theory to distinguish it from its generalization to fuzzy set theory, an object is either completely in or completely outside of a set. In the former case, the degree of membership of the object is designated as 1 and as 0 in the latter case. Equivalently, the range of the characteristic function of a crisp set consists of the two values 0 and 1. A fuzzy set is a generalization of a crisp set that allows objects to be partially in a set. The membership function of a fuzzy set provides a degree of membership that can range from 0 to 1. The more the object belongs to the fuzzy set, the higher the degree of membership. This chapter briefly presents the notation and terminology of fuzzy set theory that will be used throughout this book. A thorough introduction to fuzzy set theory may be found in a number of books including [61, 123, 263, 246, 183].

Processing information in fuzzy rule-based systems generally employs one of three patterns of inference: composition, compatibility modification, or interpolative reasoning. Compositional inference originated as a generalization of binary logical deduction to fuzzy logic. Compatibility modification was developed to facilitate the evaluation of rules by separating the evaluation of the input from the generation of the output. The first step in compatibility modification inference is to assess the degree to which the input matches the antecedent of a rule. The result of this assessment is then combined with the consequent of the rule to produce the output. Interpolative and analogical reasoning consider rules to define paradigmatic examples and inference is based on proximity to these examples. Interpolative techniques have been developed to compensate for sparse data and to produce small rule bases. The premises underlying these approaches and their dependence on compatibility assessment are presented in the following sections. Various aspects of the role of compatibility measurement in fuzzy inference have been examined in [53, 70, 247, 26] and their use in fuzzy rule base simplification in [191].

Fuzzy approximate reasoning, or simply approximate reasoning, may be viewed as a collection of techniques for analyzing information and generating inferences when the information under consideration is fuzzy rather than exact or deterministic [260]. Fuzzy rule-based expert systems provide a well known example of approximate reasoning using fuzzy information.

The class of set-theoretic compatibility measures has its roots in the content model of psychology [76] and the binary presence-absence similarity coefficients for both taxonomic and biassociational interpretations [39, 196]. As previously noted, Jaccard’s unparameterized ratio model (Section 3.1) was extended by Tversky’s parameterized ratio model of similarity where X and Y represent objects and f is an additive function on disjoint sets (see Section 3.2). The intersection X ⋂ Y consists of features that are common to both objects, X − Y the features that belong to X but not to Y, and Y − X the features that belong to Y but not to X. The function f is assumed to satisfy feature additivity; f (X ⋃ Y) = f (X) + f(Y) whenever X and Y are disjoint.

Tversky [221] noted that “most theoretical and empirical analyses of similarity assume that objects can be adequately represented as points in some coordinate space and that dissimilarity behaves like a distance function.” While Tversky’s observation concerned objects as crisp values, the notion of proximity defining similarity can also be used to assess the similarity of fuzzy sets. For fuzzy sets, the distance is not between points but rather between membership functions. In this chapter we consider three methods for producing metric based similarity measures.

Logic-based compatibility measures [97, 98] use the interpretation the membership function of a fuzzy set as indicating a degree of truth of a proposition represented by the fuzzy set (see Section 4.4). We will consider two methods for generating compatibility measures from a logical interpretation of membership. The first approach, which has been employed as a basis for inference in fuzzy rule-based expert systems, uses fuzzy truth values to represent the degree of compatibility between two fuzzy sets. The second approach defines an elementwise degree of equality between membership functions of two fuzzy sets.

The assessment of the similarity or compatibility analyzes common features or properties and encapsulates the degree of compatibility in a predetermined format. In the preceding chapters, the measurement of the compatibility of fuzzy sets has been scalar-valued. The summarization of similarity as fuzzy sets rather than single numbers was suggested by Dubois and Prade [63] and is analogous to the use of fuzzy truth values in reasoning with linguistic variables as discussed in the preceding chapter.

