Suyun Zhao

Building a Rule-Based Classifier—A Fuzzy-Rough Set Approach

The fuzzy-rough set (FRS) methodology, as a useful tool to handle discernibility and fuzziness, has been widely studied. Some researchers studied on the rough approximation of fuzzy sets, while some others focused on studying one application of FRS: attribute reduction (i.e., feature selection). However, constructing classifier by using FRS, as another application of FRS, has been less studied. In this paper, we build a rule-based classifier by using one generalized FRS model after proposing a new concept named as ¿consistence degree¿ which is used as the critical value to keep the discernibility information invariant in the processing of rule induction. First, we generalized the existing FRS to a robust model with respect to misclassification and perturbation by incorporating one controlled threshold into knowledge representation of FRS. Second, we propose a concept named as ¿consistence degree¿ and by the strict mathematical reasoning, we show that this concept is reasonable as a critical value to reduce redundant attribute values in database. By employing this concept, we then design a discernibility vector to develop the algorithms of rule induction. The induced rule set can function as a classifier. Finally, the experimental results show that the proposed rule-based classifier is feasible and effective on noisy data.

Rule Induction Based on Fuzzy Rough Sets

In this paper, we propose one method of rule induction based on fuzzy rough set. First, the consistence degree is proposed as the basic concept to induce rules based on fuzzy rough sets. The concepts of rule induction, such as value reduct, reduct rule and so on, are then proposed based on the definition of consistence degree. Second, a discernibility array is constructed, and then an algorithm to find the reduct rule using the discernibility array is designed. Finally, the numerical experimental results demonstrate that the method of rule induction proposed in this paper is feasible. The key idea of this paper is that the value reduct (i.e. reduct rule) keeps the consistence degree invariant. The main contribution of this paper is introduction of rule induction based on fuzzy rough sets using the concept of fuzzy lower and upper approximation.

Attribute Reduction Based on Fuzzy Rough Sets

In Ti¾?fuzzy rough sets a fuzzy Ti¾?similarity relation is employed to describe the similar degree between two objects and to construct lower and upper approximations for arbitrary fuzzy sets. The existing researches on Ti¾?fuzzy rough sets mainly concentrate on constructive and axiomatic approaches of lower and upper approximation operators. In this paper we define attribute reduction based on Ti¾?fuzzy rough sets. The structure of proposed attribute reduction is investigated in detail by the approach of discernibility matrix. At last an example is proposed to illustrate our idea in this paper.

The Model of Fuzzy Variable Precision Rough Sets

One limitation of the fuzzy rough sets is its sensitivity to the perturbation of original numerical data. In this paper we construct a model of fuzzy variable precision rough sets (FVPRS) by combining the fuzzy rough sets and variable precision rough sets, which is non-sensitive to the perturbation of the original numerical data. First, the fuzzy lower and upper approximations of FVPRS model are defined, and their properties are described. Second, the concepts of attributes reduction of FVPRS model, such as attributes reduct, core and positive region, etc, are defined. Third, a discernibility matrix is adopted to develop an algorithm to obtain all the attributes reduction of FVPRS. By the strict mathematical reasoning, we prove that the results obtained by the algorithm based on the discernibility matrix are the exact attributes reducts of FVPRS. Finally, the experimental results demonstrate that the model of FVPRS is feasible and effective in the real problems.

Fuzzy Matrix Computation for Fuzzy Information System to Reduce Attributes

Recently, many methods based on fuzzy rough sets are proposed to reduce fuzzy attributes. The common characteristic of these methods is that all of them are based on fuzzy equivalence relation. In other words, the underlying concept of rough sets, indispensability relation, is generalized to fuzzy equivalence relation. Here fuzzy equivalence relation is the binary relation, which is reflexive, symmetric and transitive. This paper tries to generalize the fuzzy equivalence relation to fuzzy similarity relation, which is more helpful to keeping the fuzzy information of initial data than fuzzy equivalence relation. Based on the fuzzy similarity relation, fuzzy matrix computation for information system is proposed which can be used to reduce fuzzy attributes. Firstly, fuzzy similarity relation who is isomorphic with the fuzzy similarity matrix is given as fuzzy indispensability relation. Then all the information of initial data, such as the similarity among objects and fuzzy inconsistence degree between two objects, can be represented by fuzzy similarity matrix. Secondly, by considering that the small perturbation of the fuzzy similarity matrix can be ignorable, we propose some basic concepts of knowledge reduction such as fuzzy attributes reduct, core and fuzzy significance of attributes etc in this paper. Thirdly, a heuristic algorithm based on the fuzzy significance of attributes is proposed to find close-to-minimal fuzzy attributes reduct. Finally, experimental comparisons with other methods of attributes reduction are given. The experimental results show that our method is feasible and effective

Learning from an incomplete information system with continuous-valued attributes by a rough set technique

Many methods based on rough sets to deal with incomplete information system have been proposed in recent years. However, they are only suitable for the nominal datasets. So far only a few methods based on rough sets to deal with incomplete information system with continuous-valued attributes have been proposed. In this paper we propose one generalized model of rough sets to reduce continuous-valued attributes and learn some rules in an incomplete information system. The definition of a relative discernible measure is firstly proposed, which is the underlying concept to redefine the concepts of knowledge reduction such as the reduct and core. We extend a number of underlying concepts of knowledge reduction (such as the reduct and core), and finally propose a heuristic algorithm to generate fuzzy reduct from initial data. The main contribution of this paper is that the underlying relationship between the reduct and core of rough sets is proved to be still correct after our extension. The advantage of the proposed method is that instead of preprocessing continuous data by discretization or fuzzification, we can reduce an incomplete information system with continuous-valued attributes directly based on the generalized model of rough sets, Finally a numerical example is given to show the feasibility of our proposed method.

On the application of rough sets to data mining in economic practice

Mathematical models play an important role in the studies of modern economics. But in many fields of economics, it is difficult to build mathematical models for complex phenomena. So data mining is getting more and more popular in discovering the potential pattern of economic knowledge from databases. As a powerful tool for data mining, rough set theory has been widely used. In this research, we draw guidelines from several cases of rough set application in economic practice. Furthermore, to avoid the drawbacks of the existing methods, we develop a methodology for rough analysis in economic sector by combining the advantages of the fuzzy variable precision rough set model.

Hybridization of Fuzzy and Rough Sets: Present and Future

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.

A fuzzy model of rough sets

In order to improve the reasoning ability of rough sets, this paper generalizes the Pawlarks rough set model to the fuzzy environment. Fuzzy indiscernibility relation is proposed in this paper to replace Pawlarks underlining indiscernibility relation. A number of fundamental concepts of Pawlarks rough set theory such as the kernel and reduct are extended. Furthermore, this paper proposes the fuzzy discernibility matrix. Based on this matrix, a fast and simple approach to obtaining the fuzzy kernel and reduct is given.