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Abstract

Evidence theory is widely used in real applications such as target recognition because of its efficiency in evidential sensor data fusing. However, counter-intuitive results may be obtained in the situation when evidence highly conflicts with each other. Recent researches show that weighting the evidences with the consideration of its corresponding credibility is an efficient methodology. As a result, how to determine the weight is an important issue. In this article, a new divergence measure of BPA is proposed based on geometric mean of Deng relative entropy. The weight of each evidence is determined by the proposed divergence measure and information volume. Compared with the existing belief Jensen–Shannon divergence, the proposed method has a better performance in the convergence to the correct target. The result shows that the proposed method outperforms other related methods, giving the highest belief value 98.98% to the correct target.

Keywords

Dempster–Shafer evidence theory belief function divergence measure relative belief entropy Deng entropy target recognition

Introduction

Multi-sensor data fusion technology is widely used in real world,^1,2 playing an important part in multi-sensor data fusing,^3–5 decision-making,^6,7 fault diagnosis,^8–11 risk and reliability analyzing,^12–16 failure mode and effective analyses,^12,17 and pattern recognition.^18,19

Due to the invertible influence from the bad weather or sensor fault, uncertain and inaccurate information may emerge and then cause an incorrect conclusion in decision-making.²⁰ To solve this problem, a great number of theories such as evidence theory, fuzzy set theory,^21–25 D number,^26–29 rough sets,³⁰ and some improved methods had been proposed.^31,32 Dempster–Shafer evidence theory,³³ the basic theory in information fusion, is considered as an extension of Bayesian theory for an incomplete model without prior probabilities.

However, the counterintuitive results may be generated when fusing the highly conflicting evidences.³⁴ It is still an open issue about how to model and handle imprecise and uncertain information.^35,36 Some modified and improved method are proposed^26,31,37 to solve the problem of conflicting and inaccurate data fusing in real applications.³⁸ Scholars such as Murphy,³⁹ Zhang et al.,⁴⁰ Wang and Deng,⁴¹ Deng et al.,⁴² and Xiao^43,44 have done a lot of researches in this field, and many of them believe that an effective way to solve this problem is to give different weights to different evidences by considering the credibility of different evidences other than revising the combining rule. Larger credibility shows more supports, and this evidence should be more important for the final decision.

How to determine the weight of each evidence is still an open issue.⁴⁵ An average approach is first proposed by Murphy³⁹ and then modified by weighted average.⁴⁶ If all the evidence is available at the same time, one can average the masses and calculate the combined masses by combining the average values multiple times. Recently, a new method for conflict management based on belief Jensen–Shannon divergence (BJS) is proposed⁴³ making use of the divergence among evidences and the uncertain degree⁴⁴ of evidence. In this article, the BJS divergence is proved as the arithmetic mean of two Deng relative entropies. Then, a new divergence measure of BPA is proposed based on geometric mean of Deng relative entropy. The proposed divergence measure, combined with the information volume, is used to determine the weight of each evidence. Finally, a new weighted evidence combination algorithm is developed. The application in target recognition is illustrated to show the efficiency of the proposed method.

The remainder of this article is organized as follows. Some preliminaries of D-S theory and Deng relative entropy are mentioned in section “Preliminaries.” A new method named belief relative entropy and the method to determine the weighted BPA is proposed in section “A novel divergence measure.” In section “Target recognition,” a case study in target recognition is used to show the effectiveness of this new method. Conclusions are given in the final part.

Preliminaries

In this section, Dempster–Shafer evidence theory, Deng entropy, and belief relative entropy are introduced.

Dempster–Shafer evidence theory

Dempster–Shafer evidence theory, as the extension of the Bayesian theory, is applied to deal with uncertain information with weaker conditions and greatly express the uncertain degree as well as combine pieces of evidence to obtain a more convincing evidence. In this section, some basic concepts and functions^33,47 are introduced as follows.

