A Weighted Evidence Combination Method Based on the Pignistic Probability Distance and Deng Entropy

ABSTRACT:

The Dempster-Shafer (D-S) theory is widely applied in various fields involved with multi-sensor information fusion for radar target tracking, which offers a useful tool for decision-making. However, the application of D-S evidence theory has some limitations when evidences are conflicting. This paper proposed a new method combining the Pignistic probability distance and the Deng entropy to address the problem. First, the Pignistic probability distance is applied to measure the conflict degree of evidences. Then, the uncertain information is measured by introducing the Deng entropy. Finally, the evidence correction factor is calculated for modifying the bodies of evidence, and the Dempster’s combination rule is adopted for evidence fusion. Simulation experiments illustrate the effectiveness of the proposed method dealing with conflicting evidences.

KEYWORDS: Multi-sensor information fusion; Conflicting evidence; D-S evidence theory; Pignistic probability distance; Deng entropy

INTRODUCTION

In complex battlefield environments, modern combat systems need to identify air targets accurately and quickly. However, the information obtained by a single information source is likely to be inaccurate, incomplete and unreliable because it usually fails to meet the operational requirement (Xiao and Qin 2018; Song and Deng 2019). Multi-sensor systems can obtain more abundant, precise and reliable information through information fusion, and they may overcome the limitations of single sensor systems. Unfortunately, data collected from different sensors are often inaccurate and uncertain (Sarabi-Jamab and Araabi 2018; Song and Deng 2019). How to model and process uncertain information is still an open issue.

The Dempster-Shafer (D-S) theory can manage uncertain information and offer a useful fusion tool for decision-making (Chen et al. 2017; Li et al. 2017; Xiao 2019; Xiao 2019b). Dempster put forward the evidence theory in 1967 (Schafer 196), and then Shafer further studied the theory in 1976 (Dempster (1967). However, the quality of evidence combination is affected by the conflicting information especially when the sources of evidence are unreliable (Klein et al. 2016). In addition, the counter-intuitive results may be generated by the combination of Dempster’s rule, which is first highlighted by Zadeh (1986).

To solve the above problems, domestic and foreign scholars have proposed a number of improved methods, which are generally divided into two types (Han et al. 2011; An et al. 2019): modification to fusion rules and pre-processing for evidence sources. The former considers that the irrationality conclusion under high-level evidence is generated in the normalization step of the Dempster combination rule, and the key to solving this problem is how to redistribute the conflict between evidence, i.e., which focal elements of the conflict should be reallocated and how to determine the proportional coefficient of assignment. In an earlier study, Yager (1987) regared all the information contained in the conflict as unknown and assigned all the conflict factors to the identification framework. Since Yager’s method is too conservative, it was further improved by Smets (1990). He believed that conflicts are caused by the incompleteness of the identification framework, and proposed a new composition rule in which conflicting items are allocated to empty set. Dubois and Prade (1988) suggested assigning the highly conflicting mass to the whole set or a particular set. However, when the evidences are highly conflict, the worse fusion results may be obtained. In fact, the good nature of the combination rules is often destroyed due to the modification. In addition, if the counter-intuitive result is caused by sensor failure, this modification is considered unreasonable. Therefore, to solve the combination problem of evidence conflict, researchers tend to adopt the second type of method (i.e., pre-process the subject of evidence).

These methods believe that the Dempster combination rule has a solid mathematical foundation and has no problem in itself. But it ignores the fact that each piece of evidence has different reliability. When dealing with high-conflict evidence, the conflict evidence should be pre-processed before using the Dempster combination rule. For example, Murphy (2000) proposed a method of averaging evidence, the idea is to modify the original evidence without changing the Dempster combination rule, i.e., to calculate the Basic Probability Assignment (BPA) of each evidence before evidence fusion. In this work, the same reliability is averaging for multiple sets of evidence, but the correlations between individual evidence were not considered. Han et al. (2004) showed that the Murphy’s method can be further improved by adding a distance function, which measures the degree of similarity between the evidence and the determined weight of evidences. Although this method has some improvement on the evidence of high degree of conflict, the distance formula cannot be used to describe the degree of mutual support between different evidence, thus the effect of the evidence itself was ignored in the target identification process. Zhang et al. (2014) applied the idea of distance-based evidence conflict analysis and proposed a new method based on the law of cosines to identify and represent conflict data. However, it ignores the influence of evidence on the correction coefficient, so the method in this paper introduces Deng entropy (Zhang and Deng 2019; Kang and Deng 2019; Gao and Deng 2020; Gao and Deng 2019) to improve the performance of information fusion.

