Abstract
The information volume of mass function based on extropy is proposed in this paper. Although the information volume of the probability distribution can be calculated by Shannon entropy, how to calculate the information of the mass function is still being explored. Recently, the concept of extropy was proposed by Lad et al. Based on extropy, the information volume of mass function is proposed in this paper. For a basic probability assignment function (BPA), if the focal elements of the frame of discernment (FOD) are all single elements, the information volume proposed in this paper is equal to the corresponding extropy. Otherwise, the information volume is greater than the corresponding extropy. Besides, when the cardinality of the FOD is identical, both the total uncertainty case and the mass function distribution of the maximum extropy have the same information volume. More precisely, the distribution of the latter can be regarded as the former obtained by decomposing the BPA once. Finally, the experiment proves that the maximum information volume increases with the increase in the cardinality of the FOD, and has the same limit value log\(_2e\) as the maximum extropy. Some numerical examples are given to prove the nature of the information volume.
Similar content being viewed by others
Availability of data and materials
The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.
Code Availability Statement
Not applicable.
References
Al-Labadi L, Berry S (2020) Bayesian estimation of extropy and goodness of fit tests. J Appl Stat. https://doi.org/10.1080/02664763.2020.1812545
Almarashi AM, Algarni A, Abdel-Khalek S, Abd-Elmougod GA, Raqab MZ (2020) Quantum extropy and statistical properties of the radiation field for photonic binomial and even binomial distributions. J Russ Laser Res 41(4):334–343
Becerra A, Ismael de la Rosa J, Gonzalez E, David A, Iracemi Escalante N (2018) Training deep neural networks with non-uniform frame-level cost function for automatic speech recognition. Multimed Tools Appl 77(20):27231–27267
Becerra A, Ismael de la Rosa J, Gonzalez E, Pedroza AD, Iracemi Escalante N, Santos E (2020) A comparative case study of neural network training by using frame-level cost functions for automatic speech recognition purposes in Spanish. Multimed Tools Appl 79(27–28):19669–19715
Bortot S, Pereira RAM, Stamatopoulou A (2020) Shapley and super Shapley aggregation emerging from consensus dynamics in the multicriteria Choquet framework. Decis Econ Finance 43(2):583–611
Buono F, Longobardi M (2020) A dual measure of uncertainty: the Deng extropy. Entropy 22(5):582
Cao Z, Ding W, Wang Y, Hussain FK, Al-Jumaily A, Lin CT (2020) Effects of repetitive SSVEPS on EEG complexity using multiscale inherent fuzzy entropy. Neurocomputing 389:198–206
Chen L, Deng Y, Cheong KH (2021) Probability transformation of mass function: a weighted network method based on the ordered visibility graph. Eng Appl Artif Intell 105:104438. https://doi.org/10.1016/j.engappai.2021.104438
Dempster AP (1967) Upper and lower probabilities induced by a multivalued mapping. Ann Math Stat 38(2):325–339
Deng Y (2020a) Information volume of mass function. Int J Comput Commun Control 15(6):3983
Deng Y (2020b) Uncertainty measure in evidence theory. Sci China Inf Sci 63(11):210201
Deng J, Deng Y (2021) Information volume of fuzzy membership function. Int J Comput Commun Control. https://doi.org/10.15837/ijccc.2021.1.4106
Deng X, Jiang W (2019) A total uncertainty measure for D numbers based on belief intervals. Int J Intell Syst 34(12):3302–3316
Gao X, Su X, Qian H, Pan X (2021) Dependence assessment in human reliability analysis under uncertain and dynamic situations. Nuclear Eng Technol(accepted)
Gilio A, Sanfilippo G (2019) Generalized logical operations among conditional events. Appl Intell 49(1):79–102
Hohle U (1987) A general theory of fuzzy plausibility measures. J Math Anal Appl 127(2):346–364
James RG, Crutchfield JP (2017) Multivariate dependence beyond Shannon information. Entropy 19(10):531
Jahanshahi SMA, Zarei H, Khammar AH (2020) On cumulative residual extropy. Probab Eng Inf Sci 34(4):605–625
Jose J, Sathar EIA (2019) Residual extropy of k-record values. Stat Probab Lett 146:1–6
Jousselme A-L, Liu C, Grenier D, Bosse E (2005) Measuring ambiguity in the evidence theory. IEEE Trans Syst Man Cybern A Syst Hum 36(5):890–903
Kamari O, Buono F (2020) On extropy of past lifetime distribution. Ricerche di Matematica. https://doi.org/10.1007/s11587-020-00488-7
