Centroid Density of Interval Type-2 Fuzzy Sets: Comparing Stochastic and Deterministic Defuzzification

Recently, Type-2 (T2) Fuzzy Logic Systems (FLSs) gained increased attention due to their capability to better describe, model and cope with the ubiquitous dynamic uncertainties in many engineering applications. By far the most widely used type of T2 FLSs are the Interval T2 (IT2) FLSs. This paper provides a comparative analysis of two fundamentally different approaches to defuzzification of IT2 Fuzzy Sets (FSs) - the deterministic Karnik-Mendel Iterative Procedure (KMIP) and the stochastic sampling defuzzifier. As previously demonstrated by other researchers, these defuzzification algorithms do not always compute identical output values. In the presented work, the concept of centroid density of an IT2 FS is introduced in order to explain such discrepancies. It was demonstrated that the stochastic sampling defuzzification method converges towards the center of gravity of the proposed centroid density function. On the other hand, the KMIP method calculates the midpoint of the interval centroid obtained according to the extension principle. Since the information about the centroid density is removed via application of the extension principle, the two methods produce inevitably different results. As further demonstrated, this difference significantly increases in case of non-symmetric IT2 FSs.