Study on Rough Sets and Fuzzy Sets in Constructing Intelligent Information System

Since human being is not an omniscient and omnipotent being, we are actually living in an uncertain world. Uncertainty was involved and connected to every aspect of human life as a quotation from Albert Einstein said: �As far as the laws of mathematics refer to reality, they are not certain. And as far as they are certain, they do not refer to reality.� The most fundamental aspect of this connection is obviously shown in human communication. Naturally, human communication is built on the perception1-based information instead of measurement-based information in which perceptions play a central role in human cognition [Zadeh, 2000]. For example, it is naturally said in our communication that �My house is far from here.� rather than let say �My house is 12,355 m from here�. Perception-based information is a generalization of measurement-based information, where perception-based information such as �John is excellent.� is hard to represent by measurement-based version. Perceptions express human subjective view. Consequently, they tend to lead up to misunderstanding. Measurements then are needed such as defining units of length, time, etc., to provide objectivity as a means to overcome misunderstanding. Many measurers were invented along with their methods and theories of measurement. Hence, human cannot communicate with measurers including computer as a product of measurement era unless he uses measurement-based information. Perceptions are intrinsic aspect in uncertainty-based information. In this case, information may be incomplete, imprecise, fragmentary, not fully reliable, vague, contradictory, or deficient in some other way. 1In psychology, perception is understood as a process of translating sensory stimulation into an organized experience Generally, these various information deficiencies may express different types of uncertainty. It is necessary to construct a computer-based information system called intelligent information system that can process uncertainty-based information. In the future, computers are expected to be able to make communication with human in the level of perception. Many theories were proposed to express and process the types of uncertainty such as probability, possibility, fuzzy sets, rough sets, chaos theory and so on. This book extends and generalizes existing theory of rough set, fuzzy sets and granular computing for the purpose of constructing intelligent information system. The structure of this book is the following: In Chapter 2, types of uncertainty in the relation to fuzziness, probability and evidence theory (belief and plausibility measures) are briefly discussed. Rough set regarded as another generalization of crisp set is considered to represent rough event in the connection to the probability theory. Special attention will be given to formulation of fuzzy conditional probability relation generated by property of conditional probability of fuzzy event. Fuzzy conditional probability relation then is used to represent similarity degree of two fuzzy labels. Generalization of rough set induced by fuzzy conditional probability relation in terms of covering of the universe is given in Chapter 3. In the relation to fuzzy conditional probability relation, it is necessary to consider an interesting mathematical relation called weak fuzzy similarity relation as a generalization of fuzzy similarity relation proposed by Zadeh [1995]. Fuzzy rough set and generalized fuzzy rough set are proposed along with the generalization of rough membership function. Their properties are examined. Some applications of these methods in information system such as α-redundancy of object and dependency of domain attributes are discussed. In addition, multi rough sets based on multi-context of attributes in the presence of multi-contexts information system is defined and proposed in Chapter 4. In the real application, depending on the context, a given object may have different values of attributes. In other words, set of attributes might be represented based on different context, where they may provide different values for a given object. Context can be viewed as background or situation in which somehow it is necessary to group some attributes as a subset of attributes and consider the subset as a context. Finally, Chapter 5 summarizes all discussed in this book and puts forward some future topics of research

On the Relation of Probability, Fuzziness, Rough and Evidence Theory

Since the appearance of the first paper on fuzzy sets proposed by Zadeh in 1965, the relationship between probability and fuzziness in the representation of uncertainty has been discussed among many people. The question is whether probability theory itself is sufficient to deal with uncertainty. In this paper the relationship between probability and fuzziness is analyzed by the process of perception to simply understand the relationship between them. It is clear that probability and fuzziness work in different areas of uncertainty. Here, fuzzy event in the presence of probability theory provides probability of fuzzy event in which fuzzy event could be regarded as a generalization of crisp event. Moreover, in rough set theory, a rough event is proposed representing two approximate events, namely lower approximate event and upper approximate event. Similarly, in the presence of probability theory, rough event can be extended to be probability of rough event. Finally, the paper shows and discusses relation among lower-upper approximate probability (probability of rough events), belief-plausibility measures (evidence theory), classical probability measures, probability of generalized fuzzy-rough events and probability of fuzzy events

