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
4. Numerical Examples
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
- Hose, D.; Hanss, M. A universal approach to imprecise probabilities in possibility theory. Int. J. Approx. Reason 2021, 133, 133–158. [Google Scholar] [CrossRef]
- Yin, H.; Huang, J.; Chen, H. Possibility-based robust control for fuzzy mechanical systems. IEEE Trans. Fuzzy Syst. 2020, 99, 1. [Google Scholar] [CrossRef]
- Gu, Q.; Xuan, Z. A new approach for ranking fuzzy numbers based on possibility theory. J. Comput. Appl. Math. 2017, 309, 674–682. [Google Scholar] [CrossRef]
- Meng, L.; Li, L. Time-sequential hesitant fuzzy set and its application to multi-attribute decision making. J. Complex Intell. Syst. 2022, 1–20. [Google Scholar] [CrossRef]
- Moko, J.; Hurtík, P. Approximations of fuzzy soft sets by fuzzy soft relations with image processing application. Soft Comput. 2021, 25, 6915–6925. [Google Scholar] [CrossRef]
- Hja, B.; Bao, Q. A decision-theoretic fuzzy rough set in hesitant fuzzy information systems and its application in multi-attribute decision-making. Inform. Sci. 2021, 579, 103–127. [Google Scholar] [CrossRef]
- Chen, Z.; Cai, R. A novel divergence measure of mass function for conflict management. Int. J. Intell. Syst. 2022, 37, 3709–3735. [Google Scholar] [CrossRef]
- Liu, J.; Tang, Y. Conflict data fusion in a multi-agent system premised on the base basic probability assignment and evidence distance. Entropy 2021, 23, 820. [Google Scholar] [CrossRef]
- Tong, Z.; Xu, P.; Denaux, T. An evidential classifier based on Dempster–Shafer theory and deep learning. Neurocomputing 2021, 450, 275–293. [Google Scholar] [CrossRef]
- Mi, X.; Tian, Y.; Kang, B. A hybrid multi-criteria decision making approach for assessing health-care waste management technologies based on soft likelihood function and d-numbers. Appl. Intell. 2021, 2, 1–20. [Google Scholar] [CrossRef]
- Lai, H.; Liao, H. A multi-criteria decision making method based on DNMA and CRITIC with linguistic D numbers for blockchain platform evaluation. Eng. Appl. Artif. Intell. 2021, 101, 104200. [Google Scholar] [CrossRef]
- Liu, P.; Zhu, B.; Wang, P. A weighting model based on best–worst method and its application for environmental performance. Appl. Soft Comput. 2021, 103, 107168. [Google Scholar] [CrossRef]
- Jia, Q.; Hu, J. A novel method to research linguistic uncertain Z-numbers. Inform. Sci. 2022, 586, 41–58. [Google Scholar] [CrossRef]
- Hu, Z.; Lin, J. An integrated multicriteria group decision making methodology for property concealment risk assessment under Z-number environment. Expert Syst. Appl. 2022, 205, 117369. [Google Scholar] [CrossRef]
- Yousefi, S.; Valipour, M.; Gul, M. Systems failure analysis using Z-number theory-based combined compromise solution and full consistency method. Appl. Soft Comput. 2021, 113, 107902. [Google Scholar] [CrossRef]
- Yu, Z.; Wang, D.; Wang, P. A study of interrelationships between rough set model accuracy and granule cover refinement processes. Inform. Sci. 2021, 578, 116–128. [Google Scholar] [CrossRef]
- Jin, C.; Mi, J.; Li, F. A novel probabilistic hesitant fuzzy rough set based multi-criteria decision-making method. Inform. Sci. 2022, 608, 489–516. [Google Scholar] [CrossRef]
- Zhang, X.; Jiang, J. Measurement, modeling, reduction of decision-theoretic multigranulation fuzzy rough sets based on three-way decisions. Inform. Sci. 2022, 607, 1550–1582. [Google Scholar] [CrossRef]
- Wang, H.; Liu, S.; Qu, X. Field investigations on rock fragmentation under deep water through fractal theory. Measurement 2022, 199, 111521. [Google Scholar] [CrossRef]
