Simon Coupland

Geometric Type-1 and Type-2 Fuzzy Logic Systems

This paper presents a novel approach to the representation of type-1 and type-2 fuzzy sets utilising computational geometry. To achieve this our approach borrows ideas from the field of computational geometry and applies these techniques in the novel setting of fuzzy logic. We provide new algorithms for various operations on type-1 and type-2 fuzzy sets and for defuzzification. Results of experiments indicate that this approach reduces the execution speed of these operations

Type-2 Fuzzy Logic: A Historical View

Type-2 fuzzy logic offers an opportunity to model levels of uncertainty with which traditional fuzzy logic (type-1) struggles. This article provides a historical perspective of the development of type-2 fuzzy logic with a retrospective review of important developments in the field. We discuss key questions about the technology such as - when should type-2 systems be used? and why has it taken so long for this technology to emerge? The article concludes by highlighting areas where, we believe, type-2 fuzzy logic will have a significant role to play in the future

The collapsing method of defuzzification for discretised interval type-2 fuzzy sets

This paper proposes a new approach for defuzzification of interval type-2 fuzzy sets. The collapsing method converts an interval type-2 fuzzy set into a type-1 representative embedded set (RES), whose defuzzified values closely approximates that of the type-2 set. As a type-1 set, the RES can then be defuzzified straightforwardly. The novel representative embedded set approximation (RESA), to which the method is inextricably linked, is expounded, stated and proved within this paper. It is presented in two forms: Simple RESA: this approximation deals with the most simple interval FOU, in which a vertical slice is discretised into 2 points. Interval RESA: this approximation concerns the case in which a vertical slice is discretised into 2 or more points. The collapsing method (simple RESA version) was tested for accuracy and speed, with excellent results on both criteria. The collapsing method proved more accurate than the Karnik-Mendel iterative procedure (KMIP) for an asymmetric test set. For both a symmetric and an asymmetric test set, the collapsing method outperformed the KMIP in relation to speed.

Enhanced Interval Approach for Encoding Words Into Interval Type-2 Fuzzy Sets and Its Convergence Analysis

Construction of interval type-2 fuzzy set models is the first step in the perceptual computer, which is an implementation of computing with words. The interval approach (IA) has, so far, been the only systematic method to construct such models from data intervals that are collected from a survey. However, as pointed out in this paper, it has some limitations, and its performance can be further improved. This paper proposes an enhanced interval approach (EIA) and demonstrates its performance on data that are collected from a web survey. The data part of the EIA has more strict and reasonable tests than the IA, and the fuzzy set part of the EIA has an improved procedure to compute the lower membership function. We also perform a convergence analysis to answer two important questions: 1) Does the output interval type-2 fuzzy set from the EIA converge to a stable model as increasingly more data intervals are collected, and 2) if it converges, then how many data intervals are needed before the resulting interval type-2 fuzzy set is sufficiently similar to the model obtained from infinitely many data intervals? We show that the EIA converges in a mean-square sense, and generally, 30 data intervals seem to be a good compromise between cost and accuracy.

A Fast Geometric Method for Defuzzification of Type-2 Fuzzy Sets

Generalized type-2 fuzzy logic systems cannot currently be used for practical problems because the amount of computation required to defuzzify a generalized type-2 fuzzy set is too large. This paper presents a new method for defuzzifing a type-2 fuzzy set. The new much faster technique is based on geometric representations and operations. The results of a real world example contained in this paper show this new approach to be over 200,000 times faster than type-reduction. We present a new method for assessing the accuracy of the membership function of a type-2 fuzzy set. This method is used to show that the new representation used by the defuzzifier is not detrimental to the accuracy of the set. We also discuss the differences between the new approach and type-reduction, identifying the origin of this massive improvement in execution speed.

The sampling method of defuzzification for type-2 fuzzy sets: Experimental evaluation

For generalised type-2 fuzzy sets the defuzzification process has historically been slow and inefficient. This has hampered the development of type-2 Fuzzy Inferencing Systems for real applications and therefore no advantage has been taken of the ability of type-2 fuzzy sets to model higher levels of uncertainty. The research reported here provides a novel approach for improving the speed of defuzzification for discretised generalised type-2 fuzzy sets. The traditional type-reduction method requires every embedded type-2 fuzzy set to be processed. The high level of redundancy in the huge number of embedded sets inspired the development of our sampling method which randomly samples the embedded sets and processes only the sample. The paper presents detailed experimental results for defuzzification of constructed sets of known defuzzified value. The sampling defuzzifier is compared on aggregated type-2 fuzzy sets resulting from the inferencing stage of a FIS, in terms of accuracy and speed, with other methods including the exhaustive and techniques based on the @a-planes representation. The results indicate that by taking only a sample of the embedded sets we are able to dramatically reduce the time taken to process a type-2 fuzzy set with very little loss in accuracy.

New geometric inference techniques for type-2 fuzzy sets

This paper presents new techniques for performing logical operations on type-2 fuzzy sets. These techniques make significant use of geometric methods to give, for the first time, logic operators that can be implemented over continuous domains, thereby eliminating the need for discretisation. We give a full exposition of the geometric inference operations and consider computational speed and accuracy. We show this novel approach to be more accurate, although slightly slower than existing techniques.

A novel alpha-cut representation for type-2 fuzzy sets

Alpha-cuts and the extension principle form a methodology for extending mathematical concepts from crisp sets to fuzzy sets. They have been applied to many operations, and have also been extended to interval valued fuzzy sets. Recently, some researchers defined new representations of type-2 fuzzy sets, namely, the alpha-plane representation and the zSlice representation. In this paper we investigate an alternative representation utilising alpha-cuts and the extension principle for interval valued fuzzy sets and type-2 fuzzy sets.

A new and efficient method for the type-2 meet operation

Type-2 fuzzy logic systems require considerable processing resources to arrive at a conclusion. This paper shows how this can be significantly reduced, focusing specifically on the meet operation. We show how to eliminate large amounts of redundancy from the operation and then present a geometric method that is even more efficient. Results from initial experiments into performance increases are given. These results show that our method can give a significant increase in performance for certain types of secondary memberships.

Type-2 Fuzzy Sets: Geometric Defuzzification and Type-Reduction

This paper presents the geometric defuzzifier for generalised type-2 fuzzy sets. This defuzzifier can be executed in real-time and can therefore be applied to control and other real world problems. We believe this to be a significant step forward for generalised type-2 fuzzy logic systems