Fuzzy Sets and Their Extensions: Representation, Aggregation and Models

This chapter provides a review of the development of the field of type-2 fuzzy logic. We explore some underpinning philosophical arguments that support the notion of type-2 fuzzy logic. We give the fundamental definitions of type-2 fuzzy sets and basic logical operations. The key stages in development of the field are reviewed and placed in a historical context. In addition, we report an example application of type-2 fuzzy logic to mobile robot navigation, demonstrating the potential of type-2 fuzzy systems to outperform type-1 fuzzy systems.

In this chapter, some remarks are given on the history, theory, applications and research on the extension of fuzzy sets model proposed by the author in 1983.

Though fuzzy set theory has been a very popular technique to represent vagueness between sets and their elements, the approximation of a subset in a universe that contains finite objects was still not resolved until the Pawlak’s rough set theory was introduced. The concept of rough sets was introduced by Pawlak in 1982 as a formal tool for modeling and processing incomplete information in information systems. Rough sets describe the approximation of an arbitrary subset of a universe by two definable or observable subsets called lower and upper approximations. Even though Pawlak’s rough set theory has been widely applied to solve many real world problems, the problem of being not able to deal with real attribute values had been spotted and found. This problem is originated in the crispness of upper and lower approximation sets in traditional rough set theory (TRS). Under the TRS philosophy, two nearly identical real attribute values are unreasonably treated as two different values. TRS theory deals with this problem by discretizing the original dataset, which may result in unacceptable information loss for a large amount of applications. To solve the above problem, a natural way of combining fuzzy sets and rough sets has been proposed. Since 1990’s, researchers had put a lot of efforts on this area and two fuzzy rough set techniques that hybridize fuzzy and rough sets had been proposed to extend the capabilities of both fuzzy sets and rough sets. This chapter does not intend to cover all fuzzy rough set theories. Rather, it firstly gives a brief introduction of the state of the art in this research area and then goes into details to discuss two kinds of well developed hybridization approaches, i.e., constructive and axiomatic approaches. The generalization for equivalence relationships, the definitions of lower and upper approximation sets and the attribute reduction techniques based on these two hybridization frameworks are introduced in different sections. After that, to help readers apply the fuzzy rough set techniques, this chapter also introduces some applications that have successfully applied fuzzy rough set techniques. The final section of this chapter gives some remarks on the merits and problems of each fuzzy rough hybridization technique and the possible research directions in the future.

This chapter will give an overview of the label semantics framework for computing with words. Label semantics is an alternative methodology that models linguistic labels in terms of label descriptions, appropriateness measures and mass assignments. It has a clear operational semantics and is straightforward to integrate with other uncertainty formalisms. In the chapter we will introduce the theory of label semantics, discuss its compatibility with a functional calculus and show how it can be used to infer both probabilistic and possibilistic information from linguistic assertions.

In this chapter, we review some families of fuzzy measures and their use in fuzzy integrals. We will also review the determination of fuzzy measures from examples in the case of the Choquet integral.

In this chapter we examine a number of methods to construct aggregation operators of interpolatory type for specific applications. The construction is based on the desired values of the aggregation operator at certain prototypical points, and on other desired properties, such as conjunctive, disjunctive or averaging behaviour, symmetry and marginals.

Several local and global properties of (extended) aggregation functions are discussed and their relationships are examined. Some special classes of averaging, conjunctive and disjunctive aggregation functions are reviewed. A special attention is paid to the weighted aggregation functions, including some construction methods

This chapter provides a review of various techniques for identification of weights in generalized mean and ordered weighted averaging aggregation operators, as well as identification of fuzzy measures in Choquet integral based operators. Our main focus is on using empirical data to compute the weights. We present a number of practical algorithms to identify the best aggregation operator that fits the data.

