Comparison of Typical DS Conflict Evidence Improved Algorithms

Sunqing Xu, Junbao Geng, Shuhuan Wei and Kejia Wei

College of Power Engineering, Military Key Laboratory for Naval Ship Power Engineering, Wuhan 430033, China

College of Power Engineering, Naval University of Engineering, Wuhan 430033, China

xusunqing123@163.com

Keywords: Evidence theory, conflict, improved algorithm.

Abstract: Aiming at there will generate a counter-intuitive conclusion when DS evidence theory handle the highly

conflict evidence information, some of the existing methods of improvement do not solve these problems

well, and there is no unified, widely accepted scheme in the academia. Therefore, several typical improved

algorithms are introduced in this paper, and an example is given to show which method is more reasonable,

and the effect to handle conflict evidence is better. The results of the paper can provide the ideas for the

further research to solve the conflict of evidence.

1 INTRODUCTION

Dempster-Shafer (DS) theory of evidence was first

proposed by Dempster in 1967 in studying the

multi-valued mapping issues (Dempster, 1967).

Shafer further developed it into a systematic theory

of uncertain reasoning in Shafer (1976). DS

evidence theory is a common and efficient method

used to handle uncertain information. After Jeff

Barnett introduced the name “Dempster-Shafer” in

1981, the theory quickly acquired textbook status in

artificial intelligence (Barnett, 1981). The theory is

used in many branches of technology. Articles on

the theory and its applications appear in a

remarkable number of journals and recurring

conferences. Books on the theory continue to appear.

However, in some special situations, especially

when dealing with combination of the conflicting

evidences, Dempster’s combination rule may

produce the counter-intuitive result. As an inherent

problem, the rule is incapable of managing the high

conflicts from various information sources at the

step of normalization and may generate

counter-intuitive results as first highlighted by

Zadeh (1986). In the actual data processing, the

situation of evidence conflict is often encountered,

so it is necessary to try to avoid the errors caused by

the combination of conflicting evidence, otherwise

unpredictable consequences can be caused.

Therefore, it is an important research topic in this

field to study the method of combination of

conflicting evidence. By studying the improved

methods, they can be divided into two main

categories: One methodology is to modify

Dempster’s combination rule, which had more

satisfactory behaviour compared with Dempster’s

combination rule. The representative method is

Yager’s method(Yager, 1983). This method can be

also divided into two kinds, which are completely

reliable evidence and incompletely reliable evidence.

In the case of incomplete reliable evidence, the main

issue is how to allocate the conflict, including

conflict will be assigned to a subset of what

proportion. The unified belief function method

represented by Lefevre (2002) is essentially the

process of redistributing global conflicts. The above

methods, they are based on the closed framework,

when the recognition framework is not complete,

can not effectively deal with the conflict. Smets

(1990) believes that in an unknown environment

could not get a poor and complete recognition

framework, he put forward the concept of the open

framework, the transferable belief model, will be

part of the conflict assignment to the empty set. The

literature (Yager, 1987; Dubois, 1998) presented the

combination method in the open recognition

framework. But these methods only for the group

with the rest of the empty set reliability value for

processing, not considered in BBA generation will

be set into the system. On this basis, Deng Yong

(2004) systematically put forward the generalized

evidence theory. Scholars who put forward this

method believe that the cause of high conflict

evidence combination failure is due to some defects

178

Xu, S., Geng, J., Wei, S. and Wei, K.

Comparison of Typical DS Conﬂict Evidence Improved Algorithms.

In 3rd International Conference on Electromechanical Control Technology and Transportation (ICECTT 2018), pages 178-181

ISBN: 978-989-758-312-4

of the Dempster’s combination rules. However,

when Dempster’s combination rules are modified,

some new rules will usually lose the advantage of

meeting commutative law and associative law at the

same time.

