Employing zSlices based general type-2 fuzzy sets to model multi level agreement

In this paper, we introduce the concept of Multi Level Agreement (MLA) based on (zSlices based) general type-2 fuzzy sets. We define the notion of MLA and describe how it can be computed based on a series of interval type-2 fuzzy sets. We provide examples, visualizing the nature of MLA sets and discuss their properties and interpretation. Moreover, we specifically address the reason for introducing MLA in order to allow the modeling of agreement in real world applications using fuzzy sets while still maintaining an uncertainty model and show that the use of general type-2 fuzzy sets is essential for MLA as classical sets, type-1 and interval type-2 fuzzy sets do not provide a degree of freedom which could be employed to model agreement. © 2011 IEEE

On transitioning from type-1 to interval type-2 fuzzy logic systems

Capturing the uncertainty arising from system noise has been a core feature of fuzzy logic systems (FLSs) for many years. This paper builds on previous work and explores the methodological transition of type-l (Tl) to interval type-2 fuzzy sets (IT2 FSs) for given "levels" of uncertainty. Specifically, we propose to transition from Tl to IT2 FLSs through varying the size of the Footprint Of Uncertainty (FOU) of their respective FSs while maintaining the original FS shape (e.g., triangular) and keeping the size of the FOU over the FS as constant as possible. The latter is important as it enables the systematic relating of FOU size to levels of uncertainty and vice versa, while the former enables an intuitive comparison between the Tl and T2 FSs. The effectiveness of the proposed method is demonstrated through a series of experiments using the well-known Mackey-Glass (MG) time series prediction problem. The results are compared with the results of the IT2 FS creation method introduced in [1] which follows a similar methodology as the proposed approach but does not maintain the membership function (MF) shape

On the Choice of Similarity Measures for Type-2 Fuzzy Sets

Similarity measures are among the most common methods of comparing type-2 fuzzy sets and have been used in numerous applications. However, deciding how to measure similarity and choosing which existing measure to use can be difficult. Whilst some measures give results that highly correlate with each other, others give considerably different results. We evaluate all of the current similarity measures on type-2 fuzzy sets to discover which measures have common properties of similarity and, for those that do not, we discuss why the properties are different, demonstrate whether and what effect this has in applications, and discuss how a measure can avoid missing a property that is required. We analyse existing measures in the context of computing with words using a comprehensive collection of data-driven fuzzy sets. Specifically, we highlight and demonstrate how each method performs at clustering words of similar meaning

A weighted goal programming approach to fuzzy linear regression with quasi type-2 fuzzy input-output data

This study attempts to develop a regression model when both input data and output data are quasi type-2 fuzzy numbers. To estimate the crisp parameters of the regression model, a linear programming model is proposed based on goal programming. To handle the outlier problem, an omission approach is proposed. This approach examines the behavior of value changes in the objective function of proposed model when observations are omitted. In order to illustrate the proposed model, some numerical examples are presented. The applicability of the proposed method is tested on a real data set on soil science. The predictive performance of the model is examined by cross-validation.Publisher's Versio

Learning of interval and general type-2 fuzzy logic systems using simulated annealing: theory and practice

This paper reports the use of simulated annealing to design more efficient fuzzy logic systems to model problems with associated uncertainties. Simulated annealing is used within this work as a method for learning the best configurations of interval and gen- eral type-2 fuzzy logic systems to maximize their modeling ability. The combination of simulated annealing with these models is presented in the modeling of four bench- mark problems including real-world problems. The type-2 fuzzy logic system models are compared in their ability to model uncertainties associated with these problems. Issues related to this combination between simulated annealing and fuzzy logic sys- tems, including type-2 fuzzy logic systems, are discussed. The results demonstrate that learning the third dimension in type-2 fuzzy sets with a deterministic defuzzifier can add more capability to modeling than interval type-2 fuzzy logic systems. This finding can be seen as an important advance in type-2 fuzzy logic systems research and should increase the level of interest in the modeling applications of general type-2 fuzzy logic systems, despite their greater computational load

Review of Type-1 and Type-2 Fuzzy Numbers

We review type-1 and type-2 fuzzy numbers in this chapter, and propose one way of perceiving the concept of fuzzy numbers by comparing with that of round numbers. There are some definitions of fuzzy numbers, but we particularly adopt the definition often used in fuzzy analysis. Thereby, we emphasize that fuzzy number theory can be reduced to an argument for interval analysis. Moreover, we explain type-2 fuzzy sets and list two specific type-2 fuzzy numbers, one is a (triangular) perfect quasi-type-2 fuzzy number and the other is a triangular shaped type-2 fuzzy number. Finally, we mention the importance and utility of using type-2 fuzzy numbers

