On the Accuracy of Input-Output Uncertainty Modeling with Interval Type-2 Fuzzy Logic Systems

Type-2 Fuzzy Logic Systems (T2 FLSs) have been commonly attributed with the capability to model various sources of data uncertainties. The input uncertainties of an FLS were modeled using T2 Fuzzy Sets (FSs) and the type-reduced centroid of the output FS was interpreted as a measure of uncertainty associated with the terminal real-valued output. However, the accuracy of this input-output uncertainty modeling has been rarely studied. It is well established that T2 FSs can be understood as a composition of a large number of embedded T1 FSs and thus model the uncertainty of selecting a specific T1 FSs. However, whether the same can be achieved with T2 FLSs can be considered an open question. This paper contributes by presenting a study of the input-output uncertainty modeling capability of Interval T2 (IT2) FLSs. First, the Monte Carlo simulation technique is used to simulate linguistic uncertainties and to compute the aggregated output result. This simulation is then compared to the output bounds provided by the interval centroid computed with IT2 FLS. It is demonstrated that the interval output of the IT2 FLS overestimates the output uncertainty range when compared to the results of the Monte Carlo simulation. To further understand this problem the concept of Equivalent Type-1 FSs is used. Finally, a detailed example is presented to demonstrate why the IT2 fuzzy inference process overestimates the output uncertainty.