An improved α-cut approach to trans forming fuzzy membership function into basic belief assignment

Yng Yi,X.Rong Li,Hn Deqing

aSKLSVMS,School of Aerospace,Xi’an Jiaotong University,Xi’an 710049,China

bDepartment of Electrical Engineering,University of New Orleans,New Orleans,LA 70148,USA

cCenter for In formation Engineering Science Research,Xi’an Jiaotong University,Xi’an 710049,China

An improved α-cut approach to trans forming fuzzy membership function into basic belief assignment

Yang Yia,X.Rong Lib,Han Deqiangc,*

aSKLSVMS,School of Aerospace,Xi’an Jiaotong University,Xi’an 710049,China

bDepartment of Electrical Engineering,University of New Orleans,New Orleans,LA 70148,USA

cCenter for In formation Engineering Science Research,Xi’an Jiaotong University,Xi’an 710049,China

In practical applications,pieces of evidence originated from different sources might be modeled by different uncertainty theories.To implement the evidence combination under the Dempster–Shafer evidence theory(DST)framework,trans formations from the other type of uncertainty representation into the basic belief assignment are needed.α-Cut is an important approach to trans forming a fuzzy membership function into a basic belief assignment,which provides a bridge between the fuzzy set theory and the DST.Some drawbacks of the traditional α-cut approach caused by its normalization step are pointed out in this paper.An improved α-cut approach is proposed,which can counteract the drawbacks of the traditional α-cut approach and has good properties.Illustrative examples,experiments and related analyses are provided to show the rationality of the improved α-cut approach.

1.Introduction

Uncertainty modeling and reasoning is a very crucial research field in in formation fusion.Probability theory,rough set theory,1fuzzy sets theory,2and Dempster–Shafer evidence theory(DST)3are major theories and tools for dealing with various types of uncertainty.

DST is an important modeling and reasoning tool for uncertainty such as ambiguity4,5(including non-specificity and discord),and it has been widely used in many applications such aspattern recognition,6,7in formation fusion8and decision-making.9–11In DST,the basic belief assignment(BBA)is used to model the uncertainty.When multiple BBAs are available,they can be combined to reduce the uncertainty.However,in practical applications,we will usually encounter different types of in formation sources,where the uncertainty is modeled by different uncertainty theories.In such cases,how to implement the combination or fusion of different types of in formation under the framework of DST?It needs trans formations from other types of uncertainty representation into BBAs.12,13This paper focus on the trans formation from a fuzzy membership function(FMF)2into a BBA.

Many approaches6,12–16to trans forming an FMF into a BBA have emerged,where the α-cut approach14is a simple yet effective and commonly used approach.However,it includes a normalization step.This leads to drawbacks,e.g.,in some cases,it has counter-intuitive results.In some extreme cases,it can even not be executed.In this paper,we propose an improved α-cut approach without the problematic normalization.More rational results can be obtained using the improved version.Furthermore,it has some desired properties when compared with the traditional one.Experiments and simulations are provided to illustrate the new α-cut approach and show its efficiency.

2.Basics of DST

In DST,3elements in the frame of discernment(FOD)Θ are mutually exclusive and exhaustive.Assume that 2Θdenotes the power set of FOD.m:2Θ→[0,1]is called a BBA ifit satisfies

?A ? Θ,when m(A)＞0,A is called a focal element.A BBA is also called a mass function.

Belieffunction(Bel)and plausibility function(Pl)are defined by Ref.3

isalso aBBA.Eq.(4)iscalled Dempster’sruleof combination.3

Besides Dempster’s rule of combination,some other alternative combination rules,e.g.,robust combination rule(RCR),17proportional conflict redistribution rule 6(PCR6),18and the mean rule19are given as follows.

(1)RCR

In RCR,17the conjunctive rule and the disjunctive rule are jointly used.