Krishnan AS, Sunoj SM, Nair N (2020) Some reliability properties of extropy for residual and past lifetime random variables. J Korean Stat Soc 49(2):457–474
Lad F, Sanfilippo G, Agrò G (2015) Extropy: complementary dual of entropy Stat Sci 30(1):40–58
Lad F, Sanfilippo G, Agrò G (2018) The duality of entropy/extropy, and completion of the Kullback information complex. Entropy 20(8). https://doi.org/10.3390/e20080593
Maya R, Irshad MR (2019) Kernel estimation of residual extropy function under alpha-mixing dependence condition. S Afr Stat J 53(2):65–72
Noughabi HA, Jarrahiferiz J (2019) On the estimation of extropy. J Nonparametr Stat 31(1):88–99
Noughabi HA, Jarrahiferiz J (2020) Extropy of order statistics applied to testing symmetry. Commun Stat Simul Comput 1:1–11. https://doi.org/10.1080/03610918.2020.1714660
Qiu G (2017) The extropy of order statistics and record values. Stat Probab Lett 120:52–60
Qiu G, Jia K (2018) The residual extropy of order statistics. Stat Probab Lett 133:15–22
Raqab MZ, Qiu G (2019) On extropy properties of ranked set sampling. Statistics 53(1–3):210–226
Rényi A (1961) On measures of entropy and information. In: Proceedings IV Berkeley symposium on mathematical statistics and probability, vol 1, pp 547–561
Sanfilippo G, Gilio A, Over DE, Pfeifer N (2020) Probabilities of conditionals and previsions of iterated conditionals. Int J Approx Reason 121:150–173
Sathar EIA, Nair DR (2021) On dynamic survival extropy. Commun Stat Theory Methods 50(6):1295–1313
Shannon CE (1948) Mathematical theory of communication. Bell Syst Tech J 27(3):379–423
Singh S, Lalotra S, Sharma S (2019) Dual concepts in fuzzy theory: entropy and knowledge measure. Int J Intell Syst 34(5):1034–1059
Song Y, Deng Y (2021) Entropic explanation of power set. Int J Comput Commun Control 16(4):4413
Srivastava A, Kaur L (2019) Uncertainty and negation-information theoretic applications. Int J Intell Syst 34(6):1248–1260
Tahmasebi S, Toomaj A (2020) On negative cumulative extropy with applications. Commun Stat Theory Methods 5:1–23. https://doi.org/10.1080/03610926.2020.1831541
Tsallis C (1988) Possible generalization of Boltzmann–Gibbs statistics. J Stat Phys 52(1–2):479–487
Weiss CH (2019) Measures of dispersion and serial dependence in categorical time series. Econometrics 7(2):17
Xiao F (2020) On the maximum entropy negation of a complex-valued distribution. IEEE Trans Fuzzy Syst. https://doi.org/10.1109/TFUZZ.2020.3016723
Xiao F (2021a) CEQD: a complex mass function to predict interference effects. IEEE Trans Cybern. https://doi.org/10.1109/TCYB.2020.3040770
Xiao F (2021b) CaFtR: a fuzzy complex event processing method. Int J Fuzzy Syst. https://doi.org/10.1007/s40815-021-01118-6
Xiao F (2021c) GIQ: a generalized intelligent quality-based approach for fusing multi-source information. IEEE Trans Fuzzy Syst 29(7):2018–2031
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
Xue Y, Deng Y (2021a) Tsallis eXtropy. Commun Stat Theory Methods. https://doi.org/10.1080/03610926.2021.1921804
Xue Y, Deng Y (2021b) Interval-valued belief entropies for Dempster–Shafer structures. Soft Comput 25:8063–8071
Yager RR (1983) Entropy and specificity in a mathematical theory of evidence. Int J Gen Syst 9(4):249–260
Yang J, Xia W, Hu T (2019) Bounds on extropy with variational distance constraint. Probab Eng Inf Sci 33(2):186-204
Zadeh LA (1965) Fuzzy sets. Inf Control 8(3):338–353
Funding
This research is supported by the National Natural Science Foundation of China (No. 62003280). The authors greatly appreciate the reviewers’ suggestions and the editor’s encouragement.
Author information
Authors and Affiliations
Contributions
All authors contributed to the study conception and design. Material preparation, data collection and analysis were performed by FX. The first draft of the manuscript was written by JL and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.
Corresponding author
Ethics declarations
Conflict of interest
The authors state that there are no conflict of interest.
Ethics approval
This article does not contain any studies with human participants or animals performed by any of the authors.
Consent to participate
Not applicable.
Consent for publication
Not applicable.
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
Liu, J., Xiao, F. Information volume of mass function based on extropy. Soft Comput 26, 2409–2418 (2022). https://doi.org/10.1007/s00500-021-06410-z
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00500-021-06410-z