# An Improved Multi-Source Data Fusion Method Based on the Belief Entropy and Divergence Measure

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

#### 2.1. Dempster–Shafer Evidence Theory

**Definition 1**(Frame of discernment)

**.**

**Definition 2**(Basic probability assignment)

**.**

**Definition 3**(Belief function)

**.**

**Definition 4**(Plausibility function)

**.**

**Definition 5**(DS combination rule)

**.**

#### 2.2. Belief Entropy

**Example**

**1.**

#### 2.3. Belief Jenson–Shannon Divergence

## 3. The Proposed Method

#### 3.1. Calculate the Credibility Weight of Evidences

**Step 1-1.**Suppose $M=\{{m}_{1},{m}_{2},\dots ,{m}_{n}\}$ is a set of n independent BPAs on the same frame of discernment that contains N elements: $\mathsf{\Theta}=\{{F}_{1},{F}_{2},{F}_{3},\dots ,{F}_{N}\}$. The arithmetical average BPA ${m}_{a}$ is defined as:

**Step 1-2.**Calculate the BJS divergence between ${m}_{i}$ and ${m}_{a}$$(i=1,2,3,\dots ,n)$ according to Equation (9).

**Step 1-3.**Since the similarities of the pieces of evidence are negatively correlated with their divergences, if the divergence between two pieces of evidence is higher, they have lower similarity. The divergence between ${m}_{i}$ and ${m}_{a}$ is converted into their similarity as follows:

**Step 1-4.**If a piece of evidence is highly similar to the average BPA, it means that the evidence is supported by most of the other pieces of evidence and it is more reliable, thus it gains high credibility. Thus, the credibility weight of the pieces of evidence is determined by normalizing their similarity with the arithmetical average BPA. The credibility weight (${W}_{c}$) of each piece of evidence is defined as follows:

#### 3.2. Calculate Information Volume Weight of Evidence

**Step 2-1.**Calculate the belief entropy of ${m}_{i}$ ($i=1,2,3,\dots ,n$) according to Equation (8).

**Step 2-2.**To avoid assigning zero weight to the evidence whose belief entropy is zero, the information volume in Step 2-1 is modified as follows:

**Step 2-3.**Calculate the information volume weight (${W}_{iv}$) of each piece of evidence by normalizing $IV$, which is defined as:

#### 3.3. Generate the Modified Evidence and Fuse

**Step 3-1.**Based on the credibility weight and information volume weight of evidence, the weight of each piece of evidence is adjusted as follows:

**Step 3-2.**Normalize the modified weight as follows:

**Step 3-3.**Generate the modified evidence by calculating the weighted average sum of BPAs, which is defined as:

**Step 3-4.**DS combination rule is used $n-1$ times on the modified evidence based on [52], then the final combination result is obtained:

## 4. Numerical Example

#### 4.1. Example Presentation

#### 4.2. Combination by the Proposed Method

**Step 1-1.**Calculate the arithmetical average BPA.

**Step 1-2.**Calculate the BJS divergence between each piece of evidence and ${m}_{a}$.

**Step 1-3.**Calculate the similarity degree of each piece of evidence.

**Step 1-4.**Calculate the weight of credibility.

**Step 2-1.**Calculate the belief entropy of each piece of evidence.

**Step 2-2.**Adjust the information volume of each piece of evidence.

**Step 2-3.**Calculate the weight of information volume.

**Step 3-1.**Adjust the weight of each piece of evidence.

**Step 3-2.**Normalize the modified weight of each piece of evidence.

**Step 3-3.**Generate the modified evidence.

**Step 3-4.**Use the classical DS combination rule to fuse the modified evidence four times, and the result is shown in Table 2.

#### 4.3. Analysis

## 5. Application

#### 5.1. Problem Statement

#### 5.2. Fuse Evidences by the Proposed Method

**Step 1-1.**Calculate the arithmetical average BPA.

**Step 1-2.**Calculate the BJS divergence between each piece of evidence and ${m}_{a}$.

**Step 1-3.**Calculate the similarity degree of each piece of evidence.

**Step 1-4.**Calculate the weight of credibility.

**Step 2-1.**Calculate the belief entropy of each piece of evidence.

**Step 2-2.**Adjust the information volume of each piece of evidence.

**Step 2-3.**Calculate the weight of information volume.

**Step 3-1.**Adjust the weight of each piece of evidence.

**Step 3-2.**Normalize the modified weight of each piece of evidence.

**Step 3-3.**Generate the modified evidence.

**Step 3-4.**Use the classical DS combination rule to fuse the modified evidence four times, and the result is shown in Table 4.