Definition 1

$Θ$ is the set of N elements which represent mutually exclusive and exhaustive hypotheses. Let $Θ$ be the frame of discernment^33,47

Θ = {H_{1}, H_{2}, \dots, H_{i}, \dots, H_{N}}

(1)

The power set of $Θ$ is denoted by $2^{Θ}$ , and

\begin{matrix} 2^{Θ} = & {\emptyset, {H_{1}}, \dots, {H_{N}}, {H_{1}, H_{2}}, \dots, \\ {H_{1}, H_{2}, \dots, H_{i}}, \dots, Θ} \end{matrix}

(2)

where $\emptyset$ is an empty set.

Definition 2

A BPA function m is a mapping of $2^{Θ}$ to a probability interval $[0, 1]$ , formally defined by Dempster³³ and Shafer⁴⁷

m : 2^{Θ} \to [0, 1]

(3)

which satisfies the following conditions

m (\emptyset) = 0 \sum_{A \in 2^{Θ}} m (A) = 1 0 \leq m (A) \leq 1 A \in 2^{Θ}

(4)

The mass $m (A)$ represents how strongly the evidence supports A.

For the same evidence, the different $BPA s$ come from the different evidence resources. Dempster’s combination rule can be used to obtain the combined evidence. Suppose $m_{1}$ and $m_{1}$ are two mass functions, the combining result is defined as $m = m_{1} \oplus m_{2}$ ³³

{\begin{matrix} m (\emptyset) = 0 \\ m (A) = \frac{\sum_{B \cap C = A} m_{1} (B) m_{2} (C)}{1 - K} \end{matrix}

(5)

where $K = \sum_{B \cap C = \emptyset} m_{1} (B) m_{2} (C)$ .

It is markable that K is the coefficient to measure the conflict between evidence in evidence theory, and the combination rules could not be used when $K > 1$ .

Belief entropy

Entropy is widely used in many applications.^48–51 How to determine the entropy of a BPA is an hot issue.^44,52–55 A novel belief entropy which is called Deng entropy^56,57 is proposed to measure the uncertain information.⁵⁸ As the generalization of the Shannon entropy,⁵⁹ Deng relative entropy is an efficient method to measure the uncertain information.⁶⁰ When the belief value is allocated in the single element, Deng entropy degenerates to Shannon entropy. Given a BPA, its corresponding Deng entropy is defined as follows

E_{d} = - \sum_{i} m (A_{i}) \log \frac{m (A_{i})}{2^{| A_{i} |} - 1}

(6)

In this article, the information volume⁴³ making use of Deng entropy is defined as follows

I V_{i} = e^{E_{d}}

(7)

The greater Deng entropy of an evidence is, more information is contained and larger the information volume is. A large information volume of a body of evidence indicates that the evidence plays an important role for the final combining result.

Deng relative entropy

A new relative entropy named as Deng relative entropy^61,62 is proposed in order to measure the divergence between different basic probability assignments (BPAs). Deng relative entropy is defined as follows⁶¹

D_{d} (m_{1} ∥ m_{2}) = \sum_{i} m_{1} (F_{i}) \log \frac{m_{1} (F_{i})}{m_{2} (F_{i})}

(8)

The Deng relative entropy is the generalization of Kullback–Leibler divergence.⁶³ When the BPA is degenerated as probability, Deng relative entropy is equal to Kullback–Leibler divergence. It is the average of logarithmic difference between mass function $m_{1}$ and $m_{2}$ .

An existing measure of BPA divergence

In Dempster–Shafer evidence theory, how to measure the discrepancy among bodies of evidences is still an open issue, thus different methods for divergence measure are proposed. An existing method proposed by Xiao⁴³ combines JS divergence with Dempster’s theories to get the divergence measure among different evidences. The novel method is named as BJS, the JS divergence Shannon entropy which is derived from the formation of BPA. BJS⁴³ is defined as follows

BJS (m_{1}, m_{2}) = H (\frac{m_{1} + m_{2}}{2}) - \frac{1}{2} H (m_{1}) - \frac{1}{2} H (m_{2})

where

{\begin{matrix} H (m_{j}) = - \sum_{i} m_{j} (A_{i}) \log m_{i} (A_{i}) \\ i = 1, 2, \dots, M; j = 1, 2 \end{matrix}

(9)

$H (m_{j})$ is the Shannon entropy. BJS is used to measure the discrepancy and conflict degree between bodied of evidences.

