Defuzzification of the Discretised Generalised Type-2 Fuzzy Set: Experimental Evaluation

CCI - Centre for Computational Intelligence NOTICE: this is the author’s version of a work that was accepted for publication in Information Science. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version can be found by following the DOIThe work reported in this paper addresses the challenge of the efficient and accurate defuzzification of discretised generalised type-2 fuzzy sets as created by the inference stage of a Mamdani Fuzzy Inferencing System. The exhaustive method of defuzzification for type-2 fuzzy sets is extremely slow, owing to its enormous computational complexity. Several approximate methods have been devised in response to this defuzzification bottleneck. In this paper we begin by surveying the main alternative strategies for defuzzifying a generalised type-2 fuzzy set: (1) Vertical Slice Centroid Type-Reduction; (2) the sampling method; (3) the elite sampling method; and (4) the $\alpha$-planes method. We then evaluate the different methods experimentally for accuracy and efficiency. For accuracy the exhaustive method is used as the standard. The test results are analysed statistically by means of the Wilcoxon Nonparametric Test and the elite sampling method shown to be the most accurate. In regards to efficiency, Vertical Slice Centroid Type-Reduction is demonstrated to be the fastest technique

Slicing Strategies for the Generalised Type-2 Mamdani Fuzzy Inferencing System

The final publication is available at Springer via http://dx.doi.org/[insert DOI]".As a three-dimensional object, there are a number of ways of slicing a generalised type-2 fuzzy set. In the context of the Mamdani Fuzzy Inferencing System, this paper concerns three accepted slicing strategies, the vertical slice, the wavy slice, and the horizontal slice or alpha -plane. Two ways of de ning the generalised type-2 fuzzy set, vertical slices and wavy slices, are presented. Fuzzi cation and inferencing is presented in terms of vertical slices. After that, the application of all three slicing strategies to defuzzi cation is described, and their strengths and weaknesses assessed

Type-Reduced Set Structure and the Truncated Type-2 Fuzzy Set

The file attached to this record is the author's final peer reviewed version.In this paper, the Type-Reduced Set (TRS) of the continuous type-2 fuzzy set is considered as an object in its own right. The structures of the TRSs of both the interval and generalised forms of the type-2 fuzzy set are investigated. In each case the respective TRS structure is approached by first examining the TRS of the discretised set. The TRS of a continuous interval type-2 fuzzy set is demonstrated to be a continuous horizontal straight line, and that of a generalised type-2 fuzzy set, a continuous, convex curve. This analysis leads on to the concept of truncation, and the definition of the truncation grade. The truncated type-2 fuzzy set is then defined, whose TRS (and hence defuzzified value) is identical to that of the non-truncated type-2 fuzzy set. This result is termed the Type-2 Truncation Theorem, an immediate corollary of which is the Type-2 Equivalence Theorem which states that the defuzzified values of type-2 fuzzy sets that are equivalent under truncation are equal. Experimental corroboration of the equivalence of the non-truncated and truncated generalised type-2 fuzzy set is provided. The implications of these theorems for uncertainty quantification are explored. The theorem’s repercussions for type-2 defuzzification employing the α-Planes Representation are examined; it is shown that the known inaccuracies of the α-Planes Method are deeply entrenched

The Structure of the Type-Reduced Set of a Continuous Type-2 Fuzzy Set

CCIThis paper is concerned with the structure of the type-reduced set (TRS) of the continuous type-2 fuzzy set, in both its interval and generalised forms. In each case the TRS structure is approached by first investigating the discretised set. The TRS of a continuous interval type-2 fuzzy set is shown to be a continuous straight line, and that of a generalised type-2 fuzzy set, a continuous, convex curve

