Nonparametric Bayesian approach to the detection of change point in statistical process control

Abstract

This paper gives an intensive overview of nonparametric Bayesian model relevant to the determination of change point in a process control. We first introduce statistical process control and develop on it describing Bayesian parametric methods followed by the nonparametric Bayesian modeling based on Dirichlet process. This research proposes a new nonparametric Bayesian change point detection approach which in contrast to the Markov approach of Chib [6] uses the Dirichlet process prior to allow an integrative transition of probability from the posterior distribution. Although the Bayesian nonparametric technique on the mixture does not serve as an automated tool for the selection of the number of components in the finite mixture. The Bayesian nonparametric mixture shows a misspecication model properly which has been explained further in the methodology. This research shows the principal step-bystep algorithm using nonparametric Bayesian technique with the Dirichlet process prior defined on the distribution to the detection of change point. This approach can be further extended in the multi-variate change point detection which will be studied in the near future.

Keywords

• Nonparametric,Bayesian,Change point,Clustering,Mixture model,Dirichlet process

References

1. Bhattacharya Some aspects of change-point analysis,In Change Point Problems E. Carlstein, H.G. Muller and D. Siegmund (eds.), (IMS Lecture Notes-Monograph Series, 1994) 23, 28- 56.
2. Bolton, R. and Hand, D. Statistical Fraud Detection: A Review, Statistical Science, 17, 235-225, 2002.
3. Brodsky, B.E. and Darkhovsky B.S. Nonparametric methods in change-point problems, (Kluwer Academic Publ., The Netherlands, 1993).
4. Broemeling, L.D. Bayesian procedures for detecting a change in a sequence of random vari- ables, Metron , 30, 214-227, 1972.
5. Cappe, O. and Harchaoui, Z. Retrospective multiple change-point estimation with kernels, IEEE Computer Society Statistical Signal Processing, IEEE/SP Workshop on, 768-772, 2007.
6. Chib, S. Estimation and comparison of multiple change-point models, Journal of Economet- rics, 86, 221-241, 1998.
7. Cobb, G.W. The problem of the Nile, Biometrika, 65, 243-251, 1978.
8. Eckley, I.A., Fearnhead, P. and Killick, R. Analysis of Changepoint Models, (Cambridge University Press, In D Barber, AT Cemgil, S Chiappa (eds.), Bayesian Time Series Models, 2011).

9. Kim, A.Y. et al. Using labeled data to evaluate change detectors in a multivariate streaming environment, Signal Process, 89, 2529-2536, 2009.
10. Lorden, G. Procedures for Reacting to a Change in Distribution, The Annals of Mathemat- ical Statistics, 42(6), 1897-1908, 1971.
11. Lung-Yut-Fong, A., Levy-Leduc, C., and Cappe, Homogeneity and Change-Point Detection Tests for Multivariate Data Using Rank Statistics, arXiv, 1107.1971, 2011.
12. Moustakides G.V. Optimal stopping times for detecting changes in distributions, Ann. Statist. 14(4), 1379-1387, 1986.
13. Muliere P. and Scarsini M. Change-point problems: A Bayesian nonparametric approach, Aplikace Matematiky, 30, 397-402, 1985.
14. Page, E. Continuous Inspection Schemes, Biometrika, 14, 100-115, 1954.
15. Petrone S. and Raftery, A.E. A note on the Dirichlet process prior in Bayesian nonpara- metric inference with partial exchangeability, Statistics & Probability Letters, 36, 69-83, 1997.
16. Pettit, A.N. Posterior probabilities for a change-point using ranks, Biometrika, 68, 443-450, 1981.
17. Pitman, J. and Yor, M. The two-parameter Poisson-Dirichlet distribution derived from a stable subordinator, Ann. Probab., 25(2), 855-900, 1997.
18. Rigaill, G. Pruned Dynamic Programming for Optimal Multiple Change-Point Detection, arXiv:1004.0887, 2010.
19. Shiryaev, A.N. On optimum methods in quickest detection problems, Theory of Probability and Its Applications, 8, 22-46, 1963.
20. Smith, A.F.M. A Bayesian approach to inference about a change point in a sequence of random variables, Biometrika , 62, 407-416, 1975.
21. Smith, A.F.M. A Bayesian analysis of some time-varying models, (In Recent Developments in Statistics, eds. Barra, J.R. et. al., North-Holland, Amsterdam, 1977), 257-267.
22. Smith, A.F.M Change-point problems: approaches and applications, Trab. Estadist., 31, 83-98, 1980.
23. Talih M. and Hengartner, N. Structural Learning With Time-Varying Components: Track- ing the Cross-Section of Financial Time Series, Journal of the Royal Statistical Society, 67, 321-341, 2005.
24. Zacks, S. Classical and Bayesian approaches to the change-point problem: Fixed sample and sequential procedures, Stat. Anal. Donnees, 7, 48-81, 1982.