Extended Generalization of Fuzzy Rough Sets

This paper extends and generalizes the approximations of fuzzy rough sets dealing with fuzzy coverings of the universe induced by a weak fuzzy similarity relation. The weak fuzzy similarity relation is considered as a generalization of fuzzy similarity relation in representing a more realistic relationship between two objects in which it has weaker symmetric and transitive properties. Since the conditional symmetry in the weak fuzzy similarity relation is an asymmetric property, there are two distinct fuzzy similarity classes that provide two different fuzzy coverings. The generalization of fuzzy rough sets approximations is discussed based on two interpretations: object-oriented generalization and class-oriented generalization. More concepts of generalized fuzzy rough set approximations are introduced and defined, representing more alternatives to provide level-2 interval-valued fuzzy sets. Moreover, through combining several pairs of proposed approximations of the generalized fuzzy rough sets, it is possible to provide the level-2 type-2 fuzzy sets as an extension of the level-2 interval valued fuzzy sets. Some properties of the concepts are examined

Fuzzy-rough set and fuzzy ID3 decision approaches to knowledge discovery in datasets

Fuzzy rough sets are the generalization of traditional rough sets to deal with both fuzziness and vagueness in data. The existing researches on fuzzy rough sets mainly concentrate on the construction of approximation operators. Less effort has been put on the knowledge discovery in datasets with fuzzy rough sets. This paper mainly focuses on knowledge discovery in datasets with fuzzy rough sets. After analyzing the previous works on knowledge discovery with fuzzy rough sets, we introduce formal concepts of attribute reduction with fuzzy rough sets and completely study the structure of attribute reduction

Rough sets based on Galois connections

Rough set theory is an important tool to extract knowledge from relational databases. The original definitions of approximation operators are based on an indiscernibility relation, which is an equivalence one. Lately. different papers have motivated the possibility of considering arbitrary relations. Nevertheless, when those are taken into account, the original definitions given by Pawlak may lose fundamental properties. This paper proposes a possible solution to the arising problems by presenting an alternative definition of approximation operators based on the closure and interior operators obtained from an isotone Galois connection. We prove that the proposed definition satisfies interesting properties and that it also improves object classification tasks

Approximate Reasoning in the Knowledge-based Dynamic Fuzzy Sets

Intan and Mukaidono discussed that knowledge plays an important role in determining the membership function of a given fuzzy set by introducing a concept, called Knowledge-based Fuzzy Sets (KFS) in 2002. Here, the membership degree of an element given a fuzzy set is subjectively determined by the knowledge. Every knowledge may have each different membership degree of the element given the fuzzy set. In 1988, Wang et al. extended the concept of fuzzy set, called Dynamic Fuzzy Sets (DFS) by considering that the membership degree of an element given a fuzzy set might be dynamically changeable over the time. Both generalized concepts, KFS and DFS, were hybridized by Intan et al. to be a Knowledge-based Dynamic Fuzzy Set (KDFS). As usually happened in the real-world application, the KDFS showed that a membership function of a given fuzzy set subjectively determined by a certain knowledge may be dynamically changeable over time. Moreover, the concept of fuzzy granularity was discussed dealing with the KDFS. Related to the concept of fuzzy granularity in KDFS, this paper discusses the concept of approximate reasoning of KDFS in representing fuzzy production rules as generally applied in the fuzzy expert system. Ke

Approximate Reasoning in the Knowledge-Based Dynamic Fuzzy Sets

Intan and Mukaidono discussed that knowledge plays an important role in determining the membership function of a given fuzzy set by introducing a concept, called Knowledge-based Fuzzy Sets (KFS) in 2002. Here, the membership degree of an element given a fuzzy set is subjectively determined by the knowledge. Every knowledge may have each different membership degree of the element given the fuzzy set. In 1988, Wang et al. extended the concept of fuzzy set, called Dynamic Fuzzy Sets (DFS) by considering that the membership degree of an element given a fuzzy set might be dynamically changeable over the time. Both generalized concepts, KFS and DFS, were hybridized by Intan et al. to be a Knowledge-based Dynamic Fuzzy Set (KDFS). As usually happened in the real-world application, the KDFS showed that a membership function of a given fuzzy set subjectively determined by a certain knowledge may be dynamically changeable over time. Moreover, the concept of fuzzy granularity was discussed dealing with the KDFS. Related to the concept of fuzzy granularity in KDFS, this paper discusses the concept of approximate reasoning of KDFS in representing fuzzy production rules as generally applied in the fuzzy expert system

New Generalized Definitions of Rough Membership Relations and Functions from Topological Point of View

In this paper, we shall integrate some ideas in terms of concepts in topology. In fact, we introduce two different views to define generalized membership relations and functions as mathematical tools to classify the sets and help for measuring exactness and roughness of sets. Moreover, we define several types of fuzzy sets. Comparisons between the induced operations were discussed. Finally, many results, examples and counter examples to indicate connections are investigated