#### 5.3. Discussion

## 6. Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## References

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**Table 1.**A numerical example in [55].

$\left\{\mathit{A}\right\}$ | $\left\{\mathit{B}\right\}$ | $\left\{\mathit{C}\right\}$ | $\{\mathit{A},\mathit{C}\}$ | |
---|---|---|---|---|

${m}_{1}$ | 0.41 | 0.29 | 0.30 | 0 |

${m}_{2}$ | 0 | 0.90 | 0.10 | 0 |

${m}_{3}$ | 0.58 | 0.07 | 0 | 0.35 |

${m}_{4}$ | 0.55 | 0.10 | 0 | 0.35 |

${m}_{5}$ | 0.60 | 0.10 | 0 | 0.30 |

Method | ${\mathit{m}}_{1,2}$ | ${\mathit{m}}_{1,2,3}$ | ${\mathit{m}}_{1,2,3,4}$ | ${\mathit{m}}_{1,2,3,4,5}$ |
---|---|---|---|---|

DS [38] | $m(A)=0.0000$ $m(B)=0.8969$ $m(C)=0.1031$ | $m(A)=0.0000$ $m(B)=0.6575$ $m(C)=0.3425$ | $m(A)=0.0000$ $m(B)=0.3321$ $m(C)=0.6679$ | $m(A)=0.0000$ $m(B)=0.1422$ $m(C)=0.8578$ |

Yager [47] | $m(A)=0.0000$ $m(B)=0.2610$ $m(C)=0.0300$ $m(A,C)=0.0000$ $m(\mathsf{\Theta})=0.7090$ | $m(A)=0.4112$ $m(B)=0.0679$ $m(C)=0.0105$ $m(A,C)=0.2481$ $m(\mathsf{\Theta})=0.2622$ | $m(A)=0.6508$ $m(B)=0.0330$ $m(C)=0.0037$ $m(A,C)=0.1786$ $m(\mathsf{\Theta})=0.1339$ | $m(A)=0.7732$ $m(B)=0.0167$ $m(C)=0.0011$ $m(A,C)=0.0938$ $m(\mathsf{\Theta})=0.1152$ |

Murphy [52] | $m(A)=0.0964$ $m(B)=0.8119$ $m(C)=0.0917$ $m(A,C)=0.0000$ | $m(A)=0.4619$ $m(B)=0.4497$ $m(C)=0.0794$ $m(A,C)=0.0090$ | $m(A)=0.8362$ $m(B)=0.1147$ $m(C)=0.0410$ $m(A,C)=0.0081$ | $m(A)=0.9620$ $m(B)=0.0210$ $m(C)=0.0138$ $m(A,C)=0.0032$ |

Deng et al. [53] | $m(A)=0.0964$ $m(B)=0.8119$ $m(C)=0.0917$ $m(A,C)=0.0000$ | $m(A)=0.4974$ $m(B)=0.4054$ $m(C)=0.0888$ $m(A,C)=0.0084$ | $m(A)=0.9089$ $m(B)=0.0444$ $m(C)=0.0379$ $m(A,C)=0.0089$ | $m(A)=0.9820$ $m(B)=0.0039$ $m(C)=0.0107$ $m(A,C)=0.0034$ |

Sun et al. [50] | $m(A)=0.0937$ $m(B)=0.5330$ $m(C)=0.1214$ $m(A,C)=0.0000$ $m(\mathsf{\Theta})=0.2519$ | $m(A)=0.4544$ $m(B)=0.2639$ $m(C)=0.1107$ $m(A,C)=0.1124$ $m(\mathsf{\Theta})=0.0585$ | $m(A)=0.6907$ $m(B)=0.1324$ $m(C)=0.0636$ $m(A,C)=0.1088$ $m(\mathsf{\Theta})=0.0045$ | $m(A)=0.8202$ $m(B)=0.0711$ $m(C)=0.0309$ $m(A,C)=0.0760$ $m(\mathsf{\Theta})=0.0018$ |

