# A Self-Adaptive Combination Method in Evidence Theory Based on the Power Pignistic Probability Distance

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Definition**

**1.**

**Definition**

**2.**

## 3. Classification of Evidence Sets and Adaptive Fusion

#### 3.1. Classification Criteria for Non-Conflict/Conflict Evidence Sets

**Definition**

**3.**

**Definition**

**4.**

_{1}and m

_{2}is defined as

**Definition**

**5.**

#### 3.2. Selection of Fusion Rules

_{i}and m

_{j}be any two of these BPAs.

_{i}and m

_{j}, and define the corresponding similarity measure as

_{i}as follows:

_{i}as follows:

_{i}.

## 4. Experimental Examples and Analysis

_{1}, S

_{2}, S

_{3}, S

_{4}and S

_{5}, where S

_{1}, S

_{2}, S

_{3}are radars, and the other two sensors are electro-optical sensors. Based on the information collecting by each sensor, the type of the target can be evaluated initially. The evaluation results can be transformed into BPA in evidence theory, which can reflect the probability of the target belonging to each type. In these examples, the type of the target to be recognized will be one of three types, i.e., warhead, decoy, and debris. So the discernment frame of this problem is Ω = {A (warhead), B (decoy), C (debris)}. The BPA obtained based on the information of each sensor is expressed as m

_{i}, $(i=1,2,\dots ,5)$. The BPA indicates the probability to which the target is recognized as each type. For example, m

_{1}(A) = 0.6 says that based on the information collected by S

_{1}, the target is recognized as a warhead with probability 0.6; m

_{1}(AC) = 0.2 indicated that based on the information collected by S

_{1}, the target is recognized as a warhead or debris following probability 0.2. The evidence set formed by the information acquired by each sensor is $M=\left\{{m}_{i}\right\},(i=1,2,\dots ,5)$.

_{i}}, i = 1, 2, …, 5, are listed in Table 1. As can be seen from Table 1, all five BPAs assign a greater level of confidence to A, and reasonable fusion results should likewise assign the greatest confidence to A. The method proposed in this paper yields a value of $\mathrm{max}({d}_{PBet})=0.2<\mu =0.3$ for this evidence set. The evidence set is therefore determined to be a non-conflicting evidence set, and Dempster’s rule should be applicable for conducting the fusion process. The results obtained by the proposed method and the seven other methods are listed in Table 2.

_{i}}, i = 1, 2, 3, 4, 5, are listed in Table 3. It can be seen from Table 3 that m

_{1}, m

_{3}, m

_{4}, and m

_{5}assign the greatest degree of confidence to A, and only m

_{2}has assigned the greatest degree of confidence to B, in conflict with the other sensors. After comprehensive consideration of the information provided by the five BPAs, the final reasonable fusion result should assign the maximum confidence level to A. The method proposed in this paper yields a value of $\mathrm{max}({d}_{PBet})=0.8>\mu =0.3$ for this evidence set. The evidence set is therefore determined to be a conflict evidence set, and the fusion process should be conducted using the weighted average combination method based on the power pignistic probability distance. The results obtained by this method and the seven other methods are listed in Table 4.

_{1}are A, B, and AB, the focus elements of m

_{2}are B, C, and BC, and the focus elements of m

_{3}are A, C, and AC. The greatest confidence of 0.8 is respectively assigned to A, B, and C, and the low confidence of 0.1 is assigned to the two remaining focus elements in conjunction with relatively high pair-wise conflicts. Because the BPAs have very similar focus element compositions and confidence distributions, and the degree of pair-wise conflicts are also the same, it is impossible to determine which of these has a lower reliability. Therefore, a reasonable fusion result should assign the same confidence level to A, B, and C. The method proposed in this paper yields a value of $\mathrm{max}({d}_{PBet})=0.8333>\mu =0.3$ for this evidence set. The evidence set is therefore determined to be a conflicting evidence set, and fusion should be conducted using the weighted average combination method based on the power pignistic probability distance. The results obtained by this method and the seven other methods are listed in Table 6.

_{1}is gradually changed by adding a small value θ that varies over the range [−0.1, 0.1] in increments of 0.02. Ideally, we would wish to apply all eight methods to calculate the fusion results for ${m}_{1}$, ${m}_{2}$, and ${m}_{3}$ and plot the variation curve of the fusion results with respect to θ. However, we note that Example 4 is equivalent to Example 3 when θ = 0, and the five evidence discounting methods fail to provide a fusion result. In this case, fusion results are obtained for $\theta =-0.001$ and $\theta =0.001$ as a supplement to ensure the continuity of the trends. Accordingly, the evidence set formed by the information acquired by each of the sensors is $M=\left\{{m}_{i}\right\},(i=1,2,3)$, and the details regarding the three BPAs are listed in Table 7.

_{1}changes when θ changes, but the degree of confidence assigned to these elements by ${m}_{2}$ and ${m}_{3}$ remain constant. Under normal circumstances, the fusion results obtained from ${m}_{1}$, ${m}_{2}$, and ${m}_{3}$ will vary in accordance with the value of θ. However, the degree of confidence given to each focus element by the fusion result should change only gradually because the change in θ is small.

_{1}is also very small. Therefore, the change in the fusion result should also be very small. This demonstrates that the evidence discounting methods lack robustness when applied to conflict conditions like those adopted in Example 4.

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Flow chart of the proposed adaptive evidence fusion algorithm based on the power pignistic probability distance.

