A Novel Conflict Measurement in Decision-Making and Its Application in Fault Diagnosis

Dempster–Shafer evidence (DSE) theory, which allows combining pieces of evidence from different data sources to derive a degree of belief function that is a type of fuzzy measure, is a general framework for reasoning with uncertainty. In this framework, how to optimally manage the conflicts of multiple pieces of evidence in DSE remains an open issue to support decision making. The existing conflict measurement approaches can achieve acceptable outcomes but do not fully consider the optimization at the decision-making level using the novel measurement of conflicts. In this article, we propose a novel evidential correlation coefficient (ECC) for belief functions by measuring the conflict between two pieces of evidence in decision making. Then, we investigate the properties of our proposed evidential correlation and conflict coefficients, which are all proven to satisfy the desirable properties for conflict measurement, including nonnegativity, symmetry, boundedness, extreme consistency, and insensitivity to refinement. We also present several examples and comparisons to demonstrate the superiority of our proposed ECC method. Finally, we apply the proposed ECC in a decision-making application of motor rotor fault diagnosis, which verifies the practicability and effectiveness of our proposed novel measurement.

have been presented for reasoning with and managing uncertainty, including the extended intuitionistic fuzzy sets [9], rough sets [10], Z numbers [11], evidence theory [12], [13], evidential reasoning [14], D numbers [15], R sets and numbers [16], [17], and other hybrid methods [18]. These theories are applied broadly in various fields, such as image classification [19], medical diagnosis [20], [21], information fusion [22], and decision making [23], [24]. In these fuzziness-related approaches, one of the most useful tools to handle uncertainty is Dempster-Shafer evidence (DSE) theory [25], [26], which has posed several attractive advantages as follows: 1) quantitatively modeling uncertainty by means of a basic probability assignment (BBA) [27]; 2) the belief function is a type of fuzzy measure that provides partial information in terms of the appropriate fuzzy measure in relation to an uncertain variable [28], [29]; 3) Dempster's combination rule (DCR) satisfies the commutative and associative laws [30], [31]; 4) the results generated by the DCR have the characteristic of fault-tolerance and relieve the uncertainty level by the DCR [32], [33]. Consequently, DSE theory can be of benefit for supporting decision making [34] and has been extensively investigated in extracting the information quality of BBA [35] and evidence reliability evaluation [36], [37].
According to previous studies of evidence theory, considering optimal management of conflicts may improve the accuracy performance at the decision-making level in data science applications [38]- [40]. Therefore, how to measure the conflict of multiple pieces of evidence has attracted considerable research attention in recent years [41]- [43], and many related definitions have been presented [44], which can be used for fuzzy system-based industrial application areas. Although the outcomes of current conflict management methods are acceptable in DSE theory, we assume there still remains room for improving decision-making performance at the decision level in terms of the measure and management of conflicts.
Therefore, in this article, we explored a novel conflict measurement in decision making. Here, we proposed a new evidential correlation coefficient (ECC), inspired by Jiang's [45] method, to measure the correlation between BBAs in DSE theory, which could be proved, analyzed, and applied in the decision making of data science applications. Specifically, we proposed a new evidential conflict coefficient based on ECC to measure the conflict degree between BBAs. Then, we analyzed and proved that the newly defined evidential conflict coefficient has the desirable properties for conflict measurement, including nonnegativity, symmetry, boundedness, extreme consistency, and insensitivity to refinement. Furthermore, we compared the proposed evidential conflict coefficient with well-known methods and demonstrated a motor rotor fault diagnosis application devised based on the ECC.
The rest of this article is organized as follows. Section III and IV briefly introduce the preliminaries of evidence theory and some existing conflict measures, respectively. Section V proposes new evidential correlation and conflict coefficients and their properties are analyzed and proved. Section VI compares various conflict measures to demonstrate the superiority of the proposed method. In Section VII, a fault diagnosis algorithm is devised based on the new correlation coefficient measure; then, the algorithm is applied to solve a motor rotor fault diagnosis problem. Finally, Section VIII concludes this article.