In a word, the above improved methods based on redistribution of conflicting evidence do not fully take into account the fact that each piece of evidence has different degrees of reliability. To solve this problem, a new combination method for multi-sensor conflicting information is proposed in this paper. Compared with the Jousselme distance used by most scholars, the Pignistic distance can better judge the conflict between the evidences and has lower complexity (Liu 2006). Only the distance between the evidence is it not a good measure of the conflict of evidence. Deng entropy is used to quantify the uncertainty of different evidences, which not only can better measure evidence conflicts, but also solves the problem of non-convergence in calculation. This paper introduces Pignistic probability distance and Deng entropy to compute the evidence correction factor, and then the bodies of evidence are modified before using Dempster’s combination rule. By correcting the evidence, reasonable and effective fusion results can be obtained. The simulation results and analyses demonstrate that the proposed method can not only achieve accurate fusion results with low conflicting evidence, but also obtain reliable performance under high conflicting information compared with several existing methods. Hence, the novelty and practicability of the proposed method are verified. As the reliability and accuracy of information fusion are both solved, the effective application of multi-sensor systems is further guaranteed.

D-S EVIDENCE THEORY

As the generalization of the probability theory and Bayesian reasoning, the D-S evidence theory can obtain fusing results without the requirement of prior knowledge and conditional probability. Based on the accumulation of evidences, the effective and accurate multi-sensor fusion results can be obtained. The basic concepts are introduced as below.

FRAME OF DISCERNMENT

In D-S evidence theory, a sample space is called a frame of discernment, represented by Θ, which is composed of M objects. The objects are mutually exclusive and contain the entire object to be identified. The frame of discernment defined as

(1) $\Theta =\left\{{\mathit{A}}_1,{\mathit{A}}_2\mathrm{,...}{\mathit{A}}_{\mathit{M}}\right\}$

Accordingly, we can derive the power set A ∈ 2^Θ of D-S evidence theory.

(2) $2^{\Theta }=\left\{\varnothing ,\left\{{\mathit{A}}_1\right\},\left\{{\mathit{A}}_2\right\}\mathrm{,...}\left\{{\mathit{A}}_M\right\},\left\{{\mathit{A}}_1,{\mathit{A}}_2\right\}\mathrm{,...}\left\{{\mathit{A}}_1,{\mathit{A}}_M\right\}\mathrm{,...}\left\{{\mathit{A}}_1,{\mathit{A}}_2\mathrm{,...}{\mathit{A}}_M\right\}\right\}$

where ∅ is the empty set. The power set 2^Θ is composed with 2^M propositions from (2), and any proposition A ⊆ Θ satisfies A ∈ 2^Θ.

MASS FUNCTION

In D-S theory evidence, evidences are obtained through multi-sensor information. If the function m: 2^Θ → [0, 1] satisfies equation (3) and (4), it is called the basic probability assignment (BPA, also called mass function). BPA reflects the degree of evidence support for propositions in the frame of discernment, namely m(A).

(3) $\mathit{m}\left(\varnothing \right)=0$

(4) $\sum _{\mathit{A}\subseteq \Theta }\mathit{m}\left(\mathit{A}\right)=1$

for A ⊆ Θ, A ∈ 2^Θ

UNCERTAINTY REPRESENTATION

For a proposition A ⊆ Θ, the sum of BPA corresponding to all subsets in Θ is called belief function. The belief function Bel: 2^Θ → [0, 1] is defined as

(5) $\mathit{Bel}\left(\mathit{A}\right)=\sum _{\mathit{\alpha }\subseteq \mathit{A}}\mathit{m}\left(\mathit{\alpha }\right)$

(6) $\mathit{P}\left(\mathit{A}\right)=\sum _{\mathit{A}\bigcap \mathit{\alpha }\cancel{=}\varnothing }\mathit{m}\left(\mathit{\alpha }\right)=1-\mathit{Bel}\left(\overline{\mathit{A}}\right)$

where α is any subset of the set A. The function Bel(A) and Pl(A) separately reflects the lower and upper bounds of limit function of proposition A.

D-S COMBINATION RULE

D-S combination rule can synthesize multi-sensor information to obtain effective and accurate decision-making results. Assuming that the frame of a multi-sensor system is Θ={A₁ ,A₂ , . . . , A_M}, the D-S combination rule of two evidences m₁ and m₂ is defined as

(7) $\begin{array}{c}\mathit{m}\left(\mathit{A}\right)=\frac{1}{1-\mathit{K}}\sum _{A_i\bigcap A_j=A}{\mathit{m}}_1\left({\mathit{A}}_i\right){\mathit{m}}_2\left({\mathit{A}}_j\right),\mathit{A}\cancel{=}0\hfill \\ \mathit{m}\left(\varnothing \right)=0\hfill \end{array}$

where K is the conflicting factor that reflexes the conflicting degree of evidences m₁ and ${\mathit{m}}_2.\frac{1}{1-\mathit{K}}$ is the normalized factor that ensures the unity property of fused mass.