The other methodology is to pre-process

evidence, without changing the Dempster’s

combination rule. The idea is that the modified

combination rule is not a good solution to the

conflict of evidence fusion, but some of the

advantages of evidence theory cannot be preserved,

so using the modified data model, fusion method

retains the combination rule to solve the conflict of

evidence. This method is mainly divided into the

weighted average discount method and evidence.

Modify the model proposed by Murphy(2000) is

simple average of evidence, weighted average

method is more classic, but did not consider the

different between evidence. Jousselme(2003)

proposed a function of distance between evidences,

measure the relevance of evidence. Deng Yong

(2011) proposed an effective method for the

combination of conflict evidence based on the

distance between the evidence put forward by

Jousselme. In addition, Paper (Yao, 2012)

According to the correlation between evidence,

redistribute BPA. The evidence discount model was

first proposed by Shafer, take the evidence on the

discount factor and endow the remaining credibility

with the complete set. Due to the discount method,

the remaining reliability of the discount is allocated

to the whole set, which increases the uncertainty.

When the credibility of the evidence source is low or

the credibility information is not available, the

conflict problem cannot be well handled. The

scholars who are in favour of this kind of

methodology believe that the high conflict is due to

the unreliable evidence rather than Dempster’s rule

itself.

In this paper, we refer to the research at home

and abroad, and introduce the research status of

conflict evidence, several typical improved

algorithms are introduced in this paper, and an

example is given to show which method is more

reasonable, and the effect to handle conflict

evidence is better. The conclusions of the paper

provides new content and consumption services for

further study of evidence theory, and enhances the

ability of reasoning decision-making by using

conflicting evidence.

2 DS EVIDENCE THEORY AND

ITS DEFICIENCY

In this section, the basic concepts and shortcomings

of DS evidence theory are introduced.

2.1 DS evidence theory

In theory of evidence, all of the objects of the study

are called the frame of discernment, here are some

basic concepts.

Definition 1: Let



be the frame of discernment,



is incompatible focal element. A basic probability

assignment (BPA) is a function m mapping from



to [0, 1], which satisfies the following conditions

0)( 



(1)

1)( 



A

(2)

Definition 2: Suppose m

,…,m

be n BPAs



, then the Dempster’s combination rule can be

defined as:























BmAm

ABA

)()(

)(



(3)

Where K=







BmAm



)()(

(0 < K < 1) is

the conflict coefficient reflecting the degree of

conflict between the two sources of evidence.

2.2 The deficiency of DS evidence theory

When the conflict of evidence is small ,the DS

combination rule of evidence theory can centrally set

the credibility of evidence to a higher certainty. But

in larger evidence conflict or completely opposite,

because the DS theory discarded all the conflicts and

lost its fusion ability, the combination conclusion

was often contrary to the actual situation. The

following examples are illustrated.

Example 1. Consider



= {A, B, C} and two

experts opinions given by m1(A) =0.9, m1(B) =

0,m1(C) = 0.1, and m2(A) =0, m2(B) = 0.9, m2(C) =

0.1.

According to the results of combination, we can

get table 1.

Comparison of Typical DS Conﬂict Evidence Improved Algorithms

179

Table 1: combination results of DS theory.

m 2

m 1

A (0.9)

B (0)

C (0.1)

A (0)

A(0)

 

0

 

0

B (0.9)

 

0.81

B(0)

 

0.09

C (0.1)

 

0.09

 

0

C (0.01)

It can be seen that K＝0.09+0.81+0.09=0.99.

evidence m1 and m2 are highly conflicting, highly

supported by A and B, respectively. However, the

BBA resulting in the combination using Dempster’s

rule is :

m (A)=(0)/(1-K)=0；

m(B)=(0)/(1-K)=0;

m (C)=(0.001)/(1-K)=1.

It is the counter-intuitive result that m(C) = 1..

The combination is failed.

