Abstract
We first introduce the Dempster–Shafer belief structure and highlight its role in the representation of information about a random variable for which our knowledge of the probabilities is interval-valued. We investigate the formation of the cumulative distribution function (CDF) for these types of variables. It is noted that this is also interval-valued and is expressible in terms of plausibility and belief measures. The class of aggregation operators known as copulas are introduced and a number of their properties are provided. We discuss Sklar’s theorem, which provides for the use of copulas in the formulation of joint CDFs from the marginal CDFs of classic random variables. We then look to extend these ideas to the case of joining the marginal CDFs associated with Dempster–Shafer belief structures. Finally we look at the formulation CDFs obtained from functions of multiple D–S belief structures.
Similar content being viewed by others
References
Cherubini, U., Luciano, E., & Vecchiato, W. (2004). Copula methods in finance. New York: Wiley.
Dempster, A. P. (1967). Upper and lower probabilities induced by a multi-valued mapping. The Annals of Mathematical Statistics, 38, 325–339.
Dempster, A. P. (2008). The Dempster–Shafer calculus for statisticians. International Journal of Approximate Reasoning, 48, 365–377.
Denoeux, T., & Zouhal, L. M. (2001). Handling possibilistic labels in pattern classification using evidential reasoning. Fuzzy Sets and Systems, 122, 47–62.
Durante, F., & Sempi, C. (2010). Copula theory: An introduction. In P. Jaworski, F. Durante, W. Hardle, & T. Rychlik (Eds.), Copula theory and its applications (pp. 3–31). Berlin: Springer.
Fu, C., & Yang, S. L. (2011). Analyzing the applicability of Dempster’s rule to the combination of interval-valued belief structures. Expert Systems with Applications, 38, 4291–4301.
Jaffray, J. Y. (1994). Dynamic decision making with belief functions. In R. R. Yager, J. Kacprzyk, & M. Fedrizzi (Eds.), Advances in the Dempster–Shafer theory of evidence (pp. 331–352). New York: Wiley.
Janssens, S., De Baets, B., & De Meyer, H. (2004). Bell-type inequalities for quasi-copulas. Fuzzy Sets and Systems, 148, 263–278.
Jaworski, P., Durante, F., Hardle, W. K., & Rychlik, T. (2010). Copula theory and its application. Berlin: Springer.
Klement, E. P., Mesiar, R., & Pap, E. (2000). Triangular norms. Dordrecht: Kluwer.
Liu, L., & Yager, R. R. (2008). Classic works of the Dempster–Shafer theory of belief functions: An introduction. In R. R. Yager & L. Liu (Eds.), Classic works of the Dempster–Shafer theory of belief functions (pp. 1–34). Heidelberg: Springer.
Llinas, J., Nagi, R., Hall, D. L., & Lavery, J. (2010). A multi-disciplinary university research initiative in hard and soft information fusion: Overview, research strategies and initial results. In Proceedings of the 13th international conference on information fusion (Fusion 2010). Edinburgh, UK: Unpaginated.
Masson, M. H., & Denoeux, T. (2011). Ensemble clustering in the belief functions framework. International Journal of Approximate Reasoning, 52, 92–109.
McNeil, A. J., Frey, R., & Embrechts, P. (2005). Quantitative risk management. Princeton: Princeton University Press.
Moore, R. E. (1966). Interval analysis. Englewood Cliff, NJ: Prentice-Hall.
Nelsen, R. B. (1999). An introduction to copulas. New York: Springer.
Papoulis, A. (1965). Probability, random variables and stochastic processes. New York: McGraw-Hill.
Shafer, G. (1976). A mathematical theory of evidence. Princeton, NJ: Princeton University Press.
Sklar, A. (1959). Fonctions dérepartition à n dimensions et leurs marges. Publications of the Institute Statistics University Paris, 8, 229–231.
Sklar, A. (1973). Random variables, joint distributions and copulas. Kybernetica, 9, 449–460.
Smets, P., & Kennes, R. (1994). The transferable belief model. Artificial Intelligence, 66, 191–234.
Trivedi, P. K., & Zimmer, D. M. (2007). Copula modeling: An introduction for practitioners. Boston: Now.
Yager, R. R. (2004). Cumulative distribution functions from Dempster–Shafer belief structures. IEEE Transactions on Systems, Man and Cybernetics, Part B, 34, 2080–2087.
Yager, R. R. (2006). Modeling holistic fuzzy implication operators using co-copulas. Fuzzy Optimization and Decision Making, 5, 207–226.
Yager, R. R., & Liu, L. (2008). Classic works of the Dempster–Shafer theory of belief functions. Heidelberg: Springer.
Yager, R. R., Kacprzyk, J., & Fedrizzi, M. (1994). Advances in the Dempster–Shafer theory of evidence. New York: Wiley.
Yamada, K. (2008). A new combination of evidence based on compromise. Fuzzy Sets and Systems, 159, 1689–1708.
Acknowledgments
This work has been supported by a Multidisciplinary University Research Initiative (MURI) grant (Number W911NF-09-1-0392) for “Unified Research on Network-based Hard/Soft Information Fusion”, issued by the US Army Research Office (ARO). This work has also been supported by an ONR grant for “Human Behavior Modeling Using Fuzzy and Soft Technologies”, award number N000141010121. We gratefully appreciate this support
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Yager, R.R. Joint cumulative distribution functions for Dempster–Shafer belief structures using copulas. Fuzzy Optim Decis Making 12, 393–414 (2013). https://doi.org/10.1007/s10700-013-9163-z
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10700-013-9163-z