Bayesian Analysis

Bayesian Multiple Changepoint Detection for Stochastic Models in Continuous Time

Lu Shaochuan

Advance publication

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Abstract

A multiple changepoint model in continuous time is formulated as a continuous-time hidden Markov model, defined on a countable infinite state space. The new formulation of the multiple changepoint model allows the model complexities, i.e. the number of changepoints, to accrue unboundedly upon the arrivals of new data. Inference on the number of changepoints and their locations is based on a collapsed Gibbs sampler. We suggest a new version of forward-filtering backward-sampling (FFBS) algorithm in continuous time for simulating the full trajectory of the latent Markov chain, i.e. the changepoints. The FFBS algorithm is based on a randomized time-discretization for the latent Markov chain through uniformization schemes, combined with a discrete-time version of FFBS algorithm. It is shown that, desirably, both the computational cost and the memory cost of the FFBS algorithm are only quadratic to the number of changepoints. The new formulation of the multiple changepoint models allows varying scale of run lengths of changepoints to be characterized. We demonstrate the methods through simulations and a real data example for earthquakes.

Article information

Source
Bayesian Anal., Advance publication (2020), 24 pages.

Dates
First available in Project Euclid: 16 June 2020

Permanent link to this document
https://projecteuclid.org/euclid.ba/1592272905

Digital Object Identifier
doi:10.1214/20-BA1218

Keywords
continuous-time forward-filtering backward-sampling algorithms uniformization infinite hidden Markov Models Markov jump processes Poisson processes

Rights
Creative Commons Attribution 4.0 International License.

Citation

Shaochuan, Lu. Bayesian Multiple Changepoint Detection for Stochastic Models in Continuous Time. Bayesian Anal., advance publication, 16 June 2020. doi:10.1214/20-BA1218. https://projecteuclid.org/euclid.ba/1592272905


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References

  • Arnesen P., Holsclaw T. and Smyth P. (2016) “Bayesian Detection of Change points in Finite-State Markov Chains for Multiple Sequences.” Technometrics, 58(2):205–213.
  • Barry D. and Hartigan J. A. (1992) “Product partition models for change point problems.” The Annals of Statistics, 20(1):260–279.
  • Bebbington M. S. (2007) “Identifying volcanic regimes using Hidden Markov Models.” Geophysical Journal International, 171(2):921–942.
  • Chib S. (1998) “Estimation and Comparison of Multiple Change-Point Models.” Journal of Econometrics, 86(2):221–241.
  • Chopin N. (2007) “Dynamic detection of change points in long time series.” Annals of the Institute of Statistical Mathematics, 59(2):349–366.
  • Fearnhead P. (2006) “Exact and Efficient Bayesian Inference for Multiple Changepoint Problems.” Statistics and Computing, 16(2):203–213.
  • Fearnhead P. and Liu Z. (2007) “On-line inference for multiple changepoint problems.” Journal of the Royal Statistical Society B, 69(4):589–605.
  • Frick K., Munk A. and Sieling H. (2014) “Multiscale change point inference.” Journal of the Royal Statistical Society B, 76(3):495–580.
  • Fryzlewicz P. (2014) “Wild binary segmentation for multiple change-point detection.” Annals of Statistics, 42(6):2243–2281.
  • Galeano P. (2007) “The use of cumulative sums for detection of changepoints in the rate parameter of a Poisson Process.” Computational Statistics and Data Analysis, 51(12):6151–6165.
  • Green P. J. (1995) “Reversible Jump Markov Chain Monte Carlo Computation and Bayesian Model Determination.” Biometrika, 82(4):711–732.
  • Jensen A. (1953) “Markoff chains as an aid in the study of markoff processes.” Scandinavian Actuarial Journal, 1953(Supp 1), 87–91.
  • Killick R., Fearnhead P. and Eckley I. A. (2012) “Optimal Detection of Changepoints With a Linear Computational Cost.” Journal of the American Statistical Association, 107(500):1590–1598.
  • Ko S., Chong T. and Ghosh P. (2015) “Dirichlet process hidden Markov multiple change-point Model.” Bayesian Analysis, 10(2):275–296.
  • Lavielle M. and Lebarbier E. (2001) “An Application of MCMC Methods for the Multiple Change-Points Problem.” Signal Processing, 81(1):39–53.
  • Lewis P. A. W. and Shedler G. S. (1979) “Simulation of nonhomogeneous Poisson processes by thinning.” Naval Research Logistics Quarterly, 26(3):403–413.
  • Liu J. S. (1994) “The collapsed gibbs sampler in Bayesian computations with applications to a gene regulation problem.” Journal of the American Statistical Association, 89(427):958–966.
  • Lu S. (2019) “A Bayesian multiple changepoint model for marked Poisson processes with applications to deep earthquakes.” Stochastic Environmental Research and Risk Assessment, 33(1):59–72.
  • Lu S. (2020) “Bayesian multiple changepoint detection for Markov jump processes.” Computational Statistics.
  • Martinez A. F. and Mena R. H. (2014) “On a nonparametric change point detection model in Markovian regimes.” Bayesian Analysis, 9(4):823–858.
  • Polansky A. M. (2007) “Detecting change-points in Markov chains.” Computational Statistics and Data Analysis, 51(12):6013–6026.
  • Rao V. and Teh Y. W. (2013) “Fast MCMC Sampling for Markov Jump Processes and Extensions.” Journal of Machine Learning Research, 14(1):3295–3320.
  • Ruggieri E. and Lawrence C. E. (2014) “The Bayesian Change Point and Variable Selection Algorithm: Application to the ${\delta }$18O Proxy Record of the Plio-Pleistocene.” Journal of Computational and Graphical Statistics, 23(1):87–110,
  • Scott S. L. (2002) “Bayesian methods for hidden Markov models: Recursive computing in the 21st century.” Journal of the American Statistical Association, 97(457):337–351.
  • Stephens D. A. (1994) “Bayesian Retrospective Multiple-Changepoint Identification.” Applied Statistics, 43(1):159–178.
  • Van Dijk N. M. (1992) “Uniformization for nonhomogeneous Markov chains.” Operations Research Letters, 12(5): 283–291.
  • Van Dijk N. M., Van Brummelen S. P. and Boucherie R. J. (2018) “Uniformization: Basics, extensions and applications.” Performance Evaluation, 118:8–32.
  • Viterbi A. J. (1967) “Error bounds for convolutional codes and an asymptotically optimum decoding algorithm.” IEEE Transactions on Information Theory, 13(2):260–269.
  • Xing H., Sun N. and Chen Y. (2012) “Credit Rating Dynamics in the Presence of Unknown Structural Breaks.” Journal of Banking and Finance, 36(1):78–89.
  • Yang T. Y. and Kuo L. (2001) “Bayesian binary segmentation procedure for a Poisson Process with multiple changepoints.” Journal of Computational and Graphical Statistics, 10(4):772–785.
  • Yildirim S., Singh S. S. and Doucet A. (2013) “An online expectation–maximization algorithm for changepoint models.” Journal of Computational and Graphical Statistics, 22(4), 906–926.