General Type 2 Fuzzy

In this paper, a presented definition of type-2 fuzzy sets and type-2 fuzzy set operation on it was given. The aim of this work was to introduce the concept of general topological spaces were extended in type-2 fuzzy sets with the structural properties such as open sets, closed sets, interior, closure and neighborhoods in topological spaces were extended to general type-2 fuzzy topological spaces and many related theorems are proved.

. Other work using an alpha-planes representation has been applied, e.g. as a method for edge-detection [35] and a learning method to forecast Mackey-Glass time-series [41]. The latter showed a better performance of general type-2 fuzzy logic systems using a simpler model known as “triangle quasi- type-2 fuzzy logic system” first presented in [40]. Some other researchers used some neural network concepts or classification algorithms such as: type 2 Adaptive Network Based Fuzzy Inference System (ANFIS) [28], general type-2 fuzzy neural network (GT2FNN) [24] and fuzzy C-means algorithm with a model known as “efficient tri- angular type-2 fuzzy logic system” [43]. To the best of the authors’ knowledge, no attempt to employ a learning method to general type-2 fuzzy logic systems using the vertical-slices representation has been reported. To achieve this objective, apart from using a practical type-reducer, some kinds of parametrization are needed for general type-2 sets to allow learning or optimization techniques to deal with these parame- ters easily rather than having all the secondary grades or membership functions chosen manually. The parametrization method should preserve most of the freedom associated with GT2FLS.

Abstract. Centroid of general type-2 fuzzy set can be used as a measure of uncertainty in highly uncertain environments. Computing centroid of general type-2 fuzzy set has received an increasing research attention during recent years. Although computation complexity of such sets is higher than that of interval type-2 fuzzy sets, with the advent of new representation techniques, e.g. -planes and z-Slices, computation eorts needed to deal with general type-2 fuzzy sets have decremented. A very rst method to calculate the centroid of a general type-2 fuzzy set was to use Karnik-Mendel algorithm on each - plane, independently. Because of the iterative nature of this method, running time in this approach is rather high. To tackle such a drawback, several emerging algorithms such as Sampling method, Centroid-Flow algorithm, and, recently, Monotone Centroid- Flow algorithm have been proposed. The aim of this paper is to present a new method to calculate centroid intervals of each -plane, independently, while reducing convergence time compared with other algorithms like iterative use of Karnik-Mendel algorithm on each -plane. The proposed approach is based on estimating an initial switch point for each -plane. Exhaustive computations demonstrate that the proposed method is considerably faster than independent implementation of existing iterative methods on each -plane. © 2015 Sharif University of Technology. All rights reserved.

Abstract—Similarity measures provide one of the core tools that enable reasoning about fuzzy sets. While many types of similarity measures exist for type-1 and interval type-2 fuzzy sets, there are very few similarity measures that enable the comparison of general type-2 fuzzy sets. In this paper, we introduce a general method for extending existing interval type-2 similarity measures to similarity measures for general type-2 fuzzy sets. Specifically, we show how similarity measures for interval type-2 fuzzy sets can be employed in conjunction with the zSlices based general type-2 representation for fuzzy sets to provide measures of similarity which preserve all the common properties (i.e. reflexivity, symmetry, transitivity and overlapping) of the original interval type-2 similarity measure. We demonstrate examples of such extended fuzzy measures and provide comparisons between (different types of) interval and general type-2 fuzzy measures.

Fuzzy logic allows handling the uncertainty associated with the noisy signals encountered in realistic real-time scenarios and BMI applications. Indeed, general type-2 (GT2) fuzzy rules have been shown to provide higher robustness to noisy data and unexpected signal patterns such as extreme values and outliers [9, 10]. As EEG signals are non-stationary and the user’s brain behavior is highly unpredictable, general type-2 fuzzy inference systems (GT2 FISs) offer an excellent framework for learning using multiple models (i.e. rules in this case) in this remarkably noisy and changing BMI context. At the time when this article was written, we were not aware of a previous implementation of GT2 FISs in BMI. Additionally, rather than just tuning the model parameters, we propose a learning mechanism for GT2 FISs that provides a higher level of adaptation enabling the online creation of new models, merging or re-scaling them in order to adapt to the unpredictable nature and unknown of brain activity.

