Communications for Statistical Applications and Methods

The Korean Statistical Society (한국통계학회)
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Bayesian Multiple Change-Point Estimation and Segmentation

  • Kim, Jaehee (Department of Statistics, Duksung Women's University) ;
  • Cheon, Sooyoung (Department of Informational Statistics, Korea University)
  • Received : 2013.07.12
  • Accepted : 2013.09.30
  • Published : 2013.11.30
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Abstract

This study presents a Bayesian multiple change-point detection approach to segment and classify the observations that no longer come from an initial population after a certain time. Inferences are based on the multiple change-points in a sequence of random variables where the probability distribution changes. Bayesian multiple change-point estimation is classifies each observation into a segment. We use a truncated Poisson distribution for the number of change-points and conjugate prior for the exponential family distributions. The Bayesian method can lead the unsupervised classification of discrete, continuous variables and multivariate vectors based on latent class models; therefore, the solution for change-points corresponds to the stochastic partitions of observed data. We demonstrate segmentation with real data.

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References

  1. Adams, R. and Mackay, D. (2007). Bayesian Online Changepoint Detection, University of Cambridge Technical Report.
  2. Barry, D. and Hartigan, J. (1993). A Bayesian analysis for change-point problems, Journal of the American Statistical Association, 88, 309-319.
  3. Bartels, R. (1982). The rank version of von Neumann's Ratio test for randomness, Journal of the American Statistical Association, 77, 40-46. https://doi.org/10.1080/01621459.1982.10477764
  4. Belisle, P., Joseph, L., MacGibbon, B., Wolfson, D. and du Berger, R. (1998). Change-point analysis of neuron spike train data, Biometrics, 54, 113-123. https://doi.org/10.2307/2534000
  5. Carlin, B., Gelfand, A. and Smith, A. F. M. (1992). Hierarchical Bayesian analysis of change point problems, Applied Statistics, 41, 389-405. https://doi.org/10.2307/2347570
  6. Cheon, S. and Kim, J. (2010). Multiple change-point detection of multivariate mean vectors with Bayesian approach, Computational Statistics & Data Analysis, 54, 406-415. https://doi.org/10.1016/j.csda.2009.09.003
  7. Chernoff, H. and Zacks, S. (1964). Estimating the current mean of a normal distribution which is subject to changes in time, Annals of Mathematical Statistics, 35, 999-1018. https://doi.org/10.1214/aoms/1177700517
  8. Chib, S. (1998). Estimation and comparison of multiple change-point models, Journal of Econometrics, 86, 221-241. https://doi.org/10.1016/S0304-4076(97)00115-2
  9. Corander, J., Gyllenberg, M. and Koski, T. (2009). Bayesian unsupervised classification framework based on stochastic partitions of data and a parallel search strategy, Advanced Data Analysis and Classification, 3, 3-24. https://doi.org/10.1007/s11634-009-0036-9
  10. Cox, D. and Lewis, P. (1966). Methuen's Monographs on Applied Probability and Statistics, In The Statistical Analysis of Series of Events, John Wiley, London.
  11. Fearnhead, P. (2006). Exact and efficient Bayesian inference for multiple changepoint problems, Statistics and Computing, 16, 203-213. https://doi.org/10.1007/s11222-006-8450-8
  12. Fearnhead, P. and Clifford, P. (2003). Online inference for hidden Markov models, Journal of the Royal Statistical Society Series B, 65, 887-899. https://doi.org/10.1111/1467-9868.00421
  13. Gelfand, A. and Smith, A. (1990). Sampling-based approaches to calculating marginal densities, Journal of the American Statistical Association, 85, 398-409. https://doi.org/10.1080/01621459.1990.10476213
  14. Gelman, A., Carlin, J., Stern, H. and Rubin, D. (2003). Bayesian Data Analysis, Chapman and Hall, New York.
  15. Green, P. (1995). Reversible jump Markov chain Monte Carlo computation and Bayesian model determination, Biometrika, 82, 711-732. https://doi.org/10.1093/biomet/82.4.711
  16. Hastings, W. K. (1970). Monte Carlo sampling methods using Markov Chains and their applications, Biometrika, 57, 97-109. https://doi.org/10.1093/biomet/57.1.97
