Uncertainty in Interval Type-2 Fuzzy Systems

This paper studies uncertainty and its effect on system response displacement. The paper also describes how IT2MFs (interval type-2 membership functions) differentiate from T1MFs (type-1 membership functions) by adding uncertainty. The effect of uncertainty is modeled clearly by introducing a technique that describes how uncertainty causes membership degree reduction and changing the fuzzy word meanings in fuzzy logic controllers (FLCs). Several criteria are discussed for the measurement of the imbalance rate of internal uncertainty and its effect on system behavior. Uncertainty removal is introduced to observe the effect of uncertainty on the output. The theorem of uncertainty avoidance is presented for describing the role of uncertainty in interval type-2 fuzzy systems (IT2FSs). Another objective of this paper is to derive a novel uncertainty measure for IT2MFs with lower complexity and clearer presentation. Finally, for proving the affectivity of novel interpretation of uncertainty in IT2FSs, several investigations are done

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

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