Weighted Evidence Combination Rule Based on Evidence Distance and Uncertainty Measure: An Application in Fault Diagnosis

Conflict management in Dempster-Shafer theory (D-S theory) is a hot topic in information fusion. In this paper, a novel weighted evidence combination rule based on evidence distance and uncertainty measure is proposed. The proposed approach consists of two steps. First, the weight is determined based on the evidence distance.Then, the weight value obtained in first step is modified by taking advantage of uncertainty. Our proposed method can efficiently handle high conflicting evidences with better performance of convergence. A numerical example and an application based on sensor fusion in fault diagnosis are given to demonstrate the efficiency of our proposed method.


Introduction
Information fusion technology (IFT) is utilized to analyze multisource uncertain information comprehensively.Through reserving the common information, IFT can decrease indeterminacy greatly.The Dempster-Shafer theory of evidence [1] (D-S theory, also known as evidence theory or theory of belief functions) is regarded as an efficient model to fuse information in intelligent systems [2].And Dempster's combination rule is the most crucial instrument of D-S theory.The theory was firstly proposed by Dempster in 1967 [1] and then developed to its present form by Shafer et al. in 1976 [3].The Dempster's combination rule possesses several interesting mathematical properties, such as commutativity and associativity, and it plays a very significant role in evidence theory [4].Nowadays, the evidence theory is applied widely in many fields, like supplier selection ( [5,6]), target recognition ( [7][8][9]), decision making ([4, 10, 11]), reliability analysis ( [12][13][14][15]), and so on.
Although D-S theory has a lot of advantages, there also exist some basic problems that still are not completely clarified.One of the most significant issues is that D-S theory will become invalid when using it to fuse highly conflicting evidences, and the counter-intuitive results ( [10,16,17]) will be generated.To solve such a problem, two major methodologies are popular.One is to preprocess the bodies of evidence (BOEs) ( [18][19][20]), and the other is to modify the combined rule ( [21][22][23]).There are mainly three alternative combination rules belonging to the second type and they are, respectively, Dubois and Prade's disjunctive combination rule [24], Smets' unnormalized combination rule [25], and Yager's combination rule [21].These three alternatives mentioned above are examined and they all propose a general combination framework.The main work of preprocessing bodies of evidences (BOEs) includes Murphy's simple average in [19], Yong et al. 's weighted average on the basis of distance of evidence in [26], and Han et al. 's modified weighted average in [27].In [19], a simple averaging approach of the primitive BOEs is proposed, and in that case all BOEs are seen equally important, which is unreasonable in practice.Yong et al. [26] get a better combination result according to combining the weight average of the masses for  − 1 times.The approach proposed by Han et al. [27] is a novel weighted evidence combination approach based on the evidence distance and ambiguity measure (AM), which actually modifies Yong et al. 's work [26].
In this paper, a novel weighted evidence combination rule based on evidence distance and uncertainty measure is proposed to address the combination of conflicting evidences.The numerical example and an application in fault diagnosis are given to sufficiently prove the efficiency of our proposed method.
The remaining paper is organized as follows: Section 2 starts with a brief introduction of the Dempster-Shafer theory and evidence distance; the proposed method is presented in Section 3; Sections 4 and 5 give a numerical example and an application in fault diagnosis, respectively, to prove the efficiency of our proposed approach; finally, the conclusion is made in Section 6.

Preliminaries
In this section, some preliminaries are briefly introduced below.

Basics of Evidence Theory.
Dempster-Shafer theory of evidence (D-S theory) is used for dealing with uncertainty information as an efficient mathematical model in intelligent systems [1].In 1967, The definition, D-S theory was proposed by Dempster [1] and then his student Shafer et al. developed this theory [3] in 1976.
In Dempster-Shafer theory of evidence [3], a basic probability assignment (BPA) is a mapping: If () > 0,  is called a focal element, and the set consisting of all the focal elements is called one body of evidences (BOEs).When there are more than one independent body of evidences, Dempster's combination rule, (2) which is a powerful and crucial tool in D-S theory, can be utilized to combine these evidences.
where  = ∑  1 ∩ 2 =0  1 ( 1 ) 2 ( 2 ) stands for conflict degree, also called normalization constant.What is noted is that if  ⊕ (0) = 1, this combination rule will make no sense.Here, we give a specific example about combination rule and show the corresponding results in Table 1.
Example 1. Suppose that the frame of discernment Ω = { 1 ,  2 ,  3 } is complete and there are two BOEs listed as follows: In the frame of discernment Ω, there are two BOES,  1 and  2 .When  1 and  2 are both reliable, to generate a new BPA, we can conjunctive rule denoted (3) [31].When only one of them is totally reliable and we are not sure about another one, then we should apply disjunctive combination rule [31] denoted (4).
For any proposition  1 , its uncertainty can be represented by using its belief function and plausibility function as (9).
Here, we give an example about belief function and plausibility function and show its results in Table 2.
But the classical Dempster combination rule is not efficient all the time.When BOEs are in high conflict, illogical results will be generated [31].Nowadays, there are mainly two kinds of methodologies.One is to modify the combined rule, and the other is to preprocess evidences.Smets' unnormalized combination rule [25], Dubois and Prade's disjunctive combination rule [24], and Yager's combination rule [21] belong to the first category.These three alternatives mentioned above are examined and they all proposed a general combination framework.To preprocess data, Murphy's simple average in [19], Yong et al. 's weighted average [26], and Han et al. 's modified weighted average in [27] are popular.In [19], a simple averaging approach of the primitive BOEs is proposed.And in that case all BOEs are seen equally important, which is unreasonable in real life.In [26], Yong et al. can get a better combination result by combining the weight average of the masses for −1 times.In [27], a novel weighted evidence combination approach based on the distance of evidence and AM is proposed, which is based on Yong et al. 's work [26] actually.

