Attribute Reduction in Multi-Source Fuzzy Information System

(1.School of Mathematics,Harbin Institute of Technology,Harbin 150001,Heilongjiang,China;2.School of Mathematical Sciences,Dalian University of Technology,Dalian 116024,Liaoning,China)

Abstract: People get a lot of fuzzy information from multiple channels, so it is very important to fuse the obtained information to get more accurate information. In this paper, a fusion model of multi-source fuzzy information system was constructed by using conditional entropy. The new conditional entropy was used to fuse multiple fuzzy information systems into a fuzzy information table. Then, the similarity between two fuzzy relations was constructed by using the paste progress between any two fuzzy sets, and three kinds of fuzzy relation similarity were established. In addition, based on the fused fuzzy information table of multi-source fuzzy information system, the attribute reduction of the fused fuzzy information table was carried out according to three kinds of fuzzy relation similarity, and then the attribute reduction of multi-source fuzzy information system was realized. Finally, the correctness and effectiveness of the proposed model was verified by experiments. The results show that the reduction method of multi-source fuzzy information system proposed in this paper has a certain value in dealing with fuzzy data and enriches the theoretical basis of knowledge discovery.

Key words:attribute reduction;fuzzy information system;multi-source information fusion;rough set

0 Introduction

With the progress of society,some information has gradually penetrated into every area of our life.Human beings are using information technology to enjoy convenient life,but also have to deal with complex information.So,extracting useful information is getting more and more important.

Information fusion is to obtain more accurate and more definite information from the provided data.In general,the information fusion is a process of multi-information sources integrated in a single information source,and serval definitions have been proposed in corresponding literatures[1- 6].The information fusion theory was first used in the military field.To gain more reliable information is the basic objective of the fusion.Information fusion can also obtain the accurate status and identification of battlefield situation,and carry out the complete and timely assessment to the threat.Along with the progress of the times,the information fusion technology has become more and more important in the field of information service.For multisource information fusion,a lot of scholars have done some researches.Lin et al[7]proposed optimistic and pessimistic fusion functions on account of multigranulation rough sets.Ma[8]investigated prediction model in view of multisource fusion.In multisource information system,Zhou et al[9]provided a variation source identification method on account of evidence theory.Cai et al[10]studied that the multisource fusion of hitch diacrisis of heat pump through Bayesian network.Ribeiro et al[11]studied an algorithm for data information fusion,which included the concepts of multi-criteria decision-making and computational intelligence,especially fuzzy multi-criteria decision-making and hybrid aggregation operators with weighted functions.

In real-life,it is very difficult to get accurate information,and the collected data are usually noisy and fuzzy in many fields of social science and engineering technology.To describe this kind of information precisely,fuzzy set theory is needed.In 1965,Zadeh first proposed the fuzzy set theory(FST) in Ref.[12].It is an effective tool to deal with uncertainty.Compared with probability theory,rough set and evidence theory,it has its own advantages in the fields of medical diagnosis,knowledge discovery,and image processing and so on.The fuzzy set theory is an extension of classical crisp set theory,and it is also a study of the intelligent systems characterized by fuzzy set.To handle fuzzy information source situation,fuzzy rough set was proposed by Dubois and Prade[13].It is proved that it is useful in the fields of decision support,power system,machine learning and other fields.Recently,some scholars have studied rough and fuzzy set.For instance,Dai et al[14]studied an uncertainty measure for incomplete decision tables and its applications.Yang et al[15]investigated a fuzzy covering-based rough set model and its generalization over fuzzy lattice.Ma[16]discussed two fuzzy covering rough set models and their generalizations over fuzzy lattices.Ngu-yen[17]put forward a new knowledge-based intuitionistic fuzzy sets measurement and its application.

In recent years,there have been more and more ways for people to obtain information.Information is usually collected in different ways,which may have different results for an information source.The information source can be used to construct an information system.If all information sources are fuzzy information,then they can construct multiple fuzzy information systems.Nevertheless,the entire attribute is not all necessary for each information source.In the big data era of information explosion,attribute reduction is particularly important.A number of scholars have done research on the reduction of the information source.For example,Cornelis et al[18]investigated data reduction method based on attribute selection and tolerance relation.Huang et al[19]researched attribute reduction method based on intuitionistic fuzzy information system of interval value.Chen and Zhao[20]stu-died the local reduction of fuzzy rough sets and the concept of decision making system.However,at present,there is little research on the reduction of multi-source information system.Multi-source information systems are often encountered in real life.Therefore,it is very meaningful for the study of the reduction method of multiple information sources.In this paper,attribute reduction of an information box[21]will be discussed,which is formed by multiple information sources overlapping together under the fuzzy context through delimiting closeness degree and similarity degree.

