Similarity measures using fuzzy sets

Distances in these spaces should be based upon an Zj or city-block metric (see Eq. 2.18) and not the Z2 or Euclidean metric typically used in many applications. The reasons for this are the same as those discussed in Subheading 2.2.1. for binary vectors. Set-based similarity measures can be adapted from those based on bit vectors using an ansatz borrowed from fuzzy set theory (41,42). For example, the Tanimoto similarity coefficient becomes... 

The fuzzy Hausdorff-type similarity measures can be used for a direct comparison of a fuzzy set A and its various R sets and / -deficient sets, providing alternative fuzzy symmetry deficiency measures. 

Valid measures of approximate symmetry are given for a set A, with respect to some symmetry element R and some center c, using the fuzzy scaling-nesting dissimilarity metric FSNDSM d J A,B) or the scalingnesting similarity measure SNSM fs, expressed between A and /av on the one hand, and between A and F, on the other hand ... 

Pictures of high resolution appear crisp, whereas pictures of low resolution appear fuzzy. A decrease of resolution is accompanied by an increase of fuzziness. Consequently, similarity measures based on the minimum level of resolution required to distinguish objects can be formulated in terms of the maximum level of fuzziness at which the objects are distinguishable. Similarity can be regarded as fuzzy equivalence. This principle provides an alternative mathematical basis for using the methods of topological resolution [262] in similarity analysis the theory of fuzzy sets [382-385]. 

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