Here mDisis the BBA obtained by the disjunctive rule,mConjis the BBA obtained by the conjunctive rule,and α(K),β(K)are the weights satisfying

where K is the conflict coefficient.Robust combination rule can be considered as a weighted sum of the BBAs obtained using the disjunctive rule and the conjunctive rule,respectively.

(2)PCR6

Considers BBA sdenoted bym1(˙s),m2(˙s),...,ms(˙s).PCR6,18which redistributes the partial conflicting mass values to the elements involved,is defined as

It should be noted that PCR6 coincides with PCR5 when combining two sources,18but differs from PCR5 when combining more than two sources altogether and PCR6 is considered more efficient than PCR5 because it is compatible with classical frequentist probability estimate.18

(3)Mean rule

Mean rule19aims to find the average of the BBAs to be combined as

After the combination,we can use the pignistic probability trans formation(PPT)20in Eq.(9):

to make a probabilistic decision,where|A|denotes the cardinality of the focal element A.

Besides the DST,there are also many other theories of uncertainty,where fuzzy set theory and its fuzzy membership function(FMF)2are widely used in many applications.For the DST,the BBA is essentially defined using random sets,which is a unified framework for almost all the existing uncertainty theories.To combine the in formation in terms of the BBA and that in terms of the FMF,the FMF should be trans formed into a BBA.There are many available trans formations of an FMF into a BBA.6,12–16α-cut approach14is simple and commonly used.

3.Traditional α-cut approach

3.1.Basics of fuzzy sets and fuzzy membership function

A fuzzy set2is a set to describe the fuzzy concepts,which are not crisp.A fuzzy set is of ten defined by an FMF（θ）:Θ?［0，1］,quantifying the degree of membership of the element θ to thefuzzy setGiven α ∈∈[0,1],an α-cut of a fuzzy setis a crisp set(subset of Θ)such that= ｛θ∈ Θ｜μ（θ）≥ α｝.

μA～（θ）is briefly denoted by μ(θ)below when no confusion arises.

3.2.α-Cut approach

Suppose that the FOD is Θ ={θ1,θ2,...,θn}and the FMF is μ(θi),i=1,2,...,n,the corresponding BBA generated using M α-cuts14(0= α0＜ α1＜ α2＜ ···＜ αM≤ 1), where M ≤ ｜Θ｜=n.

where Bj, for j=1,2,...,M,(M ≤ ｜Θ｜)denotes thefocal element.α-Cut approach is illustrated in Example 1.

3.3.Example 1:Illustrative example of α-cut approach

A given FMF is μ(θ1)=0.2, μ(θ2)=0.6, μ(θ3)=0.4,μ(θ4)=0.8 and α1=0.2, α2=0.4, α3=0.6, α4=0.8.According to Eq.(10),the BBA trans formed from the FMF is given in Table 1.

The details of the trans formation are given as follows.

Step 1.Since all the μ（θi）≥ α1=0.2,i∈{1,2,3,4},B1={θ1,θ2,θ3,θ4}, αM=0.8,then m(B1)=(α1- α0)/αM=0.25.

Step 2.Since μ（θi）≥ α2=0.4,where i∈{2,3,4},B2={θ2,θ3,θ4},then,m(B2)=(α2- α1)/αM=0.25.

Step 3.Since μ（θi）≥ α3=0.6, where i∈{2,4}, B3={θ2,θ4},then,m(B3)=(α3- α2)/αM=0.25.

Step 4.Since μ（θi）≥ α4=0.8,where i=4,B4={θ4},then,m(B4)=(α4- α3)/αM=0.25.

The procedure can also be illustrated in Fig.1.

Note that α could be unequal to the given FMF.For example,when α1=0.3,α2=0.5,the BBA obtained from the given FMF using the α-cut approach is shown in Table 2.

There are less focal elements,since the length of α is small.

When α1=0.1, α2=0.3, α3=0.5, α4=0.8,the BBA obtained using the α-cut approach is shown in Table 3.