A novel divergence measure

There are three main works described in this section. First, the relationship between the BJS divergence in Xiao⁴³ and the arithmetic mean of Deng relative entropy is derived. Second, a new divergence measure of BPA is presented, based on geometry mean of Deng relative entropy. Third, a new weighted combination model is proposed.

The arithmetic mean representation of BJS

The definition of BJS⁴³ can be derived as follows

Given two BPAs m₁ and m₂ in the same frame of discernment, the BJS divergence between them is defined as

\begin{matrix} BJS (m_{1}, m_{2}) & = \frac{1}{2} [H (\frac{m_{1} + m_{2}}{2}) - \frac{1}{2} H (m_{1}) - \frac{1}{2} H (m_{2})] \\ = \frac{1}{2} [\sum_{i} m_{1} (A_{i}) \log (\frac{2 m_{1} (A_{i})}{m_{1} (A_{i}) + m_{2} (A_{i})}) \\ + \sum_{i} m_{2} (A_{i}) \log (\frac{2 m_{2} (A_{i})}{m_{1} (A_{i}) + m_{2} (A_{i})})] \\ = \frac{1}{2} [D_{d} (m_{1} ∥ \frac{m_{1} + m_{2}}{2}) + D_{d} (m_{2} ∥ \frac{m_{1} + m_{2}}{2})] \end{matrix}

(10)

where $D_{d} (m_{1} | | (m_{2} + m_{1}) / 2)$ is the Deng relative entropy between $m_{1}$ and $(m_{2} + m_{1}) / 2$ (the arithmetic mean between $m_{1}$ and $m_{2}$ ). As a result, the BJS is the arithmetic mean of $D_{d} (m_{1} | | (m_{2} + m_{1}) / 2)$ and $D_{d} (m_{2} | | (m_{2} + m_{1}) / 2)$ .

New divergence measure of BPA

Inspired by the arithmetic mean of Deng relative entropy shown as BJS,⁴³ a geometrical mean of Deng relative entropy is proposed as follows.

Definition 3

Given two BPAs m₁ and m₂ in the same frame of discernment, the BRE divergence between them is defined as

\begin{matrix} BRE (m_{1}, m_{2}) = \\ \sqrt{\sum_{i} m_{1} (A_{i}) \log \frac{m_{1} (A_{i})}{\sqrt{m_{1} (A_{i}) m_{2} (A_{i})}} \sum_{i} m_{2} (A_{i}) \log \frac{m_{2} (A_{i})}{\sqrt{m_{1} (A_{i}) m_{2} (A_{i})}}} \end{matrix}

(11)

BRE divergence can be represented as the geometrical mean between $D_{d} (m_{1} ∥ \sqrt{m_{1} m_{2}})$ and $D_{d} (m_{2} ∥ \sqrt{m_{1} m_{2}})$ . Here, we give some examples and properties of the proposed divergence measure. Assuming a complete frame of discernment is $Θ = {A, B, C}$ and some $BPA s$ supporting by data resources are given as follows.

Example 3.1

\begin{matrix} m_{1} (A) = 0.5, m_{1} (B) = 0.4, m_{1} (A, B, C) = 0.1 \\ m_{2} (A) = 0.6, m_{2} (B) = 0.3, m_{2} (A, B, C) = 0.1 \\ BRE (m_{1}, m_{2}) = BRE (m_{2}, m_{1}) = 0.0117 \end{matrix}

From Example 3.1, we could know that the BRE between $m_{1}$ and $m_{2}$ is equal to that between $m_{2}$ and $m_{1}$ . So, $BRE (m_{1}, m_{2})$ is symmetric and always well defined.

