A New Divergence Based on the Belief Bhattacharyya Coefficient with an Application in Risk Evaluation of Aircraft Turbine Rotor Blades

,


Introduction
Multisource information fusion is an invaluable information processing technology to achieve precise decisions by analyzing heterogeneous data from multiple sensors [1][2][3].However, in practical scenarios, owing to the infuence of diferent factors, such as bad weather conditions, mechanical failures, and wireless communication problems, the information collected from various sensors may be imprecise, incomplete, and ambiguous [4].Te Dempster-Shafer (D-S) evidence theory possesses the ability to directly express uncertain information by means of assigning basic probability assignment (BPA) to multielement sets, and can fuse evidence without the consideration of prior information to diminish the uncertainty of the system and improve its performance [5,6].So, it has been broadly applied in risk evaluation [7][8][9], output control [10], image processing [11,12], and multicriteria decision-making [13,14].Nevertheless, when confronted with highly conficting evidence, the D-S evidence theory yields counter-intuitive results [15]; then, a misdirected consequence may be brought about in a system.Terefore, how to deal with confict is still an urgent problem in evidence theory.
To address this issue, existing research methods are primarily conducted by modifying Dempster's combination rule or preprocessing the bodies of evidence before combination [16][17][18][19][20][21][22].In this article, we focus on the latter.In the study of evidence preprocessing, it is discovered that uncertainty and discrepancy measures have been extensively studied [23][24][25][26][27][28], and can be further utilized to investigate the evidence confict.Specifcally, a total uncertainty measure, based on the Euclidean distance between the belief interval of the singleton subset and the most uncertain interval, was employed to settle the confict [29].Cui et al. presented a plausibility entropy to measure the uncertainty of BPA [30].Xiao presented an evidential confict coefcient to measure the confict between evidence [31].Based on Tanimoto measurement, Deng devised an evidential similarity measurement to describe the evidence inconsistency [32].Deng et al. exploited the evidence distance to manage confict [33].Xiao proposed belief Jensen-Shannon (BJS) divergence to measure the discrepancy between evidence, but it is found that BJS divergence ignores the efect of multielement sets and produces measurement error [34].Zhu et al. put forward the belief Hellinger (BH) distance and overcame the defciency of BJS divergence [35].Yet, BH distance diferentiates multielement sets and singletons just by the cardinality of multielement sets.As for those multielement sets with same cardinality, BH distance cannot embody their diference.Specially, it is worth mentioning that Florea and Bossé gave a corrected extension of the Bhattacharyya distance based on the Bhattacharyya coefcient in probability theory to the D-S evidence theory [36]; however, the extensive form is still not mature enough to adequately refect the diversity of evidence.
Te main motivation of this study lies in the following points: (1) In [36], Florea and Bossé's distance ignores the uncertainty of multielement sets and the interrelationship of sets, which are also the existing drawbacks of BJS divergence and BH distance, respectively.It means that a new belief divergence should be constructed to handle the uncertainty of evidence.
(2) It is signifcant to boost the performance of the fusion system for achieving efcient decisionmaking.Terefore, it is necessary to design a new algorithm to improve the accuracy of fusion.
However, there are still several challenges in this study: (1) How to accurately refect the uncertainty characteristics of evidence is a challenge.(2) Designing an efcient algorithm to obtain better fusion results is a complex and challenging task.
In this paper, an enhanced belief divergence, named as EBD, is proposed to quantify the discrepancy.It fully considers the impact of multielement sets and the relationship between sets to ofer a more valid solution for discrepancy measurement.Te EBD satisfes the properties of boundedness, nondegeneracy, and symmetry.Based on the EBD, a new multisource information fusion algorithm is devised for confict resolution, where the EBD determines the weights of evidence to better refect their reliability and importance, and meanwhile, the information volume of evidence is also considered.Te fusion algorithm is applied in target recognition and iris classifcation to evaluate its performance.Finally, the proposed method is exploited to make the risk evaluation of the rotor blades of an aircraft turbine, and verifed efective and practical.
Te main contributions of this study are summarized as follows: (1) Te belief Bhattacharyya coefcient and adjustment function are defned.Te belief Bhattacharyya coefcient considers the uncertainty of multielement sets by the cardinality of subsets.Te adjustment function can describe the correlation between different subsets.Based on the belief Bhattacharyya coefcient and infuenced by the adjustment function, a new belief divergence BD is proposed and takes the uncertainty of the evidence into account.(2) Te enhancement factor is defned to promote the performance of the BD.After improvement, the enhanced belief divergence, called EBD, is presented.Compared with the other divergence and distance measures, the EBD performs discrepancy refection more efectively.(3) To settle confict, an EBD-based multisource information fusion algorithm is designed, in which the EBD is employed to decide the weight of evidence.
With the experiments of target recognition and iris classifcation to evaluate the performance and effectiveness, it demonstrates that the algorithm can achieve a more precise decision.Furthermore, the algorithm is applicable to the risk evaluation of the rotor blades of an aircraft turbine.
Te paper is organized as follows.In Section 2, the preliminaries of this paper are briefy introduced.In Section 3, the belief Bhattacharyya coefcient and adjustment function are defned, and a new belief divergence BD is presented.In Section 4, based on the enhancement factor, an enhanced belief divergence EBD is proposed and its properties are proven.Furthermore, a comparative analysis is given to illustrate the validity of the EBD.In Section 5, the EBD-based multisource information fusion algorithm is designed.In Section 6, two experiments are utilized to demonstrate the efectiveness of the algorithm.In Section 7, an application in the risk priority evaluation of the failure modes of the rotor blades of an aircraft turbine demonstrates the practicality of the EBD-based fusion algorithm.Finally, conclusions are drawn in Section 8.