The rest of this article is as follows.Some fundamentals will be introduced in Section 1.Multi-source fuzzy information fusion by three kinds of closeness degree and one kind of similarity degree will be studied in Section 2.In Section 3,a case to assess the applicant will be studied.Section 4 is the conclusion and the further studies of this topic.

1 Preliminaries

Some fundamentals of rough set theory(RST)[22],fuzzy set theory[12,23]and closeness degree[24]will be simply reviewed.

1.1 Rough Sets

A four tupleI=(U,A,V,f) is an information system.Uis called the universe.Ais attribute set.Fora∈A,Vis the set of attruibute valuesVa.fis a total function such thatf(u,a)⊆Va,u∈U.For anyB⊆A,an equivalence relation IND(B) can be defined as

IND(B)={(u,v)∈U×U:a(u)=a(v),∀a∈B}.LetRis an equivalence relation andX⊆U,the lower and upper approximation ofXcan be defined as

∪{[u]R|[u]R∩X≠∅}

(1)

∪{[u]R|[u]R⊆X}

(2)

And,under attribute setA,the approximation accuracy and roughness of the setXis defined as

(3)

ρ(X)=1-α(X)

(4)

Through applying the corresponding attribute set,the approximation accuracy proposed by Pawlak furnishes percentage of correct decisions.LetI=(U,A∪d,V,f) be a decision system,U/d={B1,B2,…,Bh} be a classification,B1,B2,…,Bhare some class andRbe an attribute set.Correspondingly,lower and upper approximations ofU/dare defined as

(5)

(6)

(7)

S(U/d)=1-αR(U/d)

(8)

αR(U/d) andS(U/d) are known as approximation accuracy and corresponding approximation roughness,respectively.

1.2 Fuzzy Sets

Fuzzy set theory,proposed by Zadeh[12],further extends the classical rough set theory.LetUbe a nonempty set.Then,

(9)

(10)

(11)

(12)

Where “∨” and “∧”are the maximum operation and minimum operation,respectively.

1.3 Closeness Degree and Similarity Degree

Closeness degree can be used to describe the level of similarity of two fuzzy sets,which is a quantitative index.The closeness degree is defined as:

Definition1(see Ref.[24]) Let mappingN:F(X)×F(X)→[0,1] meet the following conditions:

(1)∀A1∈F(X),N(A1,A1)=1;

(2)∀A1,A2∈F(X),N(A1,A2)=N(A2,A1);

(3)∀A1,A2,A3∈F(X) satisfies the condition |A1(u)-A3(u)|≥|A1(u)-A2(u)|,whereu∈X,thenN(A1,A3)≤N(A1,A2).The mappingNis called the closeness degree onF(X).TheN(A1,A2) be called as the closeness degree betweenA1andA2.

The closeness degree has a lot of representation forms,and three kinds of forms among them are given below.

In the Ref.[23],let mappingN:F(X)×F(X)→[0,1],∀A1,A2∈F(X),X={u1,u2,…,un},command

(13)

N1(A1,A2) of this definition can be verified to meet the three conditions defined by the closeness degree.It is called the maximum-minimum closeness degree.

Let mappingN:F(X)×F(X)→[0,1],∀A1,

A2∈F(X),X={u1,u2,…,un},command

(14)

N2(A1,A2) is called the Hamming closeness degree.

Let mappingN:F(X)×F(X)→[0,1],∀A1,

A2∈F(X),X={u1,u2,…,un},command

(15)

N3(A1,A2) is called the Euclid closeness degree.

(16)

or

(17)

or

(18)

Definition2(see Ref.[25]) LetR1,R2are two fuzzy relationships onX2,then similarity degree betweenR1andR2is defined as:

(19)

Where“↔” is the raint-norm corresponding equivalence.

The raint-norm:u⊗v=u∧v;

The raint-norm corresponding equivalence:

2 Attribute Reduction of Multi-Source Fuzzy Information System

Let us consider the scenario when we obtain information regarding a set of objects from different sources.Information from each source is collected in the form of the above information system,and thus a family of the single information system with the same domain is obtained and called a multi-source information system,which is formulated as follows(see Ref.[26]).

A multi-source information system(MS) can be defined as

MS={Ii|Ii=(U,Ai,{(Va)a∈Ai},fi)} where,

(1)Uis a finite non-empty set of objects;

(2)Aiis a finite non-empty set of attributes of each subsystem;

(3){Va} is the value of the attributea∈Ai;

(4)fi:U×Ai→{(Va)a∈Ai} such that for allx∈Uanda∈Ai,fi(x,a)∈Va.

In particular,a multi-source decision information system is given by MS,whereDis a finite non-empty set of decision attributes andgd:U→Vdfor anyd∈DwithVdbeing the domain of a decision attributed.

We can use the information box to indicate multi-source information system.The information system overlapping together can form an information box,which has levels as shown in Fig.1.

