Type-2 Sets Research Articles

Fuzzy Subsethood for Fuzzy Sets of Type-2 and Generalized Type-<formula formulatype="inline"><tex Notation="TeX"> ${n}$</tex></formula>

In this paper, we use Zadeh's extension principle to extend Kosko's definition of the fuzzy subsethood measure S(G, H) to type-2 fuzzy sets defined on any set X equipped with a measure. Subsethood is itself a fuzzy set that is a crisp interval when G and H are interval type-2 sets. We show how to compute this interval and then use the result to compute subsethood for general type-2 fuzzy sets. A definition of subsethood for arbitrary fuzzy sets of type-n > 2 is then developed. This subsethood is a type-(n - 1) fuzzy set, and we provide a procedure to compute subsethood of interval type-3 fuzzy sets.

A collaborative care model for patients with Type-2 diabetes.

The research on diabetes has shown the need to move from a traditional medical model to a patient-team orientation. This has led to a number of collaborative programs targeting the patient and family's comprehensive needs. This paper details one collaborative care program for underserved patients with Type-2 diabetes set in rural, eastern North Carolina. Roles of the therapeutic team are incorporated along with a case example highlighting the bio-psychosocial-spiritual model in action. Sustainability, challenges, and strengths are included to facilitate a realistic appraisal of the program.

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.

A Type-2 Fuzzy Approach to Linguistic Summarization of Data

This paper introduces an application of type-2 fuzzy sets in data linguistic summarization. The original approach by Yager (1982) based on representing natural language statements via type-1, i.e., the Zadeh fuzzy sets, is generalized with type-2 fuzzy sets applied as models of linguistically expressed quantities and/or properties of objects. Type-2 sets extend the known summarization procedures by handling fuzzy values stored in databases, and allow to represent a linguistic term via a few different membership functions (e.g., provided by different experts), which makes the method more general and human-consistent. Furthermore, quality measures for type-2 summaries are discussed in order to evaluate the informativeness of the messages generated. Finally, two prototype applications are presented and the success of the new method is discussed.

Detection of hard cuts and gradual transitions from video using fuzzy logic

In this paper, we introduce a novel approach for the detection and classification of shot boundaries based on interval type-2 fuzzy logic. We consider fuzzy low level features (Fuzzy Histogram (FH) and Fuzzy Co-occurrence Matrix (FCM)) and evaluate a fuzzy rule base modelled by interval type-2 sets to detect cut boundaries. After detection of cuts, a recovery mechanism is used to reduce false positives caused by illumination variation (e.g., camera flash light). The fuzzy rules used for gradual transition form a unified approach to detect various types gradual transitions like dissolves, fades and wipes, by a single method. Test results indicate that very high precision and recall could be achieved for the test videos, particularly the recall value of gradual transitions.

Higher Order Vagueness in a Dynamic Landscape: Multi-Resolution Morphometric Analysis of a Coastal Dunefield

Many environmental phenomena can be conceptualised as vague and so may be suitable for storage as fuzzy sets and analysis by fuzzy logic. Fuzzy sets directly address the vagueness in the information, but many consider that any statement about a vague phenomenon must itself be vague. This is known as higher order vagueness, and is handled in fuzzy set theory by type-2 and, by extension, type-n fuzzy sets. In this paper we use the recognition of, and change in, a system of coastal sand dunes as an environmental example in which to explore the use of type-2 fuzzy sets. The crests and troughs of the dunes are identified as fuzzy sets from geomorphometric analysis of high resolution digital elevation models from two years (1998 and 2000). By varying the parameters of the morphometric extraction, multiple instances of type-1 fuzzy sets can be defined, and these can be summarised to yield type-2 fuzzy sets. The logic of change analysis is presented, and two alternative approaches to change analysis of type-2 fuzzy sets implemented. In one approach changes in the multiple instances of type-1 fuzzy sets are analysed and summarised as type-2 sets. The second approach directly examines change in the parameters of type-2 fuzzy sets, viewing the results for the different parameters as separate instances of change. All analyses produce satisfactory results which, although they arc hard to verify, make sense, yielding a range of possible but small degrees of fuzzy change. In a rather sedentary dune system (as is usual in a coastal location in mid-latitudes) this is to be expected. Change analysis in most applications of type-2 fuzzy sets, which might be based on expert advice for defining memberships, would have to rely on this second approach, and it is therefore interesting to note that this yields the largest range of possible change results.

