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2009 | Buch

Fundamentals Kapitel lesen Erstes Kapitel lesen

Axiomatic Fuzzy Set Theory and Its Applications

verfasst von: Xiaodong Liu, Witold Pedrycz

Verlag: Springer Berlin Heidelberg

Buchreihe : Studies in Fuzziness and Soft Computing

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

It is well known that “fuzziness”—informationgranulesand fuzzy sets as one of its formal manifestations— is one of important characteristics of human cognitionandcomprehensionofreality. Fuzzy phenomena existinnature and are encountered quite vividly within human society. The notion of a fuzzy set has been introduced by L. A. , Zadeh in 1965 in order to formalize human concepts, in connection with the representation of human natural language and computing with words. Fuzzy sets and fuzzy logic are used for mod- ing imprecise modes of reasoning that play a pivotal role in the remarkable human abilities to make rational decisions in an environment a?ected by - certainty and imprecision. A growing number of applications of fuzzy sets originated from the “empirical-semantic” approach. From this perspective, we were focused on some practical interpretations of fuzzy sets rather than being oriented towards investigations of the underlying mathematical str- tures of fuzzy sets themselves. For instance, in the context of control theory where fuzzy sets have played an interesting and practically relevant function, the practical facet of fuzzy sets has been stressed quite signi?cantly. However, fuzzy sets can be sought as an abstract concept with all formal underpinnings stemming from this more formal perspective. In the context of applications, it is worth underlying that membership functions do not convey the same meaning at the operational level when being cast in various contexts.

Inhaltsverzeichnis

Frontmatter

Required Preliminary Mathematical Knowledge

Frontmatter
Fundamentals
Abstract
The main objective of this chapter is to introduce some preliminaries regarding essential mathematical notions and mathematical structures that have been commonly encountered in the theory of topological molecular lattices, fuzzy matrices, AFS (Axiomatic Fuzzy Set) structures and AFS algebras. The proofs of some theorems or propositions which are not too difficult to be proved are left to the reader as exercises.
Xiaodong Liu, Witold Pedrycz
Lattices
Abstract
This chapter offers a concise introduction to lattices, Boolean algebras, topological molecular lattices and shows main relations between them. For details, the readers may refer to [1, 3, 2]. Our purpose is to familiarize the readers with the concepts and fundamental results, which will be exploited in further discussion. Some results listed without proofs is left to the reader.
Xiaodong Liu, Witold Pedrycz

Methodology and Mathematical Framework of AFS Theory

Frontmatter
Boolean Matrices and Binary Relations
Abstract
Considering the three types of information-driven tasks where graded membership plays a role: classification and data analysis, decision-making problems, and approximate reasoning, Dubois gave the corresponding semantics of the membership grades, expressed in terms of similarity, preference, and uncertainty [1]. For a fuzzy concept ξ in the universe of discourse X, by comparison of the graded membership (Dubois interpretation of membership degree), an “empirical relational membership structure ” \( \left<X, R_{\xi } \right>\) is induced [5,6], where R ξ  ⊆ X×X is a binary relation on X, (x, y) ∈ R ξ if and only if an observer, an expert, judges that “x belongs to ξ at some extent and the degree of x belonging to ξ is at least as large as that of y. The fundamental measurement of the gradual-set membership function can be formulated as the construction of homomorphisms from an “empirical relational membership structure”, \(\left<X, R_{\xi }\right>\), to a “numerical relational membership structure”, \(\left<{\{} \mu _{\xi }(x) \ \vert \ x \in X {\}}, \le \right>\).
Xiaodong Liu, Witold Pedrycz
AFS Logic, AFS Structure and Coherence Membership Functions
Abstract
In this chapter, we start with an introduction to EI algebra and AFS structure. Then the coherence membership functions of fuzzy concepts for AFS fuzzy logic for the AFS structure are proposed and a new framework of determining coherence membership functions is developed by taking both fuzziness (subjective imprecision) and randomness (objective uncertainty) into account. Singpurwalla’s measure of the fuzzy events in a probability space has been applied to explore the proposed framework. Finally, the consistency, stability, efficiency and practicability of the proposed methodology are illustrated and studied via various numeric experiments. The investigations in this chapter open a door to explore the deep statistic properties of fuzzy sets. In this sense, they may offer further insights as to the to a role of natural languages in probability theory.
The aim of this chapter is to develop a practical and effective framework supporting the development of membership functions of fuzzy concepts based on semantics and statistics completed with regard to fuzzy data. We show that the investigations concur with the main results of the Singpurwalla’s theory [44].
Xiaodong Liu, Witold Pedrycz
AFS Algebras and Their Representations of Membership Degrees
Abstract
In this chapter, first we construct some lattices—AFS algebras using sets X and M over an AFS structure (M,τ,X) for the representation of the membership degrees of each sample x ∈ X belonging to the fuzzy concepts in EM. Then the mathematical properties and structures of AFS algebras are exhaustively discussed. Finally, the relations, advantages and drawbacks of various kinds of AFS representations for fuzzy concepts in EM are analyzed. Some results listed without proofs are left for the reader as exercises.
Xiaodong Liu, Witold Pedrycz