II. RELATED WORKS
As we know, the traditional Dempster's conflict coefficient K [25] combines the mass allocated to the empty set, accounting for the conflict among focal elements, but it ignores the global consistency between different pieces of evidence.
To overcome this limitation, George and Pal [46], Jousselme et al. [47], and Cheng and Xiao [48] considered the conflict measure from the nonintersecting parts between different pieces of evidence. Another group of researchers quantified the measure of conflict from an alternative perspective. For instance, Liu [49] designed a two-dimensional conflict model by combining Dempster's conflict coefficient and pignistic probability distance. Daniel [50] considered the plausibility conflict of evidence. Lefevre and Elouedi [51] studied measured conflict by means of the distance between pieces of evidence and the mass of an empty set. Furthermore, some novel strategies, such as divergence measures, have also been leveraged to measure evidential consistency [52]- [54]. For example, Ma and An [52] quantified the divergence grade of evidence by fuzzy nearness and a correlation coefficient. Xiao [53] measured the divergence of evidence by means of Jensen-Shannon divergence. In addition, some researchers investigated conflict measurement from the perspective of correlation coefficients [45], [55], [56]. For instance, Song et al. [55] defined a correlation coefficient [57] as the cosine of the angle between two vectors of pieces of evidence. Pan and Deng [56] developed a correlation coefficient [58] on the basis of Deng entropy [59]. Jiang [45] discussed the conflict measure by taking into account the nonintersection and the difference among focal elements [60].
In this article, inspired by Jiang's [45] method, we propose a novel conflict measurement in decision making and apply it in fault diagnosis, which can improve decision-making performance at the decision level.
Definition 1: (Frame of discernment) Let Ω be a set of mutually exclusive and collective nonempty events defined by [25], [26] where Ω is a frame of discernment (FOD). The power set of Ω is denoted as 2 Ω where ∅ represents an empty set.
satisfying m(∅) = 0, and In DSE theory, m is also called a BBA. For A i ⊆ Ω, if m(A i ) is greater than zero, A i is called a focal element. Since a BBA can effectively express the uncertainty, various BBA operations have been devised, including negation [71], [72] and an entropy function [73].
Definition 3: (Belief function) The belief function of A i ⊆ Ω, denoted as Bel(A i ), is defined as [25], [26] Bel Definition 4: (Plausibility function) The plausibility function of A i ⊆ Ω, denoted as Pl(A i ), is defined as [25], [26] Pl Bel(A i ) and Pl(A i ) represent the lower and upper bound functions of A i , respectively. An interval-valued belief structure can be used for an uncertainty measure [74], [75].
Definition 5: (Dempster's combination rule) Let m 1 and m 2 be two independent BBAs in FOD Ω. DCR, represented in the form m = m 1 ⊕ m 2 , is defined as [25], [26] m(A i ) = where A h , A k ⊆ Ω and K is the coefficient of conflict between BBAs m 1 and m 2 .

IV. EXISTING CONFLICT MEASURES
In this section, some existing conflict measures for belief functions are briefly introduced.
Let m 1 and m 2 be two BBAs with hypotheses A i and A j , respectively, on the same FOD Ω = {F 1 , . . . , F i , . . . , F n }.
where − → m 1 and − → m 2 are the BBAs in vector notation and D is a 2 n × 2 n matrix with elements in which | · | represents the cardinality function. Definition 7: Lefèvre and Elouedi's adapted conflict [51] where m ∩ (∅) is equal to K in (8) and d JGB is (9).
in which m is defined as where D is defined in (10). Song et al.'s conflict coefficient: Definition 9: Jiang's correlation coefficient [45] where c(m 1 , m 2 ) is defined as Jiang's conflict coefficient Definition 10: Cheng and Xiao's [48] distance where D α is a 2 n × 2 n matrix with elements