Preliminaries
In this section, some concise knowledge, including Dempster-Shafer evidence theory, base belief function, and Deng entropy, is introduced.Besides, several belief divergence and distance measures are investigated, and their inadequacies are pointed out by the examples.

Dempster-Shafer Evidence Teory.
As an efective method to model and process uncertain information, the Dempster-Shafer evidence theory is primitively presented by Dempster and perfected by Shafer.[5,6] Te core concepts of it are introduced in the following.
Defnition 1 (Frame of discernment).Let Θ be a fnite and complete set which is composed of N mutually exclusive and collectively exhaustive hypotheses.Θ is called a frame of discernment [5,6].
2 International Journal of Intelligent Systems Te power set of Θ, consisting of all subsets of Θ, is defned as follows: (2) m is called a basic probability assignment (BPA) or mass function, [5,6] where m(A) is the support degree to proposition A. If m(A) ≠ 0, A is called a focal element.
Defnition 3 (Dempster's combination rule).Let m 1 and m 2 be two independent BPAs on Θ, m � m 1 ⊕ m 2 is a new evidence after combination between m 1 and m 2 .Dempster's combination rule is defned as follows [5,6]: where

Base Belief Function.
Base belief function is primarily proposed to address the fusion problem of highly conficting evidence [37].In this paper, base belief function is fexibly utilized to solve zero belief value by modifying the evidence.
Defnition 4 (Base belief function).Let Θ be a frame of discernment, composed of N mutually exclusive and collectively exhaustive hypotheses.Te power set of Θ contains 2 N propositions, for every proposition A i (i � 1, . . ., 2 N ) in 2 Θ except ∅, base belief function is defned as follows [37]:

Deng Entropy.
Te uncertainty of BPA is considered benefcial for handling confict, therefore, a quantity of uncertainty measures have been explored from diferent perspectives [38,39].As the extension of Shannon entropy, the Deng entropy is proposed to represent the uncertainty of evidence.Te Deng entropy is denoted as in [40].
where A ⊆ Θ and |A| is cardinal number of A.

Divergence and Distance Measures.
In D-S evidence theory, how to choose an appropriate method to determine the diference between evidence is still an open issue.To date, a score of discrepancy measures have been developed [41,42].In this section, Bhattacharyya distance, Florea and Bossé's distance, belief Jenson-Shannon divergence, and belief Hellinger distance are introduced.
In statistics, the Bhattacharyya distance is utilized to measure the similarity between two probability distributions, and it is closely related to the Bhattacharyya coefcient which is used to calculate the overlap degree between samples [43].Bhattacharyya distance is defned as follows.
Ristic and Smets put forward the extension of Bhattacharyya distance from probability theory to D-S evidence theory [44].By correcting the above distance, Florea and Bossé's distance is given by [36] where m 1 and m 2 are two independent BPAs defned on Θ and p could be any positive number.
In this paper, we only pay attention to the case of ] 1/2 .Nevertheless, Florea and Bossé's distance is immature to accurately refect evidence diference.In other words, Florea and Bossé's distance is unable to diferentiate between probability distribution and BPA without considering the uncertainty carried by BPA.
Xiao incorporated Jensen-Shannon divergence into evidence theory and proposed a novel belief divergence, which is presented as follows [34].
International Journal of Intelligent Systems Defnition 6 (Belief Jensen-Shannon divergence).Given two independent BPAs m 1 and m 2 defned on Θ, belief Jensen-Shannon (BJS) divergence between m 1 and m 2 is defned as follows [34]: where  i m j (A i ) � 1, (i � 1, . . ., n; j � 1, 2).BJS divergence has a preferable efect on describing the deviation between evidence, but it fails to recognize the multielement sets.Tis restriction is illustrated by Example 1.
Intuitively, m 1 strongly supports A { } and m 2 strongly supports B { }, so m 1 and m 2 are highly conficting.m 3 supports the proposition A, B { }, which represents a uncertain state to support A or B. Terefore, the divergence between m 1 and m 2 is the largest amongst all evidence, the divergence between m 3 and m 2 is identical with that between m 3 and m 1 .However, according to equation ( 9), the results are calculated as follows: Obviously, it is not consistent with the intuition.Te reason for such counter-intuitive results is that BJS divergence neglects the multielement subset A, B { } with uncertainty by treating it as a singleton.
Generalized from the Hellinger distance of probability theory, belief Hellinger distance, which overcomes the defect of BJS divergence, is defned as follows [35].
Defnition 7 (Belief Hellinger distance).Let m 1 and m 2 be two independent BPAs defned on Θ, belief Hellinger (BH) distance between m 1 and m 2 is defned as follows [35]: where From (12), BH distance takes the cardinal number of subset into account, so multielement set can be distinguished from singleton by the size diference.Recalculate Example 1 by BH distance, the results are obtained as follows: In comparison with BJS divergence, the above results by BH distance are more reasonable.Nevertheless, BH cannot embody the correlation between hypotheses contained in diferent subsets.Explicitly, if the cardinality of all multielement sets is designed as the same, BH distance is unable to identify the diference between these sets, which have the same cardinality but diferent elements.Tis situation will be vividly illustrated by Example 2.
It can be seen that m Apparently, BH distance does not change, which is insufcient as an evidence distance measure.Terefore, it is needed to fnd a more reliable and stable belief discrepancy measure.