Linguistic aggregation operators are a powerful tool to aggregate linguistic information, which have been studied and applied in a wide variety of areas, including engineering, decision making, artificial intelligence, data mining, and soft computing. In this chapter, we provide a comprehensive survey of the existing main linguistic aggregation operators, and briefly discuss their characteristics and applications. Finally, we roughly classify all these linguistic aggregation operators and conclude with a discussion of some interesting further research directions.

In this chapter we give an overview of some recent advances on aggregation operators on

L

I

, where

L

I

is the underlying lattice of interval-valued fuzzy set theory (which is equivalent to Atanassov’s intuitionistic fuzzy set theory). We discuss some special classes of t-norms on

L

I

and their properties. We show that the t-representable t-norms, which are constructed as a pair of t-norms on [0,1], are not the t-norms with the most interesting properties. We study additive generators of t-norms on

L

I

, uninorms on

L

I

and generators of uninorms on

L

I

. We give the general definition and some special classes of aggregation operators on

L

I

. Finally we discuss the generalization of Yager’s OWA operators to interval-valued fuzzy set theory.

The construction of fuzzy strict preference, indifference and incomparability relations from a fuzzy large preference relation is usually cast into an axiomatic framework based on t-norms. In this contribution, we show that this construction is essentially characterized by the choice of an indifference generator, a symmetrical mapping located between the L ukasiewicz t-norm and the minimum operator. Interesting constructions are obtained by choosing as indifference generator a commutative quasi-copula, an ordinal sum of Frank t-norms or a particular Frank t-norm.

A group selection of one alternative from a set of feasible ones should be based on the preferences of individuals in the group. Decision making procedures are usually based on pair comparisons, in the sense that processes are linked to some degree of credibility of preference. The main advantage of pairwise comparison is that of focusing exclusively on two alternatives at a time and on how they are related. However, it generates more information that needed and therefore inconsistent information may be generated. This paper addresses both preference representation and consistency of preferences issues in group decision making.

Different preference representation formats individuals can use to model or present their preferences on a set of alternatives in a group decision making situation are reviewed. The results regarding the relationships between these preference representation formats mean that the fuzzy preference relation “is preferred to” representing the strength of preference of one alternative over another in the scale [0,1] can be used as the base element to integrate these different preference representation formats in group decision making situations.

Due to the complexity of most decision making problems, individuals’ preferences may not satisfy formal properties that fuzzy preference relations are required to verify. Consistency is one of them, and it is associated with the

transitivity property.

Many properties have been suggested to model transitivity of fuzzy preference relations. As aforementioned, this paper provides an overview of the main results published in this area.

Rough set theory has been proposed by Pawlak in the early 80s to deal with inconsistency problems following from information granulation. It operates on an information table composed of a set

U

of objects described by a set

Q

of condition and decision attributes. Decision attributes make a partition of

U

into decision classes. Basic concepts of rough set theory are: indiscernibility relation on

U

, lower and upper (rough) approximations of decision classes, dependence and reduction of attributes from

Q

, and decision rules induced from rough approximations of decision classes. The original rough set idea was failing, however, to handle preferential ordering of domains of attributes (scales of criteria), as well as preferential ordering of decision classes. In order to deal with multiple criteria decision problems a number of methodological changes to the original rough set theory were necessary. The main change is the substitution of the indiscernibility relation by a dominance relation, which permits approximation of ordered sets. In multiple criteria decision context, the information table is composed of decision examples given by a decision maker. The Dominance-based Rough Set Approach (DRSA) applied to this information table results with a set of decision rules, being a preference model of the decision maker. It is more general than the classical multiple attribute utility model or outranking model, and it is more understandable because of its natural syntax. In this chapter, after recalling the classical rough set approach and DRSA, we review their fuzzy set extensions. Moreover, we characterize the dominance-based rough approximation of a fuzzy set, and we show that the classical rough approximation of a crisp set is its particular case. In this sense, DRSA is also relevant in the case where preferences are not considered, but just a kind of monotonicity relating values of different attributes is meaningful for the analysis of data at hand. In general terms, monotonicity concerns relationship between different aspects of a phenomenon described by data: for example, “the larger the house, the higher its price” or “the closer the house to the city centre, the higher its price”. In this perspective, DRSA gives a very general framework for reasoning about data using only monotonicity relationships.

