Oxidation of propylene to acrolein over Cu-on-Al 2O 3 catalysts modified by molybdenum oxides

Abstract

Adaptive neuro-fuzzy inference system (ANFIS) is a well-known neuro-fuzzy model for approximating highly complex non-linear systems. ANFIS uses precise fuzzy modelling concept that aims at accuracy of a fuzzy model being designed than on interpretability. But interpretability of a fuzzy system is an equally important aspect of fuzzy modelling as accuracy. So far the research based on ANFIS has been mostly application based due to which the various issues related to the interpretability of ANFIS have not been dealt with. The rule base of ANFIS is typically obtained using data driven clustering algorithms. But this process introduces redundancy in the system in terms of similar fuzzy sets and redundant fuzzy rules which unnecessarily increases system complexity. This in turn reduces both the interpretability as well as generalization capability of ANFIS. Additionally in case of ANFIS, unconstrained gradient descent based learning algorithms are used to fine-tune the membership function parameters which usually result in a rule base with inconsistent, excessively overlapping and indistinguishable membership functions for input variables and thus the interpretability of the final optimized system is not guaranteed. This paper is based on addressing the issue of rule base redundancy in ANFIS to reduce complexity and enforcing constraints during learning phase to ensure interpretability of the final optimized system. Rule base redundancy is removed using similarity analysis based rule base simplification in which similar fuzzy sets are merged and subsequently resulting fuzzy rules with equal premises are combined. Hybrid learning technique which is an efficient parameter tuning method for ANFIS is constrained to prevent inconsistency, excessive overlapping and inclusion of membership functions so that the final fuzzy partitions of inputs stay interpretable. The empirical analysis of the impact of rule base simplification and constrained learning on ANFIS is done by application to two well-known benchmark problems and a real world stock price prediction problem. The introduction of rule base simplification and constrained learning in ANFIS modeling has shown better results in terms of obtaining a desired accuracy-interpretability tradeoff than conventional ANFIS.