A New Belief Divergence Measure
In this section, the belief Bhattacharyya coefcient is presented.In addition, an adjustment function is defned.Based on the belief Bhattacharyya coefcient and afected by the adjustment function, a new belief divergence is proposed to signify the discrepancy between evidence.
In order to refect the impact of multielement sets, it is considered that the cardinality factor can tell the multielement sets from singletons.Attributed to this peculiarity, the belief Bhattacharyya coefcient is proposed as follows.
In addition, the Bhattacharyya coefcient can be denoted as fdelity; the physical signifcance of fdelity is the inner product of two probability vectors on a sphere, representing the similarity between two probability distributions [45].Te belief Bhattacharyya coefcient can be seen as a generalization of fdelity in Dempster-Shafer theory.In [45], another extension of fdelity called FBIP has been proposed.To demonstrate the meaningfulness and value of our extension, a comparative experiment between BBC and FBIP is conducted in Example 3.
Figure 1 depicts the change of the BBC and FBIP as x uniformly increases from 0 to 1 with an increment of ∆ � 0.01.As x gradually increases, m 1 and m 2 become more similar, leading to an increase in both BBC and FBIP.Specifcally, at x � 0, m 1 and m 2 are completely conficting, resulting in a similarity of 0; at x � 1, the evidence distributions are identical, yielding a similarity of 1. Overall, the change of BBC is relatively uniform, and the change of FBIP becomes signifcantly less pronounced after x > 0.6.Terefore, the BBC exhibits better measurement characteristics.
In order to reveal the relationship between sets, it is learned that the transformation factor G in [46] can measure the intersection relationship between focal elements, [46] but we fnd that, the sum of the transformed mass function may be greater than 1, which spoils the nature of BPA.Hence, by the normalization of each row of the transformation factor, a new adjustment function is defned as follows.
Defnition 9 (adjustment function).Let m 1 and m 2 be two mass functions on Θ including N hypotheses.
is a set of n focal elements, adjustment function is defned as Adjustment function Γ is the ratio of the interaction degree between A i and A j to the whole sum of that between A i and all focal elements, which implies the importance of A j in all interaction relationships of A i .In other words, Γ A i ,A j is the ratio of the belief that A i assigns to A j relative to the belief of A i itself.Terefore, Γ A i ,A j not only retains the primary property of BPA, namely, the sum of modifed BPA with Γ is 1 but also has the ability to express the contribution diference of diferent subsets to the supportive proposition of evidence.
In Example 2, intuitively, the supportive proposition in m 1 and } in m 3 is 0, which is in conformity with the intuition.
Based on the BBC and Γ, a new belief divergence, considering the uncertainty of multielement sets and the correlation between subsets, is proposed.Its detailed defnition is as follows.
Defnition 10 (the belief divergence BD).Given two independent BPAs m 1 and m 2 defned on where Γ m 1 and Γ m 2 are new BPAs after m 1 and m 2 are modifed by adjustment function, the concrete modifcation process is displayed as follows: Here, we provide additional explanations to illustrate the divergence.Te divergence measure is utilized to refect the variation of information distribution and can be seen as an extension of the uncertainty measure.Te information quality IQ (p) proposed by Yager is the negation of the Gini index; both of them are uncertainty measures [47,48].Te Bhattacharyya coefcient BC can be seen as a similarity measure of the IQ (p) in [47] or Gini index in [48].Li extended IQ (p) to the information quality IQ (m) under the framework of evidence theory [49].Te belief Bhattacharyya coefcient BBC, a generalization of the Bhattacharyya coefcient, can be seen as a similarity measure of the IQ (m).Terefore, the BD based on the BBC can be seen as a belief divergence of the IQ (m).
Recall Examples 1 and 2, where the divergence measure among m 1 , m 2 , and m 3 is recalculated by BD, which is compared with BJS divergence and BH distance in Table 1 Te BD is more accurate to measure the confict between evidence.