Li et al. [51] | $m(A)=0.1453$ $m(B)=0.6829$ $m(C)=0.1718$ $m(A,C)=0.0000$ | $m(A)=0.4711$ $m(B)=0.3007$ $m(C)=0.1266$ $m(A,C)=0.1016$ | $m(A)=0.6925$ $m(B)=0.1350$ $m(C)=0.0650$ $m(A,C)=0.1074$ | $m(A)=0.8210$ $m(B)=0.0719$ $m(C)=0.0314$ $m(A,C)=0.0757$ |

Li and Guo [89] | $m(A)=0.1453$ $m(B)=0.6829$ $m(C)=0.1718$ $m(A,C)=0.0000$ | $m(A)=0.4801$ $m(B)=0.2821$ $m(C)=0.1512$ $m(A,C)=0.0866$ | $m(A)=0.7480$ $m(B)=0.0850$ $m(C)=0.0630$ $m(A,C)=0.1040$ | $m(A)=0.8558$ $m(B)=0.0425$ $m(C)=0.0267$ $m(A,C)=0.0750$ |

Jiang et al. [54] | $m(A)=0.2849$ $m(B)=0.5306$ $m(C)=0.1845$ $m(A,C)=0.0000$ | $m(A)=0.8295$ $m(B)=0.0680$ $m(C)=0.0854$ $m(A,C)=0.0171$ | $m(A)=0.9531$ $m(B)=0.0074$ $m(C)=0.0292$ $m(A,C)=0.0103$ | $m(A)=0.9867$ $m(B)=0.0008$ $m(C)=0.0089$ $m(A,C)=0.0036$ |

Zhang et al. [55] | $m(A)=0.0964$ $m(B)=0.8119$ $m(C)=0.0917$ $m(A,C)=0.0000$ | $m(A)=0.5681$ $m(B)=0.3319$ $m(C)=0.0929$ $m(A,C)=0.0084$ | $m(A)=0.9142$ $m(B)=0.0395$ $m(C)=0.0399$ $m(A,C)=0.0083$ | $m(A)=0.9820$ $m(B)=0.0034$ $m(C)=0.0115$ $m(A,C)=0.0032$ |

Lin et al. [56] | $m(A)=0.0964$ $m(B)=0.8119$ $m(C)=0.0917$ $m(A,C)=0.0000$ | $m(A)=0.5382$ $m(B)=0.3599$ $m(C)=0.0927$ $m(A,C)=0.0076$ | $m(A)=0.8949$ $m(B)=0.0558$ $m(C)=0.0413$ $m(A,C)=0.0080$ | $m(A)=0.9792$ $m(B)=0.0057$ $m(C)=0.0119$ $m(A,C)=0.0032$ |

Proposed method | $m(A)=0.2751$ $m(B)=0.5446$ $m(C)=0.1803$ $m(A,C)=0.0000$ | $m(A)=0.8213$ $m(B)=0.0738$ $m(C)=0.0890$ $m(A,C)=0.0159$ | $m(A)=0.9550$ $m(B)=0.0063$ $m(C)=0.0281$ $m(A,C)=0.0106$ | $m(A)=0.9874$ $m(B)=0.0006$ $m(C)=0.0082$ $m(A,C)=0.0037$ |

${\mathit{F}}_{1}$ | ${\mathit{F}}_{2}$ | ${\mathit{F}}_{3}$ | $\mathsf{\Theta}$ | |
---|---|---|---|---|

${m}_{1}$ | 0.70 | 0.10 | 0 | 0.20 |

${m}_{2}$ | 0.70 | 0 | 0 | 0.30 |

${m}_{3}$ | 0.65 | 0.15 | 0 | 0.20 |

${m}_{4}$ | 0.75 | 0 | 0.05 | 0.20 |

${m}_{5}$ | 0 | 0.20 | 0.80 | 0 |

Method | ${\mathit{m}}_{1,2}$ | ${\mathit{m}}_{1,2,3}$ | ${\mathit{m}}_{1,2,3,4}$ | ${\mathit{m}}_{1,2,3,4,5}$ | Recognized Fault |
---|---|---|---|---|---|

