Type-2 fuzzy inferencing for generalised, discretised type-2 fuzzy sets has been impeded by the computational complexity of the defuzzification stage of the fuzzy inferencing system. Indeed this stage is so complex computationally that it has come to be known as the defuzzification bottleneck. The computational complexity derives from the enormous number of embedded sets that have to be individually processed in order to effect defuzzification. Two new approaches to type-2 defuzzification are presented, the sampling method and the Greenfield-Chiclana Collapsing Defuzzifier. The sampling method and its variant, elite sampling, are techniques for the defuzzification of generalised type-2 fuzzy sets. In these methods a relatively small sample of the totality of embedded sets is randomly selected and processed. The small sample size drastically reduces the computational complexity of the defuzzification process, so that it may be speedily accomplished. The Greenfield-Chiclana Collapsing Defuzzifier relies upon the concept of the representative embedded set, which is an embedded set having the same defuzzified value as the type-2 fuzzy set that is to be defuzzified. By a process termed collapsing the type-2 fuzzy set is converted into a type-1 fuzzy set which, as an approximation to the representative embedded set, is known as the representative embedded set approximation. This type-1 fuzzy set is easily defuzzified to give the defuzzified value of the original type-2 fuzzy set. By this method the computational complexity of type-2 defuzzification is reduced enormously, since the representative embedded set approximation replaces the entire collection of embedded sets. The strategy was conceived as a generalised method, but so far only the interval version has been derived mathematically. The grid method of discretisation for type-2 fuzzy sets is also introduced in this thesis. Work on the defuzzification of type-2 fuzzy sets began around the turn of the millennium. Since that time a number of investigators have contributed methods in this area. These different approaches are surveyed, and the major methods implemented in code prior to their experimental evaluation. In these comparative experiments the grid method of defuzzification is employed. The experimental results show beyond doubt that the collapsing method performs the best of the interval alternatives. However, though the sampling method performs well experimentally, the results do not demonstrate it to be the best performing generalised technique.

Type reduction does the work of computing the centroid of a type-2 fuzzy set. The result is a type-1 fuzzy set from which a corresponding crisp number can then be obtained through defuzzification. Type reduction is one of the major operations involved in type-2 fuzzy inference. Therefore, making type reduction efficient is a significant task in the application of type-2 fuzzy systems. Liu introduced a horizontal slice representation, called the £\-plane representation, and proposed a type reduction method for a type-2 fuzzy set. By exploring some useful properties of the £\-plane representation and of the type reduction for interval type-2 fuzzy sets, we develop a fast method for computing the centroid of a type-2 fuzzy set. The number of computations and comparisons involved is greatly reduced. As a result, type reduction can be done much more efficiently. The effectiveness of the proposed method is analyzed mathematically and demonstrated by experimental results.

We propose a novel approach for building a type-2 neuro-fuzzy system from a given set of input-output training data. For an input pattern, a corresponding crisp output of the system is obtained by combining the inferred results of all the rules into a type-2 fuzzy set which is then defuzzified by applying a type reduction algorithm. Karnik and Mendel proposed an algorithm, called KM algorithm, to compute the centroid of an interval type-2 fuzzy set efficiently. Based on this algorithm, Liu developed a centroid type-reduction strategy to do type reduction for type-2 fuzzy sets. A type-2 fuzzy set is decomposed into a collection of interval type-2 fuzzy sets by £\-cuts. Then the KM algorithm is called for each interval type-2 fuzzy set iteratively. However, the initialization of the switch point in each application of the KM algorithm is not a good one. In this thesis, we present an improvement to Liu's algorithm. We employ the result previously obtained to construct the starting values in the current application of the KM algorithm. Convergence in each iteration except the first one can then speed up and type reduction for type-2 fuzzy sets can be done faster. The efficiency of the improved algorithm is analyzed mathematically and demonstrated by experimental results. Constructing a type-2 neuro-fuzzy system involves two major phases, structure identification and parameter identification. We propose a method which incorporates self-constructing fuzzy clustering algorithm and a SVD-based least squares estimator for structure identification of type-2 neuro-fuzzy modeling. The self-constructing fuzzy clustering method is used to partition the training data set into clusters through input-similarity and output-similarity tests. The membership function associated with each cluster is defined with the mean and deviation of the data points included in the cluster. Then applying SVD-based least squares estimator, a type-2 fuzzy TSK IF-THEN rule is derived from each cluster to form a fuzzy rule base. After that a fuzzy neural network is constructed. In the parameter identification phase, the parameters associated with the rules are then refined through learning. We propose a hybrid learning algorithm which incorporates particle swarm optimization and a SVD-based least squares estimator to refine the antecedent parameters and the consequent parameters, respectively. We demonstrate the effectiveness of our proposed approach in constructing type-2 neuro-fuzzy systems by showing the results for two nonlinear functions and two real-world benchmark datasets. Besides, we use the proposed approach to construct a type-2 neuro-fuzzy system to forecast the daily Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX). Experimental results show that our forecasting system performs better than other methods.