# A Novel Method for Conflict Data Fusion Using an Improved Belief Divergence Measure in Dempster–Shafer Evidence Theory

Authors: Yifan Liu; Tiantian Bao; Huiyun Sang; Zhaokun Wei
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

Dempster–Shafer (D-S) evidence theory plays an important role in multisource data fusion. Due to the nature of the Dempster combination rule, there can be counterintuitive results when fusing highly conflicting evidence data. To date, conflict management in D-S evidence theory is still an open issue. Inspired by evidence modification considering internal indeterminacy and external support, a novel method for conflict data fusion is proposed based on an improved belief divergence, evidence distance, and belief entropy. First, an improved belief divergence measure is defined to characterize the discrepancy and conflict between bodies of evidence (BOEs). Second, evidence credibility is generated to describe the external support based on the complementary advantages of the improved belief divergence and evidence distance. Third, belief entropy is utilized to quantify the internal indeterminacy and further determine evidence weight. Lastly, the classical Dempster combination rule is applied to fuse the BOEs modified by their credibility degrees and weights. As the results of numerical examples and an application show, the proposed divergence measure can overcome the invalidity of the existing measures in some special cases. Additionally, the proposed fusion method recognizes the correct target with the highest belief value of 98.96%, which outperforms other related methods in conflict management. The proposed fusion method also displays better convergence, validity, and robustness.


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## Body

## 1. Introduction

Multisource data fusion is widely used in the real-world applications, such as target recognition [1, 2], fault diagnosis [3, 4], and multicriteria decision-making [5, 6]. Because the characteristics of uncertainty, imprecision, and imperfection are inevitable in practical problems, determining how to model and handle such multisource and uncertain data is still an open issue [7–9]. To address this issue, many related theories and methods have been proposed and developed, such as probability theory [10], fuzzy set theory [11], rough set theory [12, 13], and evidence theory [14–16]. As an uncertainty reasoning method, Dempster–Shafer (D-S) evidence theory has attracted wide attention from many scholars [17]. It is known for handling uncertainty in a flexible and effective manner without prior probabilities, where uncertainty can be expressed by basic belief assignments (BBAs) assigned to singleton sets or multielement sets in the frame of discernment. In addition, D-S evidence theory manifests some good properties, such as commutativity and associativity [18]. However, there are still some open issues that require further research and breakthroughs. When bodies of evidence (BOEs) are highly conflicting, the Dempster combination rule, which is the core of D-S evidence theory, can produce counterintuitive results, such as the famous Zadeh paradox [19]. Therefore, it is vital to effectively combine highly conflicting evidence in D-S evidence theory.In terms of conflict management in D-S evidence theory, a substantial amount of research has been conducted from two major viewpoints, including the modification of the Dempster combination rule and the preprocessing of the initial BOEs [20, 21]. On the one hand, the first viewpoint holds that the reason for the counterintuitive issue lies in the irrational normalization process of the conflict degree in the Dempster combination rule [18]. As a result, some scholars and experts focus on modified combination rules by reallocating conflict degree, such as the Yager rule [22], the Dubois and Prade rule [23], and the Smets rule [24]. Although these modified combination rules are conducive to conflict data fusion, most of them are not as inherently associative and commutative as the classical Dempster combination rule [18]. On the other hand, many researchers point out that the reliability of the BOE in D-S evidence theory is ignored, while unreliable BOEs can cause high conflict and further affect the rationality of the final fusion results [18, 25]. Therefore, the second viewpoint of conflict management is to preprocess and modify the initial BOEs before fusing evidence, such as the Shafer discounting method [15], the Murphy simple average approach [26], the Deng weighted average approach [27], and other extended methods [1, 20]. Because the methods for the second viewpoint retain the good characteristics of the Dempster combination rule [28, 29], this paper prefers to modify the initial BOEs to settle the fusion problems that involve highly conflicting evidence.The basic idea of evidence modification is to preprocess the initial BOEs based on the corresponding discounting or weighting factors and then apply the Dempster combination rule to fuse the modified BOEs [30]. There is no doubt that the estimation of discounting factors is the key point in the process of evidence modification. Recently, many studies on discounting methods in D-S evidence theory have emerged based on belief entropy [31, 32], dissimilarity measures [18, 30], evidence distance [33–35], base belief function [20, 36], evidence angle [8], correlation coefficient [34], and belief divergence [1, 2]. Based on the aforementioned works, the initial BOEs can usually be preprocessed in consideration of internal indeterminacy and external support, which are quite distinct notions and measurements in D-S evidence theory [30]. As for internal indeterminacy, belief entropy has been widely studied and applied to measure information volume and evidence uncertainty [37, 38]. Hence, evidence weight with larger information volume will be greater since its indeterminacy and uncertainty should be lower based on belief entropy [39]. As the second aspect that determines the discounting factor, external support degrees can be generated based on the measurement of evidence similarity, which is negatively correlated with conflict degree or evidence distance [8, 33]. According to the principle of majority, if the BOE has greater degrees of similarity with other BOEs, the higher the corresponding support degree and the greater the evidence credibility. To date, many measurements of evidence similarity or dissimilarity have been put forward. Specifically, they can be divided to three categories [40]: (1) the mass-based measures, such as evidence distance proposed by Jousselme [41]; (2) the probability-based measures, such as conflict measure proposed by Liu [42]; (3) the combined measures that consider two or more factors, such as the methods propsoed by Chen et al. [8] and Liu et al. [30]. However, each measure has pros and cons, and there is still no definite method for quantifying the similarity among BOEs. For instance, Jousselme distance [41] cannot effectively distinguish the BBAs with singleton sets and those with multielement sets in some cases.As a recent improvement with good performance in conflict management, Xiao proposed a measure of belief divergence in D-S evidence theory, called belief Jensen–Shannon (BJS) divergence, to determine the external support and evidence credibility [1]. Divergence is one of the most significant information-theoretic measures, which is commonly applied for determining the discrepancy and discrimination between two information sets. Nowadays, divergence measures have become a vigorous area of research in many applications, such as multisensor data fusion [1, 2], medical diagnosis [43, 44], multicriteria decision-making [45], and renewable energy sources selection [46]. The well-known measurement is Kullback–Leibler (KL) divergence, which was firstly proposed to distinguish two probability distributions [47]. Afterwards, an influential extended measure in probability theory, called Jensen–Shannon (JS) divergence, is derived based on Jensen’s inequality and the Shannon entropy [48]. In the past decades, many researchers have been working on the extension and generalization of divergence measure in different information environments to deal with more complex data, such as fuzzy divergence measures [49], intuitionistic fuzzy divergence measures [45, 50], Pythagorean fuzzy divergence measure [43] and triangular fuzzy divergence measure [46]. In D-S evidence theory, the divergence measurement of BBAs is still an open issue. Followed by the BJS divergence [1], some belief divergence measures have been put forward based on KL divergence [51] and Pythagorean fuzzy divergence [44]. However, these measures directly utilize the belief function instead of the probability function, which is the cornerstone of the classical KL divergence and JS divergence, and could have no definition when the BBA is zero [2]. These deficiencies will affect the rationality of the belief divergence in some practical applications. Therefore, inspired by probabilistic transformation in D-S evidence theory and cross entropy in fuzzy set theory, an improved belief divergence is devised to more reasonably measure the discrepancy and conflict between BOEs. In addition, the improved belief divergence can be verified as a complementary criterion of evidence distance to better capture different aspects of dissimilarity degrees between BOEs. Therefore, it is a natural way to combine the improved belief divergence and evidence distance in the elaboration of evidence similarity, so as to further determine the support degree and credibility of each BOE in the process of evidence modification.The aim of this paper is to effectively address the conflict data fusion problems based on D-S evidence theory and to avoid counterintuitive fusion results. In this paper, we define an improved belief divergence measure in D-S evidence theory. Based on the improved belief divergence, evidence distance, and belief entropy, a novel method for conflict data fusion is proposed by taking into account both evidence credibility and evidence weight. Numerical examples are provided to verify the feasibility and efficiency of the proposed divergence measure and fusion method. The main motivations of this paper are summarized as follows. First, the existing measures of belief divergence have the “no definition” issues and will generate irrational results in some special cases. Hence, the belief divergence needs to be extended to effectively characterize the discrepancy and conflict between BOEs. Second, because the proposed belief divergence and evidence distance are highly complementary to each other, it is necessary to quantify the combined similarity degrees between BOEs by considering these two measures when determining the external support and evidence credibility. Third, most of the current research on conflict management is to modify the initial BOEs in the light of external support and internal indeterminacy. Following this strategy, there is need to account for evidence weight based on belief entropy and evidence credibility estimated by the combined similarity, which is conductive to the study of a more effective and proficient method for conflict data fusion.The remainder of this paper is organized as follows. Section2 briefly introduces the preliminaries for the proposed method. In Section 3, an improved belief divergence measure is defined, and its properties and performance are discussed. Section 4 proposes a novel method for conflict data fusion in D-S evidence theory based on the improved belief divergence measure, evidence distance, and belief entropy. In Section 5, an application in target recognition is presented to illustrate the validity and superiority of the proposed method. Finally, the conclusions of this paper are given in Section 6.


