Abstract
Uncertainties in fuzzy Markov chains can be treated in different ways. The use of interval type-2 fuzzy sets (IT2FS) allows describing the distributional behavior of an uncertain discrete-time Markov process through infinite type-1 fuzzy sets embedded in its Footprint of Uncertainty. In this way, a finite state fuzzy Markov chain process is defined in an interval type-2 fuzzy environment. To do so, its limiting properties and its type-reduced behavior are defined and applied to two explanatory examples.
Keywords
- Fuzzy Markov Chains
- Interval Type-2 Fuzzy Sets (IT2FS)
- Type-reduction Strategy
- Fuzzy Matrix
- Secondary Membership Function
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
As a function of the \(i_{th}\) state e.g. \(x(i)\).
- 2.
This matrix is also known as the Fuzzy Distribution of \(x\).
- 3.
Here, \(\oint \) denotes crisp integration.
References
Araiza, R., Xiang, G., Kosheleva, O., Skulj, D.: Under interval and fuzzy uncertainty, symmetric markov chains are more difficult to predict. In: 2007 Annual Meeting of the North American Fuzzy Information Processing Society (NAFIPS), vol. 26. IEEE, Los Alamitos, pp. 526–531 (2007)
Avrachenkov, K.E., Sanchez, E.: Fuzzy markov chains: specifities and properties. In: IEEE (ed.) 8th IPMU’2000 Conference, Madrid, Spain, pp. 1851–1856 (2000)
Avrachenkov, K.E., Sanchez, E.: Fuzzy markov chains and decision-making. Fuzzy Optim. Decis. Making 1(2), 143–159 (2002)
Buckley, J., Eslami, E.: Fuzzy markov chains: uncertain probabilities. Mathware Soft Comput. 9(1), 33–41 (2002)
Campos, M.A., Dimuro, G.P., da Rocha Costa, A.C., Kreinovich, V.: Computing 2-step predictions for interval-valued finite stationary markov chains. Technical Report UTEP-CS-03-20 (2003)
Figueroa, J.C.: Interval type-2 fuzzy markov chains: an approach. In: 2010 Annual Meeting of the IEEE North American Fuzzy Information Processing Society (NAFIPS), IEEE, pp. 1–6 (2010)
Figueroa, J.C., Kalenatic, D., Lopéz, C.A.: A simulation study on fuzzy markov chains. Commun. Comput. Inf. Sci. 15(1), 109–117 (2008)
Gavalec, M.: Reaching matrix period is NP-complete. Tatra Mt. Math. Publ. 12(1), 81–88 (1997)
Gavalec, M.: Periods of special fuzzy matrices. Tatra Mt. Math. Publ. 16(1), 47–60 (1999)
Gavalec, M.: Computing orbit period in max-min algebra. Discrete Appl. Math. 100(1), 49–65 (2000)
Gordon, P.: Cadenas de Markov Finitas y sus Aplicaciones. Editorial Hispano Europea, Barcelona (1967)
Grimmet, G., Stirzaker, D.: Probability and Random Processes. Oxford University Press, UK (2001)
Hillier, F., Lieberman, G.: Introduction to Operations Research. Mc Graw Hill, New York (2001)
Karnik, N.N., Mendel, J.M.: Type-2 fuzzy logic systems. IEEE Trans. Fuzzy Syst. 7(6), 643–658 (1999)
Karnik, N.N., Mendel, J.M.: Centroid of a type-2 fuzzy set. Inf. Sci. 132(1), 195–220 (2001)
Karnik, N.N., Mendel, J.M.: Operations on type-2 fuzzy sets. Fuzzy Sets Syst. 122, 327–348 (2001)
Kurano, M., Yasuda, M., Nakagami, J.: Interval methods for uncertain markov decision processes. In: International Workshop on Markov Processes and Controlled Markov Chains (1999)
Kurano, M., Yasuda, M., Nakagami, J., Yoshida, Y.: A fuzzy approach to markov decision processes with uncertain transition probabilities. Fuzzy Sets Syst. 157(1), 2674–2682 (2006)
Liang, Q., Mendel, J.M.: Interval type-2 fuzzy logic systems: theory and design. IEEE Trans. Fuzzy Syst. 8(5), 535–550 (2000)
