Abstract
Recently, a new type of set, called random permutation set (RPS), is proposed by considering all the permutations of elements in a certain set. For measuring the uncertainty of RPS, the entropy of RPS is presented. However, the maximum entropy principle of RPS entropy has not been discussed. To address this issue, this paper presents the maximum entropy of RPS. The analytical solution of maximum RPS entropy and its PMF condition are proven and discussed. Besides, numerical examples are used to illustrate the maximum RPS entropy. The results show that the maximum RPS entropy is compatible with the maximum Deng entropy and the maximum Shannon entropy. Moreover, in order to further apply RPS entropy and maximum RPS entropy in practical fields, a comparative analysis of the choice of using Shannon entropy, Deng entropy, and RPS entropy is also carried out.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Explore related subjects
Discover the latest articles, news and stories from top researchers in related subjects.Data availability
All data and materials generated or analyzed during this study are included in this article.
Code availability
The code of the current study is available from the corresponding author on reasonable request.
References
Babajanyan S, Allahverdyan A, Cheong KH (2020) Energy and entropy: Path from game theory to statistical mechanics. Physical Review Research 2(4):043055
Balakrishnan N, Buono F, Longobardi M (2022) On cumulative entropies in terms of moments of order statistics. Methodology and Computing in Applied Probability 24(1):345–359
Balakrishnan N, Buono F, Longobardi M (2022) On tsallis extropy with an application to pattern recognition. Statistics & Probability Letters 180:109241
Balakrishnan N, Buono F, Longobardi M (2022) A unified formulation of entropy and its application. Physica A: Statistical Mechanics and its Applications 127214
Buckley JJ (2005) Maximum entropy principle with imprecise side-conditions. Soft Comput 9(7):507–511
Buono F, Longobardi M (2020) A dual measure of uncertainty: The deng extropy. Entropy, 22(5). https://www.mdpi.com/1099-4300/22/5/582https://doi.org/10.3390/e22050582
Chen L, Deng Y (2021) Entropy of random permutation set. ChinaXiv:202112.00129. https://doi.org/10.12074/202112.00129
Chen X, Wang T, Ying R, Cao Z (2021) A fault diagnosis method considering meteorological factors for transmission networks based on P systems. Entropy 23(8):1008
Cheng C, Xiao F (2021) A distance for belief functions of orderable set. Pattern Recognit Lett 145:165–170
Cheong KH, Koh JM, Jones MC (2019) Paradoxical survival: Examining the parrondo effect across biology. BioEssays 41(6):1900027
Cui H, Zhou L, Li Y, Kang B (2022) Belief entropy-of-entropy and its application in the cardiac interbeat interval time series analysis. Chaos Solitons Fractal 155:111736. https://doi.org/10.1016/j.chaos.2021.111736
Dempster AP (1967) 04) Upper and lower probabilities induced by a multivalued mapping. Ann Math Stat 38(2):325–339
Deng Y (2020) Uncertainty measure in evidence theory. Sci China Inform Sci 63(11):1–19
Deng Y (2022) Random permutation set. Int J Comput Commun Control 17(1):4542. https://doi.org/10.15837/ijccc.2022.1.4542
Di Nola A, Dvurečenskij A, Hyčko M, Manara C (2005) Entropy on effect algebras with the riesz decomposition property i: Basic properties. Kybernetika 41(2):143–160
Feng G, Lu W, Pedrycz W, Yang J, Liu X (2019) The learning of fuzzy cognitive maps with noisy data: A rapid and robust learning method with maximum entropy. IEEE transact on cybern 51(4):2080–2092
Gao Q, Wen T, Deng Y (2021) Information volume fractal dimension. Fractals 29(8):2150263. https://doi.org/10.1142/S0218348X21502637
Gao X, Su X, Qian H, Pan X (2021) Dependence assessment in Human Reliability Analysis under uncertain and dynamic situations. Nuclear Engineering and Technology https://doi.org/10.1016/j.net.2021.09.045
Huang Z, Wang T, Liu W, Valencia-Cabrera L, Perez-Jimenez MJ, Li P (2021) A fault analysis method for three-phase induction motors based on spiking neural p systems. Complexity2021
Jech T (2013) Set theory. Springer Science & Business Media
Kazemi MR, Tahmasebi S, Buono F, Longobardi M (2021) Fractional deng entropy and extropy and some applications. Entropy 23(5). https://www.mdpi.com/1099-4300/23/5/623https://doi.org/10.3390/e23050623
Lai JW, Chang J, Ang LK, Cheong KH (2020) Multi-level information fusion to alleviate network congestion. Inform Fusion 63:248–255
Lee P (1980) Probability theory. Bull London Math Soc 12(4):318–319
Liu Z, Huang L, Zhou K, Denoeux T (2021) Combination of transferable classification with multisource domain adaptation based on evidential reasoning. IEEE Transact on Neural Netw Learn Syst 32(5):2015–2029
Liu Z, Zhang X, Niu J, Dezert J (2021) Combination of classifiers with different frames of discernment based on belief functions. IEEE Transacts on Fuzzy Syst 29(7):1764–1774
Ma G (2021) A remark on the maximum entropy principle in uncertainty theory. Soft Comput 25(22):13911–13920
Pan Y, Zhang L, Wu X, Skibniewski MJ (2020) Multi-classifier information fusion in risk analysis. Inf Fusion 60:121–136. https://doi.org/10.1016/j.inffus.2020.02.003
Pawlak Z (1982) Rough sets. Int j comput inf sci 11(5):341–356
Qiang C, Deng Y, Cheong KH (2022) Information fractal dimension of mass function. Fractals 30:2250110. https://doi.org/10.1142/S0218348X22501109