  17. Hawkins, D. (2001). Finding multiple change-point models to data, Computational Statistics & Data Analysis, 37, 323-341. https://doi.org/10.1016/S0167-9473(00)00068-2
  18. Hinkley, D. (1970). Inference about the change-point in a sequence of random variables, Biometrika, 57, 1-17. https://doi.org/10.1093/biomet/57.1.1
  19. Hinkley, D. (1972). Time-ordered classification, Biometrika, 59, 509-523. https://doi.org/10.1093/biomet/59.3.509
  20. Holmes, D. and Mergen, A. (1993). Improving the performance of the $T^2$ control chart, Quality Engineering, 5, 619-625. https://doi.org/10.1080/08982119308919004
  21. Hsu, D. (1979). Detecting shifts of parameter in gamma sequences with applications to stock price and air traffic flow analysis, Journal of the American Statistical Association, 74, 31-40. https://doi.org/10.1080/01621459.1979.10481604
  22. Jarrett, R. (1979). A note on the intervals between coal-mining disasters, Biometrika, 66, 191-193. https://doi.org/10.1093/biomet/66.1.191
  23. Kass, R. and Raftery, A. (1995). Bayes factors, Journal of the American Statistical Association, 90, 773-795. https://doi.org/10.1080/01621459.1995.10476572
  24. Kim, J. and Cheon, S. (2010). Bayesian multiple change-point estimation with annealing stochastic approximation Monte Carlo, Computational Statistics, 25, 215-239. https://doi.org/10.1007/s00180-009-0172-x
  25. Kojadinovic, I. and Holmes, M. (2009). Tests of independence among continuous random vectors based on Cramer-von Mises functionals of the empirical copula process, Journal of Multivariate Analysis, 100, 1137-1154. https://doi.org/10.1016/j.jmva.2008.10.013
  26. Lee, C.-B. (1998). Bayesian analysis of a change-point in exponential families with applications, Computational Statistics & Data Analysis, 27, 195-208. https://doi.org/10.1016/S0167-9473(98)00009-7
  27. Liang, F., Liu, R. and Carroll, R. (2007). Stochastic approximation in Monte Carlo computation, Journal of the American Statistical Association, 102, 305-320. https://doi.org/10.1198/016214506000001202
  28. Maguire, B., Pearson, E. and Wynn, A. (1952). The time intervals between industrial accidents, Biometrika, 38, 168-180.
  29. Metropolis, N., Rosenbluth, A. W., Rosenbluth, M. N., Teller, A. H. and Teller, E. (1953). Equations of state calculations by fast computing machines, Journal of Chemical Physics, 21, 1087-1091. https://doi.org/10.1063/1.1699114
  30. O Ruanaidh, J. and Fitzgerald, W. (1996). Bayesian Online Changepoint Detection, Springer, New York.
  31. Raftery, A. (1995). Bayesian model selection in social research, In Marsden V (Ed) Sociological methodology, 25, 111-163. https://doi.org/10.2307/271063
  32. Raftery, A. and Akman, V. (1986). Bayesian analysis of a Poisson process with a change-point, Biometrika, 73, 85-89. https://doi.org/10.1093/biomet/73.1.85
  33. Robbins, H. and Monro, S. (1951). A stochastic approximation method, Annals of Mathematical Statistics, 22, 400-407. https://doi.org/10.1214/aoms/1177729586
  34. Siegmund D. (1988). Confidence sets in change point problem, International Statistical Review, 56, 31-48. https://doi.org/10.2307/1403360
  35. Smith, A. F. M. (1975). A Bayesian approach to inference about a change-point in a sequence of random variables, Biometrika, 62, 407-416. https://doi.org/10.1093/biomet/62.2.407
  36. Stephens, D. (1994). Bayesian retrospective multiple-change-point identification, Applied Statistics, 43, 159-178. https://doi.org/10.2307/2986119
  37. Sullivan, J. and Woodall, W. (2000). Change-point detection of mean vector or covariance matrix shifts using multivariate individual observations, IIE Transactions, 32, 537-549.
  38. Venter, J. and Steel, S. (1996). Finding multiple abrupt change-points, Computational Statistics & Data Analysis, 22, 481-504. https://doi.org/10.1016/0167-9473(96)00007-2
  39. Wehrens, R., Buydens, L. M. C., Fraley, C. and Raftery, A. E. (2004). Model-based clustering for image segmentation and large datasets via sampling, Journal of Classification, 21, 231-253. https://doi.org/10.1007/s00357-004-0018-8
  40. Wu, Y. (2005). Inference for change point and post change means after a CUSUM test, Lecture notes in Statistics, 180.
  41. Yao, Y. (1984). Estimation of a noisy discrete-time step function: Bayes and empirical Bayes approaches, Annals of Statistics, 12, 1434-1447. https://doi.org/10.1214/aos/1176346802

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