Fig.1 Multi-source information system

Correspondingly,we can get the multi-source fuzzy decision system(MFDS) is a multi-source fuzzy decision system,which represents that fuzzy decision information system have the same universe,attributes and different attribute values.For example,the four information sources with the same decision attribute are as follows,which can constitute a multi-source fuzzy decision system(MFDS).

How to reduce the multi-source fuzzy information system is a very important problem.In the following,the reduction method of multi-source fuzzy information system will be given by using the conditional entropy information fusion.

(20)

wherebis a given constant,which is called the thre-shold.

(21)

(22)

Fig.2 shows how to get a new information table(NI),that is to say,the process of information fusion,which signifies that a multi-source information box withsfuzzy systems hasnobjects andmattri-butes.The rough lines signify the smallest conditional entropy for all information source under given attribute.Then,all the selected attributes form a new information table.

Fig.2 Multi-source information fusion

and for anyA2⊆A1⊆A,

does not hold,thenA1is a reduction of multi-source fuzzy decision system(MFDS).

According to three kinds of closeness degree,three kinds of fuzzy relation matrix can be constructed.Then according to the definition 2 and definition 4,three kinds of different reduction methods are provided for multi-source fuzzy information system.

Due to the differences between the various information sources,it is known that direct reduction of the multi-source fuzzy information system is not possible.So,a method is proposed to reduce the information system after fusion.For the reduction of the multi-source fuzzy information system,we regard the information system reduction after fusion as the reduction of the multi-source fuzzy information system.

3 Experiment

In this section,some simple experiments will be done according to the definition.The effectiveness of the suggested definition that we discussed previously will be verified through experiments.

Assume the Fund Committee invites ten experts to assess the applicant in some fuzzy criteria.Every expert gives a score as the membership degree in one attribute fuzzy set and the value of score is belong to[0,1].We conduct an example based on the data of Ref.[27].We use the preceding example as the ten information source.

Table 1 Information source 1

Table 2 Information source 2

Table 3 Information source 3

Table 4 Information source 4

Table 5 Information source 5

Table 6 Information source 6

Table 7 Information source 7

Table 8 Information source 8

Table 9 Information source 9

Table 10 Information source 10

Since we want to use the decision condition entropy to carry on the reduction of the multi-source fuzzy information system,each information source has the same decision attribute in the multi-source fuzzy information system,and classification standard of the same decision attributes is the same.For the same decision attribute,we let experts determine theU/Dpartition before a given decision.

Suppose that the experts give a partitionU/D={Y1,Y2}={{u1,u4,u5},{u2,u3}},and we takeb=0.2.

Thirdly,according to the,we can count thekth(k=1,2,3,4) system is the most important under attributea.

Table 11 Multi-source conditional entropy

Table 12 The result of conditional entropy fusion(NI)

Case1 The reduction of information system after fusion is calculated by maximum-minimum closeness degree.

Firstly,according to the maximum-minimum closeness degree,we can construct a fuzzy relationship matrixR=(rij)|A|×|A|based on the closeness degree,and we take

Then,for the information system after fusion,that is to say,for Table 12,according to the definition 4,we can get the reduction of the new information table.The reduction is {a1,a2} and {a2,a5}.Its core is {a2}.

Case2 The reduction of information system after fusion is calculated by Hamming closeness degree.

Firstly,according to the maximum-minimum closeness degree,we can construct a fuzzy relationship matrixR=(rij)|A|×|A|based on the closeness degree,and we take

Then,for the information system after fusion,that is to say,for Table 12,according to the definition 4,we can get the reduction of the new information table.{a1,a2,a3} and {a2,a3,a5}.Its core is {a2,a3}.

Case3 The reduction of information system after fusion is calculated by Euclid closeness degree.

Firstly,according to the maximum-minimum closeness degree,we can construct a fuzzy relationship matrixR=(rij)|A|×|A|based on the closeness degree,and we take

Then,for the information system after fusion,that is to say,for Table 12,according to the definition 4,we can get the reduction of the new information table.The reduction is {a1,a3} and {a2,a3,a5}.Its core is {a3}.

From the cases of the above three kinds of reduction,we can see that,though the requirements of the ten experts on each attribute is different,the attributea2anda3are very important standard for the applicant to apply for foundation.

4 Conclusion

The fuzzy rough set is an important expansion of classical rough set and has been applied into many fields.In our study,fuzzy rough set for attribute reduction is used in fuzzy context.The main contribution of this paper is that:three closeness degrees are defined;three fuzzy relationship matrixesR=(rij)|A|×|A|based on the closeness degree are constructed respectively;the similarity degree is defined through the fuzzy relationship.And the attribute reduction of multi-source fuzzy information system is studied through similarity degree based on multi-source fuzzy information system fusion.In this paper,the classical rough set theory is extended from the viewpoint of granular computing(GrC) and compared with the classical rough set theory(RST).In the next step,we will study the specific reduction methods and use these methods to solve practical problems.