A new computationally efficient mamdani interval type-2 fuzzy modelling framework

A modified center-of-sums (mCoS) type-reduction technique is proposed in this paper for constructing a data-driven Mamdani interval type-2 fuzzy modelling (MIT2FM) framework. The mCoS type-reducer is an extension of its type-1 counterpart, the center-of-sums defuzzification, which takes both the area of the scaled consequent membership function of each fired rule and its associated geometric center into account for computing the final output. Contrary to the normal center-of-sums type-reduction, the proposed approach considers the full area under the scaled consequent membership functions even if such area extends beyond the range of the output variable. This enables the commonly used Gaussian interval type-2 membership functions to be utilised in MIT2FM, the area of which has to be calculated via the improper integrals over the whole real line. Moreover, the mCoS method can make use of the mean of Gaussian membership functions directly, instead of computing the geometric center for each rule, so as to further reduce the computational burden of type-reduction. Compared with the state-of-the-art type-reducers, mCoS is shown to be more efficient and, therefore, makes the interval type-2 based fuzzy logic systems more competitive for data-driven fuzzy modelling applications. In order to test the validity of mCoS type-reduction and the elicited fuzzy modelling scheme, experiments are conducted on a benchmark problem of non-linear time series, where collected data are disturbed by noise, and on the real-world application, namely the prediction of mechanical properties of alloy steels. The mCoS based interval type-2 fuzzy modelling approach is shown to handle uncertainties very well and to provide desired generalisation capability when addressing large high-dimensional data sets.