(8) $\mathit{K}=\sum _{{\mathit{A}}_{\mathit{i}}\bigcap {\mathit{A}}_{\mathit{i}}=\varnothing }{\mathit{m}}_1\left({\mathit{A}}_{\mathit{i}}\right){\mathit{m}}_2\left({\mathit{A}}_{\mathit{j}}\right)$

Obviously, the D-S combination rule satisfies both commutative law and associate law, which are shown separately as follows:

(9) ${\mathit{m}}_1\oplus {\mathit{m}}_2={\mathit{m}}_2\oplus {\mathit{m}}_1$

(10) $\left({\mathit{m}}_1\oplus {\mathit{m}}_2\right)\oplus {\mathit{m}}_3={\mathit{m}}_1\oplus \left({\mathit{m}}_2\oplus {\mathit{m}}_3\right)$

A NEW METHOD FOR MODIFYING COMBINATION RULES

In this paper, a new combination method based on the Pignistic distance and Deng entropy is proposed to deal with the evidence conflict problem. Compared with the Jousselme distance, the Pignistic distance can better measure the difference of evidence. Based on the Pignistic distance, the evidence support can be derived to describe the reliability of evidence. However, only using the Pignistic distance between the evidence is not necessarily effective for describe the conflict degree of evidences. Thus Deng entropy is introduced and used to measure the information volume of evidence, e.g., the greater the amount of information, the greater the uncertainty of the evidence. In other words, Deng entropy is adopted to quantify the uncertainty of different evidences. Based on the Pignistic distance and Deng entropy, the proposed method not only can better measure evidence conflicts but also solves the computational divergence problem.

PIGNISTIC PROBABILITY FUNCTION

The representative methods of measuring conflict evidence mainly include: Pignistic probability distance, Jousselme distance correlation coefficient, compatibility coefficient, etc. Recently, some new measurement methods have also appeared one after another. For example, Yu et al. (2015) proposed support probability distance, and Smets (2007) presented a similarity measure combining Pignistic probability distance and Tanimoto measure. Tessem (1993) proposed the Pignistic probability distance based on the Pignistic probability function, which is adopted in this paper and specifically defined as follows.

Let m be a BPA on Θ. Its associated Pignistic probability function ${\mathit{BetP}}_{\mathit{m}}:\Theta \to \left[0,1\right]$ is defined as:

(11) ${\mathit{BetP}}_{\mathit{m}}\left(\mathit{\omega }\right)=\sum _{\mathit{A}\subseteq \Theta ,\mathit{\omega }\varepsilon \mathit{A}}\frac{1}{\left|\mathit{A}\right|}\frac{\mathit{m}\left(\mathit{A}\right)}{1-\mathit{m}\left(\varnothing \right)},\mathit{m}\left(\varnothing \right)\cancel{=}1$

where ω∈Θ, |A| is the cardinality of subset A.

Let m₁ and m₂ be two BPA on frame Θ and let BetP_m1 and BetP_m2 be the results of two Pignistic transformations from them respectively. Then:

(12) ${\mathit{difBetP}}_{\mathit{m}2}^{\mathit{m}1}={\max }_{\mathit{A}\subseteq \Theta }\left(\left|{\mathit{Bet}}_{\mathit{m}1}\left(\mathit{A}\right)-{\mathit{Bet}}_{\mathit{m}2}\left(\mathit{A}\right)\right|\right)$

Obviously, difBetP^m1_m2 =0 when m₁=m₂, i.e., when any two pieces of evidence are the same, their BPA’s betting commitment distance is always 0.

DENG ENTROPY

Due to the complex natural environment and human interference, the information obtained by some detection sensors is disturbed and conflicts with other sensors (Lin et al. 2016; Wang et al. 2018; Sun et al. 2018; Gong et al. 2018). Therefore, effective analysis of the detection information is required. In thermodynamics, the entropy concept of the disordered state size of the system, Claude Shannon defined the information entropy in information theory, and estimated the redundancy or uncertainty of the information according to (Zhang et al. 2017; Jiang et al. 2017). But it is not applicable to evidence theory, because there is a multi-subset proposition in evidence theory.

A new type of belief entropy, known as the Deng entropy, was first proposed by Deng (2016). The basic concepts are introduced as follows:

(13) ${\mathit{E}}_{\mathit{d}}\left(\mathit{m}\right)=-\sum _{\mathit{A}\subseteq \Theta }\mathit{m}\left(\mathit{A}\right){\log }_2\frac{\mathit{m}\left(\mathit{A}\right)}{2^{\left|\mathit{A}\right|}-1}$

where m is a mass function defined on the frame of discernment Θ, and A is the focal element of m, |A| is the cardinality of A, i.e., the number of elements in A.

Deng entropy reflects the amount of information contained in the evidence. The more Deng entropy of evidence is, the more information it contains (i.e., the more uncertainty it has); On the contrary, the less Deng entropy of evidence is, the less information it contains (i.e., the less uncertainty it has). However, only using Deng entropy to calculate the weight may increase the weight of interference evidence in fusion, which may lead to unreasonable results. In other words, Deng entropy cannot judge whether there is conflict between evidences. Fortunately, the pignistic distance of evidence is an effective tool to reflect the conflicted degree between evidences. Combination of Deng entropy and the pignistic distance can greatly improve the effect of evidence fusion and the recognition rate of evidence.