Shafer G (1976) A math theory evid, vol 1. Princeton University Press Princeton
Shannon CE (1948) A mathematical theory of communication. Bell Syst Tech J 27(4):379–423
Song M, Sun C, Cai D, Hong S, Li H (2022) Classifying vaguely labeled data based on evidential fusion. Inf Sci 583:159–173
Song X, Xiao F (2022) Combining time-series evidence: A complex network model based on a visibility graph and belief entropy. Appl Intel. https://doi.org/10.1007/s10489-021-02956-5
Song Y, Deng Y (2021) Entropic explanation of power set. Int J Comput Commun Control 16(4):4413. https://doi.org/10.15837/ijccc.2021.4.4413
Tsallis C (1988) Possible generalization of boltzmann-gibbs statistics. J stat phys 52(1–2):479–487
Wang H, Abdin AF, Fang Y-P, Zio E (2021) Resilience assessment of electrified road networks subject to charging station failures. Comput-Aided Civil Infrastruct Eng. https://doi.org/10.1111/mice.12736
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 Transact on Intell Transpo Syst 22(1):142–155. https://doi.org/10.1109/TITS.2019.2955359
Wang H, Fang Y-P, Zio E (2022) Resilience-oriented optimal post-disruption reconfiguration for coupled traffic-power systems. Reliab Eng Syst Safety 222:108408
Wang T, Liu W, Zhao J, Guo X, Terzijae V (2020) A rough set-based bio-inspired fault diagnosis method for electrical substations. Int J Electr Power Energy Syst 119:105961
Wang T, Wei X, Wang J, Huang T, Peng H, Song X, Perez-Jimenez MJ (2020) A weighted corrective fuzzy reasoning spiking neural p system for fault diagnosis in power systems with variable topologies. Eng Appl Artif Intel 92:103680
Wen T, Cheong KH (2021) The fractal dimension of complex networks: A review. Inf Fusion 73:87–102
Wen Z, Liu Z, Zhang S, Pan Q (2021) Rotation awareness based self-supervised learning for SAR target recognition with limited training samples. IEEE Transact on Image Process 30:7266–7279
Wu Q, Deng Y, Xiong N (2022) Exponential negation of a probability distribution. Soft Comput 26(5):2147–2156. https://doi.org/10.1007/s00500-021-06658-5
Xiao F (2021) CEQD: A complex mass function to predict interference effects. IEEE Trans on Cybern. https://doi.org/10.1109/TCYB.2020.3040770
Xiao F (2021) A distance measure for intuitionistic fuzzy sets and its application to pattern classification problems. IEEE Trans on Syst Man Cybern Syst 51(6):3980–3992
Xiao F (2022) CaFtR: A fuzzy complex event processing method. Int J Fuzzy Syst 24(2):1098–1111. https://doi.org/10.1007/s40815-021-01118-6
Xiao, F., Pedrycz, W (2022) Negation of the quantum mass function for multisource quantum information fusion with its application to pattern classification. IEEE Transactions on Pattern Analysis and Machine Intelligence. https://doi.org/10.1109/TPAMI.2022.3167045
Xie D, Xiao F, Pedrycz W (2021) Information quality for intuitionistic fuzzy values with its application in decision making. Eng Appl Artificial Intell. https://doi.org/10.1016/j.engappai.2021.104568
Xiong L, Su X, Qian H (2021) Conflicting evidence combination from the perspective of networks. Inf Sci 580:408–418. https://doi.org/10.1016/j.ins.2021.08.088
Yager RR (2009) Weighted maximum entropy owa aggregation with applications to decision making under risk. IEEE Trans on Syst Man Cybern Part A: Syst Hum 39(3):555–564
Yager RR (2014) On the maximum entropy negation of a probability distribution. IEEE Trans on Fuzzy Syst 23(5):1899–1902
Zadeh LA (1965) Fuzzy sets. Inf control 8(3):338–353
Zadeh LA (2011) A note on z-numbers. Inf Sci 181(14):2923–2932
Zhou Q, Deng Y (2021) Belief extropy: Measure uncertainty from negation. Commun Stat Theory Method. https://doi.org/10.1080/03610926.2021.1980049
Zhou Q, Deng Y (2022) Higher order information volume of mass function. Inf Sci 586:501–513
Zhou Q, Deng Y, Pedrycz W (2022) Information dimension of galton board. Fractals. https://doi.org/10.1142/S0218348X22500797
Acknowledgements
The authors greatly appreciate the editor’s encouragement and the reviews’ suggestions. The work was partially supported by National Natural Science Foundation of China (Grant No. 61973332) and JSPS Invitational Fellowships for Research in Japan (Short-term).
Funding
The work was partially supported by National Natural Science Foundation of China (Grant No. 61973332) and JSPS Invitational Fellowships for Research in Japan (Short-term).
Author information
Authors and Affiliations
Contributions
All authors contributed to the study conception and design. All authors performed material preparation, data collection, and analysis. Jixiang Deng wrote the first draft of the paper. All authors contributed to the revisions of the paper. All authors read and approved the final manuscript.
Corresponding author
Ethics declarations
Conflict of interest
All the authors certify that there is no conflict of interest with any individual or organization for this work.
Ethics approval
This article does not contain any studies with human participants or animals performed by any of the authors.
Consent to participate
Informed consent was obtained from all individual participants included in the study.
Consent for publication
The participant has consented to the submission of the case report to the journal.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Deng, J., Deng, Y. Maximum entropy of random permutation set. Soft Comput 26, 11265–11275 (2022). https://doi.org/10.1007/s00500-022-07351-x
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00500-022-07351-x