Evidence Distance.
With D-S theory applying widely, the study about evidence distance has attracted more and more interests [34].The dissimilarity measure of evidences can represent the lack of similarity between two BOEs.Performance evaluation [35], reliability evaluation [36], conflict evidence combination [26], target association ( [37,38]), and a lot of methods regarding evidence distance are brought up as an appropriate measure of the difference.And several definitions of distance in evidence theory are also proposed, like Jousselme et al. distance [39], Wen et al. 's cosine similarity [40], Ristic and Smets' transferable belief model (TBM) global distance measure ( [37,38]), Sunberg and Rogers's belief function distance metric [41], and so on.Among those definitions on distance of evidence, the most frequently used is Jousselme et al. 's distance [39].The Jousselme et al. distance [39] is identified based on Cuzzolin's the geometric interpretation of evidence theory [42].The power set of the frame of discernment 2 Ω is regarded as a 2  -linear space.A distance and vectors are defined with the BPA as a particular case of vectors.The Jousselme et al. distance is defined as follows: ,   are two BPAs under the frame of discernment Ω, and  is a 2  × 2  matrix.The element in  is defined as The value inside the BOE vectors  →  1 and  →  2 and the distance matrix  are given by It follows that
where   stands for the corresponding weight degree of each BOE   and  WAE is the weighted average BPA of  BOEs.By means of the classical Dempster's rule ( [1,3]) to combine  WAE for −1 times, we can get the final combined result.But, to find an appropriate weight   is a little difficult.Actually there are many related works including Yong et al. [26] and Han et al. 's approach [27].In this paper, we present a new modified weighted evidence combination rule on the basis of evidence distance and a novel uncertainty measure.The flow graph of our proposed method is described in Figure 1.
Step 1.In this step, based on evidence distance, we can determine all weights of the primitive BOEs.
As seen from the definition of evidence distance, the less the distance between two BOEs is, the more the similarity of those two is.The similarity measure SIM  between two BOEs is defined [20] in After getting all the degrees of similarity between BOEs, we can construct a similarity measure matrix (Smm) [26].The Smm will give us insight into the agreement between BOEs.
We define the support degree of each BOE   ( = 1, 2, . . ., ) [26] as follows: Then, we can obtain the credibility degree of each BOE   ( = 1, 2, . . ., ) based on SUP(  ) [26]: In fact, for ∑  =1 CRD  = 1 it is evident that we can regard the credibility degree CRD  as a weight directly.So, CRD  can be used to replace   in (13) to modify original BPAs.As mentioned in [27], the uncertainty degree can also be utilized to construct weights.A combination result can come up by using AM and modifying CRD  obtained based on evidence Step 2. In this step, the weight will be modified on the basis of uncertainty.Supposing that one of some BOEs with relatively high credibility degree generated in the first step has less uncertainty degree than the others, we believe that the BOE is more credible and it should possess more weight because of its good quality.On the contrary, if a BOE has both more uncertainty degree and a low credibility degree, such a BOE is relatively incredible and even causes a wrong result perhaps.Thus, such a BOE should possess less weight.
Based on the thoughts mentioned above, the weight determined based on evidence distance will be modified through the following steps.
Step 2-5.The modified weighted averaged BOE denoted  WAE is obtained as If  pieces of evidence are supplied, we can use the classical Dempster's rule [1] to combine  WAE for  − 1 times [19].After that, we could get a better combined outcome to make a better decision.

Experiments
In this section, a numerical example is illustrated to demonstrate the efficiency of our proposed method.
Step 2. Through this step, we can get the modified weight.
Then by making use of ( 19) and ( 20), the final modified weight CRD  ( = 1, 2, . . ., 5) of each BOE   ( = 1, 2, . . ., 5) can be obtained.The results are shown in Table 6.Similarly, because the final weights of good evidences are increased (e.g., in experiment section, consider the cases  1 ,  2 ,  3 ; the total weight of credible evidences  1 and  3 adds to 0.8743 from 0.5), the effect of these credible evidences on the final combined results is strengthened extremely.The numerical example illustrates adequately that our proposed method is efficient.

