Attribute Reduction in Interval and Set-valued Decision Information Systems

Copyright © 2013 Hong Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. ABSTRACT In many practical situation, some of the attribute values for an object may be interval and set-valued. This paper introduces the interval and set-valued information systems and decision systems. According to the semantic relation of attribute values, interval and set-valued information systems can be classified into two categories: disjunctive (Type 1) and conjunctive (Type 2) systems. In this paper, we mainly focus on semantic interpretation of Type 1. Then, we define a new fuzzy preference relation and construct a fuzzy rough set model for interval and set-valued information systems. Moreover, based on the new fuzzy preference relation, the concepts of the significance measure of condition attributes and the relative significance measure of condition attributes are given in interval and set-valued decision information systems by the introduction of fuzzy positive region and the dependency degree. And on this basis, a heuristic algorithm for calculating fuzzy positive region reduction in interval and set-valued decision information systems is given. Finally, we give an illustrative example to substantiate the theoretical arguments. The results will help us to gain much more insights into the meaning of fuzzy rough set theory. Furthermore, it has provided a new perspective to study the attribute reduction problem in decision systems.


Introduction
Rough set theory, introduced by Pawlak in 1982, is a useful mathematic approach for dealing with uncertain, imprecise and incomplete information [1].It has attracted much attention from researchers.
In Pawlak's original rough set theory, partition or equivalence (indiscernibility) relation is an important and primitive concept.But partition or equivalence relation is still restrictive for many applications.It is unsuitable for handing incomplete information systems or incomplete decision systems.To address this issue, several interesting and meaningful extensions to equivalence relation have been proposed in the past, such as tolerance relations [2-5], dominance relations [6] and others [7][8][9][10][11][12][13][14][15][16][17][18][19].Particularly, in 1990, Dubois and Prade combined fuzzy sets with rough sets in a fruitful way by defining rough fuzzy sets and fuzzy rough sets.
Fuzzy rough sets were first proposed by Dubois and Prade to extend crisp rough set models [20,21].Fuzzy rough sets encapsulate the related but distinct concepts of vagueness and indiscernibility, both of which occur as a result of knowledge uncertainty.Fuzzy rough set models have been a popular topic in recent years.In this paper, we introduce the interval and set-valued information systems and decision systems.Interval and set-valued information systems are important type of data tables, and generalized models of set-valued information systems and interval-valued information systems.Several authors have studied about interval and set-valued information systems.Lin et al. [6] introduced interval and set-valued information systems and presented a dominance-based rough set model for the interval and set-valued information systems.However, interval and set-valued information systems have not been investigated under the framework of fuzzy rough set model.The main objective of this paper is to introduce a fuzzy rough set model for interval and set-valued information systems by defining a fuzzy preference relation for interval and set-valued information systems.
In rough set theory, an important concept is attribute reduction [2, 13,14,[16][17][18]22,23], which can be considered as a kind of specific feature selection.In other words, based on rough set theory, one can select useful features from a given data set.Recently, more attention has been focused on the area of attribute reduction and many scholars have studied attribute reduction based on fuzzy rough sets [2, [16][17][18]22,23].Dai et al.
[2] proposed a fuzzy rough set model for set-valued data and investigated the attribute reduction in set-valued information systems based on discernibility matrices and functions.Yao et al. [16] proposed an attribute reduction approach based on generalized fuzzy evidence theory in fuzzy decision system.Shen et al. [17] studied an attribute reduction method based on fuzzy rough sets.Hu et al. [18] also proposed an attribute reduction approach by using information entropy as a tool to measure the significance of attributes.Rajen B. Bhatt and M. Gopal [22] put forward the concept of fuzzy rough sets on compact computational domain based on the properties of fuzzy tnorm and t-conorm operators and build improved feature selection algorithm.Zhao et al. [23] revisited attribute reductions based on fuzzy rough sets, and then presented and proved some theorems which describe the impacts of fuzzy approximation operators on attribute reduction.However, attribute reduction based on fuzzy rough set in interval and set-valued decision information systems has not been reported.In this paper, a fuzzy preference relation is defined and the upper and lower approximations of decision classes based on the fuzzy preference relation are given.Moreover, the definition of the significance measure of condition attributes and the relative significance measure of condition attributes are given in interval and set-valued decision information systems by the introduction of fuzzy positive region and the dependency degree.And on this basis, a heuristic algorithm for calculating fuzzy positive region reduction in interval and set-valued decision information systems is given.
The remainder of this paper is organized as follows.In Section 2, we give a brief introduction to interval and set-valued information systems and fuzzy preference relation.In Section 3, we propose a fuzzy rough set model for interval and set-valued information systems by defining a new fuzzy preference relation.In Section 4, fuzzy positive region reduction in interval and set-valued decision information systems is introduced into interval and set-valued decision information systems.To substantiate the theoretical arguments, an illustrative example is given in Section 5.In Section 6, we conclude this paper.