IMPROVED NEW METHODS

D-S evidence theory is widely used in the field of multi-sensor target recognition, which can deal with the uncertain information fusion problem. However, the application of D-S evidence theory has some limitations when evidences are conflicting. Traditional evidence theory fusion rules usually may not distribute evidence conflicts reasonably, which makes the result of decision fusion is often contrary to the facts. Besides, most existed methods only redistribute the conflicted evidence without considering their credibility. In other words, they do not fully take into account the fact that each piece of evidence has different degrees of reliability. Thus this paper a weighted evidence combination method based on the Pignistic probability distance and Deng entropy, in which the uncertainty of evidence not only can be reflected and the conflict degree can be descried. It has not only the better identification performance and faster convergence speed, but also the less risk of decision-making. Even if there exist high conflicts between evidences, the proposed method can also make correct identification more rapidly than other approaches.

Step 1: Calculate the credibility of evidence.

1. Equations (11)-(13) are used to calculate the Pignistic probability distance between n evidence pairs collected in difBetP^m1_m2.

2. Calculate the support and reliability of the evidence. Suppose the similarity degree S_ij between m_i and m_j is:

   (14) ${\mathit{S}}_{\mathit{ij}}=1-{\mathit{difBetP}}_{\mathit{ij}}$

The greater the distance difbetP_ij between the evidences, the greater the similarity s_ij is. In other words, the higher evidence conflicts make them less similar. Based on the definition of the similarity degree, the support and credibility of the evidence are calculated separately

(15) ${\mathit{Sup}}_{\mathit{i}}=\sum _{\mathit{j}=1,\mathit{j}\cancel{=}\mathit{i}}^{\mathit{N}}{\mathit{S}}_{\mathit{ij}}$

(16) ${\mathit{Cre}}_{\mathit{i}}=\frac{\sup }{\sum _{\mathit{i}=1}^{\mathit{s}}{\sup }_{\mathit{i}}}$

Step 2: Calculate the information entropy of the evidence.

1. Set thresholds and select credible evidence. After verification and comparison of experimental data, the threshold rate is defined as 10%, and the threshold is calculated as:

   (17) $\mathit{\phi }=\sum _{\mathit{i}=1}^{\mathit{N}}{\mathit{Cre}}_{\mathit{i}}10\%$

2. Calculate the information entropy of credible evidence.

When the credibility of the evidence is higher than the threshold, they are regarded as credible evidences. Otherwise, they are incredible evidences.

Select all credible evidences E_l (l=1,2L S) and calculate their respective Deng entropy through equation (14). When the focal element in the identification framework is monad set, equation (13) can be simplified as in [35]:

(18) ${\mathit{I}}_{\mathit{i}}^{\text{'}}=\sum _{\mathit{t}=1}^{\mathit{M}}\mathit{m}\left({\mathit{A}}_{\mathit{t}}\right)\log {\mathit{m}}_{\mathit{i}}\left({\mathit{A}}_{\mathit{t}}\right)\left(\mathit{t}-1,2,\mathit{L},\mathit{M}\right)$

After the information entropy is normalized, it can be obtained:

(19) $\mathit{I}=\frac{{\mathit{I}}_{\mathit{i}}^{\text{'}}}{\sum _{\mathit{i}=1}^{\mathit{S}}{\mathit{I}}_{\mathit{i}}^{\text{'}}}$

Step 3: calculate the correction coefficient of evidence.

1. The correction coefficient of the evidence in the selected evidence set is obtained, and the correction coefficient of the t-th evidence is expressed as:

   (20) ${\mathit{\omega }}_{\mathit{i}}^{\text{'}}=\left(1-\mathit{l}\right){\mathit{e}}^{-\mathit{I}}$

2. If the correction coefficients of untrustworthy evidence are replaced with credibility, the correction coefficients of all evidence are normalized to obtain:

   (21) ${\mathit{\omega }}_{\mathit{i}}=\frac{{\mathit{\omega }}_{\mathit{i}}^{\text{'}}}{\sum _{\mathit{i}=1}^{\mathit{N}}{\mathit{\omega }}_{\mathit{i}}^{\text{'}}}$

Step 4: Calculate the correction coefficient of each evidence according to the above algorithm, and perform weighted average on the basic probability distribution of all evidence. The Dempster rule is used to fuse the number of iterations (n-1) to obtain the fusion result (Fig. 1).

Figure 1
Flow chart of the proposed method.

NUMERICAL EXPERIMENTS AND ANALYSES

To verify the validity and superiority of the proposed D-S combination method, three numerical simulations are conducted. Two kinds of multi-sensor data are adopted respectively, where low and high conflicting information are included. The methods in D-S (Shafer 1976), Yager (1987) and Yuan et al. (2016) are compared with the presented method using D-S combination rule. It contains multi-sensor data with low conflict information and high conflict information, and the method in D-S, Yager and Yuan is compared with the method using d-s combination rule.

LOW CONFLICTING INFORMATION

Example 1. In the multi-sensor system, assume that there are 5 evidences in the framework Θ={A,B,C}, and proposition A is the true; the low conflicting evidences are exhibited in Table 1.

