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Top1. Introduction
System modelling is the act of approaching a real system in order to fabricate a theoretical design that eases the understanding of the real system. The goal is the design of comprehensible and reliable models that enable to explain, simulate, control, predict or improve the real system. In this regard, fuzzy modelling is a well-known approach to model a real system using descriptive language based on fuzzy logic (Casillas and Cordon, 2003). Fuzzy models offer unique benefits in system modelling. These systems have the advantage of explicitly representing expert knowledge in the form of fuzzy if-then rules. This enables to model various aspects of human knowledge and reasoning process without using precise qualitative analysis (Jang, 1993). These systems are therefore useful in modelling uncertain and ill-defined systems. Also, fuzzy if-then rules result in better transparency as it is easier to interpret the system information allowing an in-depth understanding of its functionality. Like multi-layer perceptron (MLP), fuzzy models are also capable of universal approximation (Castro, 1995) but standalone fuzzy systems lack learning ability. As a result, neuro-fuzzy systems which combine the advantages of fuzzy models in terms of interpretability and learning capability of artificial neural networks (ANN) were proposed. In case expert knowledge is used to build a fuzzy model it is easier to ensure that the system remains interpretable. On the other hand if automated data driven approaches are used to construct fuzzy rules, the interpretability aspect is not necessarily guaranteed as it usually results in a fuzzy model with poor transparency. In fact there are two main but contradictory goals of designing fuzzy systems which are also used to access the quality of fuzzy models: (1) accuracy which is the ability of the system to faithfully represent the real system; (2) interpretability which is the ability to express the behavior of the system in a comprehensible manner (Casillas and Cordon, 2003). In practical data driven fuzzy modelling one of these two properties prevails over the other, increasing one usually decreases the other. During 1990’ significant proliferation in the research on fuzzy modelling happened but the focus was on improving accuracy as much as possible with no attention paid towards the main motivation of using fuzzy systems which is their descriptive power (Ishibuchi and Nojima, 2007). Various techniques were proposed that increased accuracy of these systems which usually increased model complexity also. Nonetheless, lately there is a shift in the fuzzy modelling research towards achieving a tradeoff between accuracy and interpretability (Yen and Wang, 1999).
The neuro-fuzzy systems based on TSK fuzzy modelling technique (Sugeno et al., 1988) are one of the major areas in theoretical and practical fuzzy system literature. These have found significant practical applications in prediction, control and inference. Widely used neuro-fuzzy models like the one proposed by Takagi and Hayashi (1991) and adaptive neuro-fuzzy inference system (ANFIS) (Jang, 1993) are based on TSK model. But TSK based neuro-fuzzy models have been used in practical applications mainly to replace other non-linear models with focus on how accurately the model approximates a real system, ignoring the interpretability aspect which is the core motivation for using fuzzy systems. This is mainly due to the assumption that a fuzzy model is implicitly interpretable in the form of fuzzy if-then rules which is not essentially true. Therefore, these systems have been used in much the same way as other black box techniques like ANNs with approximation accuracy as the main goal which is questionable as indicated in Nauck and Kruse (1999). However, lately the research trend in fuzzy system modelling has shifted towards obtaining models with high accuracy using various approaches such that interpretability is not compromised. This paper investigates the potential of TSK based neuro-fuzzy systems in developing interpretable models of complex real systems. The methodology is based on employing techniques that ensure interpretability during the two main stages of fuzzy modelling viz. structure learning and parameter learning.