Example 3.2

\begin{matrix} m_{1} (A) = 0.6, m_{1} (B) = 0.1, m_{1} (A, B, C) = 0.3 \\ m_{2} (A) = 0.6, m_{2} (B) = 0.1, m_{2} (A, B, C) = 0.3 \\ BRE (m_{1}, m_{2}) = 0.0000 \end{matrix}

From Example 3.2, for the same BPA, its corresponding BRE is zero.

Example 3.3

\begin{matrix} m_{1} (A) = 0.5, m_{1} (B) = 0.5 \\ m_{2} (A) = x, m_{2} (B) = 1 - x \end{matrix}

As the parameter x varies from 0 to 0.5, the BRE between $m_{1}$ and $m_{2}$ is shown in Figure 1.

Figure 1.

BRE divergence with changing parameter x.

It is clear that when x increases from 0 to 0.5, BRE decreases gradually, dropping to 0 at $x = 0.5$ .

Weighted combination model

The flow chart of the proposed weighted combination model is illustrated and the detailed steps are shown in Figure 2 as follows.

Figure 2.

The weighted combination model.

Calculate the divergence measure matrix

Use equation (11) to calculate the divergence measure matrix $(DMM)$ as follows

DMM = (\begin{matrix} 0 \dots & BR E_{1 i} \dots & BR E_{1 k} \\ BR E_{i 1} \dots & 0 \dots & BR E_{ik} \\ BR E_{k 1} \dots & BR E_{ki} \dots & 0 \end{matrix})

(12)

where $BR E_{ki}$ means the divergence of $m_{k}$ and $m_{i}$ .

Calculate the supporting degree of evidence

Calculate the supporting degree of the body of evidence of $m_{i}$ which is defined as follows

Su p_{i} = \frac{1}{\frac{\sum_{j = 1, j \neq i}^{k} BR E_{ij}}{k - 1}}

(13)

Calculate the information volume of each evidence

I V_{i} = e^{E_{d}} = e^{\sum_{i} m (A_{i}) \log \frac{m (A_{i})}{2^{| A_{i} |} - 1}} 1 \leq i \leq k

(14)

where $E_{d}$ is the Deng entropy of the evidence, which is calculated by equation (6).

Obtain the credibility of each evidence

Given an evidence $m_{i}$ , multiple the supporting degree $Su p_{i}$ and the information volume $I V_{i}$ to obtain the credibility of each evidence as follows

Cr d_{i} = Su p_{i} \times I V_{i}

(15)

Obtain the final weight of each evidence

Normalize the credibility $Cr d_{i}$ to obtain the final weight of each evidence ${\tilde{W}}_{i}$

{\tilde{W}}_{i} = \frac{Cr d_{i}}{\sum_{k} Cr d_{i}}

(16)

Obtain the weighted average evidence and combine them

Weight the collected N pieces of evidence to obtain the weighted average evidence. Then, use Dempster rules to combine the weighted evidence $N - 1$ times

M (A_{j}) = \sum_{i} m_{i} (A_{j}) • {\tilde{W}}_{i}

(17)

where $m_{i} (A_{j})$ is the BPA from sensor i about the object $A_{j}$ .

Target recognition

Evidence theory is widely used in real life in the data fusing process.⁶⁴ In this section, an application based on multi-sensor data in Zhang et al.⁴⁰ is given to show the efficiency of the proposed method.

Assume that there are five sensors (evidence resources) in a target recognition system sending $BPA s$ from its collected information. BPAs are given in Table 1, and the frame of discernment is established as $Θ = {A, B, C}$ .

Table 1.

The BPAs for a multi-sensor-based target recognition from Zhang et al.⁴⁰

BPA	{A}	{B}	{C}	{A, C}
$S_{1} : m_{1} (•)$	0.41	0.29	0.30	0.00
$S_{2} : m_{2} (•)$	0.00	0.90	0.10	0.00
$S_{3} : m_{3} (•)$	0.58	0.07	0.00	0.35
$S_{1} : m_{4} (•)$	0.55	0.10	0.00	0.35
$S_{5} : m_{5} (•)$	0.60	0.10	0.00	0.30

The calculating steps

The steps of the proposed method are detailed as follows.