V. NEW EVIDENTIAL CORRELATION AND CONFLICT COEFFICIENTS
For developing an effective conflict measurement, firstly our proposed a new method aims to satisfy the properties of a conflict measurement. Second, we consider determining how conflict identification between BBAs for improving performance. Third, for two arbitrary BBAs m 1 and m 2 , we explore the conflict from the view of m 1 to m 2 , as well as the conflict from the view of m 2 to m 1 . Based on the abovementioned context, inspired by Jiang's work [45], we design the evidential correlation and conflict coefficients, and specifically address an ECC for measuring the correlation between BBAs. We, then, analyze and prove the properties of ECC. Furthermore, we define an evidential conflict coefficient and discuss desirable properties for conflict management.
Definition 11: (ECC measure between BBAs) Let m 1 and m 2 be two BBAs on Ω = {F 1 , . . . , F i , . . . , F n }, where A i and A j are hypotheses of BBAs. The ECC between BBAs m 1 and m 2 , denoted as ECC(m 1 , m 2 ), is defined as In (20), cos Θ is a cosine angle function between − → m 1 and − → m 2 which has a mathematical formula similar to (12) [55]; it can be seen that Hence, (20) can be expressed in another form (25) Theorem 1: The ECC has the properties of nonnegativity, nondegeneracy, symmetry, and boundedness [45].
Clearly, ECC(m a , m b ) ≥ 0 can be conducted, which proves the property of nonnegativity of the ECC.
(P1.2) Consider two arbitrary BBAs m a = m b in FOD Ω with the hypotheses of A i and A j ; we have Then, we obtain ⎡ Hence, ECC(m a , m b ) = 1 ⇐⇒ m a = m b , which proves the property of nondegeneracy of the ECC.
(P1.3) Consider two arbitrary BBAs m a and m b in FOD Ω with the hypotheses of A i and A j .
From (26) and (27), it is clear that which proves the property of symmetry of the ECC. (P1.4) Consider two arbitrary BBAs m a and m b in FOD Ω.
Since D is a Hermitian positive-definite matrix [45], for a 2 n × 2 n lower triangular matrix G, we have Thus, .
Because u T G T Gu = (Gu) T (Gu) = Gu 2 , by applying the triangle inequality on the vector 2-norm [45], we obtain Hence, as proved in (P1.1) that ECC(m a , m b ) ≥ 0, we obtain which proves the property of boundedness of the ECC. Remark 1: Note that the larger ECC(m 1 , m 2 ) is, the greater the correlation coefficient between the BBAs. Therefore, if ECC(m 1 , m 2 ) = 1, then m 1 and m 2 are completely correlated; if ECC(m 1 , m 2 ) = 0, then m 1 and m 2 are completely uncorrelated.
Next, an example is presented to illustrate the properties of ECC(m 1 , m 2 ).
Example 1: Assume there are two BBAs, m 1 and m 2 , in Ω In Example 1, m 1 and m 2 change according to α and the subset of ϑ. Here, α is set within [0,1], and the subset ϑ is set as  Additionally, in this example, the boundedness property of the ECC, in which ECC(m 1 , m 2 ) is greater than or equal to 0 and less than or equal to 1, is verified. Furthermore, the results shown in Fig. 1 reveal the symmetry property of the ECC.
Based on Definition 11, the evidential conflict coefficient between BBAs is defined as follows.
Definition 12: (The evidential conflict coefficient between BBAs.) The evidential conflict coefficient between BBAs m 1 and m 2 , denoted as k ECC (m 1 , m 2 ), is defined as (28) Theorem 2: The k ECC has desirable properties for conflict measurement [45], including nonnegativity, symmetry, boundedness, extreme consistency, and insensitivity to refinement. P2.5 Insensitivity to refinement: for m 1 and m 2 refined from . Proof: The proofs of (P2.1)-(P2.5) are trivial. Remark 2: Note that the larger k ECC (m 1 , m 2 ) is, the greater the conflict coefficient between the BBAs. If k ECC (m 1 , m 2 ) = 1, then m 1 and m 2 are in complete conflict; if k ECC (m 1 , m 2 ) = 0, then m 1 and m 2 are in no conflict.
Next, an example is presented to illustrate the nonnegativity and boundedness properties of k ECC .
Example 2: Assume there are two BBAs m 1 and m 2 in Ω In Example 2, m 1 changes according to α and β, which are set within [0,1] and satisfy α + β ≤ 1, as shown in Fig. 2(a). Then, as α and β vary, the corresponding correlation coefficient measures are shown in Fig. 2(b)-(d). Fig. 2 verifies the nonnegativity and boundedness properties of k ECC , where k ECC ≥ 0 and 0 ≤ k ECC ≤ 1.
As shown in Fig. 2 Furthermore, Fig. 2(c) shows the variation in k ECC as α increases from 0 to 1. Clearly, when α increases from 0 to 0.7, since m 1 gradually becomes closer to m 2 , the conflict coefficient k ECC decreases. As α increases from 0.7 to 1, because m 1 tends to become dissimilar to m 2 , the conflict coefficient k ECC increases.
Moreover, Fig. 2(d) shows the variation in k ECC as β increases from 0 to 1. Similarly, when β increases from 0 to 0.3, since m 1 becomes closer to m 2 , the conflict coefficient k ECC decreases. As β increases from 0.3 to 1, because m 1 shifts farther from m 2 , the conflict coefficient k ECC increases.
Next, we present an example to illustrate the symmetry and insensitivity to refinement properties of k ECC .   As shown in Fig. 3(a)-(d), k ECC is not impacted by the variation in the FODs from Ω to Ω . Even under variation in α and γ, the value of k ECC under the FOD Ω is always the same as that under the FOD Ω . Moreover, when we change the input k ECC (m 1 , m 2 ) to k ECC (m 2 , m 1 ), the results are exactly the same. Consequently, the symmetry and insensitivity to refinement properties of k ECC are verified.  [55] k SW , Jiang's [45] k J , Cheng and Xiao's [48] d CX , and the proposed k ECC . In addition, we assess whether these conflict measures satisfy the abovementioned properties, as well as the degrees of conflict.   EXAMPLE 4 In Example 4, for the focal elements A i and A j of m 1 and m 2 , respectively, we have (∪A i ) ∩ (∪A j ) = ∅. Thus, m 1 and m 2 are in complete conflict, so the conflict grade between m 1 and m 2 is assumed to be 1.

