Monotone data samples do not always produce monotone fuzzy if-then rules: Learning with ad hoc and system identification methods

In this paper, ad hoc and system identification methods are used to generate fuzzy If-Then rules for a zero-order Takagi-Sugeno-Kang (TSK) Fuzzy Inference System (FIS) using a set of multi-attribute monotone data. Convex and normal trapezoidal fuzzy sets, with a strong fuzzy partition strategy, is employed. Our analysis shows that even with multi-attribute monotone data, non-monotone fuzzy If-Then rules can be produced using an ad hoc method. The same observation can be made, empirically, using a system identification method, e.g., a derivative-based optimization method and the genetic algorithm. This finding is important for modeling a monotone FIS model, as the result shows that even with a 'clean' data set pertaining to a monotone system, the generated fuzzy If-Then rules may need to be pre-processed, before being used for FIS modeling. As such, monotone fuzzy rule relabeling is useful. Besides that, a constrained non-linear programming method for FIS modelling is suggested, as a variant of the system identification method.