The Annals of Statistics

Tail-greedy bottom-up data decompositions and fast multiple change-point detection

Piotr Fryzlewicz

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


This article proposes a “tail-greedy”, bottom-up transform for one-dimensional data, which results in a nonlinear but conditionally orthonormal, multiscale decomposition of the data with respect to an adaptively chosen unbalanced Haar wavelet basis. The “tail-greediness” of the decomposition algorithm, whereby multiple greedy steps are taken in a single pass through the data, both enables fast computation and makes the algorithm applicable in the problem of consistent estimation of the number and locations of multiple change-points in data. The resulting agglomerative change-point detection method avoids the disadvantages of the classical divisive binary segmentation, and offers very good practical performance. It is implemented in the R package breakfast, available from CRAN.

Article information

Ann. Statist., Volume 46, Number 6B (2018), 3390-3421.

Received: March 2017
Revised: September 2017
First available in Project Euclid: 11 September 2018

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G05: Estimation

Tail-greediness bottom-up methods multiscale methods segmentation thresholding sparsity


Fryzlewicz, Piotr. Tail-greedy bottom-up data decompositions and fast multiple change-point detection. Ann. Statist. 46 (2018), no. 6B, 3390--3421. doi:10.1214/17-AOS1662.

Export citation


  • Aggarwal, R., Inclan, C. and Leal, R. (1999). Volatility in emerging stock markets. J. Financ. Quant. Anal. 34 33–55.
  • Auger, I. E. and Lawrence, C. E. (1989). Algorithms for the optimal identification of segment neighborhoods. Bull. Math. Biol. 51 39–54.
  • Bai, J. (1997). Estimating multiple breaks one at a time. Econometric Theory 13 315–352.
  • Bai, J. and Perron, P. (2003). Computation and analysis of multiple structural change models. J. Appl. Econometrics 18 1–22.
  • Baranowski, R., Chen, Y. and Fryzlewicz, P. (2016). Narrowest-Over-Threshold detection of multiple change-points and change-point-like features. Preprint.
  • Birge, L. and Massart, P. (2001). Gaussian model selection. J. Eur. Math. Soc. (JEMS) 3 203–268.
  • Boysen, L., Kempe, A., Liebscher, V., Munk, A. and Wittich, O. (2009). Consistencies and rates of convergence of jump-penalized least squares estimators. Ann. Statist. 37 157–183.
  • Braun, J., Braun, R. and Mueller, H.-G. (2000). Multiple changepoint fitting via quasilikelihood, with application to DNA sequence segmentation. Biometrika 87 301–314.
  • Braun, J. and Mueller, H.-G. (1998). Statistical methods for DNA sequence segmentation. Statist. Sci. 13 142–162.
  • Brodsky, B. and Darkhovsky, B. (1993). Nonparametric Methods in Change-Point Problems. Kluwer, Dordrecht.
  • Chen, K.-M., Cohen, A. and Sackrowitz, H. (2011). Consistent multiple testing for change points. J. Multivariate Anal. 102 1339–1343.
  • Cho, H. and Fryzlewicz, P. (2011). Multiscale interpretation of taut string estimation and its connection to Unbalanced Haar wavelets. Stat. Comput. 21 671–681.
  • Cho, H. and Fryzlewicz, P. (2012). Multiscale and multilevel technique for consistent segmentation of nonstationary time series. Statist. Sinica 22 207–229.
  • Cho, H. and Fryzlewicz, P. (2015). Multiple change-point detection for high-dimensional time series via Sparsified Binary Segmentation. J. R. Stat. Soc. Ser. B. Stat. Methodol. 77 475–507.