Application
As all know, sensor data fusion plays a crucial role in fault diagnosis.In this section, our proposed method is also applied to solve such a problem and the results show that our proposed method is as feasible and efficient as the other  existing approaches, and the most important thing is that, by utilizing our proposed method, the accuracy of fault diagnosis will be improved drastically.The example in this application section is cited from [28].Suppose a machine has three gears  1 ,  2 , and  3 , and the failure fault modes  1 ,  2 ,  3 represent that there are faults in  1 ,  2 , and  3 , respectively.The fault hypothesis set is Ω = { 1 ,  2 ,  3 }.Assume  1 ,  2 ,  3 are three types of sensors, respectively.The evidence set derived from different sensors is denoted as  = { 1 ,  2 ,  3 } and the BPAs are all listed in Table 10.Assume the sufficiency indexes of those three pieces of evidences are 1, 0.6, and 1 and the importance indexes are 1, 0.34, and 1, respectively.
According to the equation  = ∑ ∩=0  1 () 2 (), we can easily get the conflict degree of each pair of evidences and The dynamic sensor reliability is influenced by the surrounding conditions changing with time, which can be measured by comparing the consistency of the outputs with other sensors that aim at the same point.
In this application section, the statistic sensor reliability   is measured based on the evidence sufficiency index  and evidence importance index ] in Fan and Zuo's approach [28] and we define    =   × ] , .The dynamic sensor reliability   is measured based on the newly proposed method in this paper and we define    = Crd  .The comprehensive sensor reliability  =   ×   is defined to modify the highly conflicting evidences and finally through the classical Dempster's combination rule, we can obtain the final results with a high accuracy to address such a fault diagnosis problem.The detailed steps are shown as follows.
Step 1. Calculate the statistic sensor reliability   of each evidence and the results are listed in Table 11.
Step 2. Based on the newly proposed method, calculate the dynamic sensor reliability   of each evidence and the results are listed in Table 12.
Step 3.According to the equation  =   ×   , calculate the comprehensive sensor reliability  and then normalize.The results are shown in Table 13.
Step 4. The normalized comprehensive sensor reliability  is applied to replace   in (13) to modify BPAs, and through using the classical Dempster's combination rule for two times, we can obtain the final results shown in Table 14.
As seen from Table 14, based on our method, the belief degree of the fault  1 is 88.99%, while the fault  2 only has a belief degree of 7.39%.It is evident that   1 >   2 .Thus, through our approach, we can correctly find the fault  1 ; that is, Gear 1 has a fault and so our proposed method is efficient in fault diagnosis.We also compare our proposed method with others and the comparison results are shown in Table 15.In D-S evidence theory, the belief degree of the fault  1 is 0.4519, while that of  2 is 0.5048.Due to the existence of conflicting evidence  2 , D-S evidence theory comes to the counterintuitive results that ( 2 ) > ( 1 ), which will lead to a wrong decision.However, the remaining two methods can handle the conflicting evidence  2 , so that they can both reach the right result.In Fan and Zuos method,   1 = 81.19,while our newly proposed approach has a higher belief degree of 88.99.The main reason is that the proposed method takes into consideration not only the static reliability represented by evidence sufficiency and evidence importance, but also the dynamic reliability, the information volume of the sensor itself, measured by evidence distance and uncertainty measurement, which decreases the weight of conflicting evidence  2 and then makes  2 's influence on the final result less.

Conclusion
In this paper, a novel weighted evidence combination rule based on evidence distance ( [20,39]) and uncertainty measure is put forward.The proposed approach can address the combination of the conflicting evidences efficiently.By comparing other existing method, our proposed approach performs with the fastest convergence when managing high conflicting evidences.In addition, we apply our newly proposed method to the fault diagnosis.The results of the application sufficiently demonstrate the efficiency of our approach.The process of our proposed method is simple and with better convergence, so we believe our proposed method has a promising future.

Figure 1 :
Figure 1: The flow graph of the proposed method.

Table 1 :
An example of combination rule.

Table 2 :
An example of Bel and Pl.
Example 3. Assume there are two BOEs  1 and  2 .
1 ,  2 ∈ (Ω); | ⋅ | represents cardinality.This Jousselme et al. distance satisfies all four requirements (nonnegativity, nondegeneracy, symmetry, and triangle inequality) [39] of a strict distance metric.This distance is an efficient tool to quantify the dissimilarity of two BOEs.An example of the Jousselme et al. distance is shown below.

Table 5 :
The total uncertainty for each BOE.

Table 7 :
WAE on the basis of the novel modified weight averaging approach.Figure 2: The final BPAs based on the proposed method.Figure 3: The results using different combination rules when  1 ,  2 are given.Figure 4: The results using different combination rules when  1 ,  2 ,  3 are given.

Table 10 :
The basic probability assignments in the application.

Table 11 :
The statistic sensor reliability   of each evidence.

Table 12 :
The dynamic sensor reliability   of each evidence.

Table 13 :
The comprehensive reliability  of each evidence.= 0.52,  1,3 = 0.26, and  2,3 = 0.605, respectively, which obviously shows that the second evidence  2 has conflicts highly with others.To solve such a fault diagnosis problem, the statistic sensor reliability and the dynamic sensor reliability will be considered.The statistic sensor reliability mainly depends on the technical factors, which can be evaluated by experts' assessment or comparing the detection value with the real value in the long-term practice.

Table 14 :
The final results in the application.

Table 15 :
The comparison results between our proposed approach and other methods.