Statistics and Computing

, Volume 21, Issue 2, pp 217–229 | Cite as

Efficient Bayesian analysis of multiple changepoint models with dependence across segments

  • Paul FearnheadEmail author
  • Zhen Liu


We consider Bayesian analysis of a class of multiple changepoint models. While there are a variety of efficient ways to analyse these models if the parameters associated with each segment are independent, there are few general approaches for models where the parameters are dependent. Under the assumption that the dependence is Markov, we propose an efficient online algorithm for sampling from an approximation to the posterior distribution of the number and position of the changepoints. In a simulation study, we show that the approximation introduced is negligible. We illustrate the power of our approach through fitting piecewise polynomial models to data, under a model which allows for either continuity or discontinuity of the underlying curve at each changepoint. This method is competitive with, or outperform, other methods for inferring curves from noisy data; and uniquely it allows for inference of the locations of discontinuities in the underlying curve.

Changepoint detection Particle filters Sequential Monte Carlo Segmentation Wavelets Well-log 


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  1. Abramovich, F., Sapatinas, T., Silverman, B.W.: Wavelet thresholding via a Bayesian approach. J. R. Stat. Soc., Ser. B 60, 725–749 (1998) zbMATHCrossRefMathSciNetGoogle Scholar
  2. Barber, S., Nason, G.P.: Real nonparametric regression using complex wavelets. J. R. Stat. Soc., Ser. B 66, 927–939 (2004) zbMATHCrossRefMathSciNetGoogle Scholar
  3. Barber, S., Nason, G.P., Silverman, B.W.: Posterior probability intervals for wavelet thresholding. J. R. Stat. Soc., Ser. B 64, 189–205 (2002) zbMATHCrossRefMathSciNetGoogle Scholar
  4. Barry, D., Hartigan, J.A.: Product partition models for change point problems. Ann. Stat. 20, 260–279 (1992) zbMATHCrossRefMathSciNetGoogle Scholar
  5. Blom, H.A.P., Bar-Shalom, Y.: The interacting multiple model algorithm for systems with Markovian switching coefficients. IEEE Trans. Autom. Control 33, 780–783 (1988) zbMATHCrossRefGoogle Scholar
  6. Chib, S.: Calculating posterior distributions and modal estimates in Markov mixture models. J. Econom. 75, 79–98 (1996) zbMATHCrossRefMathSciNetGoogle Scholar
  7. Chib, S.: Estimation and comparison of multiple change-point models. J. Econom. 86, 221–241 (1998) zbMATHCrossRefMathSciNetGoogle Scholar
  8. Denison, D.G.T., Mallick, B.K., Smith, A.F.M.: Automatic Bayesian curve fitting. J. R. Stat. Soc., Ser. B 60, 333–350 (1998) zbMATHCrossRefMathSciNetGoogle Scholar
  9. DiMatteo, I., Genovese, C.R., Kass, R.E.: Bayesian curve-fitting with free-knot splines. Biometrika 88, 1055–1071 (2001) zbMATHCrossRefMathSciNetGoogle Scholar
  10. Dobigeon, N., Toumeret, J.Y.: Joint segmentation of wind speed and direction using a hierarchical model. Comput. Stat. Data Anal. 51, 5603–5621 (2007) zbMATHCrossRefGoogle Scholar
  11. Donoho, D.L., Johnstone, I.M.: Ideal spatial adaptation by wavelet shrinkage. Biometrika 81, 425–455 (1994) zbMATHCrossRefMathSciNetGoogle Scholar
  12. Fearnhead, P.: Exact Bayesian curve fitting and signal segmentation. IEEE Trans. Signal Process. 53, 2160–2166 (2005) CrossRefMathSciNetGoogle Scholar
