# Improvement of DS Evidence Theory for Multi-Sensor Conflicting Information

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theoretical Foundations

#### 2.1. Frame of Discernment

#### 2.2. Basic Probability Assignment

#### 2.3. Uncertainty Description

#### 2.4. DS Combination Rule

## 3. Fuse Paradoxes of DS Theory

**Case 1**: Assuming that the frame of system is $\Theta =\left\{A,B,C\right\}$, and there are two pieces of evidence that are processed by the system:

**Case 2**: Assuming that the frame of system is $\Theta =\left\{A,B,C\right\}$, and the evidence set is $E=\left\{{E}_{1},{E}_{2},{E}_{3},{E}_{4}\right\}$:

**Case 3**: Assuming that the frame of system is $\Theta =\left\{A,B,C\right\}$, and the evidence set is $E=\left\{{E}_{1},{E}_{2},{E}_{3},{E}_{4},{E}_{5}\right\}$:

**Case 2**triggers error reasoning results of $m\left(A\right)=0$. In addition, most pieces of evidence such as ${E}_{1},{E}_{3},{E}_{4},{E}_{5}$ give proposition B little support, while the synthetic results give proposition B the biggest support. This counterintuitive situation further reflects the possible error reasoning processing of conventional DS theory.

- Conflict situations commonly occur in multi-sensor fusion systems. The way to fuse conflicting information is the key to realizing multi-sensor information fusion.
- All three conflict situations have one common point—the conflict degree K is high. The way to combine highly conflicting evidence is the key to solving conflict situations.

## 4. Existing Modified Methods

#### 4.1. Combination Rule Based on Local Conflict Degrees

#### 4.2. Combination Method Based on Mahalanobis Evidence

**Step 1**: Calculate the distance degree of evidence by the introduction of the Mahalanobis distance function.

**Step 2**: After the calculation of Mahalanobis distance, the distance matrix

**D**can be defined:

**Step 3**: As ${\overline{d}}_{i}$ represents the unreliable degree of evidence ${m}_{i}$, the evidence reliability is derived:

## 5. The Improvement of DS Theory

#### 5.1. Revised Evidence Based on the Lance Distance Function

#### 5.2. Revised Evidence Based on Spectral Angle Cosine Function

#### 5.3. Improved Conflict Redistribution Strategy

#### 5.4. Flow Chart of the Proposed Algorithm

**Step****1:**- Revise original evidence by the introduction of the Lance distance function.
**Step****2:**- Revise original evidence by the introduction of the spectral angle cosine function.
**Step****3:**- Redistribute the conflict degree of two pieces of revised evidence by employing a new redistribution strategy.

## 6. Simulation Results and Analyses

#### 6.1. Multi-Sensor Fusion with Consistent Information

- In the fusion results of the DS combination rule, although the support to the true target A is the biggest, the support to target B is always 0. Through the observation of Table 1, we can conclude that the “one ballot veto” paradox described in Section 3 leads to the unreasonable reasoning, and the paradox is caused by ${m}_{1}\left(B\right)=0$.
- References [15,16] represent the former kind of methods that are discussed in Section 1, while K-L distance [23] and Mahalanobis distance [24] represent the latter kind of methods that are discussed in Section 1. These four methods all give reasonable fusion results and recognize the true target A precisely.

#### 6.2. Multi-Sensor Fusion with Conflicting Information

- The fusion result of the DS combination rule completely believes that B is the true target, which is contrary to intuition judgement. Obviously, the DS combination rule cannot achieve reliable fusion for conflicting information.
- With the combination of added evidence ${E}_{4},{E}_{5}$, the support degree of true target A in reference [16], K-L distance [23], and Mahalanobis distance [24] is growing. However, the growth is slow, which means that reference [16], K-L distance [23], and Mahalanobis distance [24] are not completely reliable combination methods under conflict situations.

^{rd}sensor ${E}_{3}$. Obviously, the novel algorithm has a better convergence effect that can deal with highly conflicting situations more effectively. Thus, it has priority in the multi-sensor fusion with highly conflicting information.