Given different α’s,the corresponding BBAs obtained from a given FMF using the α-cut approach might be different.When using α with smaller length,the BBA obtained is simpler,which can be used as an approximation.13Note that in our work,the focus is the mechanism of trans formation from an FMF to a BBA when α is given.

As we can see,there involves a normalization at each step of α-cut approach.This can assure the unity of the BBA obtained;however,it may lead to counter-intuitive results in some cases,as illustrated later.

4.Drawbacks of traditional α-cut approach

Using the traditional α-cut approach with normalization might lead to counter-intuitive results shown as follows.

4.1.Example 2:Invariance of fixed ratio FMFs

Suppose that the FOD is Θ ={θ1,θ2,θ3}.

Three FMF sare μ1(θ1)=0.01, μ1(θ2)=0.03, μ1(θ3)=0.04; μ2(θ1)=0.10, μ2(θ2)=0.30, μ2(θ3)=0.40;and μ3(θ1)=0.25,μ3(θ2)=0.75,μ3(θ3)=1.

For simplicity,we sort each μi(·)in ascending order to generate their corresponding α values.

For μ1(·), α1=0.01, α2=0.03, α3=0.04.For μ2(·)α1=0.10, α2=0.30, α3=0.40.For μ3(·), α1=0.25,α2=0.70, α3=1.Then,their corresponding BBAs using αcut approach are m1(θ3)=0.25,m1(θ2,θ3)=0.50,m1(Θ)=0.50,m2(θ3)=0.25,m2(θ2,θ3)=0.50,m2(Θ)=0.50,and m3(θ3)=0.25,m3(θ2,θ3)=0.50,m3(Θ)=0.50.

As we can see,three different FMFs correspond to the same BBA.Note that as far as the ratios between different μ(θi)are fixed,the α-cut trans formed BBAs are always the same due to the normalization involved.For μ1(s˙),the membership degree of θ3is very small(μ1(θ3)=0.04,i.e.,a close to zero possibility)while for μ3(s˙),the membership degree of θ3is much greater(μ3(θ3)=1,i.e.,almost sure)compared with μ1(θ3).However, m1({θ3})=m3({θ3})=0.25, and BetP1(θ3)=BetP3(θ3)=0.5833.For μ1(·),θ3has a very low possibility,while it has a possibility greater than 0.5 after the trans formation into BBA followed by the PPT,which is counter-intuitive.

Table 1 BBA obtained in Example 1 using α-cut approach.

Fig.1 Illustration of Example 1 using α-cut approach.

Table 2 BBA obtained in Example 1 using α-cut approach(α1=0.3,α2=0.5).

Table 3 BBA obtained in Example 1 using α-cut approach(α1=0.1,α2=0.3,α3=0.5,α4=0.8).

4.2.Example 3:Inability to handle singular cases

For an FMF μ(θi)=0, ?i=1,2,...,n,traditional α-cut approach cannot be executed due to the normalization step(αM=0).This is a limitation of α-cut,that is,it is unable to handle the singular case(all-zero case).

4.3.Example 4:Lack of discriminability

For an FMF μ(·)defined on Θ ={θ1,θ2,...,θn},μ(θi)=a ∈[0,1],and μ(θj)=0,?j≠ i,where i,j∈{1,2,...,n}.By using the traditional α-cut approach,the BBA obtained is always m({θi})=1.For example, for the two FMFs defined on Θ ={θ1,θ2}: μ1(θ1)=0.01, μ1(θ2)=0; μ2(θ1)=0.99, μ2(θ2)=0.Their corresponding BBA sarethesame:m({θ1})=1,although μ1(·)and μ2(·)are very different.So m1({θ1})and m2({θ1})cannot be discriminated.This is counter-intuitive.

4.4.Example 5:Inability to reflect the magnitude of FMF

Suppose that the FOD is Θ ={θ1,θ2}.Two FMFs are μ1（·）:μ1（θ1）=0.010，μ1（θ2）=0.012;μ2（·）:μ2（θ1）=0.450，μ2（θ2）=0.400.