The Enhanced Belief Divergence
Te BD does refect the degree of confict between evidence from diverse sources and has vanquished the demerits of BJS divergence and BH distance.Notably, it appears that the divergence measures of all groups of evidence, respectively, in the frst case of zero belief value and the second case of the same cardinality, are the same.To address this limitation, an enhancement factor is proposed, and a base belief function is used to reinforce the BD; that is, an enhanced belief divergence EBD is presented.Besides, the properties of the EBD have been discussed.Finally, a comparative evaluation is given to illustrate the validity and superiority of the EBD.

Modifcation with the Base Belief Function.
From the operational property of the BBC, it is observed that the BD may fail in measuring the discrepancy of evidence, when evidence has a zero belief value.Te nature of this phenomenon is concretely illustrated by the Case 11.
Case 11.Suppose m 1 and m 2 , m 3 and m 4 are two groups of BPAs defned on In group 1, m 1 and m 2 are conficting.In group 2, because of the uncertainty of the multielement set B, C { } in m 4 , the confict degree between m 3 and m 4 is smaller than that of  International Journal of Intelligent Systems group 1.By equation (20), both groups of modifed BPAs are obtained as follows: According to equation ( 16), we have Ten, according to equation ( 19), the BD is calculated as follows: From the above result, the degree of confict of the two groups is the same.It is counterintuitive.Actually, as for each focal element of Γ m 1 and Γ m 2 , if its belief value in Γ m 1 or Γ m 2 is 0, the BBC is 0, then the BD is always equal to 1 in this situation.
It is noticed that Liu exploited the base belief function to handle a possible zero in the denominator of the divergence [46].Inspired by it, in this paper, for resolving zero belief, we modify the Γ m by averaging it and m b to obtain Γ b m, then belief of all focal elements in Γ b m is made nonzero.Tus, the Case 11 for the BD is managed.

Te Enhancement Factor.
In addition, it is found from (19) that when all focal elements of two pieces of evidence have the same cardinality, the infuence of cardinality will be ofset by the fractional term BBC In this example, each group of evidence is highly conficting.With the cardinality of subsets enlarging, m 5 and m 6 in group 3 carry the largest ambiguity.Terefore, the divergence between m 5 and m 6 is the smallest, and that between m 1 and m 2 is the largest.Whereas, by applying the BD, we obtain the divergences as follows: Te above result is not in line with the intuitive analysis.Te reason for this is that the item 1/(2 |A i | − 1) in equation (19) appears in both the numerator and denominator of BBC When the cardinality of all focal elements in each group is the same, the fractional expression makes the infuence of |A i | eliminated.
To deal with this case, an enhancement factor is devised to perfect the measure efect of the BD, which is defned as follows.
Defnition 13 (the enhancement factor β). Given two independent BPAs m 1 and m 2 defned on the enhancement factor β is denoted as follows: As can be seen in ( 27), the enhancement factor considers the ambiguity diference by various values of c. Tus, by it, the BD can distinguish evidence confict degree of the Case 2.