## 2. Preliminaries

In this section, some related concepts and definitions are briefly introduced, including Dempster–Shafer evidence theory, probabilistic transformation, belief entropy, evidence distance, and belief divergence.


### 2.1. Dempster–Shafer Evidence Theory

As a generalization of Bayes probability theory, Dempster–Shafer (D-S) evidence theory is an efficient mathematical tool for uncertainty reasoning [14–17]. It requires weaker conditions and can directly address uncertainty without prior probability [32]. Here, the basic concepts are introduced as follows.Definition 1.
LetΘ=θ1,…,θN be the frame of discernment (FOD), which is a finite set of mutually exclusive and collectively exhaustive propositions. The power set of Θ, denoted by 2Θ, is defined as [14](1)2Θ=∅,θ1,…,θ1,θ2,…θ1,…,θN−1,Θ,where ∅ is an empty set.Definition 2.
For the FODΘ, a basic belief assignment (BBA) m·, also called a mass function, is a mapping m:2Θ⟶0,1 that satisfies [14](2)∑A⊆ΘmA=1,m∅=0.
For anyA⊆Θ, if mA>0, A is named a focal element. The tuple for all focal elements and their BBA is called a body of evidence (BOE), which is the unit for uncertain information fusion based on D-S evidence theory [8].Definition 3.
Suppose that two bodies of evidence (BOEs) on the FODΘ are represented by two independent BBAs m1 and m2, for any elements A⊆Θ, the Dempster combination rule is defined as follows [14]:(3)m⊕A=m1B⊕m2C=∑B,C⊆Θ,B∩C=Am1Bm2C1−K,A≠∅,0,A=∅,(4)K=∑B,C⊆Θ,B∩C=∅m1Bm2C,where ⊕ stands for the orthogonal sum operation and K is the normalization coefficient, which represents the conflict degree between two BOEs.
The Dempster combination rule, which is the core of D-S evidence theory, is rooted in probability theory and constitutes a conjunctive probabilistic inference process [52]. Let m1 and m2 be two BOEs on the FOD Θ=θ1,θ2. Then, Figure 1 describes the fusion process of the BOEs m1 and m2 based on the Dempster combination rule [53]. Here, line segments that represent two BOEs m1 and m2 are orthogonally combined to obtain a square standing for the fusion results, where the orange rectangles can be regarded as conflict degree K. If K=1, which means that the two BOEs are fully conflicting with each other, the Dempster combination rule will tend to infinity because in this case, the denominator of equation (3) is zero [54]. When the BOEs are highly conflicting, the rule can generate the counterintuitive results, such as the famous Zadeh paradox [19]. It is therefore fundamental to resolve the nondefinition and counterintuitive problems in D-S evidence theory.Figure 1
An example of evidence fusion based on the Dempster combination rule.


### 2.2. Probabilistic Transformation

Probabilistic transformation is a method for approximating any BBAs to probability distributions on the frame of discernment [18]. The pignistic probabilistic transformation proposed by Smets [55] has attracted the attention of many scholars, in which uncertainty in the BBA will be equally distributed among all possible propositions.Definition 4.
Letm be a BBA on the FOD Θ=θ1,…,θN; then, its associated pignistic probability function, denoted by BetP, is defined as [55](5)BetPθi=∑θi∈A,A⊆Θ1AmA1−m∅,where A is the cardinality of subset A.


### 2.3. Belief Entropy

In order to measure the uncertainty of BBAs, belief entropy in D-S evidence theory has been proposed based on the concept of information entropy [31, 32], such as Hohle’s confusion measure [56], Yager’s dissonance measure [57], Deng entropy [58], and other modified entropy [59, 60]. As a generalization of Shannon entropy in information theory, Deng entropy plays a vital role in the measurement of information volume and evidence modification when fusing conflict data [61–63].Definition 5.
Letm be a BBA on the FOD Θ=θ1,…,θN; then, Deng entropy, denoted by Ed, is defined as [58](6)Edm=−∑A⊆ΘmAlog2mA2A−1,where A is the cardinality of subset A. When the BBAs are only assigned to singleton elements θi∈Θ, Deng entropy will definitely degenerate to Shannon entropy [64], as follows:(7)Edm=−∑θi∈Θmθilog2mθi2θi−1=−∑θi∈Θmθilog2mθi.


### 2.4. Evidence Distance

To identify the dissimilarity and difference between the BOEs, many measures of evidence distance have been proposed, such as the Jousselme distance [41], Bhattacharyya distance [65], pignistic probability distance [42], and other improved methods [66]. Among them, the most well-known and frequently-used is Jousselme distance, which has been widely used to describe the characteristics of conflict [34], determine the credibility degrees [18], and derive the relative weights [35]. Moreover, the Jousselme distance satisfies four requirements of distance measurement: nonnegativity, nondegeneracy, symmetry, and triangle inequality [35].Definition 6.
Suppose that two BOEs on the FODΘ are represented by two independent BBAs m1 and m2; then, the Jousselme distance, denoted by dJ, is defined as [41](8)dJm1,m2=12m⟶1−m⟶2TD¯¯m⟶1−m⟶2,where m⟶1 and m⟶2 are vector forms of the BBAs m1 and m2, D¯¯ is a matrix of 2Θ×2Θ, whose elements are DA,B=A∩B/A∪B, ∀A,B∈2Θ, and · is the cardinality of a particular set.


### 2.5. Belief Divergence

In order to measure the discrepancy and conflict among the BOEs, Xiao [1] introduced the Jensen–Shannon divergence of probability distribution into D-S evidence theory and proposed a divergence measure for BBAs, named belief Jensen–Shannon (BJS) divergence. The main concept is defined as follows.Definition 7.
Suppose that two BOEs on the FODΘ are represented by two independent BBAs m1 and m2; then, belief Jensen–Shannon (BJS) divergence is expressed as [1](9)BJSm1,m2=12Divm1,m1+m22+Divm2,m1+m22,(10)Divmj,mj+mk2=∑A⊆ΘmjAlog22×mjAmjA+mkA,where ∑A⊆ΘmjA=1, ∑A⊆ΘmkA=1, and jandk=1and2.
Through the study and analysis of BJS divergence, it can be seen that this measure is similar to the Jensen–Shannon divergence in form. When the values of BBA are assigned only to singleton elements, the BJS divergence will degenerate as Jensen–Shannon divergence [48]. Moreover, the BJS divergence also satisfies symmetry, boundedness, and square root’s triangle inequality [1], which are the most important and useful properties in practical applications, such as target diagnosis [3] and data fusion [67].
However, there is still some room for improvement and development of belief divergence measurement. Specifically, if the value of BBA is zero, i.e.,m1A=0 or m2A=0, both measures tend to infinity because, in this case, the logarithm values of equation (10) will be undefined. In addition, these measures can lead to irrational results because they do not transform the BBAs to probability distributions and allocates BBAs to singleton elements and multiple elements in the same way [3], as discussed in example 1.Example 1.
Suppose that the frame of discernment (FOD) isΘ=θ1,θ2,…,θ7 and three bodies of evidence (BOEs) on Θ are given as follows:(11)m1:m1θ2,θ3,θ4=0.85,m1θ7=0.15,m2:m2θ5,θ6,θ7=0.85,m2θ7=0.15,m3:m3θ1,θ2,θ3,θ4,θ5=1.
In Example1, it can be seen that all three BOEs m1, m2, and m3 have uncertainty, but the subset θ1,θ2,θ3,θ4,θ5 for m3 is much closer to the subset θ2,θ3,θ4 for m1 than to the subset θ5,θ6,θ7 for m2. Intuitively, there are some discrepancies and dissimilarities among the three BOEs, and the divergence measure between m1 and m3 will be smaller than that between m2 and m3. However, the BJS divergence measure fails in this case. Here, the zero values of BBAs are replaced by a very small number 1 × 10−12 [1]. Then, it can be found that BJSm1,m3=BJSm2,m3=1, which is inconsistent with intuitive judgement. In particular, the BJS divergence believes that there are absolute conflicts and discrepancies among the three BOEs. The reason for such counterintuitive results lies in the lack of probabilistic transformation and the inappropriate solution of zero value in the literature [1]. Therefore, it is necessary and critical to investigate an improved belief divergence measure to better modify the initial BOEs and further design the method for conflict data fusion.