Step 1. Use equation (11) to obtain BRE divergence measure among pieces of evidences and then built the DMM

DMM = [\begin{matrix} 0 1.5788 4.3904 4.3577 4.0327 \\ 1.5788 0 5.5650 5.2836 5.2744 \\ 4.3904 5.5650 0 0.0031 0.0048 \\ 4.3577 5.2836 0.0031 0 0.0030 \\ 4.0327 5.2744 0.0048 0.0030 0 \end{matrix}]

Step 2. Calculate the supporting degree of the body of evidence of $m_{i}$ from the DMM above. If a body of evidence is supported by other bodies’ evidence greatly, its supporting degree is high and this evidence has more effect on the final combination results

Su p_{1} = \frac{1}{\frac{\sum_{j = 1, j \neq i}^{5} BR E_{1 j}}{5}} = 0.2786

Su p_{2} = \frac{1}{\frac{\sum_{j = 1, j \neq i}^{5} BR E_{2 j}}{5}} = 0.2260

Su p_{3} = \frac{1}{\frac{\sum_{j = 1, j \neq i}^{5} BR E_{3 j}}{5}} = 0.4015

Su p_{4} = \frac{1}{\frac{\sum_{j = 1, j \neq i}^{5} BR E_{4 j}}{5}} = 0.4146

Su p_{5} = \frac{1}{\frac{\sum_{j = 1, j \neq i}^{5} BR E_{5 j}}{5}} = 0.4294

Step 3. Calculate the information volume of each evidence based on equation (7) as follows

\begin{matrix} I V_{1} = e^{E_{d 1}} = 4.7894 \\ I V_{2} = e^{E_{d 2}} = 1.5984 \\ I V_{3} = e^{E_{d 3}} = 6.1056 \\ I V_{4} = e^{E_{d 4}} = 6.6286 \\ I V_{5} = e^{E_{d 5}} = 5.8767 \end{matrix}

Step 4. Given an evidence $m_{i}$ , multiple the supporting degree $Su p_{i}$ , and the information volume $I V_{i}$ to obtain the credibility of each evidence. If a body of evidence is supported by other bodies’ evidence greatly, its credibility degree is high and this evidence has more effect on the final combination results. The average weight for each evidence is calculated as follows

\begin{matrix} Cr d_{1} = Su p_{1} • I V_{1} = 1.3341 \\ Cr d_{2} = Su p_{2} • I V_{2} = 0.3612 \\ Cr d_{3} = Su p_{3} • I V_{3} = 2.4512 \\ Cr d_{4} = Su p_{4} • I V_{4} = 2.7483 \\ Cr d_{5} = Su p_{5} • I V_{5} = 2.5236 \end{matrix}

Step 5. Normalize the credibility $Cr d_{i}$ to obtain the final weight of each evidence ${\tilde{W}}_{i}$

\begin{matrix} {\tilde{W}}_{1} = \frac{Cr d_{1}}{\sum_{i} Cr d_{i}} = 0.1416 \\ {\tilde{W}}_{2} = \frac{Cr d_{2}}{\sum_{i} Cr d_{i}} = 0.0383 \\ {\tilde{W}}_{3} = \frac{Cr d_{3}}{\sum_{i} Cr d_{i}} = 0.2603 \\ {\tilde{W}}_{4} = \frac{Cr d_{4}}{\sum_{i} Cr d_{i}} = 0.2918 \\ {\tilde{W}}_{5} = \frac{Cr d_{5}}{\sum_{i} Cr d_{i}} = 0.2679 \end{matrix}

