Abstract
In uncertainty theory, distance is defined as the difference value between uncertain variables. A triangle inequality \(d(\xi ,\eta )\le 2(d(\xi ,\tau )+d(\tau ,\eta ))\) has been proved before. This paper shows that the triangle inequality is tight. In addition, an inequality about expected value is discussed.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Data availability
Enquiries about data availability should be directed to the authors.
References
Chen X, Li J, Xiao C, Yang P (2021) Numerical solution and parameter estimation for uncertain SIR model with application to COVID-19 pandemic. Fuzzy Optim Decis Mak 20(2):189–208
Dai W (2018) Quadratic entropy of uncertain variables. Soft Comput 22(17):5699–5706
Gao X (2009) Some properties of continuous uncertain measure. Int J Uncertain Fuzziness Knowl-Based Syst 17(3):419–426
Gao X, Jia L, Kar S (2018) A new definition of cross-entropy for uncertain variables. Soft Comput 22(17):5617–5623
Jia L, Chen W (2021) Uncertain SEIAR model for COVID-19 cases in China. Fuzzy Optim Decis Mak 20(2):243–259
Lio W, Liu B (2021) Initial value estimation of uncertain differential equations and zero-day of COVID-19 spread in China. Fuzzy Optim Decis Mak 20(2):177–188
Liu B (2007) Uncertainty theory, 2nd edn. Springer-Verlag, Berlin
Liu B (2009) Some research problems in uncertainty theory. J Uncertain Syst 3(1):3–10
Liu B (2010) Uncertainty theory: a branch of mathematics for modeling human uncertainty. Springer-Verlag, Berlin
Liu B (2012) Why is there a need for uncertainty theory. J Uncertain Syst 6(1):3–10
Liu Y (2021) Uncertain circuit equation. J Uncertain Syst 14(3):2150018
Liu Z (2021) Uncertain growth model for the cumulative number of COVID-19 infections in China. Fuzzy Optim Decis Mak 20(2):229–242
Liu Y, Ha M (2010) Expected value of function of uncertain variables. J Uncertain Syst 4(3):181–186
Liu Y, Lio W (2020) A revision of sufficient and necessary condition of uncertainty distribution. J Intell Fuzzy Syst 38(4):4845–4854
Liu Y, Liu B (2022) Residual analysis and parameter estimation of uncertain differential equations. Fuzzy Optim Decis Mak 21(4):513–530
Ma G (2021) A remark on the maximum entropy principle in uncertainty theory. Soft Comput 25(22):13911–13920
Ning Y, Ke H, Fu Z (2015) Triangular entropy of uncertain variables with application to portfolio selection. Soft Comput 19(8):2203–2209
Peng Z, Iwamura K (2010) A sufficient and necessary condition of uncertainty distribution. J Interdiscip Math 13(3):277–285
Peng Z, Iwamura K (2012) Some discussions on uncertain measure. J Uncertain Syst 6(4):263–269
Tang H, Yang X (2021) Uncertain chemical reaction equation. Appl Math Comput 411:126479
Yang X (2013) On comonotonic functions of uncertain variables. Fuzzy Optim Decis Mak 12(1):89–98
Yao K (2015) A formula to calculate the variance of uncertain variable. Soft Comput 19(10):2947–2953
Ye T, Liu B (2023) Uncertain hypothesis test for uncertain differential equations. Fuzzy Optim Decis Mak 22(2):195–211
Ye T, Yang X (2021) Analysis and prediction of confirmed cases of COVID-19 in China by uncertain time series. Fuzzy Optim Decis Mak 20(2):209–228
Ye T, Zheng H (2021) Analysis of birth rates in China with uncertain time series model, Technical Report
Zhang Z (2011) Some discussions on uncertain measure. Fuzzy Optim Decis Mak 10(1):31–43
Funding
This work was supported by National Natural Science Foundation of China (Grant No. 62203026) and the Funding of Science and Technology on Reliability and Environmental Engineering Laboratory, China (No.6142004220101).
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflicts of interest
The authors declare that they have no conflict of interest regarding the publication of this paper.
Ethical approval
This article does not contain any studies with human participants performed by the author.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Jia, Y., Lio, W. Tightness of triangle inequality in uncertainty theory. Soft Comput 27, 14621–14630 (2023). https://doi.org/10.1007/s00500-023-09045-4
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00500-023-09045-4