Backward Fuzzy Interpolation and Extrapolation with Multiple Multi-antecedent Rules

Fuzzy rule interpolation is well known for reducing the complexity of fuzzy models and making inference possible in sparse rule-based systems. However, in practical applications with inter-connected subsets of rules, situations may arise when a crucial antecedent of observation is absent, either due to human error or difficulty in obtaining data, while the associated conclusion may be derived according to alternative rules or even observed directly. If such missing antecedents were involved in the subsequent interpolation process, the final conclusion would not be deduced using conventional means. However, missing antecedents may be related to certain conclusion and therefore, may be inferred or interpolated using the known antecedents and conclusion. For this purpose, this paper presents a novel approach termed backward fuzzy rule interpolation and extrapolation. In particular, the approach supports both interpolation and extrapolation which involve multiple intertwined fuzzy rules, with each having multiple antecedents. An algorithm is given to implement the approach via the use of the scale and move transformation-based fuzzy interpolation. The algorithm makes use of trapezoidal fuzzy membership functions. Realistic application examples are provided to demonstrate the efficacy of the approach.