Using the traditional α-cut approach,their corresponding BBAs are

m1（·）:m1（｛θ2｝）=0.1667，m1（｛θ1，θ2｝）=0.8333;m2（·）:m2（｛θ1｝）=0.1111，m2（｛θ1，θ2｝）=0.8889.

Using Dempster’s rule of combination,the combined BBA is

m（｛θ1｝）=0.0943，m（｛θ2｝）=0.1509，m（｛θ1，θ2｝）=0.7547.

Using PPT,BetP(θ2)=0.5283 is the maximum one;thus,the decision results is θ2.However,μ1(θ1)is only slightly smaller than μ1(θ2)(the additive difference is 0.002)and μ2(θ1)is larger than μ2(θ2)with the additive difference 0.05.When using the averaging rule for the two FMFs,μ12(θ1)=0.23,μ12(θ2)=0.206,then,θ1is more preferred.Due to the normalization,the α-cut approach only emphasizes the relative values of the FMF between different alternatives.That is,the absolute magnitude of the values of FMF is not in formative by the α-cut approach.

For more general case,suppose that FOD is{θ1,θ2}.Two FMFs are μ1(θ1)=a,μ1(θ2)=b and μ2(θ1)=d,μ2(θ2)=c,where b ＞ a,d ＞ c.Using the traditional α-cut approach,the two corresponding BBAs are m1({θ2})=1-a/b;m1({θ1,θ2})=a/b and m2({θ1})=1-c/d;m2({θ1,θ2})=c/d.

Using Dempster’s rule,m(·)=m1(·)⊕ m2(·)is obtained.Suppose the conflict between m1(·)and m2(·)is K.Then,m({θ1})=((a/b)·(1-c/d))/K, m({θ2})=((c/d)·(1-a/b))/K.Their additive difference is So,as far as a/b ＞ c/d,m({θ1})＞ m({θ2}).That is,the decision result is determined by the ratio of alternatives’membership degree when using the traditional α-cut approach.This is caused by the normalization.

In the above examples,all the counter-intuitive results or drawbacks are caused by the normalization in the traditional α-cut approach.This approach should be modified,if we want to get rid of these drawbacks.

5.A novel improved α-cut approach

We propose an improved α-cut approach(denoted by α′-cut)without the normalization.To sum up to unity,the remaining mass values are assigned to the total set Θ.The implementation of the α′-cut is as follows.

Suppose that the FOD is Θ ={θ1,θ2,...,θn}and the FMF is μ(θi), i=1,2,...,n. First, use M α-cuts(0= α0＜ α1＜ α2＜ ···＜ αM≤ 1),where M ≤ ｜Θ｜=n:

where Bj,j=1,2,...,M,represents the focal element.Then,the summation of m(Bj)is

Since αM≤ 1,the remaining mass value is 1- αM.Add the remaining mass values 1- αMto m(Θ).

Then,m(·)is the BBA obtained using the α′-cut approach.Obviously,when αM=1,there is no difference between the α′-cut and the traditional α-cut.

5.1.Example 1 revisited

First,according to Eq.(11),we can obtain m({θ4})=0.2,m({θ2,θ4})=0.2,m({θ2,θ3,θ4})=0.2,m(Θ)=0.2.

Then according to Eq.(13),m(Θ)is modified as m(Θ):=m(Θ)+1- αM=0.2+1-0.8=0.4.TheBBA obtained using the α′-cut approach is given in Table 4.

When α1=0.3,α2=0.5,the BBA obtained from the FMF using the α′-cut approach is given in Table 5.

Note that in Table 2,the BBA obtained using traditional α-cut approach has no focal element of Θ ={θ1,θ2,θ3,θ4},while the BBA obtained using α′-cut approach has thefocal element of Θ ={θ1,θ2,θ3,θ4},since the remaining mass value is assigned to Θ.