Table 1
Mass assignments of low conflicting information.

Sensors	Targets
	A	B	C
E1:m1(·)	0.9	0	0.1
E2:m2(·)	0.88	0.01	0.11
E3:m3(·)	0.5	0.2	0.3
E4:m4(·)	0.98	0.01	0.01
E5:m5(·)	0.9	0.05	0.05

Table 2 shows the fusion results of D-S, Yager, Yuan and the proposed method. Under a low conflicting condition, it indicates that the true proposition A is identified by all methods. Based on these results, the following analyses can be obtained:

Table 2
Fusion results of different methods with low conflicting information.

Methods (Low)	Targets	E1⊕E2	E1⊕E2⊕E3	E1⊕E2⊕E3⊕E4	E1⊕E2⊕E3⊕E4⊕E5
D-S	A	0.9863	0.9917	0.9999	1
	B	0	0	0	0
	C	0.0137	0.0083	0.0001	0
	Θ	0	0	0	0
Yager	A	0.7920	0.3960	0.3881	0.3493
	B	0	0	0	0
	C	0.011	0.0033	0	0
	Θ	0.1970	0.6007	0.6119	0.6507
Kaijuan Yuan	A	0.7695	0.9533	0.9915	0.9984
	B	0.0804	0.0104	0.0011	0.0002
	C	0.1501	0.0363	0.0074	0.0014
	Θ	0	0	0	0
The proposed method	A	0.8356	0.9793	0.9976	0.9997
	B	0.0571	0.0046	0.0003	0
	C	0.1073	0.0161	0.0021	0.0003
	Θ	0	0	0	0

1. D-S evidence theory cannot effectively deal with high conflict evidence. Since m₁(A₂)=0, the proposition A₂ is completely negated. Even if there is more evidence to support the proposition A₂, the fusion result always shows that the support of the proposition A₂ is 0.

2. Our method and other three existing method identify the target by using two evidences correctly when fusing low conflicting information. It has almost the same identification performance as other method except Yager. This is because Yager assigns all evidence conflicts to unknown m(Φ). As the evidence (support for objectA₁) increases, we can see that the value ofm(Φ) increases (see Table 1).

3. The method in this paper can effectively deal with the case of interference evidence, and has a faster convergence speed. In addition, compared with other methods, the true proposition A₁ quality of this method is the largest, which verifies its validity and accuracy, as shown in Fig. 2.

   

   Figure 2
   Comparison of evidence fusion results for various method with normal evidence, example 1.
   

From the above analysis, we can draw the conclusion that the proposed method can handle the conflicting situation more precisely and efficiently.

HIGHLY CONFLICTING INFORMATION.

In order to further test the performance of the proposed method under highly conflicting conditions, we set some highly conflicting evidences as Example 2 (see Table 3). Table 4 gives the fusing results of D-S, Yager, Yuan and the proposed method.

Table 3
Mass assignments of highly conflicting information.

Sensors	Targets
	A	B	C
E1:m1(·)	0.9	0	0.1
E2:m2(·)	0	0.01	0.99
E3:m3(·)	0.5	0.2	0.3
E4:m4(·)	0.98	0.01	0.01
E5:m5(·)	0.9	0.05	0.05

Table 4
Fusion results of different methods with highly conflicting information.

Methods (High)	Targets	E1⊕E2	E1⊕E2⊕E3	E1⊕E2⊕E3⊕E4	E1⊕E2⊕E3⊕E4⊕E5
D-S	A	0	0	0	0
	B	0	0	0	0
	C	1	1	1	1
	Θ	0	0	0	0
Yager	A	0	0	0	0
	B	0	0	0	0
	C	0.0990	0.0297	0.0003	0
	Θ	0.9010	0.9703	0.9997	1
Kaijuan Yuan	A	0.7052	0.9126	0.9770	0.9940
	B	0.1016	0.0189	0.0029	0.0004
	C	0.1932	0.0685	0.0201	0.0056
	Θ	0	0	0	0
The proposed method	A	0.8685	0.9865	0.9986	0.9999
	B	0.0367	0.0018	0.0001	0
	C	0.0948	0.0118	0.0013	0.0001
	Θ	0	0	0	0

1. Evidence 2 has not support for A₁, even though the evidence supporting proposition A₁ increases, D-S evidence theory considers that proposition A₃ is a true, which is obvious contrary to the intuition. The D-S evidence theory has the wrong result, so it is unreliable in highly conflicting conditions.

2. Yager’s results are similar to D-S evidence theory. No matter how much evidence is collected in the future, Yager totally denies the proposition A₁, and the value of unknown terms is always increasing. Therefore, Yager is not completely impacted in a highly conflicting situation. Yager supports true proposition A the most. It has a relatively complex method, so in practical application, it cannot quickly identify the target. Besides, Yager’s method has worse performance than the method in this paper when fusing highly conflicting conditions. This method needs further refinement.

