Measures of specificity over continuous spaces under similarity relations

We introduce the concept of specificity and indicate its importance as a measure of uncertainty for information represented using fuzzy sets or possibility distributions. We provide formal measures of specificity for variables whose domain is an interval of the real line. Similarity relations are discussed and formalized. The class of width-based similarity relations is introduced. We extend the measure of specificity to allow for the effect of an underlying similarity relationship. The manifestation of this extension for a number of different similarity relations is investigated. Particularly notable results are obtained for width-based similarity relationships. More generally we note the connection between the size of a granule of information and its uncertainty. Motivated by this we investigate the effect of a similarity relationship on the perception of distance in the underlying space and use this to develop an extension of the specificity measure.

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