The following three sections provide a description of the generic classification system used in this study. The agreement between the domain information and the observation data, both represented by fuzzy sets, is determined by an assessment of the compatibility of the two fuzzy sets. Aggregation operators are used to combine the measures obtained by the assessment of multiple pieces of information.

Table 13.1 lists the proximity-based compatibility measures examined in this study. For each subclass, at least two compatibility measures with different normalization functions are considered. As before, the Description column provides a common name for the compatibility measure or a reference to an application in which it has been used. Several of the proximity-based compatibility measures are equivalent to set-theoretic measures already tested and they are included for completeness. In the tables showing performance results, the G _D measures have an added parameter to the subscript, either univ or supp, that indicates the size of the normalizing distance β used in the dissemblance index. The subscript supp indicates that the smallest interval surrounding the support of both fuzzy sets is used for β. For univ, the cardinality of the universe of the fuzzy sets is used. The distinction between univ and supp was described in Section 8.4. We examine the performance of the proximity-based compatibility measures with T ₃ and G _1,m aggregation. For each subclass, several of the better performing measures are tested with T ₂ as the aggregator.

The subclass of logic-based compatibility measures built using inverse truth functional modification produces a fuzzy truth value that indicates the truth of one proposition given another. To obtain a compatibility measure requires the transformation of the fuzzy truth value into a scalar value. With the preceding interpretation of the fuzzy set produced truth functional modification, either the domain attribute fuzzy set or the evidential fuzzy set could be the reference. Consequently, two separate logic compatibility measures are tested, one with the evidence as the reference L _Evid/sup and the other with the domain as the reference L _Attr/sup . If the nonreference fuzzy set has multiple elements with maximal membership value, the supremum of the membership degrees of these elements in the reference fuzzy set is used to obtain the scalar compatibility measure.

For level-1 and level-3 results with the T ₃ aggregator, \({R_{{d_1}/|A{ \cup _{{S_3}}}B|}}\) is the only R compatibility measure that performed as well as the better set-theoretic similarity measures in the majority of cases. In the ph and hp cases, the set-theoretic measures outperformed \({R_{{d_1}/|A{ \cup _{{S_3}}}B|}}\). It should be noted that \({R_{{d_1}/|A{ \cup _{{S_3}}}B|}}\) is also a set-theoretic measure since it is equivalent to \({S_{1/D_{{T_1}}^ - /{S_3}/rel}}\). K_cosθ and \({K_{bhat{t_{\bmod }}}}\) performed as well as or better (mm, mh, hh) than all the set-theoretic measures except for the ph and hp case. For these two cases, the results are comparable to the inclusion indices based on relative cardinality and the \({S_{1/D_{{T_1}}^ - /{S_3}/rel}}\) and \({S_{3/{I_{{S_1}}}/rel/{T_3}}}\) measures. \({G_{{D_*}/\operatorname{supp} }}\) and \({G_{{D_@}/\operatorname{supp} }}\) level-1 results are comparable to \({S_{3/{I_{{S_1}}}/rel/{T_3}}}\) and \({S_{3/{I_{{S_2}}}/rel/{T_3}}}\) similarity measures except for mp, mm, mh, and hp where \({S_{3/{I_{{S_1}}}/rel/{T_3}}}\) and \({S_{3/{I_{{S_2}}}/rel/{T_3}}}\) perform 10 to 20 per cent better. For level-3 results, however, these two G compatibility measures perform 10 to 20 per cent better with the precise domain knowledge base than those two similarity measures. With the high domain imprecision, they perform 10 to 20 per cent worse than those two similarity measures. When the evidential fuzzy set is the reference and the domain information is precise, L _{Evid/ sup} is able to identify the target as well as any of the set-theoretic measures using T3. Similarly when the domain fuzzy set is the reference and the evidence is precise, L _{Attr/ sup} is able to identify the target as well as any of the set-theoretic measures except for \({S_{3/{I_{S1}}/rel/{T_3}}},\) which performs about 20 per cent better for the mp case.