Te Enhanced Belief Divergence.
With the enhancement factor and base belief function, the enhanced belief divergence EBD is proposed.Defnition 14 (the enhanced belief divergence EBD).Given two independent BPAs m 1 and m 2 defned on Θ including N hypotheses.
where Γ b m 1 and Γ b m 2 are new BPAs after Γ m 1 and Γ m 2 are modifed by the base belief function.Te detailed modifcation process is showed as follows: Te EBD is an improved BD by modifying BPA; in the same way as the BD, it can also be regarded as a belief divergence of the IQ(m).In addition, the EBD maintains the excellent qualities of the BD, namely, it can diferentiate multielement sets from singletons and allow for the correlation between subsets.Furthermore, the EBD improves the performance of the BD to measure the dissimilarity of evidence more efectively.
In order to show the calculation process clearly, Example 2 is adopted.Te divergence measures among m 1 , m 2 , and m 3 are calculated by the EBD as follows.
In Example 2, as for m 1 and m 2 , there are four focal ele- According to the (27), it is obtained that β is 1.For convenient calculation, m 1 and m 2 are simplifed as follows: In accordance to (18), the adjustment function Γ A i ,A j between m 1 and m 2 is obtained in the following Table 2.
Besides, for the purpose of verifying whether the EBD have solved Case 11 and Case 12, the two cases are recalculated by the EBD and the results are compared with the BD in Table 3.
From Table 3, in Case 11, the confict degree of group 2 by the EBD is smaller than that of group 1, in Case 12, with the uncertainty of the three groups of evidence increasing, the EBD is diminishing, which conforms to the analysis of the two cases.Consequently, the EBD is more reasonable and valid than the BD for evidence discrepancy measurement.
(1) m 1 and In the light of the characteristics of the adjustment function and base belief function, Γ b m 1 and Γ b m 2 meet with 0 Ten, we obtain According to we have Terefore, As a result, Finally, Te boundedness has been proven.
(3) Given two BPAs m 1 and m 2 defned on Θ: Te symmetry has been proven.
In this example, the enhancement factor β of the EBD is 1.As t � 1, it is discovered that m 1 and m 2 , respectively, support A { } and B { }, which is highly conficting.As t � 2, A t has the element B, it increases the probable belief of B { } in m 1 , the discrepancy between m 1 and m 2 decreases.Ten, as the number of elements in A t enlarges, the divergence between m 1 and m 2 also enlarges.
However, as depicted in Figure 2(a), it is clear that Florea and Bossé's distance d B and BJS divergence keep unchanged, which is not proper.In addition, although BH distance has varying values with t, it is unreasonable to have a downward trend.We can observe that only the EBD accords with the changing tendency of confict degree between m 1 and m 2 .
Example 5. Suppose m 1 and m 2 are two independent BPAs defned on Θ � A, B, C, D, E, F, G, H, I, J { }, A t is a variable set from A { } to Θ, t: 1 ⟶ 10, the specifc variations of A t are the same as those in Example 4.

International Journal of Intelligent Systems
In this example, m 1 has the varying focal elements A t   and m 2 has the focal elements A, B { } and C { }.Because the cardinality of them is not identical, the enhancement factor is 1.When t � 1, m 1 supports the A { } and m 2 supports the A, B { }.As t � 2, m 1 completely supports the proposition A, B { } same as the supportive proposition of m 2 , thus, the value of divergence between m 1 and m 2 decreases.As t � 3, with the hypothesis C added to the A t  , m 1 has the possibility to support C { }, the confict degree between m 1 and m 2 decreases.When A t   continue to add other elements, the divergence is getting large.
As displayed in Figure 2(b), d B and BJS divergence are always kept as one except t � 2. BH distance keeps decreasing with t in general.Obviously, it infers that the EBD can perform better than the other divergences on discrepancy measurement.Example 6. Suppose m 1 and m 2 are two independent BPAs defned on Θ, A t is a variable set defned as Table 4.
In this example, the belief value distributions of m 1 and m 2 are identical, the diference between m 1 and m 2 is that between focal elements A t and A, B { }.When t � 1, the two evidence is identical, so the evidence between m 1 and m 2 is 0.
In the similar way, as t � 3, 4, it is just the hypothesis C that respectively changes to D and E, therefore, the divergence between m 1 and m 2 at t � 2, 3, 4 is the same.As t � 5, A t in m 1 is B, C { }, the intersection of A { } and A t is ∅, which decreases the possibility to support A { }, so the value of divergence is much larger than the former states.Te situation of t � 6, 7 is similar to that of t � 5, namely, m 1 and m 2 at t � 5, 6, 7 is also the same.Te remaining circumstance t � 8, 9, 10 can be concluded likewise.
However, as portrayed in Figure 3, it can be observed that d B , BJS divergence and BH distance maintain unchanged except at t � 1, which cannot refect the correlation between diferent types of subsets.Terefore, the results by the EBD show more reasonable and efective.

A New EBD-Based Multisource Information Fusion Method
Multisource information fusion refers to the means that integrate data from various sensors to generate a rational and precise result.Te data gathered from every sensor can be modeled as a piece of evidence, but the credibility of the evidence is susceptible to sensor failure or detrimental environmental factors, which have an impact on the accuracy of the result.Terefore, it is crucial to evaluate the reliability of evidence during the information fusion process.In this section, based on the EBD and Deng entropy, a new multisource information fusion approach is devised.Specifcally, the weight of evidence is decided by the divergence between the evidence and the uncertainty contained in evidence.Te EBD can allude to the extent of evidence inconsistency, where the evidence modeled by the IV Step 2.3: Obtain the information volume weight.By normalizing the IV, the information volume weight W iv (m i ) of evidence m i is denoted as follows: Step 3. Producing the weighted average evidence.
Step 3.1: Generate the fnal weight.
Combing the credibility weight and information volume weight of evidence m i , the fnal weight W(m i ) of evidence m i is acquired as follows: Step 3.2: Weight the body of evidence.Te weighted average evidence is calculated as follows: Step 4. Fuse the weighted average evidence.Te weighted average evidence is fused with the Dempster's combination rule equation ( 4) by n − 1 times, the eventual result is obtained as follows: Step 2: Form the information volume weight Step 1: Determine the credibility weight Step 1.2: Calculate the average divergence Step 1.3: Generate the support degree Step 1.4: Obtain the credibility weight Step 2.1: Calculate Deng entropy Step 2.2: Get the information volume Step 2.3: Obtain the information volume weight Step 1.1: Construct the divergence measure matrix