Applications of AFS Theory

Frontmatter
AFS Fuzzy Rough Sets
Abstract
In this chapter, in order to describe the linguistically represented concepts coming from data available in a certain information system, the concept of fuzzy rough sets are redefined and further studied in the setting of the Axiomatic Fuzzy Set (AFS) theory. These concepts will be referred to as AFS fuzzy rough sets [32]. Compared with the “conventional” fuzzy rough sets, the advantages of AFS fuzzy rough sets are twofold. They can be directly applied to data analysis present in any information system without resorting to the details concerning the choice of the implication φ, t-norm and a similarity relation S. Furthermore such rough approximations of fuzzy concepts come with a well-defined semantics and therefore offer a sound interpretation.
The underlying objective of this chapter is to demonstrate that the AFS rough sets constructed for fuzzy sets form their meaningful approximations which are endowed by the underlying semantics. At the same time, the AFS rough sets become directly reflective of the available data.
Xiaodong Liu, Witold Pedrycz
AFS Topology and Its Applications
Abstract
In this chapter, first we construct some topologies on the AFS structures, discuss the topological molecular lattice structures on EI, * EI, EII, * EII algebras, and elaborate on the main relations between these topological structures. Second, we apply the topology derived by a family of fuzzy concepts in EM, where M is a set of simple concepts, to analyze the relations among the fuzzy concepts. Thirdly, we propose the differential degrees and fuzzy similarity relations based on the topological molecular lattices generated by the fuzzy concepts on some features. Furthermore, the fuzzy clustering problems are explored using the proposed differential degrees and fuzzy similarity relations. Compared with other fuzzy clustering algorithms such as the Fuzzy C-Means and k-nearest-neighbor fuzzy clustering algorithms, the proposed fuzzy clustering algorithm can be applied to data sets with mixed feature variables such as numeric, Boolean, linguistic rating scale, sub-preference relations, and even descriptors associated with human intuition. Finally, some illustrative examples show that the proposed differential degrees are very effective in pattern recognition problems whose data sets do not form a subset of a metric space such as the Eculidean one. This approach offers a promising avenue that could be helpful in understanding mechanisms of human recognition.
Xiaodong Liu, Witold Pedrycz
AFS Formal Concept and AFS Fuzzy Formal Concept Analysis
Abstract
In this chapter, based on the original idea of Wille of formal concept analysis and the AFS (Axiomatic Fuzzy Set) theory, we presents a rigorous mathematical treatment of fuzzy formal concept analysis referred to as an AFS Formal Concept Analysis (AFSFCA). It naturally augments the existing formal concepts to fuzzy formal concepts, with the aim of deriving their mathematical properties and applying them in the exploration and development of knowledge representation. Compared with other fuzzy formal concept approaches such as the L-concept [1,2] and the fuzzy concept [48], the main advantages of AFSFCA are twofold. One is that the original data and facts are the only ones required to generate AFSFCA lattices thus human interpretation is not required to define the fuzzy relation or the fuzzy set on G×M to describe the uncertainty dependencies between the objects in G and the attributes in M. Another advantage comes with the fact that is that AFSFCA is more expedient and practical to be directly applied to real world applications.
Xiaodong Liu, Witold Pedrycz
AFS Fuzzy Clustering Analysis
Abstract
In this chapter, we apply the AFS theory to propose an elementary algorithm of fuzzy clustering. In the proposed approach, each cluster is interpreted by taking advantage of the semantics captured by the AFS logic. Within the framework of AFS theory, we develop new techniques of feature selection, concept categorization and characteristic description (i.e.,the characteristic description of an object or a group of objects using the fuzzy concepts) which are often encountered in tasks of machine learning and pattern recognition. The elementary fuzzy clustering algorithm is evolved to three more elaborate fuzzy clustering techniques by incorporating new techniques of feature selection, concept categorization and characteristic description. We show that they are simpler and produce more interpretable results when contrasted with some existing techniques. Several benchmark data and the evaluation data of 30 companies are considered to evaluate the effectiveness of the proposed AFS fuzzy clustering algorithms. We provide a detailed comparative analysis in which we compare the obtained results with those produced by some “conventional” methods such as FCM, k-means, and some newer algorithms including a two-level SOM-based clustering algorithm. The proposed algorithms can be applied to the data sets with mixed features such as sub-preference relations and even those including descriptions of human intuitive judgment. We show that the flexibility of the approach comes from the fact that the distance function and the class number need not be given beforehand. These two facets offers a far more higher flexible and contribute to a powerful framework for representing human knowledge and studying intelligent systems encountered in real world applications.
Xiaodong Liu, Witold Pedrycz
AFS Fuzzy Classifiers
Abstract
In this chapter, we introduce three design strategies of classifiers which exploit the unified usage of the AFS fuzzy logic, entropy measures and decision trees. The advantage of these classifiers is two-fold. First, they can mimic the human reasoning and in this manner offer a far more transparent and comprehensible way supporting the design of the classifiers. An important aspect is concerned with the simplicity of the design methodology and the clarity of the underlying semantics. We use three well known data to illustrate the effectiveness of the classifiers and present the relationship between the parameters of the classifiers and their performance.
Xiaodong Liu, Witold Pedrycz
Backmatter
Metadaten
Titel
Axiomatic Fuzzy Set Theory and Its Applications
verfasst von
Xiaodong Liu
Witold Pedrycz
Copyright-Jahr
2009
Verlag
Springer Berlin Heidelberg
Electronic ISBN
978-3-642-00402-5
Print ISBN
978-3-642-00401-8
DOI
https://doi.org/10.1007/978-3-642-00402-5

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