## 2.1. Dempster–Shafer Evidence Theory

As a generalization of Bayes probability theory, Dempster–Shafer (D-S) evidence theory is an efficient mathematical tool for uncertainty reasoning [14–17]. It requires weaker conditions and can directly address uncertainty without prior probability [32]. Here, the basic concepts are introduced as follows.Definition 1.
LetΘ=θ1,…,θN be the frame of discernment (FOD), which is a finite set of mutually exclusive and collectively exhaustive propositions. The power set of Θ, denoted by 2Θ, is defined as [14](1)2Θ=∅,θ1,…,θ1,θ2,…θ1,…,θN−1,Θ,where ∅ is an empty set.Definition 2.
For the FODΘ, a basic belief assignment (BBA) m·, also called a mass function, is a mapping m:2Θ⟶0,1 that satisfies [14](2)∑A⊆ΘmA=1,m∅=0.
For anyA⊆Θ, if mA>0, A is named a focal element. The tuple for all focal elements and their BBA is called a body of evidence (BOE), which is the unit for uncertain information fusion based on D-S evidence theory [8].Definition 3.
Suppose that two bodies of evidence (BOEs) on the FODΘ are represented by two independent BBAs m1 and m2, for any elements A⊆Θ, the Dempster combination rule is defined as follows [14]:(3)m⊕A=m1B⊕m2C=∑B,C⊆Θ,B∩C=Am1Bm2C1−K,A≠∅,0,A=∅,(4)K=∑B,C⊆Θ,B∩C=∅m1Bm2C,where ⊕ stands for the orthogonal sum operation and K is the normalization coefficient, which represents the conflict degree between two BOEs.
The Dempster combination rule, which is the core of D-S evidence theory, is rooted in probability theory and constitutes a conjunctive probabilistic inference process [52]. Let m1 and m2 be two BOEs on the FOD Θ=θ1,θ2. Then, Figure 1 describes the fusion process of the BOEs m1 and m2 based on the Dempster combination rule [53]. Here, line segments that represent two BOEs m1 and m2 are orthogonally combined to obtain a square standing for the fusion results, where the orange rectangles can be regarded as conflict degree K. If K=1, which means that the two BOEs are fully conflicting with each other, the Dempster combination rule will tend to infinity because in this case, the denominator of equation (3) is zero [54]. When the BOEs are highly conflicting, the rule can generate the counterintuitive results, such as the famous Zadeh paradox [19]. It is therefore fundamental to resolve the nondefinition and counterintuitive problems in D-S evidence theory.Figure 1
An example of evidence fusion based on the Dempster combination rule.


## 2.2. Probabilistic Transformation

Probabilistic transformation is a method for approximating any BBAs to probability distributions on the frame of discernment [18]. The pignistic probabilistic transformation proposed by Smets [55] has attracted the attention of many scholars, in which uncertainty in the BBA will be equally distributed among all possible propositions.Definition 4.
Letm be a BBA on the FOD Θ=θ1,…,θN; then, its associated pignistic probability function, denoted by BetP, is defined as [55](5)BetPθi=∑θi∈A,A⊆Θ1AmA1−m∅,where A is the cardinality of subset A.


## 2.3. Belief Entropy

In order to measure the uncertainty of BBAs, belief entropy in D-S evidence theory has been proposed based on the concept of information entropy [31, 32], such as Hohle’s confusion measure [56], Yager’s dissonance measure [57], Deng entropy [58], and other modified entropy [59, 60]. As a generalization of Shannon entropy in information theory, Deng entropy plays a vital role in the measurement of information volume and evidence modification when fusing conflict data [61–63].Definition 5.
Letm be a BBA on the FOD Θ=θ1,…,θN; then, Deng entropy, denoted by Ed, is defined as [58](6)Edm=−∑A⊆ΘmAlog2mA2A−1,where A is the cardinality of subset A. When the BBAs are only assigned to singleton elements θi∈Θ, Deng entropy will definitely degenerate to Shannon entropy [64], as follows:(7)Edm=−∑θi∈Θmθilog2mθi2θi−1=−∑θi∈Θmθilog2mθi.


## 2.4. Evidence Distance

To identify the dissimilarity and difference between the BOEs, many measures of evidence distance have been proposed, such as the Jousselme distance [41], Bhattacharyya distance [65], pignistic probability distance [42], and other improved methods [66]. Among them, the most well-known and frequently-used is Jousselme distance, which has been widely used to describe the characteristics of conflict [34], determine the credibility degrees [18], and derive the relative weights [35]. Moreover, the Jousselme distance satisfies four requirements of distance measurement: nonnegativity, nondegeneracy, symmetry, and triangle inequality [35].Definition 6.
Suppose that two BOEs on the FODΘ are represented by two independent BBAs m1 and m2; then, the Jousselme distance, denoted by dJ, is defined as [41](8)dJm1,m2=12m⟶1−m⟶2TD¯¯m⟶1−m⟶2,where m⟶1 and m⟶2 are vector forms of the BBAs m1 and m2, D¯¯ is a matrix of 2Θ×2Θ, whose elements are DA,B=A∩B/A∪B, ∀A,B∈2Θ, and · is the cardinality of a particular set.


## 2.5. Belief Divergence

In order to measure the discrepancy and conflict among the BOEs, Xiao [1] introduced the Jensen–Shannon divergence of probability distribution into D-S evidence theory and proposed a divergence measure for BBAs, named belief Jensen–Shannon (BJS) divergence. The main concept is defined as follows.Definition 7.
Suppose that two BOEs on the FODΘ are represented by two independent BBAs m1 and m2; then, belief Jensen–Shannon (BJS) divergence is expressed as [1](9)BJSm1,m2=12Divm1,m1+m22+Divm2,m1+m22,(10)Divmj,mj+mk2=∑A⊆ΘmjAlog22×mjAmjA+mkA,where ∑A⊆ΘmjA=1, ∑A⊆ΘmkA=1, and jandk=1and2.
Through the study and analysis of BJS divergence, it can be seen that this measure is similar to the Jensen–Shannon divergence in form. When the values of BBA are assigned only to singleton elements, the BJS divergence will degenerate as Jensen–Shannon divergence [48]. Moreover, the BJS divergence also satisfies symmetry, boundedness, and square root’s triangle inequality [1], which are the most important and useful properties in practical applications, such as target diagnosis [3] and data fusion [67].
However, there is still some room for improvement and development of belief divergence measurement. Specifically, if the value of BBA is zero, i.e.,m1A=0 or m2A=0, both measures tend to infinity because, in this case, the logarithm values of equation (10) will be undefined. In addition, these measures can lead to irrational results because they do not transform the BBAs to probability distributions and allocates BBAs to singleton elements and multiple elements in the same way [3], as discussed in example 1.Example 1.
Suppose that the frame of discernment (FOD) isΘ=θ1,θ2,…,θ7 and three bodies of evidence (BOEs) on Θ are given as follows:(11)m1:m1θ2,θ3,θ4=0.85,m1θ7=0.15,m2:m2θ5,θ6,θ7=0.85,m2θ7=0.15,m3:m3θ1,θ2,θ3,θ4,θ5=1.
In Example1, it can be seen that all three BOEs m1, m2, and m3 have uncertainty, but the subset θ1,θ2,θ3,θ4,θ5 for m3 is much closer to the subset θ2,θ3,θ4 for m1 than to the subset θ5,θ6,θ7 for m2. Intuitively, there are some discrepancies and dissimilarities among the three BOEs, and the divergence measure between m1 and m3 will be smaller than that between m2 and m3. However, the BJS divergence measure fails in this case. Here, the zero values of BBAs are replaced by a very small number 1 × 10−12 [1]. Then, it can be found that BJSm1,m3=BJSm2,m3=1, which is inconsistent with intuitive judgement. In particular, the BJS divergence believes that there are absolute conflicts and discrepancies among the three BOEs. The reason for such counterintuitive results lies in the lack of probabilistic transformation and the inappropriate solution of zero value in the literature [1]. Therefore, it is necessary and critical to investigate an improved belief divergence measure to better modify the initial BOEs and further design the method for conflict data fusion.


## 3. An Improved Belief Divergence Measure

In this section, an improved belief divergence is defined to address the aforementioned invalidity and irrationality issues of the existing divergence measures. In addition, some numerical examples are given to discuss the performance of the proposed measure.


