A vector similarity measure for linguistic approximation: Interval type-2 and type-1 fuzzy sets

Fuzzy logic is frequently used in computing with words (CWW). When input words to a CWW engine are modeled by interval type-2 fuzzy sets (IT2 FSs), the CWW engine's output can also be an IT2 FS, (A) over tilde, which needs to be mapped to a linguistic label so that it can be understood. Because each linguistic label is represented by an IT2 FS (B) over tilde (i), there is a need to compare the similarity of (A) over tilde and (B) over tilde (i) to find the (B) over tilde (i) most similar to (A) over tilde. In this paper, a vector similarity measure (VSM) is proposed for IT2 FSs, whose two elements measure the similarity in shape and proximity, respectively. A comparative study shows that the VSM gives more reasonable results than all other existing similarity measures for IT2 FSs for the linguistic approximation problem. Additionally, the VSM can also be used for type-1 FSs, which are special cases of IT2 FSs when all uncertainty disappears. (c) 2007 Elsevier Inc. All rights reserved.