Hostname: page-component-76fb5796d-r6qrq Total loading time: 0 Render date: 2024-04-26T05:04:04.879Z Has data issue: false hasContentIssue false
  • English
  • Français

On Perturbation Bounds for the Joint Stationary Distribution of Multivariate Markov Chain Models

Published online by Cambridge University Press:  28 May 2015

Wen Li ,
Lin Jiang ,
Wai-Ki Ching  and
Lu-Bin Cui
Wen Li*
Affiliation:
School of Mathematical Sciences, South China Normal University, Guangzhou, 510631, P.R. China
Lin Jiang*
Affiliation:
School of Mathematical Sciences, South China Normal University, Guangzhou, 510631, P.R. China
Wai-Ki Ching*
Affiliation:
Advanced Modeling and Applied Computing Laboratory, Department of Mathematics, The University of Hong Kong, Pokfulam Road, Hong Kong
Lu-Bin Cui*
Affiliation:
School of Mathematical Sciences, South China Normal University, Guangzhou, 510631, P.R. China
*
Corresponding author. Email: liwen@scnu.edu.cn
Corresponding author. Email: lia051112@163.com
Corresponding author. Email: wching@hku.hk
Corresponding author. Email: hnzkc@163.com
Get access

Abstract

Multivariate Markov chain models have previously been proposed in for studying dependent multiple categorical data sequences. For a given multivariate Markov chain model, an important problem is to study its joint stationary distribution. In this paper, we use two techniques to present some perturbation bounds for the joint stationary distribution vector of a multivariate Markov chain with s categorical sequences. Numerical examples demonstrate the stability of the model and the effectiveness of our perturbation bounds.

Keywords

65M1078A48Multivariate Markov chain modelsstationary distribution vectorcondition numberrelative bound
Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 3 , Issue 1 , February 2013 , pp. 1 - 17
Copyright
Copyright © Global-Science Press 2013

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Cao, X., The Relations Among Potentials, Perturbation Analysis, and Markov Decision Processes, Discrete Event Dyn. Syst.: Theory Appl., 8 (1998) 71–87.Google Scholar
[2]Ching, W., Fung, E. and Ng, M., A Multivariate Markov Chain Model for Categorical Data Sequences and Its Applications in Demand Prediction, IMA J. Manag. Math., 13 (2002) 187–199.Google Scholar
[3]Ching, W. and Ng, M., Markov Chains: Models, Algorithms and Applications, International Series on Operations Research and Management Science, Springer, New York, 2006.Google Scholar
[4]Ching, W., Ng, M. and Fung, E., Higher-order Multivariate Markov Chains and Their Applications, Lin. Alg. Appl., 428 (2008) 492–507.Google Scholar
[5]Cho, G. and Meyer, C., Markov Chain Sensitivity Measured by Mean First Passage Times, Technical Report no. 112242-0199, Series 3.25.984, March 1999, North Carolina State University, 1999.Google Scholar
[6]Funderlic, R., Meyer, C., Sensitivity of the Stationary Distribution Vector for an Ergodic Markov Chain, Lin. Alg. Appl., 76 (1986) 1–17.Google Scholar
[7]Golub, G. and Meyer, C., Using QR Factorization and Group Inversion to Compute, Differentiate, and Estimate the Sensitivity of Stationary Probabilities for Markov Chains, SIAM J. Algebra. Discrete Math., 7 (1985) 273–281.Google Scholar
[8]Funderlic, R. and Meyer, C., Perturbation Bounds for the Stationary Distribution Vector for an Ergodic Markov Chain, Lin. Alg. Appl. 76 (1986) 1–17.CrossRefGoogle Scholar
[9]Horn, R. and Johnson, C., Matrix Analysis, Cambridge University Press, Cambridge, 1985.Google Scholar
[10]Ipsen, I. and Meyer, C., Uniform Stability of Markov Chains, SIAM J. Matrix Anal. Appl., 15 (1994) 1061–1074.Google Scholar
[11]Kirkland, S., Neumann, M. and Shader, B., Applications of Paz's inequality to Perturbation Bounds for Markov Chains, Lin. Alg. Appl, 268 (1998) 183–196.CrossRefGoogle Scholar
[12]Li, W., Cui, L. and Ng, M., On Computation of the Steady-State Probability Distribution of Probabilistic Boolean Networks with Gene Perturbation, J. Comput. Appl. Math., 236 (2012) 4067–4081.Google Scholar
[13]Li, X., Li, W., Ye, Y. and Ng, M., On Multivariate Markov Chains for Common and Non-common Objects in Multiple Networks, Numer. Math. Theory Meth. Appl., 5 (2012) 384–402.Google Scholar
[14]Meyer, C., The Role of the Group Generalized Inverse in the Theory of Finite Markov Chains, SIAM Review, 17 (1975) 443–464.CrossRefGoogle Scholar
[15]Meyer, C., The Condition Number of a Finite Markov Chain and Perturbation Bounds for the Limiting Probabilities, SIAM J. Algebra. Discrete Math., 1 (1980) 273–283.CrossRefGoogle Scholar
[16]Schweitzer, P., Perturbation Theory and Finite Markov Chains, J. Appl. Prob., 5 (1968) 401–413.CrossRefGoogle Scholar
[17]Siu, T., Ching, W., Ng, M. and Fung, E., On a Multivariate Markov Chain Model for Credit Risk Measurement, Quant. Finance, 5 (2005) 543–556.Google Scholar
[18]Wei, Y., Perturbation Analysis of Singular Linear Systems with Index One, Int. J. Comput. Math., 74 (2000) 483–491.Google Scholar
[19]Xue, J., Blockwise Perturbation Theory for Nearly Uncoupled Markov Chains and Its Application, Lin. Alg. Appl., 326 (2001) 173–191.Google Scholar
[20]Zhu, D. and Ching, W., A Note on the Stationary Property of High-dimensional Markov Chain Models, Int. J. Pure Appl. Math., 66 (2011) 321–330.Google Scholar