### 3.1. The Proposed Measure

Inspired by pignistic probabilistic transformation in D-S evidence theory [55] and cross entropy in fuzzy set theory [6], we convert the BBAs into probability distributions, which is in line with the classical Jessen–Shannon divergence [48], and we introduce the parameter t to ensure the validity of the logarithm value [68]. Therefore, an improved belief divergence is defined as follows.Definition 8.
Suppose that two BOEs on the FODΘ are represented by two independent BBAs m1 and m2; then, an improved belief divergence is defined as(12)BDm1,m2=12CEPPTm1,m2+CEPPTm2,m1,(13)CEPPTmj,mk=1T∑θi∈Θ1+t×BetPjθilog22×1+t×BetPjθi1+t×BetPjθi+1+t×BetPkθi,where T=1+tlog21+t−2+tlog22+t/2, t>0, and BetPj and BetPk stand for the corresponding pignistic probability functions of the BOEs mj and mk, respectively, jandk=1and2.
LetPPT1=1+t×BetP and PPT2=1+t×BetP; then, there can be always PPT1>0 and PPT2>0 since t>0,BetP1θi≥0, and BetP2θi≥0 for any θi∈Θ. Thus, the nondefinition problems of the existing divergence measure can be solved from the perspectives of logic and mathematics. In addition, the improved belief divergence can also be expressed as follows:(14)BDm1,m2=12T∑θi∈ΘPPT1θilog22×PPT1θiPPT1θi+PPT2θi+∑θi∈ΘPPT2θilog22×PPT2θiPPT1θi+PPT2θi=1TEdPPT1+PPT22−EdPPT1+EdPPT22.
By comparing equation (14) with equation (10), it is easily seen that the improved belief divergence is similar in form to the BJS divergence. Here, the remarkable properties of the BJS divergence are still preserved [69]. Therefore, three properties of the improved belief divergence can be inferred as follows.Theorem 1.
Suppose that three BOEs on the FODΘ are represented by BBAs m1, m2, and m3; then, the improved belief divergence has three properties below for measurement.(i)
(P1): symmetry:BDm1,m2=BDm2,m1(ii)
(P2): boundedness:0≤BDm1,m2≤1(iii)
(P3): triangle inequality of its square root:BDm1,m3≤BDm1,m2+BDm2,m3Proof:.
(P1): from equations (12) and (13), there is always(15)BDm1,m2=12CEPPTm1,m2+CEPPTm2,m1=BDm2,m1.
Hence, the symmetry property of the improved belief divergence is verified.Proof:.
(P2): letp=BetP1θi and q=BetP2θi; then, the summands of the improved belief divergence can be expressed as follows:(16)Lp,q=1+tplog221+tp1+tp+1+tq+1+tqlog221+tq1+tp+1+tq=1+tplog21+tp+1+tqlog21+tq−1+tp+1+tqlog21+tp+1+tq2.
Letfx=1+txlog21+tx, 0≤x≤1. By differentiation, we can obtain that f′x=tlog21+tx+t>0 and f′'x=t2/1+tx>0. As a result, fx is a concave-up function; then, the function Lp,q in equation (16) is convex [6]. Therefore, the value of function Lp,q will increase as the value of p−q becomes larger. Since p−q=BetP1θi−BetP2θi and 0≤BetP1θi,BetP2θi≤1, it is easy to see that 0≤p−q≤1. Then, the value of Lp,q reaches its minimum at p−q=0, which is Lp,p=Lq,q=0. On the contrary, the value of Lp,q reaches its maximum at p−q=1, which means L1,0=L0,1=T. Furthermore, if BetP1θi−BetP2θi=0, the measure BDm1,m2 will attain the minimum value, which means BDm1,m2=0. In addition, when BetP1θi−BetP2θi=1, the improved belief divergence measure will obtain the maximum value, which means BDm1,m2=1/2TL1,0+L0,1=1.
Therefore, the boundedness property of the improved belief divergence is proven.Proof:.
(P3): according to equation (16), the summands of property (P3) can be represented by(17)Lp,r≤Lp,q+Lq,r,where q=BetP2θi and r=BetP3θi.
Let the right-hand side of equation (17) be rhL=Lp,q+Lq,r. Then, the partial derivative of the function rhL can be calculated by(18)δrhLδq=tlog221+tq/1+tq+1+tp2Lp,q+tlog221+tq/1+tq+1+tr2Lr,q.
According to Lemmas 1 and 2 in the literature [69], the derivative δrhL/δq will change sign at 1+tq=1+tp (i.e., q=p) and likewise at 1+tq=1+tr (i.e., q=r). Those extrema are minimum values. On the basis of property (P2), it is found that when q=p (or q=r), the corresponding result is Lp,q=0 (or Lq,r=0). Thus, the function rhL reaches the minimum value, which suggests that rhL=Lp,r. Therefore, it can be concluded that the value of rhL is equal to the left-hand side of equation (17). Then, we apply Minkowski’s inequality to the square root of the improved belief divergence, and the improved belief divergence measure can be easily deduced that BDm1,m3≤BDm1,m2+BDm2,m3.
Thus, the square root of the improved belief divergence satisfies triangle inequality.


### 3.2. Numerical Examples

Let us return to Example1. The calculation process of the improved belief divergence can be displayed as follows. First, the pignistic probability distributions (PPTs) of three BOEs on Θ in Example 1 can be obtained based on equation (5), as shown in Table 1.Table 1
Pignistic probability distributions of three BOEs onΘ in Example 1.
PPTsθ1θ2θ3θ4θ5θ6θ7BetP100.28330.28330.2834000.15BetP200000.28330.28330.4334BetP30.20.20.20.20.200Then, lett=1 in Definition 8. According to the PPTs in Table 1, we can obtain the results of the improved belief divergence between these BOEs, which are BDm1,m3=0.0813 and BDm2,m3=0.2773. It is obvious that BDm1,m3<BDm2,m3. Compared with the existing divergence, the proposed measure is more consistent with intuitive judgement.Example 2.
Suppose that the FOD isΘ=θ1,θ2,θ3; then, two BOEs m1 and m2 on Θ are given as follows:(19)m1:m1θ1=0.99−2α,m1θ2=0.01+α,m1θ3=α,m2:m2θ1=α,m2θ2=0.01+α,m2θ3=0.99−2α,where 0≤α≤0.495.
In Example2, when α=0, the BOEs m1 and m2 will assign a larger value of 0.99 to support the propositions θ1 and θ3, respectively, while they both have a lower value of 0.01 on the proposition θ2. By utilizing the Dempster combination rule in Definition 3, we can obtain the conflict degree K=0.9999 and the fusion result m⊕θ2=1. In this case, it is obvious that these two BOEs are highly conflicting, and the fusion result is counterintuitive. This circumstance is the so-called Zadeh paradox [19, 25]. In addition, when α=0.33, these two BOEs m1 and m2 will have identical BBAs, which means that m1θ1=m2θ1=0.33, m1θ2=m2θ2=0.34, and m1θ3=m2θ3=0.33. Under this situation, the conflict degree based on equation (4) is K=0.6666, which means that these BOEs still have a high degree of conflict. However, it is more reasonable that the identical BOEs are not in conflict.
In terms of the improved belief divergence, assume that the parametert varies in the range 0,100. As the parameters α and t change, the values of BDm1,m2 can be depicted in Figure 2. It can be easily seen that there is always 0≤BDm1,m2≤1, regardless of the values of α and t. On the one hand, when the value of α tends to 0, the result BDm1,m2 will be close to 1. On the other hand, when α=0.33, we can obtain that BDm1,m2=0 for ∀t∈0,100. Furthermore, it can also be found that the variation in t has a limited influence on the measurement of the improved belief divergence. Therefore, the proposed measure has better performance in modelling the conflicts between BOEs.Figure 2
Results of the improved belief divergence in Example2.Example 3.
Suppose that the FOD isΘ=θ1,θ2,⋯,θN, Θ=N; then, the BOEs m1, m2, and m3 on Θ are given as follows:(20)m1:m1θ1=m1θ2=⋯=m1θN=1N,m2:m2Θ=1,m3:m3θl=1,θl∈Θ.
In Example3, both m1 and m2 reflect the intrinsic nature of uncertainty, whereas m3 is certain evidence. Specifically, m1 corresponds to full randomness, which assigns identical values of BBAs to each singleton set of Θ, m2 represents complete ignorance, which absolutely supports Θ, and m3 has 100% belief values to support the specific proposition θl. Intuitively, it is expected that m1 is closer to m2 than to m3.
According to equations (3) and (4) and (8)–(13), the values of the Jousselme distance dJ, the BJS measure, and the improved belief divergence BD are calculated and analyzed, as shown in Figure 3. We can get BJSm1,m2=1 and BJSm1,m2>BJSm1,m3, which represent that the BOEs m1 and m2 absolutely contradict based on the BJS measure. The reason for this counterintuitive phenomenon is that the BJS measure ignores the correlations between focal elements in D-S evidence theory [2]. As for evidence distance proposed by Jousselme, we can obtain dJm1,m2=dJm1,m3, which means that the Jousselme distance cannot effectively distinguish the BBAs with singleton sets and those with multielement sets. In contrast, only the improved belief divergence is in line with the intuitive judgement, which can be represented as BDm1,m2<BDm1,m3.
Although the improved belief divergence achieves better performance, some gaps still exist in conflict management. As shown in Figure3, BDm1,m2=0, which indicates that the improved belief divergence cannot describe the full randomness (i.e., m1) and complete ignorance (i.e., m2) well. By comparison, it is more reasonable that the value of the Jousselme distance is dJm1,m2=1/21−1/N≠0. Therefore, it can be concluded that the improved belief divergence measure and Jousselme distance have complementary advantages and can be integrated to better measure the evidence conflict.Figure 3
Results of the improved belief divergence in Example3.