When α1=0.1, α2=0.3, α3=0.5, α4=0.8,the BBA obtained using the α′-cut approach is given in Table 6.Given different α’s,the BBAs obtained using the α′-cut approach from a given FMF might be different.

5.2.Example 2 revisited

Using the α′-cut approach,the BBAs obtained in Example 2 are m1(θ3)=0.01,m1(θ2,θ3)=0.02,m1(Θ)=0.97,m2(θ3)=0.10,m2(θ2,θ3)=0.20,m2(Θ)=0.70,and m3(θ3)=0.25,m3(θ2,θ3)=0.50,m3(Θ)=0.50.

Table 4 BBA obtained in Example 1 using α′-cut approach.

Table 5 BBA obtained in Example 1 using α′-cut approach(α1=0.3,α2=0.5).

Table 6 BBA obtained in Example 1 using α′-cut approach(α1=0.1,α2=0.3,α3=0.5,α4=0.8).

Three different FMFs correspond to three different BBAs,respectively,although the ratio between different μ(θi)is fixed.In BBAs obtained,the mass of Θ are large due to the reassignment of 1- αM.Θ is always non-in formative in evidence combination,so,the large mass of Θ does not matter for the later combination.

5.3.Example 3 revisited

For an FMF μ(θi)=0, ?i=1,2,...,n,the BBA obtained using the α′-cut approach is m(Θ)=1,which represents a totally unknown state.Note that the traditional α-cut approach cannot be applied to this case.

5.4.Example 4 revisited

For an FMF μ(·)defined on Θ ={θ1,θ2,...,θn},μ(θi)=a ∈[0,1],and μ(θj)=0,?j≠ i,where i,j∈ {1,2,...,n}.By using the proposed α′-cut approach,the BBA obtained is m({θi})=a,m(Θ)=1-a.For example, for the two FMFs defined on Θ ={θ1,θ2}: μ1(θ1)=0.01, μ1(θ2)=0; μ2(θ1)=0.99,μ2(θ2)=0.Their corresponding BBAs are m1({θ1})=0.01,m1(Θ)=0.99;m2({θ1})=0.99,m2(Θ)=0.01.

μ1(·)and μ2(·)are very different,and m1({θ1})and m2({θ1})are also very different.This is more intuitive than the results obtained using the traditional α-cut approach.

5.5.Example 5 revisited

Two FMFs are μ1(·):μ1(θ1)=0.010, μ1(θ2)=0.012; μ2(·):μ2(θ1)=0.450,μ2(θ2)=0.400.

Using the α′-cut approach,their corresponding BBAs are m1(·):m1({θ2})=0.0020,m1(Θ)=0.9980;m2(·):m2({θ1})=0.0500,m2(Θ)=0.9500.

As we can see,since the normalization is removed,the magnitude of the FMF is in formative by using the α′-cut approach.After using Dempster’s rule of combination,the combined BBA is

m({θ1})=0.0499,m({θ2})=0.0019,m(Θ)=0.9482.

Using PPT,BetP(θ1)=0.5240 is the maximum one.There fore,the decision results is θ1,which is intuitive and consistent with the result obtained using the averaging rule of FMFs.For a more general case,suppose that FOD={θ1,θ2}.Two FMFs are μ1(θ1)=a, μ1(θ2)=b; μ2(θ1)=d, μ2(θ2)=c,where b＞a,d＞c.

Using the α′-cut approach,the two corresponding BBAs are

Using Dempster’s rule of combination,m(˙s)=m1(˙s)⊕m2(˙s),then

where K denotes the conflict between m1(˙s)and m2(˙s).The difference between θ1and θ2is

So,when(d-c)＞ (b-a),m({θ1})＞ m({θ2}).That is,the decision result is determined by the absolute difference of alternatives’membership degree when using the α′-cut approach,which appears more intuitive.