BPA
Step 3: Produce the weighted average evidence Step 3.2: Obtain the weighted average evidence Step 3.1: Generate the final weight Step 4: Fuse the weighted average evidence and obtain the final fusion results (59)

Experiment
To demonstrate the feasibility and efectiveness of our method, two experiments, i.e., a target recognition problem and a classifcation problem are presented.Te comparison with other methods is conducted to further illustrate the superiority of the EBD-based multisource information fusion approach.
6.1.Target Recognition.In a multisensor-based target recognition system, suppose the frame of discernment, including three possible targets, is Θ � A, B, C { }, there are fve installed sensors S 1 , S 2 , S 3 , S 4 , S 5   in the system.Te data collected from the fve sensors are modeled as fve BPAs, m 1 , m 2 , m 3 , m 4 , m 5  , shown in Figure 5. Tis experimental data is based on Deng [33].
From Figure 5, it is noted that the target directivity of sensors m 1 , m 3 , m 4 , and m 5 is more oriented to A; therefore, we can infer that A is the real target, which should be allocated a high level of accuracy in the fusion results.While only m 2 assigns most of its belief to strongly support the target B, it has a diferent direction from the other sensors.As a result, it is believed that m 2 is highly conficting with other four pieces of evidence.As a comparison, the fusion results of the well-known methods and the proposed method are presented in Table 5.
As can be seen in Table 5, the comparative results indicate that A is the real target, which verifes the perception from the above analysis.Obtained by Dempster's method, the fusion result realizes C as the identifed target and distributes zero belief to A. Evidently, such a result is unreasonable.Hence, it is unsuitable to adopt Dempster's combination rule directly to combine the conficting evidence.Murphy's method can correctly determine the target type as A.Moreover, the recognized target of Deng's method is in accordance with that of Murphy's method with a higher belief.Although the two aforesaid methods are able to identify the real target, it is noteworthy that the proposed method can achieve the highest accuracy of 0.9904.From these fndings, the proposed method can make a more accurate decision result when dealing with conficting evidence.

Iris
Classifcation.An iris dataset-based classifcation experiment, containing the data without confict and with confict, is implemented here.For the sake of fairly comparing the results, the generated BPAs in Qian [50] are referred to further assess the performance of our method.

Fusion without Confict.
Tere are three types of iris fowers (Setosa, Versicolor, and Virginica), and the framework of discernment is Θ � Se, Vc, Vi { }.In addition, each type of iris fower has four attributes, namely Sepal Length (SL), Sepal Width (SW), Petal Length (PL), and Petal Width (PW).Te converted BPAs based on the iris dataset are shown in Figure 6.
From Figure 6, the BPAs of the four attributes, m 1 , m 2 , m 3 , and m 4 , all bestow a relatively high belief to the fower type Se.In other words, there is no confict between them, and the belief allocated to Se { } should be the highest after fusion.Te fusion results of the proposed method and comparative methods are presented in Table 6.From Table 6, as we expected, all methods, including our method, can identify the fower type, when evidence is not conficting.

Fusion with Confict.
To further verify the robustness of the proposed method, the data of attribute SW source is revised to serve as noisy evidence.A group of obtained evidence with confict is shown in Figure 7. From Figure 7, it infers that the SL attribute has no clear directivity toward the fower types of Se and Vc, with proximate belief values of 0.3337 and 0.3165.While the SW attribute assigns almost all beliefs to Vc. Te attributes of PL and PW believe that the test sample belongs to the fower type of Se.As a consequence, it is suggested that the correct fower type of the test sample is Se.
After conducting the proposed method and other researchers' schemes, the fusion results are shown in Table 7.As can be seen from Table 7, Dempster's method and Murphy's method trust that Vc is the real fower type of the test sample, and they give Se a low support degree, which yields a completely misleading result.Terefore, they cannot work efectively when the evidence is conficting.Both Deng's method and the proposed method can precisely recognize the real target; what's more, the proposed method endows a larger belief to Se than Deng's method dose.
Te reason why the proposed method outperforms other methods is that in Dempster's method, it directly uses the combination rule to fuse highly conficting evidence, but produces a counter-intuitive result.To a certain extent, Murphy's method can handle the confict by simply averaging the evidence.However, it makes all evidence have the same weight, which may eliminate the confict among evidence and greatly infuence the fusion result.Taking the distance between evidence into account, Deng's method distributes diferent weights to evidence, while it ignores the information volume of evidence itself.Te proposed method considers not only the divergence but also the information volume to thoroughly calculate the weight of the evidence.Terefore, it can be concluded that the proposed method has a preferable efect on decision fusion.

