Rule model simplification
Due to its high performance and comprehensibility, fuzzy modelling is becoming more and more popular in dealing with nonlinear, uncertain and complex systems for tasks such as signal processing, medical diagnosis and financial investment. However, there are no principal routine methods to obtain the optimum fuzzy rule base which is not only compact but also retains high prediction (or classification) performance. In order to achieve this, two major problems need to be addressed. First, as the number of input variables increases, the number of possible rules grows exponentially (termed curse of dimensionality). It inevitably deteriorates the transparency of the rule model and can lead to over-fitting, with the model obtaining high performance on the training data but failing to predict the unknown data successfully. Second, gaps may occur in the rule base if the problem is too compact (termed sparse rule base). As a result, it cannot be handled by conventional fuzzy inference such as Mamdani. This Ph.D. work proposes a rule base simplification method and a family of fuzzy interpolation methods to solve the aforementioned two problems. The proposed simplification method reduces the rule base complexity via Retrieving Data from Rules (RDFR). It first retrieves a collection of new data from an original rule base. Then the new data is used for re-training to build a more compact rule model. This method has four advantages: 1) It can simplify rule bases without using the original training data, but is capable of dealing with combinations of rules and data. 2) It can integrate with any rule induction or reduction schemes. 3) It implements the similarity merging and inconsistency removal approaches. 4) It can make use of rule weights. Illustrative examples have been given to demonstrate the potential of this work. The second part of the work concerns the development of a family of transformation based fuzzy interpolation methods (termed HS methods). These methods first introduce the general concept of representative values (RVs), and then use this to interpolate fuzzy rules involving arbitrary polygonal fuzzy sets, by means of scale and move transformations. This family consists of two sub-categories: namely, the original HS methods and the enhanced HS methods. The HS methods not only inherit the common advantages of fuzzy interpolative reasoning -- helping reduce rule base complexity and allowing inferences to be performed within simple and sparse rule bases -- but also have two other advantages compared to the existing fuzzy interpolation methods. Firstly, they provide a degree of freedom to choose various RV definitions to meet different application requirements. Secondly, they can handle the interpolation of multiple rules, with each rule having multiple antecedent variables associated with arbitrary polygonal fuzzy membership functions. This makes the interpolation inference a practical solution for real world applications. The enhanced HS methods are the first proposed interpolation methods which preserve piece-wise linearity, which may provide a solution to solve the interpolation problem in a very high Cartesian space in the mathematics literature. The RDFR-based simplification method has been applied to a variety of applications including nursery prediction, the Saturday morning problem and credit application. HS methods have been utilized in truck backer-upper control and computer hardware prediction. The former demonstrates the simplification potential of the HS methods, while the latter shows their capability in dealing with sparse rule bases. The RDFR-based simplification method and HS methods are further integrated into a novel model simplification framework, which has been applied to a scaled-up application (computer activity prediction). In the experimental studies, the proposed simplification framework leads to very good fuzzy rule base reductions whilst retaining, or improving, performance.