BPA | A | B | C | AC |
---|---|---|---|---|

${m}_{1}$ | 0.7 | 0.1 | 0 | 0.2 |

${m}_{2}$ | 0.6 | 0.1 | 0.2 | 0.1 |

${m}_{3}$ | 0.8 | 0.1 | 0 | 0.1 |

${m}_{4}$ | 0.7 | 0.1 | 0.1 | 0.1 |

${m}_{5}$ | 0.6 | 0.2 | 0.1 | 0.1 |

Method | ${\mathit{m}}_{\oplus}(\mathit{A})$ | ${\mathit{m}}_{\oplus}(\mathit{B})$ | ${\mathit{m}}_{\oplus}(\mathit{C})$ | ${\mathit{m}}_{\oplus}(\mathit{A}\mathit{C})$ |
---|---|---|---|---|

Yu’s method | 0.99796 | 0.00024 | 0.00156 | 0.00024 |

Wen’s method | 0.99895 | 0.00010 | 0.00086 | 0.00009 |

Bi’s method | 0.99861 | 0.00014 | 0.00110 | 0.00015 |

Hu’s method | 0.99841 | 0.00020 | 0.00113 | 0.00026 |

Luo’s method | 0.98758 | 0.00311 | 0.00376 | 0.00555 |

Murphy’s method | 0.99895 | 0.00007 | 0.00090 | 0.00008 |

Deng’s method | 0.99895 | 0.00008 | 0.00089 | 0.00008 |

The proposed method | 0.99918 | 0.00006 | 0.00069 | 0.00006 |

BPA | A | B | C | AC |
---|---|---|---|---|

${m}_{1}$ | 0.5 | 0.2 | 0.3 | 0 |

${m}_{2}$ | 0 | 0.9 | 0.1 | 0 |

${m}_{3}$ | 0.55 | 0.1 | 0 | 0.35 |

${m}_{4}$ | 0.55 | 0.1 | 0 | 0.35 |

${m}_{5}$ | 0.55 | 0.1 | 0 | 0.35 |

Method | ${\mathit{m}}_{\oplus}(\mathit{A})$ | ${\mathit{m}}_{\oplus}(\mathit{B})$ | ${\mathit{m}}_{\oplus}(\mathit{C})$ | ${\mathit{m}}_{\oplus}(\mathit{A}\mathit{C})$ |
---|---|---|---|---|

Yu’s method | 0.9550 | 0.0010 | 0.0231 | 0.0209 |

Wen’s method | 0.9586 | 0.0009 | 0.0289 | 0.0115 |

Bi’s method | 0.9619 | 0.0007 | 0.0311 | 0.0062 |

Hu’s method | 0.9434 | 0.0018 | 0.0126 | 0.0422 |

Luo’s method | 0.9639 | 0.0007 | 0.0090 | 0.0265 |

Murphy’s method | 0.9659 | 0.0155 | 0.0148 | 0.0037 |

Deng’s method | 0.9846 | 0.0010 | 0.0103 | 0.0041 |

The proposed method | 0.9849 | 0.0008 | 0.0103 | 0.0040 |

**Table 5.**Evidence set obtained by the three sensors of the target identification system (example 3).

BPA | A | B | C | AB | AC | BC |
---|---|---|---|---|---|---|

${m}_{1}$ | 0.8 | 0.1 | 0 | 0.1 | 0 | 0 |

${m}_{2}$ | 0 | 0.8 | 0.1 | 0 | 0 | 0.1 |

${m}_{3}$ | 0.1 | 0 | 0.8 | 0 | 0.1 | 0 |

Method | ${\mathit{m}}_{\oplus}(\mathit{A})$ | ${\mathit{m}}_{\oplus}(\mathit{B})$ | ${\mathit{m}}_{\oplus}(\mathit{C})$ | ${\mathit{m}}_{\oplus}(\mathit{A}\mathit{B})$ | ${\mathit{m}}_{\oplus}(\mathit{A}\mathit{C})$ | ${\mathit{m}}_{\oplus}(\mathit{B}\mathit{C})$ |
---|---|---|---|---|---|---|

Yu’s method | null | null | null | null | null | null |

Wen’s method | null | null | null | null | null | null |

Bi’s method | null | null | null | null | null | null |

Hu’s method | null | null | null | null | null | null |

Luo’s method | null | null | null | null | null | null |

Murphy’s method | 0.3331 | 0.3331 | 0.3331 | 0.0003 | 0.0003 | 0.0003 |

Deng’s method | 0.3331 | 0.3331 | 0.3331 | 0.0003 | 0.0003 | 0.0003 |

The proposed method | 0.3331 | 0.3331 | 0.3331 | 0.0003 | 0.0003 | 0.0003 |

**Table 7.**Evidence set obtained by the three sensors of the target identification system (example 4).

Evidence | A | B | C | AB | AC | BC |
---|---|---|---|---|---|---|

${m}_{1}$ | 0.8 + θ | 0.1 − θ | 0 | 0.1 | 0 | 0 |

${m}_{2}$ | 0 | 0.8 | 0.1 | 0 | 0 | 0.1 |

${m}_{3}$ | 0.1 | 0 | 0.8 | 0 | 0.1 | 0 |

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**MDPI and ACS Style**

Wang, J.; Zhu, J.-w.; Song, Y.
A Self-Adaptive Combination Method in Evidence Theory Based on the Power Pignistic Probability Distance. *Symmetry* **2020**, *12*, 526.
https://doi.org/10.3390/sym12040526

**AMA Style**

Wang J, Zhu J-w, Song Y.
A Self-Adaptive Combination Method in Evidence Theory Based on the Power Pignistic Probability Distance. *Symmetry*. 2020; 12(4):526.
https://doi.org/10.3390/sym12040526

**Chicago/Turabian Style**

Wang, Jian, Jing-wei Zhu, and Yafei Song.
2020. "A Self-Adaptive Combination Method in Evidence Theory Based on the Power Pignistic Probability Distance" *Symmetry* 12, no. 4: 526.
https://doi.org/10.3390/sym12040526