Application in Failure Mode and Effects Analysis of Aircraft Turbine Rotor Blades
Information fusion is widely applied in risk evaluation and expert system [51,52].In the aerospace feld, rotor blades, including compressor rotor blades and turbo rotor blades, are the major components of an aircraft turbine, whose reliability seriously afects the overall aircraft turbine security.In order to enhance their safety, failure mode and efects analysis (FMEA) can facilitate the identifcation of potential failures International Journal of Intelligent Systems and determine the efect of each failure to decrease failure rates and avoid hazardous accidents.However, there may be a load of failure modes with diferent risks and efects.Consequently, it is necessary to prioritize their risks.Te risk priority number (RPN) is one way to rank these failure modes.Te RPN is the product of the three factors, the probability of the occurrence of a failure mode (O), the severity of a failure efect (S), and the probability of a failure being detected (D), expressed as RPN � O × S × D. However, multiple experts may give diferent risk evaluations on three risk factors for one failure mode, which may be imprecise and uncertain.Terefore, multisource information fusion can be used to promote the accuracy of evaluation.In this section, the EBD-based multisource fusion method is adopted to calculate the new mean value of the RPN, and then determine the risk priority of multiple failure modes of aircraft turbine rotor blades, in which the EBD plays a key role in deciding the weights of experts.Furthermore, the risk ranking results are compared with other methods to determine the validity of our method..Te experts may give their diferent evaluations to the same risk factor, which are modeled as J evidence: m 1 , . . ., m J  .Consequently, there are three discernment frames respectively for O, S and D.Moreover, for the N failure modes, the total number of discernment frame is 3N.Under this circumstance, the frame of discernment of the i th risk factor of the n th failure mode can be presented as follows: where min X| X⊆Θ n i and max X| X⊆Θ n i separately means the minimum and maximum of the rank of the n th failure mode to the i th risk factor from J experts.

Implementation.
Te rotor blades of an aircraft turbine consist of two subsystems, the compressor rotor blades and the turbo rotor blades.According to the practical engineering background, there are nine potential failure modes in the compressor rotor blades and eight failure modes in the turbo rotor blades, namely, 17 recognized failure modes FM 1 , . . ., FM 17   in total [53].In this experiment, the BPAs in Yuan [54], transformed from the evaluation information of the three experts E 1 , E 2 , E 3   to O, S and D on the 17 failure modes, are referred.On the simplifed frame of discernment in (61), we  use the EBD-based information fusion algorithm in Section 5 to aggregate the BPAs of i th risk factor in the n th failure mode, the fusion results are obtained as m n i (A), A ⊆ Θ n i , i � O, S, D, where A represents the rating of the risk factors.Attributed to the axiom of additivity, m n i (A) can be regarded as the probability of A. According to [54], the mean value of RPN can be used to compare the overall risk of each failure mode.Based on the proposed fusion method, the new mean value of RPN, named as EBD RPN avg , can be obtained by For ease of understanding, the above process of calculating the EBD RPN avg is showed in the form of a fowchart as Figure 8.At the same time, the calculated EBD RPN avg s of 17 failure modes are presented in Table 8.
As shown in Table 8, among the 8 failure modes of compressor rotor blades, failure mode 2 has the largest EBD RPN avg and failure mode 5 has the least EBD RPN avg .Decided by sorting numeric size of EBD RPN avg , the risk priority order of them is FM 2 ≻ FM 6 ≻ FM 1 ≻ FM 3 ≻ FM 7 ≻ FM 4 ≻ FM 8 ≻ FM 5 .Among the 9 failure modes of turbo rotor blades, failure mode 9 has the largest EBD RPN avg and failure mode 16 has the least EBD RPN avg , the risk priority order of them is FM 9 ≻ FM 10 ≻ FM 14 ≻ FM 12 ≻ FM 11 ≻ FM 13 ≻ FM 15 ≻ FM 17 ≻ FM 16 ,≻ hints that the previous item has a higher priority.
Several comparative methods to investigate the RPN in FMEA are introduced here.In detail, AMWRPN takes into consideration of the relative weight of diferent risk factors, by measuring the ambiguity degree of the experts' assessments, to get a new ambiguity measure weighted risk priority number [55].MVRPN calculates the average of the obtained RPN values with the modifed belief function and combination rule [53].Te improved MVRPN constructs the BPA to handle the conficting evidence and refne the MVRPN [56].Te method in the literature [54] gives a new mean value of RPN based on triangular fuzzy numbers, negation of BPAs and evidence distance.Te comparison results with the above methods are shown in Table 9.
From Table 9, the results of the AMWRPN method in FM 1 − FM 17 are very close.And the values of the MVRPN method are similar to those of the improved MVRPN method.In addition, for turbo rotor blades, in the MVRPN method, failure mode 10, 13, and 14 have the same RPN 60, and failure mode 11 and 12 have the same RPN 50.In the improved MVRPN method, the RPN of failure mode 11, 12, and 13 is all the same.Remarkably, the results by our method are very close to those of RPN avg .It is worth noting that the values of EBD RPN avg are not dense and well distinguished, which contributes to the diferentiated risk ranking of multiple failure modes of rotor blades.18 International Journal of Intelligent Systems  Te rank results of failure modes of rotor blades for an aircraft turbine are shown in Figure 9.In Figure 9(a), the risk priorities for compressor rotor blades by our method are nearly consistent with other methods.In Figure 9(b), the risk priorities for turbo rotor blades by our method completely coincide with the RPN avg .Although the risk priorities by EBD RPN avg are slightly diferent with the three other methods, this is acceptable.It is the reason that the several identical RPN values, in the MVRPN method and the improved MVRPN method, lead to the sorting diference.Terefore, in the FMEA of the rotor blades of an aircraft turbine, the proposed method has efectiveness and practicality.