The IAA enables these capabilities while minimising the number and scope of assumptions during the model-creation process. In concrete terms, it generates non-parametric FS models of the interval-based data without preselecting a spe- cific FS type (such as triangular or Gaussian) and minimises the loss of information during the model creation process (from the original, potentially uncertain intervals). Finally, the IAA leverages the different degrees of freedom of zGT2 FSs to represent and to distinguish the different types of uncer- tainty. Specifically, agreement/variation over multiple interval samples (e.g., survey responses) from the same source (e.g., expert) is commonly captured using the primary memberships (y ∈ [0, 1]), with lower and upper bounds of primary member- ship modelling potential uncertainty about interval endpoints. Uncertainty originating from the agreement/variation between multiple sources (e.g., experts) of information is modelled through the secondary memberships of the zGT2 FSs (z ∈ [0, 1]). Note that the choice of in which dimension to model which type of uncertainty is arbitrary and either choice is valid - as long as data collection and the interpretation of the re- sulting sets is modified accordingly. To illustrate, consider the case where only a single sample/survey was administered to a large number of subjects; here, it is preferable to model inter- expert uncertainty using the primary membership, resulting in simpler and easier interpretable (type-1 FS) models.

Jonathan M. Garibaldi received the B.Sc (Hons) degree in Physics from Bristol University, UK in 1984, and the M.Sc. degree in Intelligent Systems and the Ph.D. degree in Uncertainty Handling in Immediate Neonatal Assessment from the University of Plymouth, UK in 1990 and 1997, respectively. He is Professor of Computer Science, and leads the Intelligent Modelling and Analysis (IMA) Research Group, in the School of Computer Science at the University of Nottingham, UK. The IMA research group undertakes research into intelligent modelling, utilising data analysis and transformation techniques to enable deeper and clearer understanding of complex problems. His main research interests are modelling uncertainty and variation in human reasoning, and in modelling and interpreting complex data to enable better decision making, particularly in medical domains. He has made many theoretical and practical contributions in fuzzy sets and systems, and in a wide range of generic machine learning techniques in real-world applications. Prof. Garibaldi has published over 200 papers on fuzzy systems and intelligent data analysis, is an Associate Editor of Soft Computing, and three other journals. He has served regularly in the organising committees and programme committees of a range of leading international conferences and workshops, such as FUZZ-IEEE, WCCI, EURO and PPSN. He is a member of the IEEE.

is the source of extra parameters, is in itself the origin of extra computational cost. The quest for a representation that allow practical systems to be implemented is a fertile field of research. There are four main representation theorems for T2FSs, in which practical applications and theoretical defi- nition have been investigated. The vertical slice, wavy slice [22], alpha-plane (or zSlices) [17], [27] and geometric [5] representations. Zadeh [33] was the first to define operations for T2FSs, utilising α-cuts of each fuzzy membership grade. Recently, Chen and Kawase [4], Tahayori et al. [26], Liu et al. [17], [19], and Wagner and Hagras [27], [28] focused their attention towards decomposing T2FSs into several IVFSs. In particular, Liu [17] defined α-planes and Wagner and Hagras [27] defined zSlices as part of their effort to calculate the Centroid of T2FSs. In his work, Liu concluded that the union, intersection and centroid of T2FSs is equal to their respective operations of its constituent α-planes. Wagner and Hagras independently concluded the same. Hamrawi and Coupland [9], [10] derived arithmetic operations and defined non-specificity for T2FSs using the same concept and stated a generalised formula in [11], [12]. In this paper we investigate the use of the concept of α-cuts and its extension principle for T2FSs. We explain, step by step, the development phases of the theory and definitions. We believe it is a significant step forward in the theory and application of T2FSs. The novel ideas provided in this paper, are themselves built upon existing theories and definitions well accepted in the literature and is an extension to available and definitions. We show how operations on general type-2 fuzzy sets can be broken down into a collection of interval type-2 or crisp interval operations. The paper is organised as follows: Section 2 provides the notations and necessary back ground for the following work; Section 3 revisits the α- plane representation and defines the α-plane extension prin- ciple; Section 4 discusses the α-cut representation of IVFSs; Section 5 defines the α-cut representation for T2FSs and the extension principle associated with this representation; Section 6 provides a conclusion.

Abstract—Facial expressions of a person representing similar emotion is not always unique. Naturally, the facial features of a subject taken from different instances of the same emotion have wide variations. In presence of two or more facial features, the variation of the attributes together makes the emotion recognition problem more complicated. This variation is the main source of uncertainty in the emotion recognition problem, which has been addressed here in two steps using type-2 fuzzy sets. First a type-2 fuzzy face-space is constructed with the background knowledge of facial features of different subjects for different emotions. Second, the emotion of an unknown facial expression is determined based on the consensus of the measured facial features with the fuzzy face-space. Both Interval and General Type-2 Fuzzy Sets have been used separately to model the fuzzy face space. The interval type-2 fuzzy set involves primary membership functions for m facial features obtained from n-subjects, each having l–instances of facial expressions for a given emotion. The general type-2 fuzzy set besides employing the primary membership functions mentioned above, also involves the secondary memberships for individual primary membership curve, which has been obtained here by formulating and solving an optimization problem. The optimization problem here attempts to minimize the difference between two decoded signals: the first one being the type-1 defuzzification of the average primary membership functions obtained from the n- subjects, while the second one refers to the type-2 defuzzified signal for a given primary membership function with secondary memberships as unknown. The uncertainty management policy adopted using general type-2 fuzzy set has resulted in a classification accuracy of 98.333% in comparison to 91.667% obtained by its interval type-2 counterpart. A small improvement (approximately 2.5%) in classification accuracy by IT2FS has been attained by pre-processing measurements using the well- known Interval Approach.