## 7. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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Sensors | Targets | ||
---|---|---|---|

$\mathit{A}$ | $\mathit{B}$ | $\mathit{C}$ | |

${E}_{1}:{m}_{1}(\xb7)$ | 0.90 | 0 | 0.10 |

${E}_{2}:{m}_{2}(\xb7)$ | 0.88 | 0.01 | 0.11 |

${E}_{3}:{m}_{3}(\xb7)$ | 0.50 | 0.20 | 0.30 |

${E}_{4}:{m}_{4}(\xb7)$ | 0.98 | 0.01 | 0.01 |

${E}_{5}:{m}_{5}(\xb7)$ | 0.90 | 0.05 | 0.05 |

Methods | Targets | ${\mathit{E}}_{1}\oplus {\mathit{E}}_{2}$ | ${\mathit{E}}_{1}\oplus {\mathit{E}}_{2}\oplus {\mathit{E}}_{3}$ | $\begin{array}{c}{\mathit{E}}_{1}\oplus {\mathit{E}}_{2}\\ \oplus {\mathit{E}}_{3}\oplus {\mathit{E}}_{4}\end{array}$ | $\begin{array}{c}{\mathit{E}}_{1}\oplus {\mathit{E}}_{2}\oplus {\mathit{E}}_{3}\\ \oplus {\mathit{E}}_{4}\oplus {\mathit{E}}_{5}\end{array}$ |
---|---|---|---|---|---|

$\mathit{K}=\mathbf{0.1970}$ | $\mathit{K}=\mathbf{0.6007}$ | $\mathit{K}=\mathbf{0.6119}$ | $\mathit{K}=\mathbf{0.6507}$ | ||