Quantification of R-Fuzzy sets

The main aim of this paper is to connect R-Fuzzy sets and type-2 fuzzy sets, so as to provide a practical means to express complex uncertainty without the associated difficulty of a type-2 fuzzy set. The paper puts forward a significance measure, to provide a means for understanding the importance of the membership values contained within an R-fuzzy set. The pairing of an R-fuzzy set and the significance measure allows for an intermediary approach to that of a type-2 fuzzy set. By inspecting the returned significance degree of a particular membership value, one is able to ascertain its true significance in relation, relative to other encapsulated membership values. An R-fuzzy set coupled with the proposed significance measure allows for a type-2 fuzzy equivalence, an intermediary, all the while retaining the underlying sentiment of individual and general perspectives, and with the adage of a significantly reduced computational burden. Several human based perception examples are presented, wherein the significance degree is implemented, from which a higher level of detail can be garnered. The results demonstrate that the proposed research method combines the high capacity in uncertainty representation of type-2 fuzzy sets, together with the simplicity and objectiveness of type-1 fuzzy sets. This in turn provides a practical means for problem domains where a type-2 fuzzy set is preferred but difficult to construct due to the subjective type-2 fuzzy membership

Fuzzy in 3-D: Two Contrasting Paradigms

DIGITS The full text of this article can be read via open access on the publisher's page.Type-2 fuzzy sets and complex fuzzy sets are both three dimensional extensions of type-1 fuzzy sets. Complex fuzzy sets come in two forms, the standard form, postulated in 2002 by Ramot et al., and the 2011 innovation of pure complex fuzzy sets, proposed by Tamir et al.. In this paper we compare and contrast both forms of complex fuzzy set with type-2 fuzzy sets, as regards their rationales, applications, definitions, and structures. In addition, pure complex fuzzy sets are compared with type-2 fuzzy sets in relation to their inferencing operations. Complex fuzzy sets and type-2 fuzzy sets differ in their roles and applications; complex fuzzy sets are pertinent to inferencing where there is seasonality, and type-2 fuzzy sets are applicable to reasoning under uncertainty. Their definitions differ also, though there is equivalence between those of a pure complex fuzzy set and a type-2 fuzzy set. Structural similarity is evident between these three- dimensional sets. Complex fuzzy sets are represented by a 3–D line, and type- 2 fuzzy sets by a 3–D surface, but a surface is simply a generalisation of a line. This similarity is particularly apparent between pure complex fuzzy sets and type- 2 fuzzy sets, which are both mappings from the domain onto the unit square. However type-2 fuzzy sets were found not to be isomorphic to pure complex fuzzy sets. The mechanisms by which complex fuzzy sets model and quantify periodicity, and type-2 fuzzy sets model and quantify uncertainty are discussed. A type-2 fuzzy set can be represented as the union of its type-2 embedded set. An embedded type-2 fuzzy set is a type-2 fuzzy set in itself, whose geomet- rical representation is a 3-D line. Thus, geometrically an embedded type-2 fuzzy set can be seen as equivalent to a pure complex fuzzy set and therefore a type-2 fuzzy set can be represented as the union of a collection pure complex fuzzy sets, which in turn can be regarded as embedded complex fuzzy sets of a type-2 fuzzy set. This relationship is exploited to provide a complex definition of a type-2 fuzzy set

Uncertainty Measurement for the Interval Type-2 Fuzzy Set

In this paper, two measures of uncertainty for interval type-2 fuzzy sets are presented, evaluated, compared and contrasted. Wu and Mendel regard the length of the type-reduced set as a measure of the uncertainty in an interval set. Green eld and John argue that the volume under the surface of the type-2 fuzzy set is a measure of the uncertainty relating to the set. For an interval type-2 fuzzy set, the volume measure is equivalent to the area of the footprint of uncertainty of the set. Experiments show that though the two measures give di erent results, there is considerable commonality between them. The concept of invariance under vertical translation is introduced; the uncertainty measure of a fuzzy set has the property of invariance under vertical translation if the value it generates remains constant under any vertical translation of the fuzzy set. It is left unresolved whether invariance under vertical translation is an essential property of a type-2 uncertainty measure

Задача оптимізації на нечіткій множині типу 2

Доповідь присвячена розв’язанню задачі максимізації функції на нечіткій множині типу 2 (НМТ-2). Побудована функція належності НМТ-2 її «оптимальних» розв’язків. Morenets V. I. Optimization problem on the type 2 fuzzy set. A report is devoted to the solution to the problem of maximizing the function on the type 2 fuzzy set (T2FS). The membership function of “optimal”solutoion T2FS is constructed