6.Experiments on pattern classification

To verify the advantages of the proposed α′-cut approach compared with the traditional α-cut approach,experimental results on pattern classification are provided below.

Here we use one of the commonly used iris dataset from UCI database.21Iris dataset has three classes,where each class has 50 samples.Suppose that θ1,θ2,θ3represent Class 1,Class 2 and Class 3,respectively.Each sample has four feature dimensions.On each experimental run,samples of each class are separated into two parts with an equal probability:training samples and test samples.So,on each run,totally 75 samples are for training and 75 samples are for testing.On each run,the 75 training samples are used to generate each class’s triangular fuzzy number[lbj(θi),mej(θi),ubj(θi)],where lbj(θi)is the minimum value,mej(θi)is the mean value,and ubj(θi)is the maximum value of j th dimension of the training samples in the class θi.Using[lbj(θi),mej(θi),ubj(θi)],each class’FMF can be obtained using Eq.(14)as illustrated in Fig.2.

where x(j)is the j th feature value of sample x.

Given a query sample xq,we use each dimension xq(i)to generate an FMF μi(·)(i=1,2,3,4,four FMFs in total)based on Eq.(14).Then,using the traditional α-cut approach,we can generate four BBAs(denoted by（·）,i=1,2,3,4 corresponding to four dimensions)of xq,and using the proposed α′-cut approach,four other BBAs can be obtained(denoted by（·）,i=1,2,3,4).Four（·）are combined using Dempster’s rule,RCR,PCR6,and the mean rule,respectively,to obtain four combined BBAs.Then,use the PPT to make a classification decision.The same operation is executed for（·）.Calculate the classification accuracies based on the α-cut and α′-cut approaches,respectively, for all the training samples on the current run.Repeat the run 100 times and then calculate the average classification accuracy.The experimental results are listed in Table 7.

As shown in Table 7,using the proposed α′-cut approach,classification accuracies are higher than those using the traditional α-cut approach judging from all four combination rules.We make a detailed analysis below.Some query samples are picked out,whose classification are correct using the α′-cut and incorrect using the α-cut approach.

(1)Case 1

The test sample is xq=[6.1000,2.6000,5.6000,1.4000]belonging to Class 3.The triangular fuzzy number of each dimension obtained based on the training samples are listed in Table 8.

Based on Eq.(14),we can use these triangular fuzzy numbers to generate each class’s FMF for each dimension of xqbelow,also as shown in Fig.3.

Fig.2 Triangular fuzzy member-based FMF.

Then,by using the traditional α-cut approach,we can generate four BBAs(denoted by（·）,i=1,2,3,,4 corresponding to four dimensions)of xq:

Due to the total conflict between mα3and mα4,Dempster’s rule cannot be executed.Using RCR,PCR6,and the mean rule,the combination could be executed;however,the classifications are incorrect based on the pignistic probability trans formation,where the classification results are class 2 as shown below:

where θ2always has the maximum pignistic probability.

Table 7 Classification accuracy comparison.

Table 8 Triangular fuzzy number model.

Fig.3 FMF generation using triangular fuzzy member for Case 1.

Using the proposed α′-cut approach,four other BBAs can be obtained(denoted by（·）,i=1,2,3,4).

Dempster’s rule of combination,RCR,PCR6 and Mean rule could all be executed when using the α′-cut approach.After the PPT,their corresponding pignistic probabilities are

where Dempster’s rule,PCR6 has correct classification results(their pignistic probability of θ3is the maximum).

(2)Case 2

The test sample is xq=[5.7000,4.4000,1.5000,0.4000]belonging to Class 3.According to the triangular fuzzy number in Table 8 and Eq.(10),the FMFs for xq(j),j=1,2,3,4 are given below,as shown in Fig.4.

By using the traditional α-cut approach,we can generate four BBAs(denoted by（·）,i=1,2,3,4 corresponding to four dimensions)of xq.