3. Comparing with other methods, our method assigns a bigger mass to true proposition A₃. Thus this method maintains the accurate fusion performance when combining highly conflicting conditions. It reduces the impact of conflicted evidences on the fusion results and strong anti-disturbance ability, as shown in Fig. 3.

   

   Figure 3
   Comparison of evidence fusion results for various method with conflict evidence, example 2.
   

To sum up, the method proposed in this paper keeps its superiority to obtain accurate and stable fusion. Apparently, along with the increasing conflicting degree among evidences, the proposed method always has the best combination effects.

Example 3. In the multi-sensor system, assume that there are 5 evidences in the framework Θ={A,B,C}, and proposition A is the true; the evidences are shown in Table 5.

Table 5
Mass assignments of multi-elements subsets.

Sensors	Targets
	A	B	C	AC
E1:m1(·)	0.41	0.29	0.3	0
E2:m2(·)	0	0.9	0.1	0
E3:m3(·)	0.58	0.07	0	0.35
E4:m4(·)	0.55	0.1	0	0.35
E5:m5(·)	0.6	0.1	0	0.3

Table 6 shows the fusion results of D-S, Yage, Yuan and the proposed method. Under a hightly conflicting condition, it indicates that the true proposition A is identified by all methods. Based on these results, the following analyses can be obtained:

Table 6
Fusion results of different methods with multi-elements subsets.

Methods (Low)	Targets	E1⊕E2	E1⊕E2⊕E3	E1⊕E2⊕E3⊕E4	E1⊕E2⊕E3⊕E4⊕E5
D-S	A	0	0	0	0
	B	0.8969	0.6575	0.3321	0.1422
	C	0.1031	0.3425	0.6679	0.8578
	Θ	0	0	0	0
Yager	A	0	0.4112	0.6508	0.7732
	B	0.2610	0.0679	0.0330	0.0167
	C	0.0300	0.0105	0.0037	0.0011
	AC	0	0.2481	0.1786	0.0938
	Θ	0.7090	0.2622	0.1339	0.1152
Kaijuan Yuan	A	0.2849	0.8274	0.9596	0.9886
	B	0.5306	0.0609	0.0032	0.0002
	C	0.1845	0.0986	0.0267	0.0072
	AC	0	0.0131	0.0106	0.0039
	Θ	0	0	0	0
The proposed method	A	0.0055	0.6128	0.9985	0.9999
	B	0.9696	0.3702	0.0004	0
	C	0.0249	0.0168	0.0009	0.0001
	AC	0	0.0002	0.0002	0
	Θ	0	0	0	0

1. Evidence 2 has not support for proposition A₁, even though the evidence supporting proposition proposition A₁ increases, D-S evidence theory comes to a wrong conclusion, so it is unreliable in highly conflicting conditions.

2. Yager’s method assigns a bigger mass to true proposition A₃, which means it fails to identify the correct target, when there are only two pieces of evidence. With the increase of effective evidence, Yager’s can make a correct decision, but it is obvious that the effect is not very good and the accuracy is not high.

3. It’s clear that the proposed method is not only efficient but also reliable. Though both Yuan’s method and the proposed method can identify the object is A, our method assigns a bigger mass to true proposition A₃, when there are only four evidences. Under the situation of five evidences, the proposed method improves the accuracy of identification to 0.9999, while Yuan’s method only has 0.98864. Therefore the proposed method can deal with conflict and make decision effectually.

In this section, we choose two different kind of evidences with low and highly conflicting information. The simulation results prove that the method in this paper effectively solves the problem of conflict evidence combination.

CONCLUSION

Potential errors in sensor measurement, uncertainty in the unknown monitoring environment, and even possible human interference can lead to ambiguity and conflicts of information in multi-sensor systems. As the conflict information is common in multi-sensor systems, how to combine them becomes the core problem of achieving reliable and accurate fusion results.

D-S evidence theory is a widely used uncertainty management method in multi-sensor fusion systems. However, the conflict phenomenon usually occurs in the application of D-S theory, so its practical application has certain limitations. In order to solve the problem of evidence conflict, a new multi-sensor conflict information combination method is proposed in this paper. First, by introducing the Pignistic probability distance function and Deng entropy, the conflict degree and uncertainty information are put forward respectively to obtain the evidence correction coefficient. Then, the body of evidence is modified before using Dempster’s combination rules. Finally, two simulation experiments are carried out, the results verify the effectiveness and accuracy of the proposed method in low-conflict evidence, and prove its stability and superiority in high-conflict evidence.

• FUNDING
  Aeronautical Science Foundation of China [http://doi.org/10.13039/501100004750] Grant #20185142003 National Defense Basic Scientific Research Program of China Grant #JCKY2018419C001 National Natural Science Foundation of China [http://doi.org/10.13039/501100001809] Grant #U1504619, #61671139, #61573020 National Thirteen-Five Equipment Pre-Research Foundation of China Grant #61403120207, #61402100203 Scientific Technology Program of Henan Province Grant #182102110397, #192102210064

ACKNOWLEDGMENTS

Editors and authors are thankful to Fundação Conrado Wessel for providing the financial support for publishing this article.