Group decision making, as meant in this chapter, is the following choice problem which proceeds in a multiperson setting. There is a group of individuals (decisionmakers, experts, ...) who provide their testimonies concerning an issue in question. These testimonies are assumed here to be individual preference relations over some set of option (alternatives, variants, ...). The problem is to find a solution, i.e. an alternative or a set of alternatives, from among the feasible ones, which best reflects the preferences of the group of individuals as a whole. We will survey main developments in group decision making under fuzziness. First, we will briefly outline some basic inconsistencies and negative results of group decision making and social choice, and show how they can be alleviated by some plausible modifications of underlying assumptions, mainly by introducing fuzzy preference relations and, to a lesser extent, a fuzzy majority. Then, we will concentrate on how to derive solutions under individual fuzzy preference relations, and a fuzzy majority equated with a fuzzy linguistic quantifier (e.g., most, almost all, ...) and dealt with in terms of a fuzzy logic based calculus of linguistically quantified statements or via the ordered weighted averaging (OWA) operators. We will briefly mention that one of solution concepts proposed can be a prototype for a wide class of group decision making choice functions. Then, we will discuss a related issue of how to define a “soft” degree of consensus in the group under individual fuzzy preference relations and a fuzzy majority. Finally, we will show how fuzzy preferences can help alleviate some voting paradoxes.

In the classical theory of social choice, there exist many voting procedures for determining a collective preference on a set of alternatives. The simplest situation happens when a group of individuals has to choose between two alternatives. In this context, some voting procedures such as simple and absolute special majorities are frequently used. However, these voting procedures do not take into account the intensity with which individuals prefer one alternative to the other. In order to consider this situation, one possibility is to allow individuals showing their preferences through values located between 0 and 1. In this case, the collective preference can be obtained by means of an aggregation operator. One of the most important matter in this context is how to choose such aggregation operator. When we consider the class of OWA operators, it is necessary to determine the associated weights. In this contribution we survey several methods for obtaining the OWA operator weights. We pay special attention to the way the weights are chosen, regarding the concrete voting system we want to obtain when individuals do not grade their preferences between the alternatives.

The evaluation is a process that analyzes elements to achieve different objectives such as quality inspection, design, marketing exploitation and other fields in industrial companies. In many of these fields the items, products, designs, etc., are evaluated according to the knowledge acquired via human senses (sight, taste, touch, smell and hearing), in such cases, the process is called

Sensory Evaluation

. In this type of evaluation process, an important problem arises as it is the modelling and management of uncertain knowledge, because the information acquired by our senses throughout human perceptions involves uncertainty, vagueness and imprecision. The Fuzzy Linguistic Approach [34] has showed its ability to deal with uncertainty, ambiguity, imprecision and vagueness, so it seems logic and suitable the use of the Fuzzy Linguistic Approach to model the information provided by the experts in sensory evaluation processes.

The decision analysis has been usually used in evaluation processes because it is a formal methodology that can help to achieve the evaluation objectives. In this chapter we present a linguistic evaluation model for sensory evaluation based on the decision analysis scheme that will use the Fuzzy Linguistic Approach and the 2-tuple fuzzy linguistic representation to model and manage the uncertainty and vagueness of the information acquired through the human perceptions in the sensory evaluation process. This model will be applied to some sensory evaluation processes of the Olive Oil.

Since decision making is omnipresent in any human activity, it is quite clear that not much later after the concept of a fuzzy set was introduced as a tool for a description and handling of imprecise concepts, a next rational step was an attempt to devise a general framework for dealing with decision making under fuzziness. Since intuitionistic fuzzy sets (in the sense of Atanassov, to be called A-IFSs, for short) provide a richer apparatus to grasp imprecision than the conventional fuzzy sets, they seem to be a promising tool for extended decision making models. We will present some of the extended models and try to show why A-IFSs make it possible to avoid some more common cognitive biases, the decision makers are prone to do, which call into question the correctness of a decision.

