Abstract
This study considers the problem of multiple change-points detection. For this problem, we develop an objective Bayesian multiple change-points detection procedure in a normal model with heterogeneous variances. Our Bayesian procedure is based on a combination of binary segmentation and the idea of the screening and ranking algorithm (Niu and Zhang in Ann Appl Stat 6:1306–1326, 2012). Using the screening and ranking algorithm, we can overcome the drawbacks of binary segmentation, as it cannot detect a small segment of structural change in the middle of a large segment or segments of structural changes with small jump magnitude. We propose a detection procedure based on a Bayesian model selection procedure to address this problem in which no subjective input is considered. We construct intrinsic priors for which the Bayes factors and model selection probabilities are well defined. We find that for large sample sizes, our method based on Bayes factors with intrinsic priors is consistent. Moreover, we compare the behavior of the proposed multiple change-points detection procedure with existing methods through a simulation study and two real data examples.
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References
Arlot S, Celisse A (2011) Segmentation of the mean of heteroscedastic data via cross-validation. Stat Comput 21:613–632
Bai J, Perron P (2003) Computation and analysis of multiple structural change models. J Appl Econometrics 18:1–22
Barry D, Hartigan J (1993) A Bayesian analysis for change-point problems. J Am Stat Assoc 88:309–319
Berger JO, Pericchi LR (1996) The intrinsic Bayes factor for model selection and prediction. J Am Stat Assoc 91:109–122
Berger JO, Pericchi LR (1997) On justification of default and intrinsic Bayes factor. In: Lee JC et al (eds) Modeling and prediction. Springer-Verlag, New York, pp 276–293
Berger JO, Pericchi LR (1998) On criticism and comparison of default Bayes factor for model selection and hypoghesis testing. In: Racugno W (ed) Proceedings of the Workshop on Model Selection. Pitagora, Bologna, pp 1–50
Braun JV, Müller HG (1998) Statistical methods for DNA sequence segmentation. Stat Sci 13:142–162
De Santis F, Spezzaferri F (1999) Methods for default and robust Bayesian model comparison: the fractional Bayes factor approach. Int Stat Rev 67:267–286
Fearnhead P (2006) Exact and efficient Bayesian inference for multiple changepoint problems. Stat Comput 16:203–213
Fearnhead P, Clifford P (2003) Online inference for well-log data. J R Stat Soc Ser B 65:887–899
Fearnhead P, Liu Z (2007) On-line inference for multiple changepoint problems. J R Stat Soc Ser B 69:589–605
Frick K, Munk A, Sieling H (2014) Multiscale change-point inference. J R Stat Soc B 76:495–580
Fryziewicz P (2014) Wild binary segmentation for multiple change-point detection. Ann Stat 42:2243–2281
Giordani P, Kohn R (2008) Efficient Bayesian inference for multiple change-point and mixture innovation models. J Bus Econ Stat 26:66–77
Hao N, Niu YS, Zhang H (2013) Multiple change-point detection via a screening and ranking algorithm. Stat Sin 23:1553–1572
Haynes K, Eckley IA, Fearnhead P (2014). Efficient penalty search for multiple changepoint problems. arXiv:1412.3617
Jeffreys H (1961) Theory of probability, 3rd edn. Oxford University Press, Oxford, UK
Jensen G (2013) Closed-form estimation of multiple change-point models. PeerJ PrePrint, 1:e90v3 https://doi.org/10.7287/peerj.preprints.90v3
Kass RE, Raftery AE (1995) Bayes factors. J Am Stat Assoc 90:773–795
Killick R, Fearnhead P, Eckley IA (2012) Optimal detection of changepoints with a linear computational cost. J Am Stat Assoc 107:1590–1598
Killick R, Eckley IA, Jonathan P (2013) A wavelet-based approach for detecting changes in second order structure within nonstationary time series. Electron J Stat 7:1167–1183
Moreno E (1997) Bayes factor for intrinsic and fractional priors in nested models: Bayesian Robustness. In: Yadolah D (ed) L1-statistical procddures and related topics, vol 31. Institute of Mathematical Statistics, Hayward, pp 257–270
Moreno E, Bertolino F, Racugno W (1998) An intrinsic limiting procedure for model selection and hypotheses testing. J Am Stat Assoc 93:1451–1460
Moreno E, Bertolino F, Racugno W (1999) Default Bayesian analysis of the Behrens-Fisher problem. J Stat Plan Inference 81:323–333
Muggeo VMR, Adelfio G (2011) Efficient change point detection for genomic sequences of continuous measurements. Bioinformatics 27:161–166
Niu YS, Zhang H (2012) The screening and ranking algorithm to detect DNA copy number variations. Ann Appl Stat 6:1306–1326
Niu YS, Hao N, Zhang H (2016) Multiple change-point detection: a selective overview. Stat Sci 31:611–623
O’Hagan A (1995) Fractional bayes factors for model comparison (with discussion). J R Stat Soc B 57:99–138
O’Hagan A (1997) Properties of intrinsic and fractional Bayes factors. Test 6:101–118
Olshen AB, Venkatraman ES, Lucito R, Wigler M (2004) Circular binary segmentation for the analysis of array-based DNA copy number data. Biostatistics 5:557–572
Pein F, Sieling H, Munk A (2017) Heterogeneous change point inference. J R Stat Soc B 79:1207–1227
Reeves J, Chen J, Wang XL, Lund R, Lu Q (2007) A review and comparison of changepoint detection techniques for climate data. J Appl Meteorol Climatol 46:900–915
Ruanaidh JJK, Fitzgerald WJ (1996) Numerical bayesion methods applied to signal processing. Springer, New York
Schwarz G (1978) Estimating the dimension of a model. Ann Stat 6:461–464
Scott AJ, Knott M (1974) A Cluster analysis method for grouping means in the analysis of variance. Biometrics 30:507–512
Tibshirani R, Wang P (2008) Spatial smoothing and hot spot detection for CGH data using the fused lasso. Biostatistics 9:18–29
Vostrikova LJ (1981) Detecting disorder in multidimensional random processes. Soviet Mathematics: Doklady 24:55–59
Whiteley N, Andrieu C, Doucet A (2011) Bayesian computational methods for inference in multiple change-points models. University of Bristol, Discussion Paper
Yao YC (1988) Estimating the number of change-points via Schwarz’ criterion. Stat Probab Lett 6:181–189
Zhang NR, Siegmund DO (2007) A modified bayes information criterion with applications to the analysis of comparative genomic hybridization data. Biometrics 63:22–32
Zhang NR, Siegmund DO (2012) Model selection for high-dimensional, multi-sequence change-point problems. Stat Sin 22:1057–1538
Acknowledgements
The Research of Yongku Kim was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (No.2018R1D1A1B07043352).
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Kang, S.G., Lee, W.D. & Kim, Y. Bayesian Multiple Change-Points Detection in a Normal Model with Heterogeneous Variances. Comput Stat 36, 1365–1390 (2021). https://doi.org/10.1007/s00180-020-01054-3
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DOI: https://doi.org/10.1007/s00180-020-01054-3