## 3.1. The Proposed Measure

Inspired by pignistic probabilistic transformation in D-S evidence theory [55] and cross entropy in fuzzy set theory [6], we convert the BBAs into probability distributions, which is in line with the classical Jessen–Shannon divergence [48], and we introduce the parameter t to ensure the validity of the logarithm value [68]. Therefore, an improved belief divergence is defined as follows.Definition 8.
Suppose that two BOEs on the FODΘ are represented by two independent BBAs m1 and m2; then, an improved belief divergence is defined as(12)BDm1,m2=12CEPPTm1,m2+CEPPTm2,m1,(13)CEPPTmj,mk=1T∑θi∈Θ1+t×BetPjθilog22×1+t×BetPjθi1+t×BetPjθi+1+t×BetPkθi,where T=1+tlog21+t−2+tlog22+t/2, t>0, and BetPj and BetPk stand for the corresponding pignistic probability functions of the BOEs mj and mk, respectively, jandk=1and2.
LetPPT1=1+t×BetP and PPT2=1+t×BetP; then, there can be always PPT1>0 and PPT2>0 since t>0,BetP1θi≥0, and BetP2θi≥0 for any θi∈Θ. Thus, the nondefinition problems of the existing divergence measure can be solved from the perspectives of logic and mathematics. In addition, the improved belief divergence can also be expressed as follows:(14)BDm1,m2=12T∑θi∈ΘPPT1θilog22×PPT1θiPPT1θi+PPT2θi+∑θi∈ΘPPT2θilog22×PPT2θiPPT1θi+PPT2θi=1TEdPPT1+PPT22−EdPPT1+EdPPT22.
By comparing equation (14) with equation (10), it is easily seen that the improved belief divergence is similar in form to the BJS divergence. Here, the remarkable properties of the BJS divergence are still preserved [69]. Therefore, three properties of the improved belief divergence can be inferred as follows.Theorem 1.
Suppose that three BOEs on the FODΘ are represented by BBAs m1, m2, and m3; then, the improved belief divergence has three properties below for measurement.(i)
(P1): symmetry:BDm1,m2=BDm2,m1(ii)
(P2): boundedness:0≤BDm1,m2≤1(iii)
(P3): triangle inequality of its square root:BDm1,m3≤BDm1,m2+BDm2,m3Proof:.
(P1): from equations (12) and (13), there is always(15)BDm1,m2=12CEPPTm1,m2+CEPPTm2,m1=BDm2,m1.
Hence, the symmetry property of the improved belief divergence is verified.Proof:.
(P2): letp=BetP1θi and q=BetP2θi; then, the summands of the improved belief divergence can be expressed as follows:(16)Lp,q=1+tplog221+tp1+tp+1+tq+1+tqlog221+tq1+tp+1+tq=1+tplog21+tp+1+tqlog21+tq−1+tp+1+tqlog21+tp+1+tq2.
Letfx=1+txlog21+tx, 0≤x≤1. By differentiation, we can obtain that f′x=tlog21+tx+t>0 and f′'x=t2/1+tx>0. As a result, fx is a concave-up function; then, the function Lp,q in equation (16) is convex [6]. Therefore, the value of function Lp,q will increase as the value of p−q becomes larger. Since p−q=BetP1θi−BetP2θi and 0≤BetP1θi,BetP2θi≤1, it is easy to see that 0≤p−q≤1. Then, the value of Lp,q reaches its minimum at p−q=0, which is Lp,p=Lq,q=0. On the contrary, the value of Lp,q reaches its maximum at p−q=1, which means L1,0=L0,1=T. Furthermore, if BetP1θi−BetP2θi=0, the measure BDm1,m2 will attain the minimum value, which means BDm1,m2=0. In addition, when BetP1θi−BetP2θi=1, the improved belief divergence measure will obtain the maximum value, which means BDm1,m2=1/2TL1,0+L0,1=1.
Therefore, the boundedness property of the improved belief divergence is proven.Proof:.
(P3): according to equation (16), the summands of property (P3) can be represented by(17)Lp,r≤Lp,q+Lq,r,where q=BetP2θi and r=BetP3θi.
Let the right-hand side of equation (17) be rhL=Lp,q+Lq,r. Then, the partial derivative of the function rhL can be calculated by(18)δrhLδq=tlog221+tq/1+tq+1+tp2Lp,q+tlog221+tq/1+tq+1+tr2Lr,q.
According to Lemmas 1 and 2 in the literature [69], the derivative δrhL/δq will change sign at 1+tq=1+tp (i.e., q=p) and likewise at 1+tq=1+tr (i.e., q=r). Those extrema are minimum values. On the basis of property (P2), it is found that when q=p (or q=r), the corresponding result is Lp,q=0 (or Lq,r=0). Thus, the function rhL reaches the minimum value, which suggests that rhL=Lp,r. Therefore, it can be concluded that the value of rhL is equal to the left-hand side of equation (17). Then, we apply Minkowski’s inequality to the square root of the improved belief divergence, and the improved belief divergence measure can be easily deduced that BDm1,m3≤BDm1,m2+BDm2,m3.
Thus, the square root of the improved belief divergence satisfies triangle inequality.


## 3.2. Numerical Examples

Let us return to Example1. The calculation process of the improved belief divergence can be displayed as follows. First, the pignistic probability distributions (PPTs) of three BOEs on Θ in Example 1 can be obtained based on equation (5), as shown in Table 1.Table 1
Pignistic probability distributions of three BOEs onΘ in Example 1.
PPTsθ1θ2θ3θ4θ5θ6θ7BetP100.28330.28330.2834000.15BetP200000.28330.28330.4334BetP30.20.20.20.20.200Then, lett=1 in Definition 8. According to the PPTs in Table 1, we can obtain the results of the improved belief divergence between these BOEs, which are BDm1,m3=0.0813 and BDm2,m3=0.2773. It is obvious that BDm1,m3<BDm2,m3. Compared with the existing divergence, the proposed measure is more consistent with intuitive judgement.Example 2.
Suppose that the FOD isΘ=θ1,θ2,θ3; then, two BOEs m1 and m2 on Θ are given as follows:(19)m1:m1θ1=0.99−2α,m1θ2=0.01+α,m1θ3=α,m2:m2θ1=α,m2θ2=0.01+α,m2θ3=0.99−2α,where 0≤α≤0.495.
In Example2, when α=0, the BOEs m1 and m2 will assign a larger value of 0.99 to support the propositions θ1 and θ3, respectively, while they both have a lower value of 0.01 on the proposition θ2. By utilizing the Dempster combination rule in Definition 3, we can obtain the conflict degree K=0.9999 and the fusion result m⊕θ2=1. In this case, it is obvious that these two BOEs are highly conflicting, and the fusion result is counterintuitive. This circumstance is the so-called Zadeh paradox [19, 25]. In addition, when α=0.33, these two BOEs m1 and m2 will have identical BBAs, which means that m1θ1=m2θ1=0.33, m1θ2=m2θ2=0.34, and m1θ3=m2θ3=0.33. Under this situation, the conflict degree based on equation (4) is K=0.6666, which means that these BOEs still have a high degree of conflict. However, it is more reasonable that the identical BOEs are not in conflict.
In terms of the improved belief divergence, assume that the parametert varies in the range 0,100. As the parameters α and t change, the values of BDm1,m2 can be depicted in Figure 2. It can be easily seen that there is always 0≤BDm1,m2≤1, regardless of the values of α and t. On the one hand, when the value of α tends to 0, the result BDm1,m2 will be close to 1. On the other hand, when α=0.33, we can obtain that BDm1,m2=0 for ∀t∈0,100. Furthermore, it can also be found that the variation in t has a limited influence on the measurement of the improved belief divergence. Therefore, the proposed measure has better performance in modelling the conflicts between BOEs.Figure 2
Results of the improved belief divergence in Example2.Example 3.
Suppose that the FOD isΘ=θ1,θ2,⋯,θN, Θ=N; then, the BOEs m1, m2, and m3 on Θ are given as follows:(20)m1:m1θ1=m1θ2=⋯=m1θN=1N,m2:m2Θ=1,m3:m3θl=1,θl∈Θ.
In Example3, both m1 and m2 reflect the intrinsic nature of uncertainty, whereas m3 is certain evidence. Specifically, m1 corresponds to full randomness, which assigns identical values of BBAs to each singleton set of Θ, m2 represents complete ignorance, which absolutely supports Θ, and m3 has 100% belief values to support the specific proposition θl. Intuitively, it is expected that m1 is closer to m2 than to m3.
According to equations (3) and (4) and (8)–(13), the values of the Jousselme distance dJ, the BJS measure, and the improved belief divergence BD are calculated and analyzed, as shown in Figure 3. We can get BJSm1,m2=1 and BJSm1,m2>BJSm1,m3, which represent that the BOEs m1 and m2 absolutely contradict based on the BJS measure. The reason for this counterintuitive phenomenon is that the BJS measure ignores the correlations between focal elements in D-S evidence theory [2]. As for evidence distance proposed by Jousselme, we can obtain dJm1,m2=dJm1,m3, which means that the Jousselme distance cannot effectively distinguish the BBAs with singleton sets and those with multielement sets. In contrast, only the improved belief divergence is in line with the intuitive judgement, which can be represented as BDm1,m2<BDm1,m3.
Although the improved belief divergence achieves better performance, some gaps still exist in conflict management. As shown in Figure3, BDm1,m2=0, which indicates that the improved belief divergence cannot describe the full randomness (i.e., m1) and complete ignorance (i.e., m2) well. By comparison, it is more reasonable that the value of the Jousselme distance is dJm1,m2=1/21−1/N≠0. Therefore, it can be concluded that the improved belief divergence measure and Jousselme distance have complementary advantages and can be integrated to better measure the evidence conflict.Figure 3
Results of the improved belief divergence in Example3.