Methods for the automated induction of models and the extraction of interesting patterns from empirical data have recently attracted considerable attention in the fuzzy set community. This chapter briefly reviews some typical applications and highlights potential contributions that fuzzy set theory can make to machine learning, data mining, and related fields. Finally, a critical consideration of recent developments is given and some suggestions regarding future research are made.

Linguistic rules are fuzzy rules described by linguistic terms such as

small

and

large

. Here we discuss pattern classification with linguistic rules. The main advantage of using linguistic rules is their high interpretability. We can construct linguistically interpretable fuzzy rule-based classification systems using linguistic rules. First we briefly explain fuzzy rules for function approximation. Next we explain fuzzy rules and fuzzy reasoning for pattern classification. Then we explain linguistic rule extraction from numerical data. Finally we show some future research topics on pattern classification with linguistic rules.

Data mining is the process of extracting desirable knowledge or interesting patterns from existing databases for specific purposes. Many types of knowledge and technology have been proposed for data mining. Among them, finding association rules from transaction data is most commonly seen. Most studies have shown how binary valued transaction data may be handled. Transaction data in real-world applications, however, usually consist of fuzzy and quantitative values, so designing sophisticated data-mining algorithms able to deal with various types of data presents a challenge to workers in this research field. This chapter thus surveys some fuzzy mining concepts and techniques related to association-rule discovery. The motivation from crisp mining to fuzzy mining will be first described. Some crisp mining techniques for handling quantitative data will then be briefly reviewed. Several fuzzy mining techniques, including mining fuzzy association rules, mining fuzzy generalized association rules, mining both membership functions and fuzzy association rules, will then be described. The advantages and the limitations of fuzzy mining will also be discussed.

Subgroup discovery can be defined as a form of supervised inductive learning in which, given a population of individuals and a specific property of individuals in which we are interested, find population subgroups that have the most unusual distributional characteristics with respect to the property of interest. Subgroup discovery algorithms aim at discovering individual rules, which must be represented in explicit symbolic form and which must be simple and understandable in order to be recognized as actionable by potential users.

A fuzzy approach for a subgroup discovery process, which considers linguistic variables with linguistic terms in descriptive fuzzy rules, lets us obtain knowledge in a similar way of the human thought process. Linguistic rules are naturally inclined towards coping with linguistic knowledge and to produce more interpretable and actionable solutions. This chapter analyzes the use of linguistic rules for modelling this problem, and shows a genetic extraction model for learning this kind of rules.

Cognitive psychology works have shown that the cognitive representation of categories is based on a typicality notion: all objects of a category do not have the same representativeness, some are more characteristic or more typical than others, and better exemplify their category. Categories are then defined in terms of prototypes, i.e. in terms of their most typical elements. Furthermore, these works showed that an object is all the more typical of its category as it shares many features with the other members of the category and few features with the members of other categories.

In this paper, we propose to profit from these principles in a machine learning framework: a formalization of the previous cognitive notions is presented, leading to a prototype building method that makes it possible to characterize data sets taking into account both common and discriminative features. Algorithms exploiting these prototypes to perform tasks such as classification or clustering are then presented.

The formalization is based on the computation of

typicality degrees

that measure the representativeness of each data point. These typicality degrees are then exploited to define

fuzzy prototypes

: in adequacy with human-like description of categories, we consider a prototype as an intrinsically imprecise notion. The fuzzy logic framework makes it possible to model sets with unsharp boundaries or vague and approximate concepts, and appears most appropriate to model prototypes. We then exploit the computed typicality degrees and the built fuzzy prototypes to perform machine learning tasks such as classification and clustering. We present several algorithms, justifying in each case the chosen parameters. We illustrate the results obtained on several data sets corresponding both to crisp and fuzzy data.