Interval and Set-Valued Information Systems
As a result of limitation of subjective and objective con-ditions and the interference of random factors, people often get an approximation of the data in data acquisition of the data in data acquisition.In an information system, it may occur that some of the attribute values for an object are similar, we often make it difficult to determine the similar values.Therefore, sometimes there are some object's attribute values in information systems can not be determined, but we can know the range, which leads to the interval and set-valued information systems.Definition 2.1.1.[6] Let and Q are ordinary sets, if the range of variable takes set as the lower limit, set as the upper limit, then the variable is called interval and set-valued variable.
, where the universe is a nonempty finite set of objects, U A is a non-empty finite set of attributes, V is the union of attribute domains is the set of all possible values for attribute and is an interval and set-valued variable, is a function that assigns particular values from attribute domains to objects, then the information system is called an interval and set-valued information system.
The semantics of interval and set-valued information systems have been studied by different approaches, which, actually, fall into two types [6]: the lowest grade of the environmental risk assessment index is 1, the highest grade may be 2 or 3. Type 2: x U   , a A  , the value of attribute a for object x is denoted by , where are the finite sets, and satisfy conditions: . The value of attribute a for object x is interpreted conjuc- tively.For example, if a denotes the oral expression indicates that there is no difference between ability, then can be inter- preted as: x can speak English, but may also speak French and German.Therefore, the value of i x and j x .
x is absolutely preferred to x j .r   may be {English}, {English, French}, {English, German} or {English, French, German}.
In this paper, we mainly focus on semantic interpretation of Type 1.

Fuzzy Preference Relation
Definition 2.2.1.[7][8][9][10][11][12] A fuzzy preference relation on a set of U is a fuzzy set on the product set , which is characterized by a membership function: If the cardinality of is finite, the preference relation can also be conveniently represented by a matrix , where In this case, the preference matrix

 
M R  is usually assumed to be an additive reciprocal.i.e.

Fuzzy Rough Set Model for Interval and Set-Valued Information Systems
Definition 3.1.Let be an interval and set-valued information system, , a fuzzy relation can be defined as: Obviously, there are some important properties of the fuzzy relation defined above: Hence, is a fuzzy preference relation.R  Definition 3.2.Let be an interval and set-valued information system, is a fuzzy preference relation on , x U  induced by the relation can be defined as:  regarded as the fuzzy information granule), where Here, "+" means the unions of elements.is a fuzzy set, and the fuzzy cardinal number of For a finite set X , j x X   , we have 1 ij r  , then the cardinality of is also finite and Give a fuzzy preference relation on .
be an interval and set-valued information system, is a fuzzy preference relation on Proof.It easy to prove according to Definition 3.1.

Fuzzy Positive Region Reduct in Interval and Set-Valued Decision Information Systems
In this section, we investigate fuzzy positive region reduct with respect to the fuzzy preference relation in interval and set-valued decision information systems.Definition 4.1.Given an information system   , , , , S U A f d g  , where the universe is a non-empty finite set of decision attributes, is a function that assigns particular values from condition attribute domains to objects, is a function that assigns particular values from decision attribute domains to objects, then the in-formation system is called an interval and set-valued decision information system.Definition 4.2.Let be an interval and set-valued decision information system, is a fuzzy preference relation on ,  , , , , , then define two fuzzy operators as follows: are called fuzzy lower approximation operator and upper approximation operator of decision class with respect to , respectively.
k Theorem 4.1.Let be an interval and set-valued decision information system, , then the following properties hold: 2) Proof.Since , according to Theorem 3.1, we know Therefore, the equation can be proved in a similar way.
Definition 4.3.Let be an interval and set-valued decision information system and , decision class  , , , , . Then the fuzzy positive region of with respect to is denoted by , the membership function is defined by With respect to , according to Theorem 4.1, we have 2) and with respect t we have , , , , U A f d g be an i l and set-va en S is called a con stem; otherwise, it is referred to as an inconsistent decision syst sitive region redu of.The necessity of obvious, we prove the sufficiency in the following: , according to Theorem 4.1 and Definition 4.3,

     
nd the relative significance measure, we can get a ca f fuzzy positive region reduct.The specific steps are written as follows: Algorithm.Fuzzy positive reduct in interval and setvalued decision information systems.
Input: An interval and set-valued decision information system   , , , , S U A f d g  .Output: Fuzzy positive reduct of interval and setvalued d Step 1. Compute the dependency degree is a zzy positive need to go to Step 3.
 , cycle the following steps: and make

trative Exam
al and set-valued ntaining information about

 
Core S (the significance measure of each attribute) According to Definition 4.8, we have 3) Compute the relative significance measure Let , according to Definition 4.9, we have

Conclusions
It is well known that attribute reduction is a basic issue in rough set theory.Recently, the attribute reduction based on fuzzy rough set in decision in In this paper, we or interval and set-valued information systems by defining a fuzzy preference relation.The concepts of the significance measure of condition attributes and the relative significance measure of condition attributes are given in interval and set-valued decision information systems by the introduction of fuzzy positive region and the dependency degree.And on this basis, a heuristic algorithm for calculating fuzzy positive region reduction in interval and set-valued decision information systems is given.
The results will help us to gain much more insights into the meaning of fuzzy rough set theory.Furthermore, it has provided a new perspective to study the attribute reduction problem in decision systems.


are the finite sets, and satisfy condition: and set-valued decision information system and B A  , then B is a fuzzy po ct of S if and only if 1) system.We can conclude t product risk is the core influencing factor and market risk, operational risk, environmental risk and financial ris th nt influencing fac rs.formation systems has attracted the attention of many scholars.introduce a fuzzy rough set model f heoretical Aspects of Reason-[2] J. H. Dai and H. W. Tian, "Fuzzy Rough Set Model Forset-Valued Dat , Vol. 229, 2013, pp.54-68.http ss.2013.03.005