Using the proposed α′-cut approach,four other BBAs can be obtained(denoted by=1,2,3,4).

Fig.4 FMF generation using triangular fuzzy member for Case 2.

Using the Dempster’s rule,RCR,PCR6,and mean rulefollowed by the PPT,their pignistic probabilities are

The pignistic probability of θ1is always the maximum using any rule here,so xq=[5.7000,4.4000,1.5000,0.4000]can all be correctly classified.

7.Property of order perseverance for uncertainty degree

Suppose that there are s FMFs μ1(·),μ2(·),...,μs(·).s BBAs m1(·),m2(·)...,ms(·)can be obtained using some trans formation Tr.We can calculate the uncertainty degree of the s FMFs denoted byand the uncertainty degree of the s BBAs denoted by.By sorting,,the ranking ΛFcan be obtained,and then by sorting,the ranking ΛBcan also be obtained.If the ranking ΛFis close to ΛB,the trans formation Tr loses less in formation.Such a desired property is called the order perseverance for the uncertainty degree.The new proposed α′-cut approach has better order perseverance for the uncertainty degree,which is verified by the simulation below.On each run,

Step 1.Suppose that the size of FOD is n.Randomly generate an integer s∈[3,20].Then,randomly generate s FMFs and calculate their fuzzy entropy22according to

where C is a normalization constant.

Sort allthe fuzzy entropy values to generate a ranking ΛF=[r(F1),r(F2),...,r(Fi),...,r(Fs)]where r(Fi),i=1,2,...,s denotes the ranking position of Fi.

Step 2.Generate s BBAs with the traditional α-cut approach denoted by）.Calculate their corresponding ambiguity measures(AM)4according to

where BetP(·)is the pignistic probability defined in Eq.(9).

AM has been criticized for the reason that it cannot satisfy the sub-additivity23;however,in our work,there is no problem for the joint BBA(Cartesian space).There fore,there is no subadditivity problem and AM can be used as an uncertainty measure for BBA here.Besides AM,aggregated uncertainty(AU)measure24is also a total uncertainty measure for belief functions;however,it has some limitations.First,AU is optimization-based.It will cause large computational cost.Second,it is insensitive to the change of BBA,4which is negative for the definition of the order perseverance for uncertainty.The non-specificity25is also an uncertainty measure in the DST,which means two or more alternatives are left unspecified and represents an imprecision degree.However,it is not a total uncertainty measure,which only describes the non-specificity part of uncertainty in a BBA.In summary,here we choose AM to calculate the uncertainty measure.

Clearly,ρ∈[0,2].ρ=0 means a total positive correlation between rankings and ρ=2 means a total negative one.To be more comprehensive,we can also use another ranking distance,i.e.,Kendall distance(Kd)27defined in Eq.(18)to calculate the distanceand=Ken）:

where

Ii,Ijare two different items in a ranking and=s（s-1）/2 is a normalization factor.

The steps above are repeated for multiple times.Average values ofare obtained.A smaller distance value is preferred within a given ranking distance.It means that the corresponding trans formation has better order perseverance for uncertainty degree.

Here,we provide an example to illustrate the procedure over a single run.Suppose that the size of FOD n=4.Three generated FMFs are

Their corresponding fuzzy entropy areUnF(μ1(·))=0.6129,UnF(μ2(·))=0.5242,UnF(μ3(·))=0.7248.

The ranking ΛF=[2,1,3](in ascending order,the same holds in the sequel).

Then using the α-cut approach,three corresponding BBAs are

Theircorresponding AM measuresareAMα(m1(·))=1.4792,AMα(m2(·))=1.4578,AMα(m3(·))=1.3759.

Their corresponding AM measures are AMα′（m1（·））=1.4849，AMα′（m2（·））=1.4605，AMα′（m3（·））=1.6402.