Then in mobile robot control by fuzzy logic, all the cited forms of uncertainties will be multiplied over fuzzi- fication, inference and defuzzification. Those errors can degrade the performance of the whole robot controller. Type-2 fuzzy logic has been used by researchers to try and overcome some of these difficulties [6-8]. And since many researchers have explored the use of type-2 fuzzy logic controllers in various applications [9-16]. In fact, type-2 fuzzy sets were initially introduced by Zadeh [17]. Firstly a general type-2 fuzzy set was defined, where it represents a 3D set in which each membership grade is a type-1 fuzzy set bounded in [0,1]. Due to the complexity of the join (OR) and meet (AND) operations performing the inference part [18] and type-reduction in the defuzzi- fication part [19], the application of general type-2 fuzzy sets has been limited. Hence, a simplified version of general type-2 set called interval type-2 fuzzy set is used more widely [6,20]. This kind of set has membership grades that are crisp interval sets bounded in [0,1]. The uncertainty here is represented as a 2D bounded region that is called the Footprint of Uncertainty. Various re- searchers have explored the advantages of interval type-2 fuzzy sets [21,22]. Moreover, a geometric approach has been introduced by Coupland and John [23] distinguish- ing between fuzzy logic over discrete and continuous do- main. But this approach is not fast enough in control ap- plications [24]. In the rest of this paper we treat only in- terval type-2 fuzzy sets over discrete domains.

Type-2 fuzzy systems, in certain applications, have often outperformed type-1 fuzzy systems. The complexity of the computations of general type-2 fuzzy sets means that many applications use interval type-2 (IT2) fuzzy sets [2]. Practi- cally, IT2 fuzzy sets are often easier to manage as compared to general type-2 fuzzy sets. When using IT2 fuzzy sets the mathematics is much less complex [3]. A type-reduction mechanism is used in type-2 fuzzy systems in order to obtain a type-1 fuzzy set - the type-reduced set [4]. The Karnik- Mendel (KM) method is a commonly used type-reduction algorithm and is used to find the centroid of IT2 fuzzy sets which in turn is a type-reduced set [5]. The main advantages of KM type-reducer are its consistency with the extension principle and its strong theoretical ground [6]. Nevertheless, the KM algorithm can suffer from the computational cost of iterations, particularly when it is used in fuzzy logic control systems [7]. There are some type reducers proposed in the literature alternative to the KM method focused on simplifying the computations and improving the performance [8], [9].

In general frequency divider categorize into two groups, analog and digital (Ching et al,. 2009). Regenerative frequency divider and the injection locking frequency divider (ILFD) belong to the analog frequency dividers while the static and dynamic logic dividers are included in digital frequency dividers. Since the dynamic logic has benefits of high speed and programmability, it is more common in the application of communication systems (Ting et al,. 2010).

In this chapter, we will present the novel applications of the IT2-FLCs into the research area of computer games. In this context, we will handle two well-known computer games, namely Flappy Bird [11], [12] and Lunar Lander [13], [14], to show the abilities of the IT2-FLCs. From a control engineering point of view, as the game player can be seen as the main controller in a feedback loop, we will transform the game logic of flappy bird into a reference tracking problem while handling the moon landing problem as a position control problem. Thus, we will construct an intelligent control system composed of three main subsystems: reference generator, the main controller, and game dynamics. In this chapter, we will design and then employ an IT2-FLC as the main controller in a feedback loop such that to have a satisfactory performance and to be able to handle the various uncertainties of the games. In this context, we will briefly present the general structure and the design methods of two IT2-FLCs which are the SIT2-FLC and DIT2-FLC. In this chapter, we will design a SIT2-FLC for the game Flappy Bird while a DIT2-FLC structure for the game moon lander. The IT2-FLCs have been designed and implemented by using the Interval Type-2 Fuzzy Logic Toolbox [28] for Matlab/Simulink. We will examine the performance of both IT2 fuzzy control systems with respect to their control system and game performances, in comparison with its T1 and conventional counterparts, to show that the presented structure can handle the uncertainties caused by the nature of the games much better.

