Cardinality, Fuzziness, Variance and Skewness of Interval Type-2 Fuzzy Sets

Abstract— Centroid, cardinality, fuzziness, variance and skewness are all important concepts for an interval type-2 fuzzy set (IT2 FS) because they are all measures of uncertainty, i.e. each of them is an interval, and the length of the interval is an indicator of the uncertainty. The centroid of an IT2 FS has been defined by Karnik and Mendel. In this paper, the other four concepts are defined. All definitions use the Mendel-John Representation Theorem for IT2 FSs. Formulas for computing the cardinality, fuzziness, variance and skewness of an IT2 FS are derived. Unlike the formulas for the centroid of an IT2 FS, which must be computed by iterative Karnik-Mendel algorithms, these new formulas have closed-form expressions, so they can be computed very fast. These definitions are useful not only for measuring the uncertainties of an IT2 FS, but also in measuring the similarity between two IT2 FSs.

• Type: Conference
• Year: 2007
• Where: FOCI
• Authors: Jerry M. Mendel, Dongrui Wu
• Keywords: Applied Computing, FOCI 2007, Mendel-John Representation Theorem, Skewness, Type-2 Fuzzy Set