# Logarithmic Negation of Basic Probability Assignment and Its Application in Target Recognition

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

#### 2.1. D–S Evidence Theory

#### 2.1.1. Frame of Discernment

#### 2.1.2. Basic Probability Assignment

#### 2.2. Dempster’s Combination Rule

#### 2.3. Shannon Entropy

#### 2.4. Deng Entropy

#### 2.5. Yin’s Negation of BPA

#### 2.6. Gao’s Negation of BPA

## 3. Proposed Negation

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

**Theorem**

**3.**

**Proof.**

**Theorem**

**4.**

**Proof.**

**Theorem**

**5.**

**Proof.**

## 4. Numerical Examples

**Example**

**1.**

**Example**

**2.**

**Example**

**3.**

**Example**

**4.**

## 5. Application

#### 5.1. Application 1

#### 5.2. Application 2

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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Number of Iterations | m(a) | m(b) | m(a,b) | Shannon Entropy |
---|---|---|---|---|

0 | 0.700 | 0.300 | 0.000 | 0.88129 |

1 | 0.300 | 0.700 | 0.000 | 0.88129 |

2 | 0.700 | 0.300 | 0.000 | 0.88129 |

3 | 0.300 | 0.700 | 0.000 | 0.88129 |

4 | 0.700 | 0.300 | 0.000 | 0.88129 |

5 | 0.300 | 0.700 | 0.000 | 0.88129 |

6 | 0.700 | 0.300 | 0.000 | 0.88129 |

7 | 0.300 | 0.700 | 0.000 | 0.88129 |

8 | 0.700 | 0.300 | 0.000 | 0.88129 |

Number of Iterations | m(a) | m(b) | m(a,b) | Shannon Entropy |
---|---|---|---|---|

0 | 0.700 | 0.300 | 0.000 | 0.88129 |

1 | 0.213 | 0.334 | 0.453 | 1.52089 |

2 | 0.376 | 0.332 | 0.292 | 1.57726 |

3 | 0.318 | 0.334 | 0.348 | 1.58402 |

4 | 0.339 | 0.333 | 0.328 | 1.58485 |

5 | 0.331 | 0.333 | 0.335 | 1.58495 |

6 | 0.334 | 0.333 | 0.333 | 1.58496 |

7 | 0.333 | 0.333 | 0.334 | 1.58496 |

8 | 0.333 | 0.333 | 0.333 | 1.58496 |

Number of Iterations | m(a) | m(b) | m(c) | m(a,b) | m(a,c) | m(b,c) | m(a,b,c) | Shannon Entropy |
---|---|---|---|---|---|---|---|---|

0 | 0.1200 | 0.1800 | 0.1100 | 0.0900 | 0.1900 | 0.2300 | 0.0800 | 2.70972 |

1 | 0.1460 | 0.1374 | 0.1475 | 0.1505 | 0.1360 | 0.1306 | 0.1520 | 2.80532 |

2 | 0.1424 | 0.1436 | 0.1422 | 0.1418 | 0.1438 | 0.1446 | 0.1415 | 2.80731 |

3 | 0.1429 | 0.1427 | 0.1430 | 0.1430 | 0.1427 | 0.1426 | 0.1430 | 2.80735 |

4 | 0.1428 | 0.1429 | 0.1428 | 0.1428 | 0.1429 | 0.1429 | 0.1428 | 2.80735 |

5 | 0.1429 | 0.1429 | 0.1429 | 0.1429 | 0.1429 | 0.1429 | 0.1429 | 2.80735 |

Number of Iterations | 0 | 1 | 2 | 3 | 4 |
---|---|---|---|---|---|

m(a) | 0.1200 | 0.0630 | 0.0669 | 0.0667 | 0.0667 |

m(b) | 0.1800 | 0.0593 | 0.0672 | 0.0666 | 0.0667 |

m(c) | 0.1100 | 0.0636 | 0.0669 | 0.0667 | 0.0667 |

m(d) | 0.0000 | 0.0711 | 0.0664 | 0.0667 | 0.0667 |

m(a,b) | 0.0900 | 0.0649 | 0.0668 | 0.0667 | 0.0667 |

m(a,c) | 0.1900 | 0.0587 | 0.0672 | 0.0666 | 0.0667 |

m(a,d) | 0.0000 | 0.0711 | 0.0664 | 0.0667 | 0.0667 |

m(b,c) | 0.2300 | 0.0563 | 0.0674 | 0.0666 | 0.0667 |

m(b,d) | 0.0000 | 0.0711 | 0.0664 | 0.0667 | 0.0667 |

m(c,d) | 0.0000 | 0.0711 | 0.0664 | 0.0667 | 0.0667 |

m(a,b,c) | 0.0800 | 0.0656 | 0.0667 | 0.0667 | 0.0667 |

m(a,b,d) | 0.0000 | 0.0711 | 0.0664 | 0.0667 | 0.0667 |

m(a,c,d) | 0.0000 | 0.0711 | 0.0664 | 0.0667 | 0.0667 |

m(b,c,d) | 0.0000 | 0.0711 | 0.0664 | 0.0667 | 0.0667 |

m(a,b,c,d) | 0.0000 | 0.0711 | 0.0664 | 0.0667 | 0.0667 |

Shannon entropy | 2.70972 | 3.90242 | 3.90687 | 3.90689 | 3.90689 |

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

${m}_{1}$ | 0.50 | 0.20 | 0 | 0.30 |

${m}_{2}$ | 0.00 | 0.90 | 0.10 | 0.00 |

${m}_{3}$ | 0.55 | 0.10 | 0.00 | 0.35 |

${m}_{4}$ | 0.55 | 0.10 | 0.00 | 0.35 |

Method | A | B | C | $\mathit{A},\mathit{C}$ |
---|---|---|---|---|

Dempster [49] | 0.0000 | 0.3288 | 0.6712 | 0.0000 |

Murphy [52] | 0.6027 | 0.2627 | 0.1346 | 0.0000 |

Deng [53] | 0.7773 | 0.0628 | 0.1600 | 0.0000 |

Li [45] | 0.8491 | 0.0112 | 0.0112 | 0.1275 |

Proposed method | 0.9653 | 0.0021 | 0.0209 | 0.0117 |

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

${m}_{1}$ | 0.40 | 0.28 | 0.30 | 0.02 |

${m}_{2}$ | 0.01 | 0.90 | 0.08 | 0.01 |

${m}_{3}$ | 0.63 | 0.06 | 0.01 | 0.30 |

${m}_{4}$ | 0.60 | 0.09 | 0.01 | 0.30 |

${m}_{5}$ | 0.60 | 0.09 | 0.01 | 0.30 |

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**MDPI and ACS Style**

Xu, S.; Hou, Y.; Deng, X.; Chen, P.; Zhou, S.
Logarithmic Negation of Basic Probability Assignment and Its Application in Target Recognition. *Information* **2022**, *13*, 387.
https://doi.org/10.3390/info13080387

**AMA Style**

Xu S, Hou Y, Deng X, Chen P, Zhou S.
Logarithmic Negation of Basic Probability Assignment and Its Application in Target Recognition. *Information*. 2022; 13(8):387.
https://doi.org/10.3390/info13080387

**Chicago/Turabian Style**

Xu, Shijun, Yi Hou, Xinpu Deng, Peibo Chen, and Shilin Zhou.
2022. "Logarithmic Negation of Basic Probability Assignment and Its Application in Target Recognition" *Information* 13, no. 8: 387.
https://doi.org/10.3390/info13080387