In our simulation,the cardinality of the FOD is set to be 5.The averaging results of the 10000 times run are=0.5598=0.5026and=0.5598，=0.5026.It means that the α′-cut approach is more preferred in terms of the order perseverance for the uncertainty degree when using two different ranking distances.

8.Conclusions

In this paper,an improved α-cut approach is proposed,which is free from some drawbacks of the traditional one.Furthermore,the robustness and accuracy for pattern classification can be improved using the new proposed approach.It also has better order perseverance for the uncertainty degree,which means the relation between the BBAs obtained is closer to the relation between the original FMFs.These have been verified by the numerical examples and simulation results provided in this paper.

In our work,the criterion forevaluating different approaches is the intuition or the rationality,which are qualitative.Quantitative evaluation criteria are needed for a more objective evaluation or design of better approaches.The order perseverance for the uncertainty degree used in this paper is our attempt on the quantitative evaluation.In our future work,we will try to design more solid performance evaluation approaches or criterion.

Acknowledgements

This work was supported by the Grant for State Key Program for Basic Research of China(No.2013CB329405),National Natural Science Foundation of China(No.61573275),Foun-dation for Innovative Research Groups of the National Natural Science Foundation of China(No.61221063),Science and Technology Project of Shaanxi Province(No.2013KJXX-46),Postdoctoral Science Foundation of China (No.2016M592790),and Fundamental Research Funds for the Central Universities of China(No.xjj2014122).

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Yang Yi received the M.S.and Ph.D.degrees in control science and engineering from Xi’an Jiaotong University in 2005 and 2010 respectively,and then became a teacher at the School of Aerospace,Xi’an Jiaotong University.Her main research interests are evidence theory,image processing and in formation fusion.

X.Rong Li received the B.S.and M.S.degrees from Zhejiang University,Hangzhou,Zhejiang,PRC,in 1982 and 1984,respectively,and the M.S.and Ph.D.degrees from the University of Connecticut,Storrs,USA,in 1990 and 1992,respectively.He joined the Department of Electrical Engineering,University of New Orleans,LA,USA,in 1994,where he is now Chancellor’s University Research Prof essor.He has authored or coauthored 4 books,10 book chapters,and more than 300 journal and conference proceedings papers.His current research interests include estimation and decision,signal and data processing,in formation fusion,target in formation processing,performance evaluation,statistical inference,and stochastic systems.Dr.Li was elected President of the International Society of In formation Fusion in 2003 and a member of Board of Directors(1998–2009);served as General Chair for several international conferences;served as Editor(1996–2003)of the IEEE Transactions on Aerospace and Electronic Systems;received a CAREER award and an RIA award from the U.S.National Science Foundation.He has given more than 150 invited seminars and taught short courses in North America,Europe,Asia,and Australia.He won several outstanding paper awards and consulted for several companies.

Han Deqiang is an associate prof essor and Ph.D.supervisor at the School of Electronic and In formation Engineering,Xi’an Jiaotong University,China.He received the Ph.D.degree from the same university in 2008.His current research interests are evidence theory,pattern classification and in formation fusion.

26 September 2015;revised 4 November 2015;accepted 7 January 2016

Available online 23 June 2016

Belieffunctions;

Evidence theory;

Fuzzy sets;

Membership functions;Uncertainty

?2016 Chinese Society of Aeronautics and Astronautics.Production and hosting by Elsevier Ltd.Thisisan open access article under the CC BY-NC-ND license(http://creativecommons.org/licenses/by-nc-nd/4.0/).

*Corresponding author.Tel.:+86 29 82667971.

E-mail address:deqhan@mail.xjtu.edu.cn(D.Han).

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http://dx.doi.org/10.1016/j.cja.2016.03.007

1000-9361?2016 Chinese Society of Aeronautics and Astronautics.Production and hosting by Elsevier Ltd.

This is an open access article under the CC BY-NC-ND license(http://creativecommons.org/licenses/by-nc-nd/4.0/).