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Möbius transformation in generalized evidence theory

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Abstract

Möbius transformation is a very important information inversion tool. Möbius transformation is sought after by many experts and scholars at home and abroad, and is a hot research topic at present. Möbius transformation can use the known information to reverse the unknown information, indicating that it has a strong ability to process information. Generalized evidence theory is an extension of classical evidence theory. When belief degree of the null subset is 0, then the generalized evidence theory will be degenerated as classical Dempster-Shafer evidence theory. However, how to apply Möbius transformation to generalized evidence theory is still an open problem. This paper proposes Möbius transformation in generalized evidence theory, which can perform function inversion of generalized evidence theory effectively. Numerical examples are used to prove the validity of Möbius transformation in generalized evidence theory. The experimental results show that the Möbius transformation in generalized evidence theory can effectively invert the generalized evidence theory and is a very effective function inversion method.

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References

  1. Demirel O (2020) The hyperbolic polygons of type (𝜖, n) and möbius transformations. Open Math 18:177–181, 03

    Article  MathSciNet  MATH  Google Scholar 

  2. Fernandez J, Bustince H, Horanská L, Mesiar R, Stupňanová A (2020) A generalization of the choquet integral defined in terms of the möbius transform. IEEE Trans Fuzzy Syst 28(10):2313–2319

    Article  Google Scholar 

  3. Reis L (2019) Möbius-like maps on irreducible polynomials and rational transformations. J Pure Appl Algebra:05

  4. Aigner M (1979) Combinatorial theory. Springer, New York, pp 01

  5. Kennes R (1992) Computational aspects of the Möbius transform of graphs. IEEE Trans Syst Man Cybern 22:201–223,04

    Article  MathSciNet  MATH  Google Scholar 

  6. Smets P (2002) The application of the matrix calculus to belief functions. Int J Approx Reason 31:1–30,10

    Article  MathSciNet  MATH  Google Scholar 

  7. Liu L (2017) Representing belief functions as random variables. IEEE Trans Syst Man Cybern Syst PP:1–10, 06

    Google Scholar 

  8. Deng Y (2015) Generalized evidence theory. Appl Intell 43:530–543

    Article  Google Scholar 

  9. Jiang W, Zhan J (2016) A modified combination rule in generalized evidence theory. Appl Intell 46:10

    Google Scholar 

  10. Koh JM, Cheong KH (2019) New doubly-anomalous parrondo’s games suggest emergent sustainability and inequality. Nonlinear Dyn 96(1):257–266

    Article  MATH  Google Scholar 

  11. Meng D, Liu M, Yang S, Zhang H, Ding R (2018) A fluid–structure analysis approach and its application in the uncertainty-based multidisciplinary design and optimization for blades. Adv Mech Eng 10(6):1687814018783410

  12. Weijia Lai J, Cheong KH (2020) Parrondo effect in quantum coin-toss simulations. Phys Rev E 101(5):052212

  13. Fei L, Feng Y, Liu L (2019) On pythagorean fuzzy decision making using soft likelihood functions. Int J Intell Syst 34(12):3317–3335

    Article  Google Scholar 

  14. Dzitac I, Filip FG, Manolescu M-j (2017) Fuzzy logic is not fuzzy World-renowned computer scientist lotfi a. zadeh. Int J Comput Commun Control 12(6):748–789

    Article  Google Scholar 

  15. Fei L, Feng Y (2020) An attitudinal nonlinear integral and applications in decision making. International Journal of Fuzzy Systems. https://doi.org/10.1007/s40815--020--00862--5

  16. Ares F, Esteve J, Falceto F, Queiroz A (2017) Entanglement entropy and möbius transformations for critical fermionic chains. J Stat Mech Theory Exper:063104

  17. Pan Y, Zhang L, Wu X, Skibniewski MJ (2020) Multi-classifier information fusion in risk analysis. Inf Fusion 60:121–136

    Article  Google Scholar 

  18. Wang H, Fang YP, Zio E (2021) Risk assessment of an electrical power system considering the influence of traffic congestion on a hypothetical scenario of electrified transportation system in new york state. IEEE Trans Intell Transp Syst 22(1):142–155

    Article  Google Scholar 

  19. Li M, Huang S, De Bock J, De Cooman G, Pižurica A (2020) A robust dynamic classifier selection approach for hyperspectral images with imprecise label information. Sensors 20(18):5262

    Article  Google Scholar 

  20. Xue Y, Deng Y (2021) Tsallis extropy. Communications in Statistics-Theory and Methods. https://doi.org/10.1080/03610926.2021.1921804

  21. Xue Y-G, Deng Y (2021) A new belief structure based on cardinality measure. Comput Appl Math 40(51). https://doi.org/10.1007/s40314--021--01452--3,

  22. Mo H (2021) A SWOT method to evaluate safety risks in life cycle of wind turbine extended by D number theory. J Intell Fuzzy Syst 40(3):4439–4452

    Article  Google Scholar 

  23. Mo H (2020) A new evaluation methodology for quality goals extended by D number theory and FAHP. Information 11(4):206