A simplified type-2 fuzzy logic controller for real-time control

Abstract Increasingly, genetic algorithms (GAs) are used to optimize the parameters of fuzzy logic controllers (FLCs). Although GAs provide a systematic design approach, the optimization process is generally performed off-line using a plant model. Differences between the model and physical plant may result in unsatisfactory control performance when the FLCs are deployed in practice. Type-2 FLCs are an attractive alternative because they can better cope with modeling uncertainties. Unfortunately, type-2 FLCs are computationally intensive. This paper presents a simplified type-2 FLC that is suitable for real-time applications. The key idea is to only replace some critical type-1 fuzzy sets by type-2 sets. Experimental results indicate that the proposed simplified type-2 FLC is as robust as a conventional type-2 FLC, while lowering the computational cost.

Operations on type-2 fuzzy sets

In this paper, we discuss set operations on type-2 fuzzy sets (including join and meet under minimum/product t-norm), algebraic operations, properties of membership grades of type-2 sets, and type-2 relations and their compositions. All this is needed to implement a type-2 fuzzy logic system (FLS).

Centroid of a type-2 fuzzy set

Abstract In this paper, we introduce the centroid and generalized centroid of a type-2 fuzzy set (both of which are essential for implementing a type-2 fuzzy logic system), and explain how to compute them. For practical use, we show how to compute the centroid of interval and Gaussian type-2 fuzzy sets. An exact computation procedure is provided for an interval type-2 set, whereas an approximation is provided for both interval and Gaussian type-2 sets. Examples are given that compare the exact computational results with the approximate results.

Uncertainty, fuzzy logic, and signal processing

In this paper we focus on model-based statistical signal processing and how some problems that are associated with it can be solved using fuzzy logic. We explain how uncertainty (which is prevalent in statistical signal processing applications) can be handled within the framework of fuzzy logic. Type-1 singleton and non-singleton fuzzy logic systems (FLSs) are reviewed. Type-2 FLSs, which are relatively new, and are very appropriate for signal processing problems, because they can handle linguistic and numerical uncertainties, are overviewed in some detail. The output of a type-2 FLS is a type-2 fuzzy set. Using a new operation called type-reduction, the type-2 set can be reduced to a type-1 set – the type-reduced set – which plays the role of a confidence interval for linguistic uncertainties. No such result can be obtained for a type-1 FLS. We demonstrate, by means of examples, that a type-2 FLS can outperform a type-1 FLS for one-step prediction of a Mackey–Glass chaotic time series whose measurements are corrupted by additive noise, and equalization of a nonlinear time-varying channel.

Fuzzy Sets of Type-2

This paper provides a detailed review of the important and growing role that fuzzy sets of type-2 play in knowledge representation and inferencing with fuzzy systems. As well as an up-to-date review of the work in this area, examples are provided that demonstrate how type-2 sets can help with both knowledge representation and inferencing. The paper also reports on the use of type-2 sets in a medical application and summarizes the other type-2 applications reported in the literature.

Processing in relational structures: Fuzzy relational equations

The paper provides an overview of methodology for processing fuzzy information in relational structures completed in terms of fuzzy relational equations. The origin and the central role of relational calculus in general, and fuzzy sets in particular, is explained with special emphasis paid to its role and representation capabilities. It is pointed out that fuzzy relational equations play a significant role as a platform for a uniform development of techniques in fuzzy sets. Many of the problems and frameworks developed in fuzzy sets so far can be easily translated into the language of fuzzy relational equations which immediately takes a significant advantage of their solid, well developed formalisms, techniques, and clarity of exposition. We will study the role of approximate solutions to fuzzy relational equations as a convenient tool to handle probabilistic type of uncertainty. Moreover those solutions can easily explain the origin and generate a way in which fuzzy sets of higher generality (such as for instance type-2 sets, interval-valued sets) can be algorithmically determined. Finally, we study the role of equations in diverse fields of applications making use of an ample framework of general systems theory.