DS [38] | $m({F}_{1})=0.9032$ $m({F}_{2})=0.0323$ $m({F}_{3})=0.0000$ $m(\mathsf{\Theta})=0.0645$ | $m({F}_{1})=0.9598$ $m({F}_{2})=0.0249$ $m({F}_{3})=0.0000$ $m(\mathsf{\Theta})=0.0153$ | $m({F}_{1})=0.9906$ $m({F}_{2})=0.0053$ $m({F}_{3})=0.0008$ $m(\mathsf{\Theta})=0.0033$ | $m({F}_{1})=0.0000$ $m({F}_{2})=0.3443$ $m({F}_{3})=0.6557$ $m(\mathsf{\Theta})=0.0000$ | Unknown |

Yager [47] | $m({F}_{1})=0.8400$ $m({F}_{2})=0.0300$ $m({F}_{3})=0.0000$ $m(\mathsf{\Theta})=0.1300$ | $m({F}_{1})=0.7530$ $m({F}_{2})=0.0195$ $m({F}_{3})=0.0000$ $m(\mathsf{\Theta})=0.2275$ | $m({F}_{1})=0.7243$ $m({F}_{2})=0.0039$ $m({F}_{3})=0.0006$ $m(\mathsf{\Theta})=0.2721$ | $m({F}_{1})=0.0000$ $m({F}_{2})=0.0013$ $m({F}_{3})=0.0024$ $m(\mathsf{\Theta})=0.9963$ | Unknown |

Murphy [52] | $m({F}_{1})=0.9032$ $m({F}_{2})=0.0296$ $m({F}_{3})=0.0000$ $m(\mathsf{\Theta})=0.0672$ | $m({F}_{1})=0.9598$ $m({F}_{2})=0.0241$ $m({F}_{3})=0.0000$ $m(\mathsf{\Theta})=0.0161$ | $m({F}_{1})=0.9899$ $m({F}_{2})=0.0058$ $m({F}_{3})=0.0008$ $m(\mathsf{\Theta})=0.0035$ | $m({F}_{1})=0.9715$ $m({F}_{2})=0.0055$ $m({F}_{3})=0.0222$ $m(\mathsf{\Theta})=0.0008$ | ${F}_{1}$ |

Deng et al. [53] | $m({F}_{1})=0.9032$ $m({F}_{2})=0.0296$ $m({F}_{3})=0.0000$ $m(\mathsf{\Theta})=0.0672$ | $m({F}_{1})=0.9597$ $m({F}_{2})=0.0243$ $m({F}_{3})=0.0000$ $m(\mathsf{\Theta})=0.0160$ | $m({F}_{1})=0.9899$ $m({F}_{2})=0.0058$ $m({F}_{3})=0.0008$ $m(\mathsf{\Theta})=0.0035$ | $m({F}_{1})=0.9933$ $m({F}_{2})=0.0030$ $m({F}_{3})=0.0028$ $m(\mathsf{\Theta})=0.0008$ | ${F}_{1}$ |

Sun et al. [50] | $m({F}_{1})=0.8826$ $m({F}_{2})=0.0330$ $m({F}_{3})=0.0000$ $m(\mathsf{\Theta})=0.0844$ | $m({F}_{1})=0.8789$ $m({F}_{2})=0.0348$ $m({F}_{3})=0.0000$ $m(\mathsf{\Theta})=0.0863$ | $m({F}_{1})=0.8897$ $m({F}_{2})=0.0187$ $m({F}_{3})=0.0036$ $m(\mathsf{\Theta})=0.0880$ | $m({F}_{1})=0.4028$ $m({F}_{2})=0.0660$ $m({F}_{3})=0.1247$ $m(\mathsf{\Theta})=0.4028$ | Unknown |

Li et al. [51] | $m({F}_{1})=0.3364$ $m({F}_{2})=0.1185$ $m({F}_{3})=0.4250$ $m(\mathsf{\Theta})=0.1201$ | $m({F}_{1})=0.9003$ $m({F}_{2})=0.0375$ $m({F}_{3})=0.0000$ $m(\mathsf{\Theta})=0.0623$ | $m({F}_{1})=0.9125$ $m({F}_{2})=0.0207$ $m({F}_{3})=0.0040$ $m(\mathsf{\Theta})=0.0629$ | $m({F}_{1})=0.5580$ $m({F}_{2})=0.0910$ $m({F}_{3})=0.1718$ $m(\mathsf{\Theta})=0.1793$ | Unknown |