  • Choi, F. Y. Y. (2000). Advances in domain independent linear text segmentation. In NAACL 2000 Proceedings of the 1st North American Chapter of the Association for Computational Linguistics Conference 26–33.
  • Chu, P.-S. and Zhao, X. (2004). Bayesian change-point analysis of tropical cyclone activity: The central north Pacific case. J. Climate 17 4893–4901.
  • Ciuperca, G. (2011). A general criterion to determine the number of change-points. Statist. Probab. Lett. 81 1267–1275.
  • Ciuperca, G. (2014). Model selection by LASSO methods in a change-point model. Statist. Papers 55 349–374.
  • Davies, P. L. and Kovac, A. (2001). Local extremes, runs, strings and multiresolution. Ann. Statist. 29 1–48.
  • Davis, R., Lee, T. and Rodriguez-Yam, G. (2006). Structural break estimation for nonstationary time series models. J. Amer. Statist. Assoc. 101 223–239.
  • Desmond, R., Weiss, H., Arani, R., Soong, S.-J., Wood, M., Fiddian, P., Gnann, J. and Whitley, R. (2002). Clinical applications for change-point analysis of herpes zoster pain. Journal of Pain and Symptom Management 23 510–516.
  • Donoho, D. L. and Johnstone, I. M. (1994). Ideal spatial adaptation by wavelet shrinkage. Biometrika 81 425–455.
  • Du, C., Kao, C.-L. and Kou, S. (2016). Stepwise signal extraction via marginal likelihood. J. Amer. Statist. Assoc. 111 314–330.
  • Efron, B., Hastie, T., Johnstone, I. and Tibshirani, R. (2004). Least angle regression. Ann. Statist. 32 407–499.
  • Eichinger, B. and Kirch, C. (2018). A MOSUM procedure for the estimation of multiple random change points. Bernoulli 24 526–564.
  • Erdman, C. and Emerson, J. (2008). A fast Bayesian change point analysis for the segmentation of microarray data. Bioinformatics 24 2143–2148.
  • Frick, K., Munk, A. and Sieling, H. (2014). Multiscale change-point inference (with discussion). J. R. Stat. Soc. Ser. B. Stat. Methodol. 76 495–580.
  • Fryzlewicz, P. (2007). Unbalanced Haar technique for nonparametric function estimation. J. Amer. Statist. Assoc. 102 1318–1327.
  • Fryzlewicz, P. (2014). Wild binary segmentation for multiple change-point detection. Ann. Statist. 42 2243–2281.
  • Fryzlewicz, P. (2018). Supplement to “Tail-greedy bottom-up data decompositions and fast multiple change-point detection”. DOI:10.1214/17-AOS1662SUPP.
  • Fryzlewicz, P. and Subba Rao, S. (2014). Multiple-change-point detection for auto-regressive conditional heteroscedastic processes. J. R. Stat. Soc. Ser. B. Stat. Methodol. 76 903–924.
  • Fryzlewicz, P. and Timmermans, C. (2016). SHAH: SHape-Adaptive Haar wavelets for image processing. J. Comput. Graph. Statist. 25 879–898.
  • Harchaoui, Z. and Lévy-Leduc, C. (2010). Multiple change-point estimation with a total variation penalty. J. Amer. Statist. Assoc. 105 1480–1493.
  • Huskova, M. and Slaby, A. (2001). Permutation tests for multiple changes. Kybernetika (Prague) 37 605–622.
  • Jackson, B., Sargle, J., Barnes, D., Arabhi, S., Alt, A., Gioumousis, P., Gwin, E., Sangtrakulcharoen, P., Tan, L. and Tsai, T. T. (2005). An algorithm for optimal partitioning of data on an interval. IEEE Signal Process. Lett. 12 105–108.
  • Jansen, M., Nason, G. and Silverman, B. (2009). Multiscale methods for data on graphs and irregular multidimensional situations. J. R. Stat. Soc. Ser. B. Stat. Methodol. 71 97–125.
  • Killick, R., Fearnhead, P. and Eckley, I. (2012). Optimal detection of changepoints with a linear computational cost. J. Amer. Statist. Assoc. 107 1590–1598.
  • Killick, R., Nam, C., Aston, J. and Eckley, I. (2012). The changepoint repository.