## 4. A Novel Method for Conflict Data Fusion in D-S Evidence Theory

In this section, a novel method for conflict data fusion is proposed based on the improved belief divergence, evidence distance, and belief entropy. Here, suppose that there areL independent bodies of evidence (BOEs) denoted by M=m1,…,mj,…,mL on the frame of discernment Θ=θ1,…θi,…θN, which contains N mutually exclusive and collectively exhaustive propositions. The main procedure of the proposed method is shown in Figure 4.Figure 4
The overall presentation of the proposed method for conflict data fusion.


### 4.1. Measurement of the BOE’s Credibility

The BOE’s credibility is able to characterize the BOE’s reliability and external support from other BOEs, and thus, it is usually adopted to preprocess conflicting evidence [18, 70]. It can be expected that the highly conflicting or unreliable BOE will receive less supports from other BOEs, and its credibility degree will be smaller, and vice versa. Due to the complementarity between the improved belief divergence and Jousselme distance, which is discussed in Example 3, they will be integrated to calculate the BOE’s credibility.First, according to equations (8)–(13), the Jousselme distance dJmj,mk and the improved belief divergence BDmj,mk can be obtained for the BOEs mjandmk∈M. In general, the greater the values of dJmj,mk and BDmj,mk, the larger their conflict degree between two BOEs mj and mk, and thus, the similarity degree of these two BOEs will be smaller. As a result, the similarity measure between BOEs mj and mk can be constructed as follows:(21)Simmj,mk=1−dJmj,mk1−BDmj,mk,j,k=1,2,…,L.Next, the support degree of the BOEmj can be determined as follows [27]:(22)Supmj=∑k=1,k≠jLSimmj,mk,j=1,2,…,L.Then, the credibility degree of the BOEmj can be defined by normalizing the support degrees of the BOEs, as follows:(23)Crdj=Supmj∑k=1LSupmk,j=1,2,…,L.


### 4.2. Determination of the BOE’s Weight

The BOE’s weight is usually regarded as evidence of importance [39]. It is believed that BOEs with various importance will have different effects on the final fusion result. Hence, it is an effective tool to modify the initial BOEs based on their weight to further resolve the counterintuitive problems in D-S evidence theory [38]. Recently, belief entropy has been widely applied to determine the evidence weight [60]. When the value of the BOE’s belief entropy is greater, which indicates that the evidence itself contains more information volume, the BOE will be allocated to a larger weight [21].First, the belief entropyEdmj of the BOE mj can be calculated by using equation (6). However, the value of the belief entropy will be zero in some cases, e.g., there is Edm3=0 in Example 3. To avoid assigning a zero value to the BOE’s weight, the information volume of the BOE mj can be measured as(24)IVmj=eEdmj,j=1,2,…,L.Then, the weight of the BOEmj can be defined through IVmj, as follows:(25)Wj=IVmj∑k=1LIVmk,j=1,2,⋯,L.


### 4.3. Generation and Fusion of the Modified BOEs

In terms of information fusion, there is a clear distinction between the concepts of evidence reliability and evidence importance [54, 71]. The former will reflect the external support degree from other BOEs, while the latter can demonstrate the inherent uncertainty of each BOE. Therefore, it is a great way to modify and discount the initial BOEs by comprehensively considering the credibility degrees and weights of the BOEs. Due to the good properties of commutativity and associativity, the classical Dempster combination rule is used to combine the modified evidence and generate the reasonable fusion results.First, the discounting factor of the BOEmj can be devised by combining credibility degree Crdj and weight Wj, as follows:(26)αj=Crdj×Wj∑k=1LCrdk×Wk,j=1,2,…,L.Subsequently, the modified average evidencem˜ can be obtained:(27)m˜=∑j=1Lαj×mj.Then, the modified average evidencem˜ can be combined based on equation (3) by L−1 times to obtain the final fusion result, as follows:(28)fm˜=m˜⊕m˜1⊕m˜2⊕⋯L−2⊕m˜L−1.


## 4.1. Measurement of the BOE’s Credibility

The BOE’s credibility is able to characterize the BOE’s reliability and external support from other BOEs, and thus, it is usually adopted to preprocess conflicting evidence [18, 70]. It can be expected that the highly conflicting or unreliable BOE will receive less supports from other BOEs, and its credibility degree will be smaller, and vice versa. Due to the complementarity between the improved belief divergence and Jousselme distance, which is discussed in Example 3, they will be integrated to calculate the BOE’s credibility.First, according to equations (8)–(13), the Jousselme distance dJmj,mk and the improved belief divergence BDmj,mk can be obtained for the BOEs mjandmk∈M. In general, the greater the values of dJmj,mk and BDmj,mk, the larger their conflict degree between two BOEs mj and mk, and thus, the similarity degree of these two BOEs will be smaller. As a result, the similarity measure between BOEs mj and mk can be constructed as follows:(21)Simmj,mk=1−dJmj,mk1−BDmj,mk,j,k=1,2,…,L.Next, the support degree of the BOEmj can be determined as follows [27]:(22)Supmj=∑k=1,k≠jLSimmj,mk,j=1,2,…,L.Then, the credibility degree of the BOEmj can be defined by normalizing the support degrees of the BOEs, as follows:(23)Crdj=Supmj∑k=1LSupmk,j=1,2,…,L.


## 4.2. Determination of the BOE’s Weight

The BOE’s weight is usually regarded as evidence of importance [39]. It is believed that BOEs with various importance will have different effects on the final fusion result. Hence, it is an effective tool to modify the initial BOEs based on their weight to further resolve the counterintuitive problems in D-S evidence theory [38]. Recently, belief entropy has been widely applied to determine the evidence weight [60]. When the value of the BOE’s belief entropy is greater, which indicates that the evidence itself contains more information volume, the BOE will be allocated to a larger weight [21].First, the belief entropyEdmj of the BOE mj can be calculated by using equation (6). However, the value of the belief entropy will be zero in some cases, e.g., there is Edm3=0 in Example 3. To avoid assigning a zero value to the BOE’s weight, the information volume of the BOE mj can be measured as(24)IVmj=eEdmj,j=1,2,…,L.Then, the weight of the BOEmj can be defined through IVmj, as follows:(25)Wj=IVmj∑k=1LIVmk,j=1,2,⋯,L.


## 4.3. Generation and Fusion of the Modified BOEs

In terms of information fusion, there is a clear distinction between the concepts of evidence reliability and evidence importance [54, 71]. The former will reflect the external support degree from other BOEs, while the latter can demonstrate the inherent uncertainty of each BOE. Therefore, it is a great way to modify and discount the initial BOEs by comprehensively considering the credibility degrees and weights of the BOEs. Due to the good properties of commutativity and associativity, the classical Dempster combination rule is used to combine the modified evidence and generate the reasonable fusion results.First, the discounting factor of the BOEmj can be devised by combining credibility degree Crdj and weight Wj, as follows:(26)αj=Crdj×Wj∑k=1LCrdk×Wk,j=1,2,…,L.Subsequently, the modified average evidencem˜ can be obtained:(27)m˜=∑j=1Lαj×mj.Then, the modified average evidencem˜ can be combined based on equation (3) by L−1 times to obtain the final fusion result, as follows:(28)fm˜=m˜⊕m˜1⊕m˜2⊕⋯L−2⊕m˜L−1.


## 5. Application

To verify the efficiency and feasibility of the conflict data fusion method proposed in this paper, an application in the literature [1] is provided to illustrate the corresponding calculation procedure step by step and conduct comparative analyses.


