Abstract - Recently, the theory regarding interval type-2 fuzzy sets and fuzzy logic systems has been further developed. In the first instance, the concepts of interval type-2 fuzzy sets were broadened, proving that this class of sets include some which are different from interval-valued sets. In a later work, the set theoretic operations of union and intersection on these sets were studied and presented under the framework of general type-2 fuzzy sets. This stimulated and motivated the further study of fuzzy logic systems using these new sets, which have been referred to as the general forms of interval type-2 fuzzy sets. However, as usual when a new fuzzy logic framework is presented, only the singleton version of these systems was introduced. In this work we aim to generalise that work, introducing the non-singleton general forms of interval type-2 fuzzy logic systems. We will present a real example on how to use these non-singleton fuzzy logic systems in real world applications.
Abstract - In the usual [0,1]-based fuzzy logic, the actual numerical value of a fuzzy degree can be different depending on a scale, what is important -- and scale-independent -- is the order between different values. To make a description of fuzziness more adequate, it is reasonable to consider interval-valued degrees instead of numerical ones. Here also, what is most important is the order between the degrees. If we have only order between the intervals, can we, based on this order, reconstruct the original numerical values -- i.e., the degenerate intervals? In this paper, we show that such a reconstruction is indeed possible, moreover, that it is possible under three different definitions of order between numerical values.
Abstract - In a previous paper, we proposed an extended ANFIS architecture and showed that interval type-2 ANFIS produced larger errors than type-1 ANFIS on the well-known IRIS classification problem. In this paper, more experiments on both synthetic and real-world data are conducted to further investigate and compare the performance of interval type-2 ANFIS and type-1 ANFIS. For each dataset, interval type-2 ANFIS is optimised in three different ways, including a strategy suggested by Mendel such that interval type-2 ANFIS would be no worse than type-1 ANFIS. Our results show that in some circumstances the performance of interval type-2 ANFIS can be improved when it is initialised with blurred optimised type-1 ANFIS parameters. However, in general, interval type-2 ANFIS does not produce a clear performance improvement compared to type-1 ANFIS, especially on Mackey-Glass data with large noise. Thus, we conclude that the choice of interval type-2 ANFIS over type-1 ANFIS should be carefully considered, since type-2 ANFIS is more computationally complex, yet significantly better performance cannot be easily obtained.
Abstract - This study addresses evolutionary structure optimization and parameter tuning processes for evolving a proposed Hierarchical interval Type-2 Beta Fuzzy System (HT2BFS). The structure learning phase is performed in a multi- objective context by applying the Multi-Objective Extended Genetic Programming (MOEGP) algorithm. This phase aims to obtain a near-optimal structure of HT2BFS taking into account the optimization of two objectives, which are the accuracy maximization and the number of rules minimization. Moreover, a second parameter tuning phase is also performed in order to refine the parameters of the obtained near-optimal structure by applying the PSO-based Update Memory for Improved Harmony Search (PSOUM-IHS) algorithm. The system's performance is validated through two classification problems. Results prove the efficiency of the proposed approach.
Abstract - Levenberge-Marquardt (LM) algorithm is a well-known optimization technique which has the advantages of the steepest descent and the Gauss-Newton methods. Unfortunately, LM algorithm-based parameter update rules, regardless of being used to tune the parameters of artificial neural networks or neuro-fuzzy systems, require the calculation of inversion of high dimensional matrices. Matrix inversions are generally computationally expensive, and it is not desired in a real-time application where the computation speed is critical. In this paper, using matrix inversion lemma, LM algorithm is modified to avoid matrix inversion calculations, and therefore lessen its computational burden. The proposed algorithm is compared with the conventional LM algorithm for the training of interval type-2 fuzzy logic systems in terms of its speed. Extensive simulation results demonstrate that that the proposed novel method can increase the speed of LM algorithm by 50\% while remaining the same performance.