Example2. Consider



= {A, B}

m1:

m1(A) =0

m1(B) = 1

m2:

m2(A) =1

m2(B) = 0

According to the rules of combination of DS

theory: K=1*1+0*0=1. It shows that evidence is

completely conflicting, because the composite

denominator is zero at this time, and the DS theory

can not fuse the data. The DS evidence theory is

invalid and can not make any decisions based on the

known evidence. When only consider the non

inclusiveness between the evidence, in the

normalization process, DS evidence theory discards

all information, and can not get ideal fusion results.

3 TYPICAL IMPROVED

METHODS

The DS evidence theory has been developed over

four decades and blossomed in various fields, but the

evidence of conflict still cannot be combined well.

The problem enlightened by the now famous

Zadeh’s example is the repartition of the global

conflict. To solve this problem, many scholars have

proposed a variety of improved methods. There has

been no uniform and widespread solution so far.

Here we introduce two classic improved algorithms.

3.1 Yager’s method

Yager (1983) proposed a modified method which

assigned the conflicting mass assignments to the

unknown state. The idea is that the paradox is due to

fusion of conflict evidence deduction of the fused

empty part of the reliability, the remaining reliability

was normalized to produce, so we need to modify

the rules of combination. The improved combination

formula is as follows:





ACB

CmBmAm



)()()(

(4)









CmBmmmm



)()()()()(

2121

(5)

3.2 Murphy’s method

In document [18], Murphy proposes a fast

convergence method. When there are N evidence in

the system, the Murphy rule first calculates all the

evidence’s average value of the propositional

support in the recognition framework, then uses DS

merging rules to iterate N-1times.

( ) ( )

m X m X X



  



(6)

(B) (C)

(A)







(7)

(B) (C)

k m m







(8)

4 COMPARISON STUDIES OF

MAIN METHODS

In this section, some numerical examples with

conflicting BOEs are given to demonstrate the

effectiveness of the different method by comparing

with Dempster’s rule, Yager’s method and Marphy’s

method.

Example3. Consider



= {A, B, C}

m1(A)=0.5

m1(B)=0.2

m1(C)=0.3

m2(A)=0

m2(B)=0.9

m2(C)=0.1

m3(A)=0.6

m3(B)=0.1

m3(C)=0.3

According to the combination rules, results are

shown in table 2 and table 3.

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180

Table 2: comparison of combined results of improved

algorithm 1.

m1、m2



0.8571

0.1429

Yager’s

0.18

0.03

0.79

Murphy’s

0.1543

0.7469

0.0988

Table 3: comparison of combined results of improved

algorithm 2.

m1、m2、



0.666

0.3334

Yager’s

0.018

0.009

0.973

Murphy’s

0.3912

0.5079

0.0988

As seen from Table 2 and Table 3, classic DS

combination rule of evidence theory for highly

conflict evidence cannot be fused because m2(A)=0,

it totally negate evidence A, even if there is a lot of

evidence support evidence A, its fusion results

always show m2(A)=0. Yager’s method is similar to

the combination rule of the classic DS evidence

theory, it can not solve the above problems

effectively, and is too conservative. The scope of the

unknown area is expanding. Although there are

many evidences, it can not get the ideal conclusion

and cannot make decisions based on it. Murphy’s

method is only a simple mean of evidence, and does

not take into account the compatibility between

evidence and conflict. But the effect to handle

conflict evidence is better than that of the two

methods mentioned above. It is proved by the

examples that Murphy’s method can effectively

compensate for the shortcomings of DS evidence

theory and Yager’s method, and it can get a more

ideal conclusion. However, due to the objectivity of

conflict, the conflict of evidence theory has not yet

been solved thoroughly, so it needs further study.

5 CONCLUSIONS

Through the comparison of the examples above,

compared to the modified combination rule, the way

to modify the body of evidence is more effectively.

Because the modification of rules often destroys the

exchange rules of the Dempster’s rule, combining

the excellent properties of law. In fact, if the

evidence conflict between sensor failure or sensor

report is not accurate, it is not reasonable to blame

the combination rule directly. Therefore the solution

of the DS conflict evidence should pay attention to

pre-process evidence, which may result in better

results.

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