VI. COMPARISON WITH EXISTING METHODS
The results in Table I indicate that the conflict degrees produced by K, k J , and k ECC are 1, in accordance with the intuitive result. By contrast, d JGB , k LE , and d CX generate a conflict value of 0.7071, and k SW has a conflict degree of 0.601. Therefore, d JGB , k LE , k SW , and d CX do not satisfy the extreme consistency property of conflict measures. In Example 5, m 1 is the same as m 2 , with the same support values for the corresponding subsets. Therefore, m 1 and m 2 are completely nonconflicting, so the conflict grade between m 1 and m 2 should be zero.    As shown in Table III, K = 0.68, d JGB = 0.6, k LE = 0.408, k J = 0.5294, d CX = 0.6, and k ECC = 0.7785, regardless of FOD Ω or Ω . This conforms to the expected result. However, k SW generates a conflict value of 0.3871 on FOD Ω and conflict of 0.3716 on FOD Ω , which does not satisfy the conflict measure property of insensitivity to refinement.
In summary, the abovementioned examples demonstrate the disadvantages of the conflict measures derived in related works. k J and the proposed k ECC satisfy the properties of conflict measures, especially the last two properties. By contrast, K, d JGB , k LE , k SW , and d CX do not satisfy the extreme consistency property, and k SW does not satisfy the property of insensitivity to refinement. To further study the effectiveness of the proposed k ECC , we discuss the following examples.
Example 7: Assume there exist two BBAs m 1 and m 2 in In Example 7, the subset ϑ i of m 1 changes from {A 1 } to {A 1 , . . . , A 19 }, as shown in Table IV. Note that m 2 has one focal element such that m 2 ({A 1 , A 2 , A 3 , A 4 , A 5 }) = 1. Then, the conflict measures between BBAs m 1 and m 2 are calculated, as shown in Fig. 4 and Table V. When i = 5, the support value of subset {A 1 , A 2 , A 3 , A 4 , A 5 } of m 1 is 0.8, which is closer to that of m 2 with the subset {A 1 , A 2 , A 3 , A 4 , A 5 } than in other cases of i. Therefore, the expected conflict grade is assumed to achieve the minimum value. The results in Fig. 4 indicate that    In Example 9, the subset δ i of m 1 changes from δ 1 to δ 19 , as shown in Table VI. When i = 1, 2, . . . , 10, δ i is the same as ϑ i . When i increases from 11 to 19, the subset δ i of m 1 is pruned from its first element until it becomes {A 10 }. Note that m 2 has one focal element such that m 2 ({A 10 }) = 1. The conflict measures between BBAs m 1 and m 2 are calculated and shown in Fig. 6. As i increases from 1 to 9, since m 1 and m 2 are highly dissimilar, the expected conflict measure is assumed to achieve the maximal value. From Fig. 6, we can see that K, d JGB , k LE , k J , d CX , and k ECC show the same trend of increasing conflict values. When i = 10, ϑ i first includes {A 10 }. All the conflict measures become smaller than those in the case where i = 1, . . . , 9. As ϑ i increases from 11 to 19, while the subset decreases from {A 2 , A 3 , A 4 , A 5 , A 6 , A 7 , A 8 , A 9 , A 10 } to {A 10 }, the d JGB , k LE , k J , d CX , and k ECC methods have decreasing conflict measures that reach minimal values of 0.1658, 0.0166, 0.0128, 0.1658, and 0.0255, respectively. By contrast, K remains unchanged at K = 0.1, which does not satisfy the intuitive result.
In summary, Examples 4 to 9 clearly show that k ECC is superior to other methods: k ECC not only satisfies all the desired properties but also provides better conflict identification. Consequently, the proposed approach is effective and suitable for measuring conflict between BBAs.

VII. ALGORITHM AND APPLICATION
Determining how to address decision-making problems has attracted considerable attention in recent years [76], [77]. In this section, a decision-making algorithm for fault diagnosis is devised based on the correlation coefficient measure. Then, a real application of motor rotor fault diagnosis from [45] is used to demonstrate the efficiency of the proposed method.