Step 6. Obtain the final average weight for each BPA based on equation (17) and weight of the collected N pieces of evidence using Dempster combination results. The average weight for each evidence is calculated as follows

\begin{matrix} M (A_{1}) = \sum_{i} m_{i} (A_{1}) • {\tilde{W}}_{i} = 0.5303 \\ M (A_{2}) = \sum_{i} m_{i} (A_{2}) • {\tilde{W}}_{i} = 0.1498 \\ M (A_{3}) = \sum_{i} m_{i} (A_{3}) • {\tilde{W}}_{i} = 0.0463 \\ M (A_{4}) = \sum_{i} m_{i} (A_{4}) • {\tilde{W}}_{i} = 0.2736 \end{matrix}

If there are N number of evidences, $A_{i} (i = 1, 2, \dots, N)$ would be fused by $N - 1$ times. Finally, the combining multi-sensor result of the system is obtained.

Result analysis and discussion

Comparing the results shown in Table 2, there are two remarkable points. First, applying Dempster combination rule recognizes the object C as the target, while other methods proposed identify A as the target. Second, the proposed method has the highest confidence to target A, showing the effectiveness of the proposed method in real applications. The method proposed by Murphy,³⁹ Zhang et al.,⁴⁰ and Xiao⁴³ obtain the same results and the belief degree of them are not less than 96.20%, showing great improvement to Dempster’s method. Compared with the method based on the arithmetic mean of Deng relative entropy,⁴³ improvement of the presented method is shown in the accuracy of target recognition. The new method has the most accurate rate in target recognition, giving the highest belief degree of 98.98%. Compared to the belief value given by Xiao,⁴³ it seems that little improvement of 0.04% is demonstrated by the proposed method. However, this improvement is significant for this example and applications in the real world since the accuracy is quite high to nearly 99%, and other methods also advance in the same degree of precision.

Table 2.

Results of BPA and target in different methods.

Method	{A}	{B}	{C}	{A, C}	Target
Dempster³³	0	0.1422	0.8578	0	C
Murphy³⁹	0.9620	0.0210	0.0138	0.0032	A
Zhang et al.⁴⁰	0.9820	0.0034	0.0115	0.0032	A
Xiao⁴³	0.9894	0.0002	0.0061	0.0043	A
Proposed method	0.9898	0.0002	0.0055	0.0045	A

Conclusion

In this article, there are two main distributions. First, the relationship between an existing divergence measure BJS and Deng relative entropy is demonstrated, and second, a new divergence measure BRE and the evidence combining the model making use of it is proposed. Inspired by the arithmetic mean of Deng relative entropy, namely, BJS, a geometrical mean of Deng relative entropy BRE divergence is presented. BRE divergence can be used to measure the supporting degree of evidences and then to determine the average weight of evidences. Furthermore, making use of BRE divergence and information volume which represents the uncertain degree of evidence, a new method is proposed to determine evidence weights, which is effective and feasible to handle the conflicting evidences. The case study in target recognition demonstrates the improvement of the accuracy of a target recognition system. The proposed divergence measure has the promising aspect in uncertain information processing, especially in evidential systems based on evidence theory.

Footnotes

Acknowledgements

The authors greatly appreciate the reviews’ suggestions and the editor’s encouragement.

Handling Editor: Daming Zhou

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work is partially supported by the National Natural Science Foundation of China (Grant Nos 61573290 and 61503237).

ORCID iD

Yong Deng

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A new method to measure the divergence in evidential sensor data fusion

Abstract

Keywords

Introduction

Preliminaries

Dempster–Shafer evidence theory

Definition 1

Definition 2

Belief entropy

Deng relative entropy

An existing measure of BPA divergence

A novel divergence measure

The arithmetic mean representation of BJS

New divergence measure of BPA

Definition 3

Example 3.1

Example 3.2

Example 3.3

Weighted combination model

Calculate the divergence measure matrix

Calculate the supporting degree of evidence

Calculate the information volume of each evidence

Obtain the credibility of each evidence

Obtain the final weight of each evidence

Obtain the weighted average evidence and combine them

Target recognition

The calculating steps

Result analysis and discussion

Conclusion

Footnotes

Acknowledgements

Declaration of conflicting interests

Funding

ORCID iD

References