### 5.1. Problem Statement

Considering the problem of target recognition, suppose that there are three potential targetsθ1,θ2,θ3 in the frame of discernment Θ. Moreover, five sensors are given to monitor and detect the potential targets. The collected reports of these sensors can be modelled by the BBAs, as shown in the following [1]:(29)m1:m1θ1=0.41,m1θ2=0.29,m1θ3=0.3,m2:m2θ2=0.9,m2θ3=0.1,m3:m3θ1=0.58,m3θ2=0.07,m3θ1,θ3=0.35,m4:m4θ1=0.55,m4θ2=0.1,m4θ1,θ3=0.35,m5:m5θ1=0.6,m5θ2=0.1,m5θ1,θ3=0.3.It can be easily found that the report of the second sensorm2 highly conflicts with those of the other sensors including m1, m3, m4, and m5. Specifically, the former strongly supports the target θ2, while the latter consisting of the majority assigns larger values to the target θ1. Thus, m2 can be regarded as unreliable evidence, whose negative effect should be diminished to avoid counterintuitive fusion results. Intuitively, θ1 will most likely become the final target by combining the reports of all five sensors.


### 5.2. Implementation Based on the Proposed Method

According to the main procedure of the proposed method in Figure4, the calculation steps are given as follows.Step 1-1: lett=1 in equations (12) and (13). The matrix of the improved belief divergence can be determined as follows:(30)BD=00.30040.09500.07820.08970.300400.65790.60870.62420.09500.657900.00010.00100.07820.60870.000100.00070.08970.62420.00100.00070.Step 1-2: the matrix of the Jousselme distance can be calculated by using equation (8):(31)dJ=00.53860.34950.32570.33110.538600.81420.78500.79060.34950.814200.03000.03740.32570.78500.030000.03540.33110.79060.03740.03540.Step 1-3: the matrix of similarity measure can be constructed by using equation (21):(32)Sim=10.32280.58870.62160.60890.322810.06360.08410.07870.58870.063610.96900.96160.62160.08410.969010.96400.60890.07870.96160.96401.Step 1-4: The support degrees of BOEs can be obtained by using equation (22):Supm1=2.1420; Supm2=0.5492; Supm3=2.5829; Supm4=2.6387; Supm5=2.6132.Step 1-5: the credibility degrees of BOEs can be measured by using equation (23):Crd1=0.2034; Crd2=0.0522; Crd3=0.2454; Crd4=0.2507; Crd5=0.2483.Step 2-1: the belief entropy of evidence can be derived by using equation (6), as follows:Edm1=1.5664; Edm2=0.4690; Edm3=1.8092; Edm4=1.8914; Edm5=1.7710.Step 2-2: the information volumes of BOEs can be calculated by using equation (24):IVm1=4.7893; IVm2=1.5984; IVm3=6.1056; IVm4=6.6286; IVm5=5.8764.Step 2-3: the weights of BOEs can be determined by using equation (25), as follows:W1=0.1916; W2=0.0639; W3=0.2442; W4=0.2652; W5=0.2351.Step 3-1: the discounting factors can be obtained by using equation (26), as follows:α1=0.1717; α2=0.0147; α3=0.2639; α4=0.2927; α5=0.2570.Step 3-2: the modified average evidence can be computed by using equation (27):m˜θ1=0.5387, m˜θ2=0.1364, m˜θ3=0.0530, and m˜θ1,θ3=0.2719.Step 3-3: after combiningm˜ by using equations (3) and (4), the fusion results can be obtained, as shown in Table 2. It is clear that the results generated with the proposed method strongly supports the target θ1 with the highest belief value, which is mθ1=0.9896.Table 2
The fusion results generated with different fusion methods.
Methodsmθ1mθ2mθ3mθ1,θ3TargetDempster [14]00.14220.85780θ3Murphy [26]0.96200.02100.01380.0032θ1Lin et al. [72]0.97920.00570.01190.0032θ1Deng et al. [27]0.98200.00390.01070.0034θ1Yan et al. [31]0.98500.00100.01100.0030θ1Jiang et al. [73]0.98670.00080.00890.0036θ1Li et al. [38]0.98810.00030.00780.0038θ1Yuan et al. [37]0.98860.00020.00720.0039θ1Xiao [1]0.98940.00020.00610.0043θ1Proposed method0.98960.00010.00610.0042θ1


### 5.3. Comparison and Discussion

To study the performance and characteristics of the proposed method, it is necessary to analyze and compare the results obtained by different fusion methods, as listed in Table2. It can be easily found that the classical Dempster method [14] cannot effectively handle highly conflicting evidence m2 and will produce counterintuitive fusion results. Based on the Dempster method [14], the final target can be recognized as the proposition θ3 with the highest belief value of 0.8578. However, the results generated by other methods all allocate the larger values to the target θ1, which is consistent with intuitive judgement. In addition, the accuracy of recognition by the proposed method is approximately 98.96%, which is higher than those by currently popular methods, such as the Murphy method [26], the Lin et al. method [72], the Deng et al. method [27], the Yan et al. method [31], the Jiang et al. method [73], the Li et al. method [38], the Yuan et al. method [37], and the Xiao method [1].Through further analysis of the reasons for the above phenomenon, Murphy [26] proposed an evidence fusion method that averages BBAs directly; Lin et al. [72] designed an improved method based on the Euclidean distance of BBAs; Deng et al. [27] proposed a modified average method to combine belief function based on evidence distance; Yan et al. [31] studied an improved belief entropy and its application in conflict management; Jiang et al. [73] devised a combination rule based on Deng entropy; Li et al. [38] proposed an evidence fusion method based on an improved Jousselme distance and Tsallis entropy; Yuan et al. [37] designed a method combining Deng entropy and Jousselme distance to address conflict management; Xiao [1] proposed a multisensor data fusion based on the BJS divergence and belief entropy. By comparison, it can be found that the proposed method not only makes full use of the complementary advantages of the improved belief divergence and the Jousselme distance to measure the external support and evidence credibility but also accounts for the internal indeterminacy and evidence weight based on the concept of belief entropy. Therefore, the highly conflicting and unreliable BOE will be greatly discounted when combined with other reliable BOEs. Consequently, the novel fusion method proposed in this paper characterizes better convergence than other methods in addressing conflict management in D-S evidence theory.Furthermore, a sensitivity analysis will be conducted to investigate the influence of parametert in Definition 8 on conflict management in D-S evidence theory. Here, we define a range of (0, 100] for the parameter t and further calculate the credibility degrees of the BOEs m1, m2, m3, m4, and m5 that represent five sensors, as shown in Figure 5. Then, it can be seen that the variation of parameter t will have a slight effect on the credibility degrees of these BOEs. In more detail, as the value of parameter t goes from near 0 to 100, the values of Crd1 and Crd2 for the BOEs m1 and m2 will decrease by 0.0012 and 0.0051, respectively, as shown in Figures 5(a) and 5(b); In contrast, the results presented in Figure 5(c) indicate that the values of Crd3, Crd4, and Crd5 for the BOEs m3, m4, and m5 will increase by 0.0016, 0.0022, and 0.0024, respectively. The reason lies in the value of t having a limited influence on the improved belief divergence, which is the key part of BOE’s credibility in equations (21)–(23), as discussed in Example 2. In addition, in view of evidence modification based on the BOE’s credibility and weight in the proposed method, the final fusion results are basically consistent under varying values of parameter t. For example, when t=100 in Definition 8, the corresponding fusion results by using the proposed method will be mθ1=0.9897, mθ2=0.0001, mθ3=0.0060, and mθ1,θ3=0.0042, which are almost the same as the fusion results in Table 2. Therefore, it can be demonstrated that the proposed fusion method has better robustness and stability for conflict management in D-S evidence theory.Figure 5
The credibility degrees of five BOEs with the variation of parametert. (a) The corresponding credibility degree Crd1 of the BOE m1. (b) The corresponding credibility degree Crd2 of the BOE m2. (c) The corresponding credibility degrees Crd3, Crd4, and Crd5 of the BOEs m3, m4, and m5.


## 5.1. Problem Statement

Considering the problem of target recognition, suppose that there are three potential targetsθ1,θ2,θ3 in the frame of discernment Θ. Moreover, five sensors are given to monitor and detect the potential targets. The collected reports of these sensors can be modelled by the BBAs, as shown in the following [1]:(29)m1:m1θ1=0.41,m1θ2=0.29,m1θ3=0.3,m2:m2θ2=0.9,m2θ3=0.1,m3:m3θ1=0.58,m3θ2=0.07,m3θ1,θ3=0.35,m4:m4θ1=0.55,m4θ2=0.1,m4θ1,θ3=0.35,m5:m5θ1=0.6,m5θ2=0.1,m5θ1,θ3=0.3.It can be easily found that the report of the second sensorm2 highly conflicts with those of the other sensors including m1, m3, m4, and m5. Specifically, the former strongly supports the target θ2, while the latter consisting of the majority assigns larger values to the target θ1. Thus, m2 can be regarded as unreliable evidence, whose negative effect should be diminished to avoid counterintuitive fusion results. Intuitively, θ1 will most likely become the final target by combining the reports of all five sensors.