A. Algorithm
Problem statement: Let {F 1 , . . ., F i , . . ., F n } be a set of fault types for a kind of machine that establishes an FOD Θ, and let M = {m 1 , . . ., m j , . . ., m k } be k pieces of evidence modeled from the collected data of the sensors. A threshold ξ can be set in advance for making a decision. The goal of the algorithm is to diagnose, which type of fault occurs according to the given BBAs {m 1 , . . ., m j , . . ., m k }, and threshold ξ.
Step 1: A correlation matrix is constructed by leveraging the ECC: Step 2: The support degree of m j is calculated as Step 3: The credibility degree of m j is calculated as: Step 4: The weighted average evidence (WAE) is obtained as Step 5: The WAE is fused k − 1 times with the DCR Fusion(m) = ((m ⊕ m) 1 ⊕ · · · ⊕ m) (k−1) . (33) Step 6: The m(F o ) with the highest value is selected Step 7: The fault type is determined as follows: This fault diagnosis based on the ECC is given in Algorithm 1.

B. Application -Fault Diagnosis
In the motor rotor fault diagnosis application [45], three types of sensors are located at different places to collect the acceleration, velocity, and displacement information for a motor rotor. Then, the collected data are modeled as BBAs, as shown in Table VII, where m 1 , m 2 , and m 3 represent three pieces of evidence from the sensors. There are four states for a motor rotor, which establishes an FOD Θ = {F 1 , F 2 , F 3 , F 4 }: F 1 represents "normal operation," F 2 represents "unbalance," F 3 represents "misalignment," and F 4 represents "pedestal looseness". In this application, the threshold for making a decision is set to 0.7 based on [45].
A decision is difficult to make based solely on the BBAs m 1 , m 2 , and m 3 . Specifically, m 1 has a value of 0.68, which indicates   2 : "unbalance"; m 2 has a value of 0.79, which indicates F 3 : "misalignment"; and m 3 has a value of 0.58, which indicates F 2 : "unbalance". Since m 1 (F 2 ) = 0.68 and m 3 (F 2 ) = 0.58, which are less than the threshold 0.7, a decision cannot be made on the basis of m 1 and m 2 , whereas according to m 3 , the diagnosis result is F 3 . As a result, conflict exists between m 1 , m 2 , and m 3 , so an accurate decision is difficult to make under such circumstances. Thus, a conflict management method is necessary to improve the decision level.
Step 1: The correlation matrix M ECC is constructed as   Step 7: Since m(F 2 ) = 0.8964, which is greater than the threshold 0.7, the fault type is F 2 .

C. Discussion
To demonstrate the effectiveness of the proposed conflict management method, we compare the proposed method with related works, including Dempster's [25], Murphy's [39], Deng et al. 's [40], and Jiang's [45] methods. The results generated by different conflict management methods are shown in Table VIII. Dempster's and Murphy's methods cannot determine the fault type because their m(F 2 ) values of 0.5230 and 0.6059, respectively, are smaller than the threshold of 0.7. On the other hand, the methods of Deng et al. and Jiang and the proposed method can diagnose the fault type of the motor rotor as "unbalance," as they obtain m(F 2 ) values of 0.7730, 0.8063, and 0.8964, respectively. Moreover, the proposed method has the highest value of 0.8964 and can, thus, diagnose the fault type with a higher rate of identification.

VIII. CONCLUSIONS
In this article, we explored a novel conflict measurement in decision making and its application in fault diagnosis. Here, a new evidential correlation coefficient, called ECC, was proposed for modeling belief functions in evidence theory to support decision making in an uncertain environment. The properties of the ECC were defined and analyzed, and the ECC was confirmed to have the properties of nonnegativity, nondegeneracy, symmetry, and boundedness. Furthermore, on the basis of the ECC, an evidential conflict coefficient was proposed to measure the conflict between two pieces of evidence. The evidential conflict coefficient was proved to have the desired properties for conflict measurement, including nonnegativity, symmetry, boundedness, extreme consistency, and insensitivity to refinement.
We provided several examples to compare our proposed ECC method with the well-known approaches to demonstrate the superiority of this novel conflict measurement. We also applied the ECC in a fault diagnosis application, and the results verified that our proposed conflict measurement is shown to more efficiently handle uncertainty compared with existing approaches. In summary, our proposed conflict measurement provides a promising way to manage conflict from multiple pieces of evidence and improve the performance of decision making, illustrating a good potential alternative to the analysis of big data from multiple sources. In future work, we intend to further study the properties of ECC as well as its application in more complex environments.