Reinforcing TSK Model Interpolation with Rule Weight Adjustment via Location View Analysis

Fuzzy rule interpolation (FRI) has been successfully applied to address real-world problems where domain knowledge is incomplete, particularly in situations where observations do not directly match existing rules. However, most of the existing research focuses on fuzzy systems based on Mamdani models. Recently, a pioneering approach has been proposed for rule interpolation within Takagi-Sugeno-Kang (TSK) fuzzy models, employing two distinct techniques that cluster neighbouring rules to facilitate interpolated outcomes. Although promising, these techniques have been developed through empirical methods, lacking formal representation and theoretical justification. This paper formalises such emerging empirical studies with the aim of strengthening the theoretical foundation of FRI. It introduces the location view of rules and rule bases regarding TSK models, analysing the influence of individual rules within FRI techniques for such models. Through geometrising sparse rule bases, comparative investigations of the empirical and theoretical results are carried out. Moreover, geometric projection of the rules is combined with a rule weight adjustment mechanism to reinforce the capability of the formalised FRI techniques. By embedding directional parameters into the rules, the resulting method increases the weighting of critical rules, strengthening the FRI algorithms' performance when many rules may be selected for interpolation. Experimental results confirm the effectiveness of the theoretical model and the consistency with the existing empirical findings. This highlights the potential and significance of formal studies in guiding algorithmic improvements on FRI with TSK models.