Melgarejo, M.: Implementing interval type-2 fuzzy processors. IEEE Comput. Intell. Mag. 2(1), 63–71 (2007)
Melgarejo, M., Bernal, H., Duran, K.: Improved iterative algorithm for computing the generalized centroid of an interval type-2 fuzzy set. In: 2008 Annual Meeting of the North American Fuzzy Information Processing Society (NAFIPS), vol. 27. IEEE, pp. 1–6 (2008)
Melgarejo, M.A.: A fast recursive method to compute the generalized centroid of an interval type-2 fuzzy set. In: Annual Meeting of the North American Fuzzy Information Processing Society (NAFIPS), IEEE, pp. 190–194 (2007)
Mendel, J.: Uncertain Rule-Based Fuzzy Logic Systems: Introduction and New Directions. Prentice Hall, New Jersey (2001)
Mendel, J.: Fuzzy sets for words: a new beginning. In: The IEEE International Conference on Fuzzy Systems, pp. 37–42 (2003)
Mendel, J.M.: Type-2 Fuzzy Sets: Some Questions and Answers. IEEE coNNectionS. A publication of the IEEE Neural Networks. Society. 8, 10–13 (2003)
Mendel, J.M., John, R.I.: Type-2 fuzzy sets made simple. IEEE Trans. Fuzzy Syst. 10(2), 117–127 (2002)
Mendel, J.M., John, R.I., Liu, F.: Interval type-2 fuzzy logic systems made simple. IEEE Trans. Fuzzy Syst. 14(6), 808–821 (2006)
Mendel, J.M., Liu, F.: Super-exponential convergence of the Karnik-Mendel algorithms for computing the centroid of an interval type-2 fuzzy set. IEEE Trans. Fuzzy Syst. 15(2), 309–320 (2007)
Pang, C.T.: On the sequence of consecutive powers of a fuzzy matrix with max-archimedean t-norms. Fuzzy Sets Syst. 138(3), 643–656 (2003)
Ross, S.M.: Stochastic Processes. Wiley, New York (1996)
Sanchez, E.: Resolution of eigen fuzzy sets equations. Fuzzy Sets Syst. 1(1), 69–74 (1978)
Sanchez, E.: Eigen fuzzy sets and fuzzy relations. J. Math. Anal. Appl. 81(1), 399–421 (1981)
Skulj, D.: Regular finite markov chains with interval probabilities. In: 5th International Symposium on Imprecise Probability: Theories and Applications, Prague, Czech Republic (2007)
Stewart, W.J.: Introduction to the Numerical Solution of Markov Chains. Princeton University Press, Princeton (1994)
Symeonaki, M., Stamou, G.: Theory of Markov systems with fuzzy states. Fuzzy Sets Syst. 143(1), 427–445 (2004)
Thomason, M.: Convergence of powers of a fuzzy matrix. J. Math. Anal. Appl. 57(1), 476–480 (1977)
Wu, D., Mendel, J.M.: Enhanced Karnik-Mendel algorithms for interval type-2 fuzzy sets and systems. In: Annual Meeting of the North American Fuzzy Information Processing Society (NAFIPS), vol. 26. IEEE, pp. 184–189 (2007)
Zeng, J., Liu, Z.Q.: Interval type-2 fuzzy hidden Markov models. In: IEEE 2004 International Conference on Fuzzy Systems, vol. 2. IEEE, pp. 1123–1128 (2004)
Zeng, J., Liu, Z.Q.: Type-2 fuzzy markov random fields to handwritten character recognition. In: Proceedings of pattern recognition, 2006. ICPR 2006. 18th international conference on ICPR 2006, vol 1. pp. 1162–1165 (2006)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer Science+Business Media New York
About this chapter
Cite this chapter
Figueroa-García, J.C. (2013). Interval Type-2 Fuzzy Markov Chains. In: Sadeghian, A., Mendel, J., Tahayori, H. (eds) Advances in Type-2 Fuzzy Sets and Systems. Studies in Fuzziness and Soft Computing, vol 301. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-6666-6_4
Download citation
DOI: https://doi.org/10.1007/978-1-4614-6666-6_4
Published:
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4614-6665-9
Online ISBN: 978-1-4614-6666-6
eBook Packages: EngineeringEngineering (R0)