REFERENCES

• An J, Hu M, Fu L, Zhan J (2019) A novel fuzzy approach for combining uncertain conflict evidences in the Dempster‐Shafer theory. IEEE Access 7:7481‐7501. https://doi.org/0.1109/access.2018.2890419
  » https://doi.org/0.1109/access.2018.2890419
• Chen J, Ye F, Jiang T, Tian Y (2017) Conflicting information fusion based on an improved D-S combination method. Symmetry 9(11):278. https://doi.org/10.3390/sym9110278
  » https://doi.org/10.3390/sym9110278
• Dempster, AP. Upper and Lower Probabilities Induced by a Multivalued Mapping. Ann. Math. Statist. 38 (1967), no. 2, 325--339. https://doi.org/10.1214/aoms/1177698950
  » https://doi.org/10.1214/aoms/1177698950
• Deng Y (2016). Deng entropy. Chaos, Solitons & Fractals 91:549-553. https://doi.org/10.1016/j.chaos.2016.07.014
  » https://doi.org/10.1016/j.chaos.2016.07.014
• Dubois D, Prade H (1988) Representation and combination of uncertainty with belief functions and possibility measures. Computational Intelligence 4(3):244-264. https://doi.org/10.1111/j.1467-8640.1988.tb00279.x
  » https://doi.org/10.1111/j.1467-8640.1988.tb00279.x
• Gao X, Deng Y (2020) The pseudo-pascal triangle of maximum deng entropy, International Journal of Computers Communications & Control 15(1): 1-10. https://doi.org/10.15837/ijccc.2020.1.3735
  » https://doi.org/10.15837/ijccc.2020.1.3735
• Gong Y, Su X, Qian H, Yang N (2018). Research on fault diagnosis methods for the reactor coolant system of nuclear power plant based on D-S evidence theory. Annals of nuclear energy 112:395-399. https://doi.org/10.1016/j.anucene.2017.10.026
  » https://doi.org/10.1016/j.anucene.2017.10.026
• Han D Q, Deng Y, Han C Z, Hou Z (2011) Weighted evidence combination based on distance of evidence and uncertainty measure. Journal of Infrared and Millimeter Waves 30(5),396-400. https://doi.org/10.3724/sp.j.1010.2011.00396
  » https://doi.org/10.3724/sp.j.1010.2011.00396
• Han D Q, Deng Y, Liu Q (2004). Combining belief functions based on distance of evidence. Decision Support Systems, 38(3), 489-493. https://doi.org/10.1016/j.dss.2004.04.015
  » https://doi.org/10.1016/j.dss.2004.04.015
• Jiang W, Wei B, Liu X, Li X, Zheng H (2017) Intuitionistic fuzzy power aggregation operator based on entropy and its application in decision making. International Journal of Intelligent Systems 33:49-67. https://doi.org/10.1002/int.21939
  » https://doi.org/10.1002/int.21939
• Kang B, Deng Y(2019) The maximum Deng entropy. IEEE Access 7: 120758-120765. https://doi.org/10.1109/access.2019.2937679
  » https://doi.org/10.1109/access.2019.2937679
• Klein J, Destercke S, Colot O (2016) Interpreting evidential distances by connecting them to partial orders:application to belief function approximation. International Journal of Approximate Reasoning 71:15-33. https://doi.org/10.1016/j.ijar.2016.01.001
  » https://doi.org/10.1016/j.ijar.2016.01.001
• Li T, Corchado J M, Sun S, Bajo J (2017) Clustering for filtering: multi-object detection and estimation using multiple/massive sensors. Information Sciences 388-389:172-190. https://doi.org/10.1016/j.ins.2017.01.028
  » https://doi.org/10.1016/j.ins.2017.01.028
• Lin Y, Wang C, Ma C, Dou Z, Ma X (2016) A new combination method for multisensor conflict information. The Journal of Supercomputing 72(7):2874-2890. https://doi.org/10.1007/s11227-016-1681-3
  » https://doi.org/10.1007/s11227-016-1681-3
• Liu WR (2006) Analyzing the degree of conflict among belief functions. Artificial Intelligence 170(11): 909-924. https://doi.org/10.1016/j.artint.2006.05.002
  » https://doi.org/10.1016/j.artint.2006.05.002
• Liu F, Gao X, Zhao J, Deng Y (2019) Generalized Belief Entropy and Its Application in Identifying Conflict Evidence. IEEE Access 7: 126625-126633. https://doi.org/10.1109/access.2019.2939332
  » https://doi.org/10.1109/access.2019.2939332
• Murphy CK (2000) Combining belief functions when evidence conflicts. Decision Support Systems 29(1):1-9. https://doi.org/10.1016/S0167-9236(99)00084-6
  » https://doi.org/10.1016/S0167-9236(99)00084-6
• Sarabi-Jamab A, Araabi BN (2018) How to decide when the sources of evidence are unreliable: A multi-criteria discounting approach in the dempster-shafer theory. Information Sciences 448-449: 233-248. https://doi.org/10.1016/j.ins.2018.03.001