The UK 2001 Output Area Classification (OAC) is a free geodemographic classification which at the highest taxonomic level is comprised of seven classes named ‘Supergroups’. This classification employed a variant of hard c-means and critically in addition to the assigned class, distances 𝑑 𝑖𝑗 are also available for all n elements and c classes. Fisher and Tate (in press) employed [Equation 4] to create possibilistic memberships for each of the seven classes for each census reporting zone (Output Area - OA) for the City of Leicester. They compared PCM outcomes with fuzzy memberships from the equivalent FCM calculation favouring the PCM approach because of the constraint change in Equations 1 and 2. Following general practice (Bezdek 1981 among others) Fisher and Tate (in press) selected a crisp value of m = 2. However, m may have any value greater than 1, and following the method of Fisher (2010; see also Hwang and Rhee, 2007) by allowing m to vary [1.1 to 3.5 in this instance] we can generate type-2 fuzzy sets for each Output Area.

Type-2 fuzzy sets were introduced by L. Zadeh aiming at modelling some settings in which fuzzy sets (usually called type-1 fuzzy sets) are not suffi- cient to reflect certain uncertainty degrees - loosely speaking, they are fuzzy sets whose membership degrees are ordinary fuzzy sets. On the other hand, fiber bundles are topological entities of extreme importance in Mathematics itself and many other scientific areas, like Physics (General Relativity, Field Theory etc.), finance modelling, and statistical inference. The present work introduces a conceptual link between the two ideas and conjectures about the potential mutual benefits that can be obtained from this viewpoint.As an ob- jective and usable product of the presented ideas, it is described a framework for defining type-2 fuzzy hedges, proper to operate on interval type-2 fuzzy sets.

Zadeh [2] initiated another important extension of the concept of type-1 fuzzy sets in the form of type-2 fuzzy sets. These sets are fuzzy sets whose membership grades themselves are type-1 fuzzy sets. Mendel and John [1] gave a simple representation for Type-2 fuzzy sets. Due to the dependence of the membership functions on available linguistic information and numerical data. Linguistic information (e.g., rules from experts), in general, does not give any information about the shapes of the membership functions when membership functions are determined or tuned based on numerical data, the uncertainty in the numerical data e.g., noise, translates into uncertainty in the membership functions. In all such cases, type-2 framework of fuzzy sets can be used to model information about the linguistic, numerical uncertainty very well [3].

Fuzzy inference systems are mechanisms that use fuzzy logic and fuzzy set theory to map inputs and outputs, which has been successfully applied in many application areas, such as decision making, robotic control, intrusion detection, and computer vision. A typical fuzzy inference system consists of a rule base and an inference engine. A number of inference engines have been developed, with the Mamdani inference approach [1] and TSK inference approach [2] being most intensively studied and widely applied. In particular, Mamdani fuzzy inference approach is more intuitive and suitable for handling human linguistic inputs, which usually leads to fuzzy outputs and thus a defuzzification process is typically required to convert the fuzzy outputs to crisp values for general system use. In contrast, crisp outputs are directly produced by the TSK approach, as polynomials (often 0- order or 1-order) are used as the rule consequences in TSK fuzzy model.

Fuzzy control is a versatile control technique that allows controlling through the descriptions of system behavior in terms of linguistic variables constituting the rule base [8]. The reason motivating us to experiment fuzzy control technique is mainly because of the appropriateness of the behaviour of the helicopter system. Furthermore, fuzzy controller can be used as an adaptive methodology as well it is combined with traditional control strategies to improve the stability, increase the robustness, and reduce the fuzzy rule base. As a general example, the combinations between Fuzzy and PID controllers (Fuzzy-PID) are widely used to control nonlinear systems by improving the control performance efficiency. While considering a feedback system with a fuzzy controller, there may be some uncertainties both in the controlled system and in the membership rules part of the fuzzy logic. However, the conventional fuzzy logic system or so-called fuzzy type-1 logic system cannot deal with such uncertainties [9], [10]. Recently, many researches have been focused to increase the performance of fuzzy logic controllers and to overcome the uncertainty problems. In order to achieve robustness, an interval fuzzy type-2 strategy was introduced, as a new generation of fuzzy logic. The main structural difference between these two types of fuzzy logic controller is in the defuzzifier composited block, where a type reduction block is used during the defuzzification in type-2 fuzzy logic [9].

Decision making is broadly defined as include any choice or selection of alternatives and is therefore of importance in many fields in both “soft” social sciences and the “hard” disciplines of natural sciences. Soft set theory [6], firstly proposed by Molodtsov, is a general mathematical tool for dealing with uncertainty. The advantage of soft set theory is that it is free from the inadequacy of the parametrization tools of existing theories. It has been demonstrated that soft set theory brings about a rich potential for applications in many fields. It was growing rapidly over the years.

In [5] the authors claim that the general definition of a T2 or IT2 FS, is too generic in certain contexts since there are no conditions on the MFs. As a consequence, the resulting FOUs and ESs that can be completely unrelated to the modelling of the concepts that they should represent. To overcome these limitations, they introduce a new kind of set, named Constrained Type-2 (CT2) fuzzy sets, by imposing the concepts of shape coherency and explicit variability on the FOU and the ES.