In this chapter we consider remotely sensed images, where land surface should be classified depending on their uses. On one hand, we discuss the advantages of the fuzzy classification model proposed by Amo et al. (European Journal of Operational Research, 2004) versus standard approaches. On the other hand, we introduce a coloring algorithm by to Gòmez et al. (Omega, to appear) in order to produce a supervised algorithm that takes into account a previous segmentation of the image that pursues the identification of possible homogeneous regions. This algorithm is applied to a real image, showing its high improvement in accuracy, which is then measured.

In this chapter we present a new generalized agent-based model to implement fuzzy logic controllers (FLCs) distributed according to the Agents Paradigm. The discussed model is presented along with FIS2JADE, a tool developed to convert Simulink®; FIS-format controllers into a Multi-Agent System adhering to the defined model, in order to improve developers efficiency when dealing with distributed control. We present the model describing first its founding elements and concepts. Then we present the model design by means of a simple and well-known case study, the stationary inverted pendulum control problem, allowing us to focus on the model definition rather than on the control problem to which it is applied. Lastly, we show snapshots from the detailed design and implementation process when choosing the JADE Agents Platform to implement the derived Multi-Agent System.

The so-called measure of approximation quality plays an important role in many applications of rough set based data analysis. In this chapter, we provide an overview on various extensions of approximation quality based on rough-fuzzy and fuzzy-rough sets, along with highlighting their potential applications as well as future directions for research in the topic.

In this contribution some applications of Fuzzy Set Theory to Information Retrieval are described, as well as the more recent outcomes of this research field. Fuzzy Set Theory is applied to Information Retrieval to the main aim to define

flexible

systems, i.e. systems that can represent and manage the vagueness and subjectivity which characterizes the process of information representation and retrieval.

In this work, a description of what is a Web Meta-search engine and the roles that fuzzy logic as a Soft Computing technique, can play to improve the search with this kind of artefacts are described. Fuzzy logic can provide tools for extracting and use knowledge from thesaurus and ontologies, formalize sentences and implement deduction capabilities in Question Answering systems, combine fuzzy values and different logics, design clustering algorithms and manage Meta-search engines architectures.

E-services involve various types, delivery systems, advanced information technologies, methodologies and applications of online services that are provided by e-government, e-business, e-commerce, e-market, e-finance, e-learning systems, to name a few. They offer great opportunities and challenges for many areas, such as government, business, commerce, marketing, finance and education. E-service intelligence is a new research field that deals with fundamental roles, social impacts and practical applications of various intelligent technologies on the Internet based e-services. This chapter aims to offer a thorough introduction and systematic overview of the new field e-service intelligence mainly based on fuzzy set related techniques. It covers the state-of-the-art of the research and development in various aspects including both theorems and applications, of e-service intelligence by applying fuzzy set theory. Moreover, it demonstrates how adaptations of existing intelligent technologies benefit from the development of e-service applications in online customer decision, personalised services, web mining, online searching/data retrieval, online pattern recognition/image processing, and web-based e-logistics/planning.

As it is known the Web is changing the information access processes. The Web is one of the most important information media. Furthermore, the Web is influencing in the development of other information media, as for example, newspapers, journals, books, libraries, etc. In this chapter we analyze its impact in the development of the University Digital Libraries (UDL). As in the Web, the growing of information is the main problem of the academic digital libraries, and similar tools could be applied in university digital libraries to facilitate the information access to the students and teachers.

Filtering systems

or

recommender systems

are tools whose objective is to evaluate and filter the great amount of information available on the Web to assist the users in their information access processes. Therefore, we present a model of fuzzy linguistic recommender system to help students and researchers to find research resources which could improve the services that render the UDL to their users.

This chapter provides an overview of fuzzy measures and fuzzy integrals, measures of fuzziness, and their application in image processing in the areas of region based segmentation, thresholding, and color retreieval. This chapter also introduces a fuzzy color image retrieval method using a new type of membership function called beta membership function with Generalized Tversky’s index as a measure of fuzziness. Lastly, the proposed method has been compared with some existing fuzzy and non-fuzzy methods.