    Article  Google Scholar 

  24. Mo H (2020) An emergency decision-making method for probabilistic linguistic term sets extended by D number theory. Symmetry 12(3):380

    Article  Google Scholar 

  25. Song Y, Fu Q, Wang Y-F, Wang X (2019) Divergence-based cross entropy and uncertainty measures of atanassov’s intuitionistic fuzzy sets with their application in decision making. Appl Soft Comput 84. https://doi.org/10.1016/j.asoc.2019.105703

  26. Garg H (2019) Intuitionistic fuzzy hamacher aggregation operators with entropy weight and their applications to multi-criteria decision-making problem. Iran J Sci Technol Trans Electr Eng 43(3):597–613

    Article  Google Scholar 

  27. Lai JW, Cheong KH (2020) Parrondo’s paradox from classical to quantum: A review. Nonlinear Dyn 100(1):849–861

    Article  MATH  Google Scholar 

  28. Liao H, Ren Z, Fang R (2020) A deng-entropy-based evidential reasoning approach for multi-expert multi-criterion decision-making with uncertainty. Int J Comput Intell Syst 13:1281–1294

    Article  Google Scholar 

  29. Özkan K (2018) Comparing shannon entropy with deng entropy and improved deng entropy for measuring biodiversity when a priori data is not clear. Forestist 68(2):136–140

    Google Scholar 

  30. Deng Y (2020) Information volume of mass function. Int J Comput Commun Control 15(6):3983

    Article  Google Scholar 

  31. Pan Q, Dezert J, Huang L, Liu Z (2020) Evidential combination of augmented multi-source of information based on domain adaptation. Sci China Inf Sci 63(11):210203

  32. Soroush MZ, Maghooli K, Setarehdan K, Nasrabadi AM (2019) Emotion recognition through eeg phase space dynamics and dempster-shafer theory. Med Hypotheses 127:03

    Google Scholar 

  33. Song Y, Zhu J, Lei L, Wang X A self-adaptive combination method for temporal evidence based on negotiation strategy. SCIENCE CHINA Information Sciences https://doi.org/10.1007/s11432--020--3045--5

  34. Xiao F (2020) CED: A distance for complex mass functions. IEEE Transactions on Neural Networks and Learning Systems. https://doi.org/10.1109/TNNLS.2020.2984918

  35. Shafer G (1976) A mathematical theory of evidence, vol 42. Princeton University Press

  36. Xiao F, Cao Z, Jolfaei A (2020) A novel conflict measurement in decision making and its application in fault diagnosis. IEEE Transactions on Fuzzy Systems. https://doi.org/10.1109/TFUZZ.2020.3002431

  37. Deng Y (2020) Uncertainty measure in evidence theory. Sci China Inf Sci 63(11):210201

  38. Song Y, Fu Q, Wang Y-F, Wang X (2019) Divergence-based cross entropy and uncertainty measures of Atanassov’s intuitionistic fuzzy sets with their application in decision making. Appl Soft Comput 84:105703

    Article  Google Scholar 

  39. Denœux T (2008) Conjunctive and disjunctive combination of belief functions induced by nondistinct bodies of evidence. Artif Intell 172:234–264,02

    Article  MathSciNet  MATH  Google Scholar 

  40. Smets P (1995) The canonical decomposition of a weighted belief, pp 1896–1901,01

  41. Smets P, Kennes R (1994) The transferable belief model. Artif Intell 66:191–234, 04

    Article  MathSciNet  MATH  Google Scholar 

  42. Hardy GH, Wright EM (1979) An introduction to the theory of numbers oxford science publications. Clarendon Press, pp 01

  43. Schroeder M (1997) Number Theory in Science and Communication 7:01

  44. Pearl J (1988) probabilistic reasoning in intelligent systems: networks of plausible inference san mateo

  45. Jiang W, Huang K, Geng J, Deng X (2020) Multi-scale metric learning for few-shot learning. IEEE Transactions on Circuits and Systems for Video Technology. https://doi.org/10.1109/TCSVT.2020.2995754

  46. Xiao F (2019) A distance measure for intuitionistic fuzzy sets and its application to pattern classification problems. IEEE Transactions on Systems, Man, and Cybernetics: Systems. https://doi.org/10.1109/TSMC.2019.2958635

  47. Fei L, Feng Y, Liu L (2019) Evidence combination using owa-based soft likelihood functions. Int J Intell Syst 34(9):2269–2290

    Article  Google Scholar 

  48. Fei L, Feng Y (2020) A novel retrieval strategy for case-based reasoning based on attitudinal choquet integral. Eng Appl Artif Intell 103791:94

    Google Scholar 

  49. Lai JW, Tan RA, Lu H, Yap ZR, Cheong KH (2020) Parrondo paradoxical walk using four-sided quantum coins. Phys Rev E 102(1):012213

  50. Mi Z, Liu X, Yang J (2017) Evidential reasoning approach for MADM based on incomplete interval value. J Intell Fuzzy Syst 33(6):3707–3721

    Article  Google Scholar 

  51. Liu Z-G, Pan Q, Dezert J, Martin A (2018) Combination of classifiers with optimal weight based on evidential reasoning. IEEE Trans Fuzzy Syst 26(3):1217–1230