- Zhou, Z.; Zhao, C.; Cai, X. Three-dimensional modeling and analysis of fractal characteristics of rupture source combined acoustic emission and fractal theory. Chaos Solitons Fractals 2022, 160, 112308. [Google Scholar] [CrossRef]
- Liu, W.; Yan, S.; Chen, T. Feature recognition of irregular pellet images by regularized Extreme Learning Machine in combination with fractal theory. Future Gener. Comp. Syst. 2022, 127, 92–108. [Google Scholar] [CrossRef]
- Wu, D.; Liu, Z.; Tang, Y. A new classification method based on the negation of a basic probability assignment in the evidence theory. Eng. Appl. Artif. Intell. 2020, 127, 92–108. [Google Scholar] [CrossRef]
- Zhao, K.; Li, L.; Chen, Z. A New Multi-classifier Ensemble Algorithm Based on D–S Evidence Theory. Neural Process. Lett. 2022. [Google Scholar] [CrossRef]
- Zhang, X.; Yang, Y.; Li, T. CMC: A Consensus Multi-view Clustering Model for Predicting Alzheimer’s Disease Progression. Comput. Meth. Prog. Biomed. 2021, 199, 105895. [Google Scholar] [CrossRef]
- Yang, F.; Wei, H.; Feng, P. A hierarchical Dempster–Shafer evidence combination framework for urban area land cover classification. Measurement 2020, 151, 105916. [Google Scholar] [CrossRef]
- Peñafiel, S.; Baloian, N.; Sanson, H. Applying Dempster–Shafer theory for developing a flexible, accurate and interpretable classifier. Expert Syst. Appl. 2020, 148, 113262. [Google Scholar] [CrossRef]
- Ji, X.; Ren, Y.; Tang, H. An intelligent fault diagnosis approach based on Dempster–Shafer theory for hydraulic valves. Measurement 2020, 165, 108129. [Google Scholar] [CrossRef]
- Verbert, K.; Babuška, R.; Schutter, B. Bayesian and Dempster–Shafer reasoning for knowledge-based fault diagnosis—A comparative study. Eng. Appl. Artif. Intell. 2017, 60, 136–150. [Google Scholar] [CrossRef]
- Gao, X.; Xiao, F. A generalized χ2 divergence for multisource information fusion and its application in fault diagnosis. Int. J. Intell. Syst. 2022, 37, 5–29. [Google Scholar] [CrossRef]
- Tingfang, Y.; Haifeng, L.; Xiangjun, Z.; Wei, Q.; Wenbin, D. Application of a combined decision model based on optimal weights in incipient faults diagnosis for power transformer. IEEE Trans. Elect. Electron. Eng. 2016, 12, 169–175. [Google Scholar] [CrossRef]
- Xu, Y.; Li, Y.; Wang, Y.; Wang, C.; Zhang, G. Integrated decision-making method for power transformer fault diagnosis via rough set and DS evidence theories. IET Gener. Transm. Distrib. 2020, 14, 5774–5781. [Google Scholar] [CrossRef]
- Dymova, L.; Sevastjanov, P. An interpretation of intuitionistic fuzzy sets in terms of evidence theory: Decision making aspect. Knowl.-Based Syst. 2010, 23, 772–782. [Google Scholar] [CrossRef]
- Li, Z.; Wen, G.; Xie, N. An approach to fuzzy soft sets in decision making based on grey relational analysis and Dempster–Shafer theory of evidence: An application in medical diagnosis. Artif. Intell. Med. 2015, 64, 161–171. [Google Scholar] [CrossRef] [PubMed]
- Liu, P.; Gao, H. Some intuitionistic fuzzy power Bonferroni mean operators in the framework of Dempster–Shafer theory and their application to multicriteria decision making. Appl. Soft Comput. 2019, 85, 105790. [Google Scholar] [CrossRef]
- Xiao, F. EFMCDM: Evidential Fuzzy Multicriteria Decision Making Based on Belief Entropy. IEEE Trans. Fuzzy Syst. 2020, 28, 1477–1491. [Google Scholar] [CrossRef]
- Xiao, F.; Cao, Z.; Jolfaei, A. A Novel Conflict Measurement in Decision-Making and Its Application in Fault Diagnosis. IEEE Trans. Fuzzy Syst. 2021, 29, 186–197. [Google Scholar] [CrossRef]
- Liu, M.; Wu, Y.; Zhao, W.; Zhang, Q.; Liao, G. Dempster–Shafer Fusion of Multiple Sparse Representation and Statistical Property for SAR Target Configuration Recognition. IEEE Geosci. Remote Sens. 2014, 11, 1106–1110. [Google Scholar] [CrossRef]