Type II fuzzy sets are high-level representation of vague data with, compared to ordinary fuzzy sets, greater capability for uncertainty management. Theoretical aspects of type II fuzzy systems have been extensively investigated, and the research is still ongoing. Many image processing tasks accompanied with different types of imperfection. In this chapter, the applications of type II fuzzy sets for image segmentation will be discussed. Global and spatial type II segmentation schemes will be systematically introduced and examples will be provided.

In this chapter, a new thresholding technique using Atanassov’s intuitionistic fuzzy sets (A-IFSs) and restricted dissimilarity functions is introduced. We interpret the intuitionistic fuzzy index of Atanassov as the degree of unknowledge/ignorance of an expert for determining whether a pixel of an image belongs to the background or the object of the image. Under these conditions we construct an algorithm on the basis of A-IFSs for detecting the threshold of an image. Then we present a method for selecting from a set of thresholds of an image the best one. This method is based on the concept of fuzzy similarity. Lastly, we prove that in most cases our algorithm for selecting the best threshold takes the threshold calculated with the algorithm constructed on the basis of A-IFSs.

Objective quality measures or measures of comparison are of great importance in the field of image processing. These measures serve as a tool to evaluate and to compare different algorithms designed to solve particular problems, such as noise reduction, deblurring, compression, ... Consequently these measures serve as a basis on which one algorithm is preferred to another. In [15, 16] we constructed several new fuzzy similarity measures for grey-scale images that outperform the classical measures of comparison, like Root Mean Square Error or Peak Signal to Noise Ratio, in the sense of image quality evaluation. In this chapter we investigate the usefulness of these similarity measures for the comparison of colour images. First of all, we discuss the component-based approach in three different colour spaces, namely the RGB, HSV and Lab colour spaces. And secondly, we discuss a vector-based approach using vector morphological operators. Both approaches are compared by means of several experiments.

There is a compelling need for automated cervical smear screening systems to improve the quality and cost/efficiency screening rate. Computer-assisted devices can reduce false negative Pap smear interpretations using computerized systems to assist the cytotechnologist in identifying Pap smear abnormalities and providing added value in their ability to consistently and objectively analyze all cells on slides without fatigue. However, automation of the process is a challenging problem due to the large variability in conventional Pap smears exhibiting no standard appearance and tremendous amount of data to be processed. Moreover, smear diagnostic may be obscured by benign conditions, overlapping cells, debris, inflammation, and no uniform staining.

Here we propose an efficient and fast Fuzzy-based Automated Cells Screening Detection System -

FACSDS

-, which can be useful for future Automatic Cells Screening System

ACSS

-. Because of detecting abnormal cells in a Pap smear can be refereed to as a “

rare

” event problem due to the normal cells and artifacts outnumber the intraepithelial lesions, the proposed algorithm has been divided into two steps. At first step the Areas of Interest

AOI

-, or best areas for screening in the smear, are detected and the degree to which these areas are interesting is given by means their evaluation goodness degree. At second step the

AOI

s are analyzed, taking into account their evaluation goodness degree, for detecting the cell nucleus. First step is carried out on monochrome images obtained using a 2.5X objective, and the results obtained have provided a high concordance degree with the cytotechnologist evaluation. The automatic system implemented at second step for detecting the nuclei is based on color information. We have considered color images because cells’ nuclei appear as dark regions within the images, hardly detected on monochrome images. Moreover, as color-order systems based on perceptual variables are somehow correlated with human being’s color perception, and their coordinates are highly independent, what makes possible to treat achromatic and chromatic information separately, the proposed algorithm first convert

RGB

into

Hue

,

Saturation

and

Intensity

(

HSI

) color components. In addition, we make use of fuzzy techniques to face up the problems due to low saturation and illumination.