Li and Guo [89] | $m({F}_{1})=0.8890$ $m({F}_{2})=0.0335$ $m({F}_{3})=0.0000$ $m(\mathsf{\Theta})=0.0775$ | $m({F}_{1})=0.8917$ $m({F}_{2})=0.0394$ $m({F}_{3})=0.0000$ $m(\mathsf{\Theta})=0.0689$ | $m({F}_{1})=0.9027$ $m({F}_{2})=0.0226$ $m({F}_{3})=0.0043$ $m(\mathsf{\Theta})=0.0704$ | $m({F}_{1})=0.6472$ $m({F}_{2})=0.0732$ $m({F}_{3})=0.0717$ $m(\mathsf{\Theta})=0.2079$ | Unknown |

Jiang et al. [54] | $m({F}_{1})=0.9032$ $m({F}_{2})=0.0910$ $m({F}_{3})=0.1718$ $m(\mathsf{\Theta})=0.1793$ | $m({F}_{1})=0.9593$ $m({F}_{2})=0.0247$ $m({F}_{3})=0.0000$ $m(\mathsf{\Theta})=0.0035$ | $m({F}_{1})=0.9895$ $m({F}_{2})=0.0062$ $m({F}_{3})=0.0008$ $m(\mathsf{\Theta})=0.0035$ | $m({F}_{1})=0.9914$ $m({F}_{2})=0.0035$ $m({F}_{3})=0.0042$ $m(\mathsf{\Theta})=0.0009$ | ${F}_{1}$ |

Zhang et al. [55] | $m({F}_{1})=0.9032$ $m({F}_{2})=0.0296$ $m({F}_{3})=0.0000$ $m(\mathsf{\Theta})=0.0672$ | $m({F}_{1})=0.9597$ $m({F}_{2})=0.0242$ $m({F}_{3})=0.0000$ $m(\mathsf{\Theta})=0.0161$ | $m({F}_{1})=0.9899$ $m({F}_{2})=0.0058$ $m({F}_{3})=0.0008$ $m(\mathsf{\Theta})=0.0035$ | $m({F}_{1})=0.9825$ $m({F}_{2})=0.0045$ $m({F}_{3})=0.0122$ $m(\mathsf{\Theta})=0.0008$ | ${F}_{1}$ |

Lin et al. [56] | $m({F}_{1})=0.9032$ $m({F}_{2})=0.0296$ $m({F}_{3})=0.0000$ $m(\mathsf{\Theta})=0.0159$ | $m({F}_{1})=0.9597$ $m({F}_{2})=0.0244$ $m({F}_{3})=0.0000$ $m(\mathsf{\Theta})=0.0159$ | $m({F}_{1})=0.9899$ $m({F}_{2})=0.0058$ $m({F}_{3})=0.0008$ $m(\mathsf{\Theta})=0.0035$ | $m({F}_{1})=0.9894$ $m({F}_{2})=0.0037$ $m({F}_{3})=0.0061$ $m(\mathsf{\Theta})=0.0008$ | ${F}_{1}$ |

Proposed method | $m({F}_{1})=0.9031$ $m({F}_{2})=0.0303$ $m({F}_{3})=0.0000$ $m(\mathsf{\Theta})=0.0667$ | $m({F}_{1})=0.9586$ $m({F}_{2})=0.0257$ $m({F}_{3})=0.0000$ $m(\mathsf{\Theta})=0.0157$ | $m({F}_{1})=0.9891$ $m({F}_{2})=0.0061$ $m({F}_{3})=0.0007$ $m(\mathsf{\Theta})=0.0035$ | $m({F}_{1})=0.9934$ $m({F}_{2})=0.0033$ $m({F}_{3})=0.0025$ $m(\mathsf{\Theta})=0.0008$ | ${F}_{1}$ |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Wang, Z.; Xiao, F.
An Improved Multi-Source Data Fusion Method Based on the Belief Entropy and Divergence Measure. *Entropy* **2019**, *21*, 611.
https://doi.org/10.3390/e21060611

**AMA Style**

Wang Z, Xiao F.
An Improved Multi-Source Data Fusion Method Based on the Belief Entropy and Divergence Measure. *Entropy*. 2019; 21(6):611.
https://doi.org/10.3390/e21060611

**Chicago/Turabian Style**

Wang, Zhe, and Fuyuan Xiao.
2019. "An Improved Multi-Source Data Fusion Method Based on the Belief Entropy and Divergence Measure" *Entropy* 21, no. 6: 611.
https://doi.org/10.3390/e21060611