  13. Fearnhead, P.: Exact and efficient inference for multiple changepoint problems. Stat. Comput. 16, 203–213 (2006) CrossRefMathSciNetGoogle Scholar
  14. Fearnhead, P.: Computational methods for complex stochastic systems: a review of some alternatives to MCMC. Stat. Comput. 18, 151–171 (2008) CrossRefMathSciNetGoogle Scholar
  15. Fearnhead, P., Clifford, P.: Online inference for hidden Markov models. J. R. Stat. Soc., Ser. B 65, 887–899 (2003) zbMATHCrossRefMathSciNetGoogle Scholar
  16. Fearnhead, P., Liu, Z.: Online inference for multiple changepoint problems. J. R. Stat. Soc., Ser. B 69, 589–605 (2007) CrossRefMathSciNetGoogle Scholar
  17. Fearnhead, P., Vasileiou, D.: Bayesian analysis of isochores. J. Am. Stat. Assoc. 485, 132–141 (2009) CrossRefGoogle Scholar
  18. Kim, S., Shephard, N., Chib, S.: Stochastic volatility: Likelihood inference and comparison with arch models. Rev. Econ. Stud. 65, 361–393 (1998) zbMATHCrossRefGoogle Scholar
  19. Lavielle, M., Lebarbier, E.: An application of MCMC methods for the multiple change-points problem. Signal Process. 81, 39–53 (2001) zbMATHCrossRefGoogle Scholar
  20. Liu, Z.: Direct simulation methods for multiple changepoint problems. Ph.D. thesis, Department of Mathematics and Statistics, Lancaster University (2007) Google Scholar
  21. Liu, J.S., Lawrence, C.E.: Bayesian inference on biopolymer models. Bioinformatics 15, 38–52 (1999) CrossRefGoogle Scholar
  22. Liu, J.S., Chen, R., Wong, W.H.: Rejection control and sequential importance sampling. J. Am. Stat. Soc. 93, 1022–1031 (1998) zbMATHMathSciNetGoogle Scholar
  23. McVean, G.A.T., Myers, S.R., Hunt, S., Deloukas, P., Bentley, D.R., Donnelly, P.: The fine-scale structure of recombination rate variation in the human genome. Science 304, 581–584 (2004) CrossRefGoogle Scholar
  24. Ó Ruanaidh, J.J.K., Fitzgerald, W.J.: Numerical Bayesian Methods Applied to Signal Processing. Springer, New York (1996) zbMATHGoogle Scholar
  25. Punskaya, E., Andrieu, C., Doucet, A., Fitzgerald, W.J.: Bayesian curve fitting using MCMC with applications to signal segmentation. IEEE Trans. Signal Process. 50, 747–758 (2002) CrossRefGoogle Scholar
  26. Rue, H., Held, L.: Gaussian Markov Random Fields: Theory and Applications. CRC Press/Chapman and Hall, Boca Raton/London (2005) zbMATHCrossRefGoogle Scholar
  27. Rue, H., Martino, S., Chopin, N.: Approximate Bayesian inference for latent Gaussian models by using integrated nested Laplace approximations (with discussion). J. R. Stat. Soc., Ser. B 71, 319–392 (2009) CrossRefGoogle Scholar
  28. Seidou, O., Ouarda, T.B.M.J.: Recursion-based multiple changepoint detection in multiple linear regression and application to river streamflows. Water Resour. Res. 43, W07404 (2007) CrossRefGoogle Scholar
  29. Stephens, D.A.: Bayesian retrospective multiple-changepoint identification. Appl. Stat. 43, 159–178 (1994) zbMATHCrossRefGoogle Scholar
  30. Tugnait, J.K.: Detection and estimation for abruptly changing systems. Automatica 18, 607–615 (1982) zbMATHCrossRefGoogle Scholar
  31. West, M., Harrison, J.: Bayesian Forecasting and Dynamic Models. Springer, New York (1989) zbMATHGoogle Scholar
  32. Yao, Y.: Estimation of a noisy discrete-time step function: Bayes and empirical Bayes approaches. Ann. Stat. 12, 1434–1447 (1984) zbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsLancaster UniversityLancasterUK

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