Abstract
We first recall the concept of Z-numbers introduced by Zadeh. These objects consist of an ordered pair (A, B) of fuzzy numbers. We then use these Z-numbers to provide information about an uncertain variable V in the form of a Z-valuation, which expresses the knowledge that the probability that V is A is equal to B. We show that these Z-valuations essentially induce a possibility distribution over probability distributions associated with V. We provide a simple illustration of a Z-valuation. We show how we can use this representation to make decisions and answer questions. We show how to manipulate and combine multiple Z-valuations. We show the relationship between Z-numbers and linguistic summaries. Finally we provide for a representation of Z-valuations in terms of Dempster-Shafer belief structures, which makes use of type-2 fuzzy sets.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Zadeh, L.A.: A note on Z-numbers. Information Science 181, 2923–2932 (2011)
Yager, R.R.: A new approach to the summarization of data. Information Sciences 28, 69–86 (1982)
Yager, R.R.: On linguistic summaries of data. In: Piatetsky-Shapiro, G., Frawley, B. (eds.) Knowledge Discovery in Databases, pp. 347–363. MIT Press, Cambridge (1991)
Zadeh, L.A.: Probability measures of fuzzy events. Journal of Mathematical Analysis and Applications 10, 421–427 (1968)
Ross, S.M.: Introduction to Probability and Statistics for Engineers and Scientists. Harcourt, San Diego (2000)
Zadeh, L.A.: Computing with Words-Principal Concepts and Idea. Springer, Berlin (to appear)
Yager, R.R.: On ordered weighted averaging aggregation operators in multi-criteria decision making. IEEE Transactions on Systems, Man and Cybernetics 18, 183–190 (1988)
Yager, R.R., Kacprzyk, J., Beliakov, G.: Recent Developments in the Ordered Weighted Averaging Operators: Theory and Practice. Springer, Berlin (2011)
Papoulis, A.: Probability, Random Variables and Stochastic Processes. McGraw-Hill, New York (1965)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Yager, R.R. (2012). On a View of Zadeh’s Z-Numbers. In: Greco, S., Bouchon-Meunier, B., Coletti, G., Fedrizzi, M., Matarazzo, B., Yager, R.R. (eds) Advances in Computational Intelligence. IPMU 2012. Communications in Computer and Information Science, vol 299. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-31718-7_10
Download citation
DOI: https://doi.org/10.1007/978-3-642-31718-7_10
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-31717-0
Online ISBN: 978-3-642-31718-7
eBook Packages: Computer ScienceComputer Science (R0)