DS combination | A | 0.9863 | 0.9917 | 0.9999 | 1 |

B | 0 | 0 | 0 | 0 | |

C | 0.0137 | 0.0083 | 0.0001 | 0 | |

$\Theta $ | 0 | 0 | 0 | 0 | |

Reference [15] | A | 0.9360 | 0.6978 | 0.7454 | 0.7492 |

B | 0.0008 | 0.0278 | 0.0241 | 0.0260 | |

C | 0.0280 | 0.0708 | 0.0570 | 0.0547 | |

$\Theta $ | 0.0352 | 0.2036 | 0.1735 | 0.1701 | |

Reference [16] | A | 0.9673 | 0.8525 | 0.8868 | 0.8480 |

B | 0.0010 | 0.0421 | 0.0337 | 0.0337 | |

C | 0.0317 | 0.1054 | 0.0795 | 0.0795 | |

$\Theta $ | 0 | 0 | 0 | 0 | |

K-L distance [23] | A | 0.9673 | 0.5365 | 0.5236 | 0.5799 |

B | 0.0010 | 0.0246 | 0.0165 | 0.0216 | |

C | 0.0317 | 0.0825 | 0.0606 | 0.0561 | |

$\Theta $ | 0 | 0.3564 | 0.3993 | 0.3423 | |

Mahalanobis distance [24] | A | 0.9605 | 0.6738 | 0.6671 | 0.7128 |

B | 0.0011 | 0.0206 | 0.0177 | 0.0236 | |

C | 0.0340 | 0.0883 | 0.0700 | 0.0658 | |

$\Theta $ | 0.0044 | 0.2173 | 0.2452 | 0.1978 | |

Proposed | A | 0.9878 | 0.9446 | 0.9668 | 0.9729 |

B | 0.0006 | 0.0159 | 0.0097 | 0.0086 | |

C | 0.0116 | 0.0395 | 0.0235 | 0.0185 | |

$\Theta $ | 0 | 0 | 0 | 0 |

Sensors | Targets | ||
---|---|---|---|

$\mathit{A}$ | $\mathit{B}$ | $\mathit{C}$ | |

${E}_{1}:{m}_{1}(\xb7)$ | 0.90 | 0 | 0.10 |

${E}_{2}:{m}_{2}(\xb7)$ | 0 | 0.01 | 0.99 |

${E}_{3}:{m}_{3}(\xb7)$ | 0.50 | 0.20 | 0.30 |

${E}_{4}:{m}_{4}(\xb7)$ | 0.98 | 0.01 | 0.01 |

${E}_{5}:{m}_{5}(\xb7)$ | 0.90 | 0.05 | 0.05 |

Methods | Targets | ${\mathit{E}}_{1}\oplus {\mathit{E}}_{2}$ | ${\mathit{E}}_{1}\oplus {\mathit{E}}_{2}\oplus {\mathit{E}}_{3}$ | $\begin{array}{c}{\mathit{E}}_{1}\oplus {\mathit{E}}_{2}\\ \oplus {\mathit{E}}_{3}\oplus {\mathit{E}}_{4}\end{array}$ | $\begin{array}{c}{\mathit{E}}_{1}\oplus {\mathit{E}}_{2}\oplus {\mathit{E}}_{3}\\ \oplus {\mathit{E}}_{4}\oplus {\mathit{E}}_{5}\end{array}$ |
---|---|---|---|---|---|

$\mathit{K}=\mathbf{0.9010}$ | $\mathit{K}=\mathbf{0.9703}$ | $\mathit{K}=\mathbf{0.9997}$ | $\mathit{K}=\mathbf{1}$ | ||

DS combination | A | 0 | 0 | 0 | 0 |

B | 0 | 0 | 0 | 0 | |

C | 1 | 1 | 1 | 1 | |

$\Theta $ | 0 | 0 | 0 | 0 | |

Reference [15] | A | 0.1647 | 0.2232 | 0.3192 | 0.3781 |

B | 0.0019 | 0.0335 | 0.0295 | 0.0311 | |

C | 0.2984 | 0.2513 | 0.1881 | 0.1671 | |

$\Theta $ | 0.5350 | 0.4920 | 0.4632 | 0.4237 | |

Reference [16] | A | 0.4055 | 0.4528 | 0.5948 | 0.5951 |

B | 0.0045 | 0.0679 | 0.0550 | 0.0550 | |

C | 0.5900 | 0.4793 | 0.3502 | 0.3499 | |

$\Theta $ | 0 | 0 | 0 | 0 | |

K-L distance [23] | A | 0.4055 | 0.3502 | 0.4446 | 0.5249 |

B | 0.0045 | 0.0677 | 0.0523 | 0.0503 | |

C | 0.5900 | 0.2850 | 0.1754 | 0.1366 | |

$\Theta $ | 0 | 0.2971 | 0.3277 | 0.2822 | |

Mahalanobis distance [24] | A | 0.4317 | 0.4276 | 0.5359 | 0.6077 |

B | 0.0020 | 0.0535 | 0.0427 | 0.0442 | |

C | 0.2841 | 0.2555 | 0.1884 | 0.1606 | |

$\Theta $ | 0.2822 | 0.2634 | 0.2330 | 0.1875 | |

Proposed | A | 0.5171 | 0.6036 | 0.8753 | 0.9206 |

B | 0 | 0.0068 | 0.0105 | 0.0081 | |

C | 0.4829 | 0.3896 | 0.1142 | 0.0713 | |

$\Theta $ | 0 | 0 | 0 | 0 |

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Ye, F.; Chen, J.; Li, Y.
Improvement of DS Evidence Theory for Multi-Sensor Conflicting Information. *Symmetry* **2017**, *9*, 69.
https://doi.org/10.3390/sym9050069

**AMA Style**

Ye F, Chen J, Li Y.
Improvement of DS Evidence Theory for Multi-Sensor Conflicting Information. *Symmetry*. 2017; 9(5):69.
https://doi.org/10.3390/sym9050069

**Chicago/Turabian Style**

Ye, Fang, Jie Chen, and Yibing Li.
2017. "Improvement of DS Evidence Theory for Multi-Sensor Conflicting Information" *Symmetry* 9, no. 5: 69.
https://doi.org/10.3390/sym9050069