- Wang, J.; Liu, F. Temporal evidence combination method for multi-sensor target recognition based on DS theory and IFS. J. Syst. Eng. Electron. 2017, 28, 1114–1125. [Google Scholar] [CrossRef]
- Pan, L.; Deng, Y. A new complex evidence theory. Inform. Sci. 2022, 608, 251–261. [Google Scholar] [CrossRef]
- Zhu, C.; Xiao, F. A belief Hellinger distance for D–S evidence theory and its application in pattern recognition. Eng. Appl. Artif. Intell. 2021, 106, 104452. [Google Scholar] [CrossRef]
- Luo, Z.; Deng, Y. A Matrix Method of Basic Belief Assignment’s Negation in Dempster–Shafer Theory. IEEE Trans. Fuzzy Syst. 2020, 28, 2270–2276. [Google Scholar] [CrossRef]
- Gao, X.; Deng, Y. The Negation of Basic Probability Assignment. IEEE Access 2019, 7, 107006–107014. [Google Scholar] [CrossRef]
- Xie, D.; Xiao, F. Negation of Basic Probability Assignment: Trends of Dissimilarity and Dispersion. IEEE Access 2019, 7, 111315–111323. [Google Scholar] [CrossRef]
- Yin, L.; Deng, X.; Deng, Y. The Negation of a Basic Probability Assignment. IEEE Trans. Fuzzy Syst. 2019, 27, 135–143. [Google Scholar] [CrossRef]
- Li, S.; Xiao, F.; Abawajy, J. Conflict Management of Evidence Theory Based on Belief Entropy and Negation. IEEE Access 2020, 8, 37766–37774. [Google Scholar] [CrossRef]
- Yang, J.; Xu, D. Evidential reasoning rule for evidence combination. Artif. Intell. 2013, 205, 1–29. [Google Scholar] [CrossRef]
- Xu, H.; Deng, Y. Dependent evidence combination based on decision-making trial and evaluation laboratory method. Int. J. Intell. Syst. 2019, 34, 1555–1571. [Google Scholar] [CrossRef]
- He, Z.; Jiang, W. An evidential Markov decision making model. Inform. Sci. 2018, 467, 357–372. [Google Scholar] [CrossRef]
- Dempster, A.P. Upper and lower probabilities induced by a multi-valued mapping. Ann. Math. Stat. 1967, 38, 325–339. [Google Scholar] [CrossRef]
- Yan, H.; Deng, Y. An Improved Belief Entropy in Evidence Theory. IEEE Access 2020, 8, 57505–57575. [Google Scholar] [CrossRef]
- Deng, Y. Deng entropy. Chaos 2016, 46, 93–108. [Google Scholar] [CrossRef]
- Murphy, K. Combining belief functions when evidence conflicts. Decis. Support Syst. 2000, 29, 1–9. [Google Scholar] [CrossRef]
- Yong, D.; Wen, S.; Qi, L. Combining belief functions based on distance of evidence. Decis. Support Syst. 2004, 38, 489–493. [Google Scholar] [CrossRef]
- Xiao, F. A new divergence measure for belief functions in D–S evidence theory for multisensor data fusion. Inform. Fusion 2020, 514, 462–483. [Google Scholar] [CrossRef]
- Wang, H.; Deng, X.; Jiang, W.; Geng, J. A new belief divergence measure for Dempster–Shafer theory based on belief and plausibility function and its application in multi-source data fusion. Eng. Appl. Artif. Intell. 2021, 97, 104030. [Google Scholar] [CrossRef]
- Xiao, F. Multi-sensor data fusion based on the belief divergence measure of evidences and the belief entropy. Inform. Fusion 2019, 46, 23–32. [Google Scholar] [CrossRef]
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 | ||
---|---|---|---|---|
0.50 | 0.20 | 0 | 0.30 | |
0.00 | 0.90 | 0.10 | 0.00 | |
0.55 | 0.10 | 0.00 | 0.35 | |
0.55 | 0.10 | 0.00 | 0.35 |
Method | A | B | 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 |
0.40 | 0.28 | 0.30 | 0.02 | |
0.01 | 0.90 | 0.08 | 0.01 | |
0.63 | 0.06 | 0.01 | 0.30 | |
0.60 | 0.09 | 0.01 | 0.30 | |
0.60 | 0.09 | 0.01 | 0.30 |
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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
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 StyleXu, 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