Abstract
The characteristics of a fuzzy set are decided by its membership function. This work aims to provide a geometric approach for enhancing the design and performance of fuzzy systems. Similarity Estimator (SimE) evaluates the membership functions of fuzzy sets on Euclidean space based on geometric area. The overlapping regions between the sets are partitioned into geometric structures. The area of overlapping is computed by summing the area of polygons and integrating the area under curves. Similarity between fuzzy sets is directly proportional to the area of overlapping between them. SimE was tested over a range of real numbers with finite intervals. Fuzzy sets using different membership functions were created for the same data distribution. From the test results it can be inferred that fuzzy sets defined using triangular membership functions have a minimum overlapping area when compared to fuzzy sets defined using other membership function. Optimal overlapping area of fuzzy sets improves the semantic representation and the performance of the system. SimE can be used by knowledge engineers to design efficient fuzzy systems.
Highlights
Large amount of data are generated and manipulated everyday but often due to human errors and system failures the data may become noisy, ambiguous and redundant
Fuzzy set theory and soft computing techniques are capable of handling imprecise, vague and incomplete data in automated tasks (Isermann, 1998)
This study proposes Similarity Estimator (SimE); an approach where fuzzy sets are said to be similar based on their overlapping region
Summary
Large amount of data are generated and manipulated everyday but often due to human errors and system failures the data may become noisy, ambiguous and redundant. Both fuzzy sets are exactly same in shape but the Euclidean space enclosed by the sets is not similar, there is no overlapping between the sets. Fuzzy set ‘A’ on the universe of discourse X is characterized by a membership function, which associates with each element ‘x’ a real number in. Similarity plays a major role in decision making, classification and clustering It is a major criterion in deciding the number of fuzzy (Jager and Benz, 2000; Setnes et al, 1998) sets for a fuzzy system design. The following section reviews some of the works in literature that use different membership functions and fuzzy set similarity measures
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