Measuring the directional or non-directional distance between type-1 and type-2 fuzzy sets with complex membership functions

Fuzzy sets may have complex, non-normal or non-convex membership functions that occur, for example, in the output of a fuzzy logic system or when automatically generating fuzzy sets from data. Measuring the distance between such non-standard fuzzy sets can be challenging as there is no clear correct method of comparison and limited research currently exists that systematically compares existing distance measures for these fuzzy sets. It is useful to know the distance between these sets, which can tell us how much the results of a system change when the inputs differ, or the amount of disagreement between individual's perceptions or opinions on different concepts. In addition, understanding the direction of difference between such fuzzy sets further enables us to rank them, learning if one represents a higher output or higher ratings than another. This paper picks up previous functions of measuring directional distance and, for the first time, presents methods of measuring the directional distance between any type-1 and type-2 fuzzy sets with both normal/non-normal and convex/non-convex membership functions. In real-world applications where data-driven, non-convex, non-normal fuzzy sets are the norm, the proposed approaches for measuring the distance enables us to systematically reason about the real-world objects captured by the fuzzy sets

General type-2 radial basis function neural network: a data-driven fuzzy model

This paper proposes a new General Type-2 Radial Basis Function Neural Network (GT2-RBFNN) that is functionally equivalent to a GT2 Fuzzy Logic System (FLS) of either Takagi-Sugeno-Kang (TSK) or Mamdani type. The neural structure of the GT2-RBFNN is based on the alpha-planes representation, in which the antecedent and consequent part of each fuzzy rule uses GT2 Fuzzy Sets (FSs). To reduce the iterative nature of the Karnik-Mendel algorithm, the Enhaned-Karnik-Mendel (EKM) type-reduction and three popular direct-defuzzification methods, namely the 1) Nie-Tan approach (NT), the 2) Wu-Mendel uncertain bounds method (WU) and the 3) Biglarbegian-Melek-Mendel algorithm (BMM) are employed. For that reason, this paper provides four different neural structures of the GT2-RBFNN and their structural and parametric optimisation. Such optimisation is a two-stage methodology that first implements an Iterative Information Granulation approach to estimate the antecedent parameters of each fuzzy rule. Secondly, each consequent part and the fuzzy rule base of the GT2-RBFNN is trained and optimised using an Adaptive Gradient Descent method (AGD) respectively. Several benchmark data sets, including a problem of identification of a nonlinear system and a chaotic time series are considered. The reported comparative analysis of experimental results is used to evaluate the performance of the suggested GT2 RBFNN with respect to other popular methodologies

Learning of Type-2 Fuzzy Logic Systems using Simulated Annealing.

This thesis reports the work of using simulated annealing to design more efficient fuzzy logic systems to model problems with associated uncertainties. Simulated annealing is used within this work as a method for learning the best configurations of type-1 and type-2 fuzzy logic systems to maximise their modelling ability. Therefore, it presents the combination of simulated annealing with three models, type-1 fuzzy logic systems, interval type-2 fuzzy logic systems and general type-2 fuzzy logic systems to model four bench-mark problems including real-world problems. These problems are: noise-free Mackey-Glass time series forecasting, noisy Mackey-Glass time series forecasting and two real world problems which are: the estimation of the low voltage electrical line length in rural towns and the estimation of the medium voltage electrical line maintenance cost. The type-1 and type-2 fuzzy logic systems models are compared in their abilities to model uncertainties associated with these problems. Also, issues related to this combination between simulated annealing and fuzzy logic systems including type-2 fuzzy logic systems are discussed. The thesis contributes to knowledge by presenting novel contributions. The first is a novel approach to design interval type-2 fuzzy logic systems using the simulated annealing algorithm. Another novelty is related to the first automatic design of general type-2 fuzzy logic system using the vertical slice representation and a novel method to overcome some parametrisation difficulties when learning general type-2 fuzzy logic systems. The work shows that interval type-2 fuzzy logic systems added more abilities to modelling information and handling uncertainties than type-1 fuzzy logic systems but with a cost of more computations and time. For general type-2 fuzzy logic systems, the clear conclusion that learning the third dimension can add more abilities to modelling is an important advance in type-2 fuzzy logic systems research and should open the doors for more promising research and practical works on using general type-2 fuzzy logic systems to modelling applications despite the more computations associated with it