Conclusion
In this paper, an enhanced belief divergence, named as EBD, is proposed to measure the discrepancy between evidence.Te proposed EBD can distinguish between singletons and multielement sets and express the intersection relationship among subsets.Some important properties of the EBD are inferred.In addition, the comparison interprets that the EBD has a preferable efect on confict measurement.Next, an EBD-based multisource information fusion method is devised.In the applications of target recognition and iris classifcation, the proposed method can efectively handle uncertainty and confict with higher accuracy values.Specially, the basic belief assignment of the true target in target recognition achieves 0.9904.Finally, in the risk priority evaluation of the failure modes of the rotor blades of an aircraft turbine, the risk ranking results by the proposed method are almost consistent with other methods, demonstrating the applicability of the proposed method.
In our future work, we intend to further study the performance of the proposed method to handle nonconficting information.Also, we can broaden the proposed approach to solve other practical problems, such as image processing problems.Besides, we will deepen our research on fusion method when the BPA is an interval value.

Example 3 .
Suppose m 1 and m 2 are two independent BPAs defned on Θ � A, B, C, D { }, where x ranges from 0 to 1.

Figure 1 :
Figure 1: Te comparison of the BBC and FBIP.

8 International Journal of Intelligent Systems Theorem 15 .
Te EBD has the properties of boundedness, nondegeneracy, and symmetry.Property 16.Let m 1 and m 2 be two BPAs defned on the frame of discernment Θ:

Figure 2 :
Figure 2: Te comparison with the d B , BJS divergence, and BH distance in Examples 4 and 5, (a) Example 4, (b) Example 5.

Figure 4 :
Figure 4: Te fowchart of the EBD-based multisource information fusion algorithm.

Figure 5 :
Figure 5: Te fve BPAs from sensors in target recognition.

O 1 Figure 8 :
Figure 8: Te fowchart for prioritizing risk based on the EBD RPN avg .

Figure 9 :
Figure 9: Te risk ranking consequences of failure modes of rotor blades for an aircraft turbine.(a) Compressor rotor blades.(b) Turbo rotor blades.
. From Table 1, diferent with the unreasonable results by BJS divergence and BH distance, it is uncovered that in Example 1 Tis is interpreted in detail by Case 12. Case 12. Suppose m 1 and m 2 , m 3 and m 4 , and m 5 and m 6 are three groups of BPAs defned on Θ � θ 1 , θ 2 , θ 3 , θ 4 , θ 5 , θ 6  .

Table 1 :
Te results comparison with BJS divergence in Example 1 and BH distance in Example 2.

Table 2 :
Te adjustment function Γ A i ,A j between m 1 and m 2 .

Table 3 :
Te results comparison with the BD in Case 11 and Case 12.

Table 5 :
Te fusion results of the four methods in target recognition.

Table 6 :
Te fusion results of non-confict data by diferent methods in iris classifcation.Figure 7: Te BPAs from four attributes with confict in the iris classifcation.

Table 7 :
Te fusion results of confict data by diferent methods in iris classifcation.