    Article  Google Scholar 

  52. Xu X, Zheng J, Yang J-b, Xu D-l, Chen Y-w (2017) Data classification using evidence reasoning rule. Knowl-Based Syst 116:144–151

    Article  Google Scholar 

  53. Fu Chao, Hou B, Chang W, Feng N, Yang S (2020) Comparison of evidential reasoning algorithm with linear combination in decision making. Int J Fuzzy Syst 22(2):686–711

    Article  Google Scholar 

  54. Ares F, Esteve J, Falceto F, Queiroz A (2016) On the möbius transformation in the entanglement entropy of fermionic chains. Journal of Statistical Mechanics: Theory and Experiment

  55. Liu Z-g, Li G, Mercier G, He Y, Pan Q (2018) Change detection in heterogenous remote sensing images via homogeneous pixel transformation. IEEE Trans Image Process 27:1822–1834

    Article  MathSciNet  MATH  Google Scholar 

  56. Deng J, Deng Y (2021) Information volume of fuzzy membership function. Int J Comput Commun Control 16(1):4106

    Article  Google Scholar 

  57. Xiao F (2020) On the maximum entropy negation of a complex-valued distribution. IEEE Transactions on Fuzzy Systems. https://doi.org/10.1109/TFUZZ.2020.3016723

  58. Xiao F (2019) EFMCDM: Evidential fuzzy multicriteria decision making based on belief entropy. IEEE Transactions on Fuzzy Systems. https://doi.org/10.1109/TFUZZ.2019.2936368

  59. Zhang H, Deng Y (2021) Entropy measure for orderable sets. Inf Sci 561:141–151

    Article  MathSciNet  Google Scholar 

  60. Fan L, Deng Y (2021) Determine the number of unknown targets in Open World based on Elbow method. IEEE Trans Fuzzy Syst 29(5):986–995

    Article  Google Scholar 

  61. Kock J (2020) From möbius inversion to renormalisation. Commun Number Theory Phys 14:171–198,01

    Article  MathSciNet  MATH  Google Scholar 

  62. Tian Y, Liu L, Mi X, Kang B (2020) ZSLF: A new soft likelihood function based on Z-numbers and its application in expert decision system. IEEE Transactions on Fuzzy Systems. https://doi.org/10.1109/TFUZZ.2020.2997328

  63. Xiao F (2020) GIQ: A generalized intelligent quality-based approach for fusing multi-source information. IEEE Transactions on Fuzzy Systems. https://doi.org/10.1109/TFUZZ.2020.2991296

  64. Cao Z, Ding W, Wang Y-K, Hussain F, Al-Jumaily A, Lin C-T (2019) Effects of Repetitive SSVEPs on EEG Complexity using Multiscale Inherent Fuzzy Entropy. Neurocomputing. https://doi.org/10.1016/j.neucom.2018.08.091

  65. Bayes T (1763) An Essay Toward Solving a Problem in the Doctrine of Chances 53:01

  66. Ye Y, Hang XR, Koh JM, Miszczak JA, Cheong KH, Xie N-g (2020) Passive network evolution promotes group welfare in complex networks. Chaos Solitons Fractals 130:109464

    Article  MathSciNet  Google Scholar 

  67. Akram M, Ilyas F, Garg H (2020) Multi-criteria group decision making based on electre i method in pythagorean fuzzy information. Soft Comput 24(5):3425–3453

    Article  Google Scholar 

  68. Liu Z, Pan Q, Dezert J, Han J-W, He Y (2018) Classifier fusion with contextual reliability evaluation. IEEE Trans Cybern 48:1605–1618

    Article  Google Scholar 

  69. Ullah K, Garg H, Mahmood T, Jan N, Ali Z (2020) Correlation coefficients for t-spherical fuzzy sets and their applications in clustering and multi-attribute decision making. Soft Comput 24(3):1647–1659

    Article  MATH  Google Scholar 

  70. Ullah K, Hassan N, Mahmood T, Jan N, Hassan M (2019) Evaluation of investment policy based on multi-attribute decision-making using interval valued t-spherical fuzzy aggregation operators. Symmetry 11(3):357

    Article  Google Scholar 

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Acknowledgments

The authors greatly appreciate the reviews’ suggestions and the editor’s encouragement. The work is partially supported by National Natural Science Foundation of China (Grant No. 61973332), JSPS Invitational Fellowships for Research in Japan (Short-term).

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Authors and Affiliations

  1. Institute of Fundamental and Frontier Sciences, University of Electronic Science and Technology of China, Chengdu, 610054, China

    Yige Xue & Yong Deng

  2. School of Eduction, Shaanxi Normal University, Xi’an, 710062, China

    Yong Deng

  3. School of Knowledge Science, Japan Advanced Institute of Science and Technology, Nomi, Ishikawa, 923-1211, Japan

    Yong Deng

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  1. Yige Xue
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Correspondence to Yong Deng.

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Xue, Y., Deng, Y. Möbius transformation in generalized evidence theory. Appl Intell 52, 7818–7831 (2022). https://doi.org/10.1007/s10489-021-02827-z

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