## 5.2. Implementation Based on the Proposed Method

According to the main procedure of the proposed method in Figure4, the calculation steps are given as follows.Step 1-1: lett=1 in equations (12) and (13). The matrix of the improved belief divergence can be determined as follows:(30)BD=00.30040.09500.07820.08970.300400.65790.60870.62420.09500.657900.00010.00100.07820.60870.000100.00070.08970.62420.00100.00070.Step 1-2: the matrix of the Jousselme distance can be calculated by using equation (8):(31)dJ=00.53860.34950.32570.33110.538600.81420.78500.79060.34950.814200.03000.03740.32570.78500.030000.03540.33110.79060.03740.03540.Step 1-3: the matrix of similarity measure can be constructed by using equation (21):(32)Sim=10.32280.58870.62160.60890.322810.06360.08410.07870.58870.063610.96900.96160.62160.08410.969010.96400.60890.07870.96160.96401.Step 1-4: The support degrees of BOEs can be obtained by using equation (22):Supm1=2.1420; Supm2=0.5492; Supm3=2.5829; Supm4=2.6387; Supm5=2.6132.Step 1-5: the credibility degrees of BOEs can be measured by using equation (23):Crd1=0.2034; Crd2=0.0522; Crd3=0.2454; Crd4=0.2507; Crd5=0.2483.Step 2-1: the belief entropy of evidence can be derived by using equation (6), as follows:Edm1=1.5664; Edm2=0.4690; Edm3=1.8092; Edm4=1.8914; Edm5=1.7710.Step 2-2: the information volumes of BOEs can be calculated by using equation (24):IVm1=4.7893; IVm2=1.5984; IVm3=6.1056; IVm4=6.6286; IVm5=5.8764.Step 2-3: the weights of BOEs can be determined by using equation (25), as follows:W1=0.1916; W2=0.0639; W3=0.2442; W4=0.2652; W5=0.2351.Step 3-1: the discounting factors can be obtained by using equation (26), as follows:α1=0.1717; α2=0.0147; α3=0.2639; α4=0.2927; α5=0.2570.Step 3-2: the modified average evidence can be computed by using equation (27):m˜θ1=0.5387, m˜θ2=0.1364, m˜θ3=0.0530, and m˜θ1,θ3=0.2719.Step 3-3: after combiningm˜ by using equations (3) and (4), the fusion results can be obtained, as shown in Table 2. It is clear that the results generated with the proposed method strongly supports the target θ1 with the highest belief value, which is mθ1=0.9896.Table 2
The fusion results generated with different fusion methods.
Methodsmθ1mθ2mθ3mθ1,θ3TargetDempster [14]00.14220.85780θ3Murphy [26]0.96200.02100.01380.0032θ1Lin et al. [72]0.97920.00570.01190.0032θ1Deng et al. [27]0.98200.00390.01070.0034θ1Yan et al. [31]0.98500.00100.01100.0030θ1Jiang et al. [73]0.98670.00080.00890.0036θ1Li et al. [38]0.98810.00030.00780.0038θ1Yuan et al. [37]0.98860.00020.00720.0039θ1Xiao [1]0.98940.00020.00610.0043θ1Proposed method0.98960.00010.00610.0042θ1


## 5.3. Comparison and Discussion

To study the performance and characteristics of the proposed method, it is necessary to analyze and compare the results obtained by different fusion methods, as listed in Table2. It can be easily found that the classical Dempster method [14] cannot effectively handle highly conflicting evidence m2 and will produce counterintuitive fusion results. Based on the Dempster method [14], the final target can be recognized as the proposition θ3 with the highest belief value of 0.8578. However, the results generated by other methods all allocate the larger values to the target θ1, which is consistent with intuitive judgement. In addition, the accuracy of recognition by the proposed method is approximately 98.96%, which is higher than those by currently popular methods, such as the Murphy method [26], the Lin et al. method [72], the Deng et al. method [27], the Yan et al. method [31], the Jiang et al. method [73], the Li et al. method [38], the Yuan et al. method [37], and the Xiao method [1].Through further analysis of the reasons for the above phenomenon, Murphy [26] proposed an evidence fusion method that averages BBAs directly; Lin et al. [72] designed an improved method based on the Euclidean distance of BBAs; Deng et al. [27] proposed a modified average method to combine belief function based on evidence distance; Yan et al. [31] studied an improved belief entropy and its application in conflict management; Jiang et al. [73] devised a combination rule based on Deng entropy; Li et al. [38] proposed an evidence fusion method based on an improved Jousselme distance and Tsallis entropy; Yuan et al. [37] designed a method combining Deng entropy and Jousselme distance to address conflict management; Xiao [1] proposed a multisensor data fusion based on the BJS divergence and belief entropy. By comparison, it can be found that the proposed method not only makes full use of the complementary advantages of the improved belief divergence and the Jousselme distance to measure the external support and evidence credibility but also accounts for the internal indeterminacy and evidence weight based on the concept of belief entropy. Therefore, the highly conflicting and unreliable BOE will be greatly discounted when combined with other reliable BOEs. Consequently, the novel fusion method proposed in this paper characterizes better convergence than other methods in addressing conflict management in D-S evidence theory.Furthermore, a sensitivity analysis will be conducted to investigate the influence of parametert in Definition 8 on conflict management in D-S evidence theory. Here, we define a range of (0, 100] for the parameter t and further calculate the credibility degrees of the BOEs m1, m2, m3, m4, and m5 that represent five sensors, as shown in Figure 5. Then, it can be seen that the variation of parameter t will have a slight effect on the credibility degrees of these BOEs. In more detail, as the value of parameter t goes from near 0 to 100, the values of Crd1 and Crd2 for the BOEs m1 and m2 will decrease by 0.0012 and 0.0051, respectively, as shown in Figures 5(a) and 5(b); In contrast, the results presented in Figure 5(c) indicate that the values of Crd3, Crd4, and Crd5 for the BOEs m3, m4, and m5 will increase by 0.0016, 0.0022, and 0.0024, respectively. The reason lies in the value of t having a limited influence on the improved belief divergence, which is the key part of BOE’s credibility in equations (21)–(23), as discussed in Example 2. In addition, in view of evidence modification based on the BOE’s credibility and weight in the proposed method, the final fusion results are basically consistent under varying values of parameter t. For example, when t=100 in Definition 8, the corresponding fusion results by using the proposed method will be mθ1=0.9897, mθ2=0.0001, mθ3=0.0060, and mθ1,θ3=0.0042, which are almost the same as the fusion results in Table 2. Therefore, it can be demonstrated that the proposed fusion method has better robustness and stability for conflict management in D-S evidence theory.Figure 5
The credibility degrees of five BOEs with the variation of parametert. (a) The corresponding credibility degree Crd1 of the BOE m1. (b) The corresponding credibility degree Crd2 of the BOE m2. (c) The corresponding credibility degrees Crd3, Crd4, and Crd5 of the BOEs m3, m4, and m5.


## 6. Conclusions

To avoid counterintuitive results when fusing highly conflicting data in Dempster–Shafer evidence theory, this paper aims to study a novel method for conflict data fusion based on an improved belief divergence, evidence distance, and belief entropy. As discussed and illustrated by numerical examples and an application in target recognition, the proposed method not only can provide an effective tool for conflict management and generate consistent fusion results but also has better superiority, validity, and robustness. Consequently, the main contributions of this paper are as follows:(1)
An improved belief divergence measure is defined based on the probabilistic transformation in D-S evidence theory and cross entropy in fuzzy set theory, which can overcome the invalidity of the existing measures in some special cases and retain some remarkable properties, including symmetry, boundedness, and triangle inequality of its square root.(2)
A novel method for conflict data fusion is proposed by considering the evidence credibility and evidence weight simultaneously, in which the former makes full use of the complementarity advantages between the improved belief divergence and the Jousselme distance, and the latter accounts for the inherent information volume of evidence via belief entropy.(3)
Compared with other related methods, the proposed belief divergence measure and information fusion method have satisfactory performance in modelling conflicts between bodies of evidence and characterize better convergence in addressing conflict management in Dempster–Shafer evidence theory.For future work, we intend to further study the generalized belief divergence measure and extend the proposed method in the open world instead of the closed-world hypothesis in this paper. In addition, we will focus on exploring and verifying the efficiency and practicality of the proposed method in more complicated and real‐environment applications.



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*Source: 6558843-2021-10-08.xml*