  » https://doi.org/10.1016/j.ins.2018.03.001
• Shafer G(1976) A mathematical theory of evidence. Princeton/NJ: Princeton University Press.
• Smets P (1990) The combination of evidence in the transferable belief model. IEEE Transactions on Pattern Analysis and Machine Intelligence 12(5):447-458. https://doi.org/10.1109/34.55104
  » https://doi.org/10.1109/34.55104
• Smets P (2007) Analyzing the combination of conflicting belief functions. Information Fusion 8(4):387-412. https://doi.org/10.1016/j.inffus.2006.04.003
  » https://doi.org/10.1016/j.inffus.2006.04.003
• Song Y, Deng Y. (2019a) A new method to measure the divergence in evidential sensor data fusion. Int J Distribut Sensor Netw 15(4):1‐8. https://doi.org/10.1177/1550147719841295
  » https://doi.org/10.1177/1550147719841295
• Song Y, Deng Y (2019b) Divergence measure of belief function and its application in data fusion. IEEE Access 7(1):107465‐107472. https://doi.org/10.1109/access.2019.2932390
  » https://doi.org/10.1109/access.2019.2932390
• Sun G D, Guan X, Yi X, Zhao J (2018) Conflict evidence measurement based on the weighted separate union kernel correlation coefficient. IEEE Access 6:30458‐30472. https://doi.org/10.1109/access.2018.2844201
  » https://doi.org/10.1109/access.2018.2844201
• Tessem B (1993) Approximations for efficient computation in the theory of evidence. Artifificial Intelligence 61 (2):315-329. https://doi.org/10.1016/0004-3702(93)90072-J
  » https://doi.org/10.1016/0004-3702(93)90072-J
• Xiao F, Qin B (2018) A Weighted Combination Method for Conflicting Evidence in Multi-Sensor Data Fusion. Sensors 18(5):1487. https://doi.org/10.3390/s18051487
  » https://doi.org/10.3390/s18051487
• Xiao F (2019) Multi-sensor data fusion based on the belief divergence measure of evidences and the belief entropy. Information Fusion, 46: 23-32. https://doi.org/10.1016/j.inffus.2018.04.003
  » https://doi.org/10.1016/j.inffus.2018.04.003
• Xiao F (2019) EFMCDM: Evidential fuzzy multicriteria decision making based on belief entropy. IEEE Transactions on Fuzzy Systems. https://doi.org/10.1109/TFUZZ.2019.2936368
  » https://doi.org/10.1109/TFUZZ.2019.2936368
• Yager R R (1987) On the dempster-shafer framework and new combination rules. Information Sciences 41(2):93-137. https://doi.org/10.1016/0020-0255(87)90007-7
  » https://doi.org/10.1016/0020-0255(87)90007-7
• Yuan K, Xiao F, Fei L, Kang B, Deng Y (2016) Conflict management based on belief function entropy in sensor fusion. SpringerPlus 5(1):2-12. https://doi.org/10.1186/s40064-016-2205-6
  » https://doi.org/10.1186/s40064-016-2205-6
• Yu C, Yang J H, YANG D B, Ma X H (2015) An improved conflicting evidence combination approach based on a new supporting probability distance. Expert Systems with Applications, 42(12):5139-5149. https://doi.org/10.1016/j.eswa.2015.02.038
  » https://doi.org/10.1016/j.eswa.2015.02.038
• Zadeh L A (1986) A simple view of the dempster-shafer theory of evidence and its implication for the rule of combination. Ai Magazine 7(2):85-90. https://doi.org/10.1609/aimag.v7i2.542
  » https://doi.org/10.1609/aimag.v7i2.542
• Zhang H, Deng Y (2019) Weighted belief function of sensor data fusion in engine fault diagnosis. Soft Computing 24(3): 2329-2339. https://doi.org/10.1007/s00500-019-04063-7
  » https://doi.org/10.1007/s00500-019-04063-7
• Zhang Q, Li M, Deng Y (2017) Measure the structure similarity of nodes in complex networks based on relative entropy. Physica A: Statistical Mechanics and its Applications 491:749-763. https://doi.org/10.1016/j.physa.2017.09.042
  » https://doi.org/10.1016/j.physa.2017.09.042
• Zhang Z, Liu T, Chen D, Zhang W (2014) Novel algorithm for identifying and fusing conflicting data in wireless sensor networks. Sensors 14(6):9562-9581. https://doi.org/10.3390/s140609562
  » https://doi.org/10.3390/s140609562
• Wang Y, Zhang K, Deng Y (2018) Base belief function: an efficient method of conflict management. Journal of Ambient Intelligence and Humanized Computing 10(9):3427-3437. https://doi.org/10.1007/s12652-018-1099-2
  » https://doi.org/10.1007/s12652-018-1099-2

Edited by

• Section editor: Alison Moraes

Publication Dates

• Publication in this collection
  10 Aug 2020
• Date of issue
  2020

History

• Received
  03 Jan 2020
• Accepted
  06 May 2020