  • Koprinska, I. and Carrato, S. (2001). Temporal video segmentation: A survey. Signal Process. Image Commun. 16 477–500.
  • Lavielle, M. (1999). Detection of multiple changes in a sequence of dependent variables. Stochastic Process. Appl. 83 79–102.
  • Lavielle, M. (2005). Using penalized contrasts for the change-point problem. Signal Process. 85 1501–1510.
  • Lavielle, M. and Moulines, E. (2000). Least-squares estimation of an unknown number of shifts in a time series. J. Time Series Anal. 21 33–59.
  • Lebarbier, E. (2005). Detecting multiple change-points in the mean of Gaussian process by model selection. Signal Process. 85 717–736.
  • Lee, C.-B. (1995). Estimating the number of change points in a sequence of independent normal random variables. Statist. Probab. Lett. 25 241–248.
  • Li, H., Munk, A. and Sieling, H. (2016). FDR-control in multiscale change-point segmentation. Electron. J. Stat. 10 918–959.
  • Lio, P. and Vanucci, M. (2000). Wavelet change-point prediction of transmembrane proteins. Bioinformatics 16 376–382.
  • Lu, L., Zhang, H.-J. and Jiang, H. (2002). Content analysis for audio classification and segmentation. IEEE Trans. Speech Audio Process. 10 504–516.
  • Mahmoud, M., Parker, P., Woodall, W. and Hawkins, D. (2007). A change point method for linear profile data. Qual. Reliab. Eng. Int. 23 247–268.
  • Maidstone, R., Hocking, T., Rigaill, G. and Fearnhead, P. (2017). On optimal multiple changepoint algorithms for large data. Stat. Comput. 27 519–533.
  • Matteson, D. and James, N. (2014). A nonparametric approach for multiple change point analysis of multivariate data. J. Amer. Statist. Assoc. 109 334–345.
  • Minin, V., Dorman, K., Fang, F. and Suchard, M. (2005). Dual multiple change-point model leads to more accurate recombination detection. Bioinformatics 21 3034–3042.
  • Olshen, A., Venkatraman, E. S., Lucito, R. and Wigler, M. (2004). Circular binary segmentation for the analysis of array-based DNA copy number data. Biostat. 5 557–572.
  • Pan, J. and Chen, J. (2006). Application of modified information criterion to multiple change point problems. J. Multivariate Anal. 97 2221–2241.
  • Rigaill, G. (2015). A pruned dynamic programming algorithm to recover the best segmentations with 1 to ${K}_{\max}$ change-points. J. SFdS 156 180–205.
  • Rinaldo, A. (2009). Properties and refinements of the fused lasso. Ann. Statist. 37 2922–2952.
  • Robbins, M., Gallagher, C., Lund, R. B. and Aue, A. (2011). Mean shift testing in correlated data. J. Time Series Anal. 32 498–511.
  • Rojas, C. and Wahlberg, B. (2014). On change point detection using the fused lasso method. Preprint.
  • Schroeder, A. L. and Fryzlewicz, P. (2013). Adaptive trend estimation in financial time series via multiscale change-point-induced basis recovery. Stat. Interface 6 449–461.
  • Shriberga, E., Stolckea, A., Hakkani-Türb, D. and Türb, G. (2000). Prosody-based automatic segmentation of speech into sentences and topics. Speech Commun. 32 127–154.
  • Tartakovsky, A., Rozovskii, B., Blazek, R. and Kim, H. (2006). A novel approach to detection of intrusions in computer networks via adaptive sequential and batch-sequential change-point detection methods. IEEE Trans. Signal Process. 54 3372–3382.
  • Tibshirani, R., Saunders, M., Rosset, S., Zhu, J. and Knight, K. (2005). Sparsity and smoothness via the fused lasso. J. R. Stat. Soc. Ser. B. Stat. Methodol. 67 91–108.
  • Venkatraman, E. S. (1992). Consistency results in multiple change-point problems. Technical Report 24, Dept. Statistics, Stanford Univ. Available from
  • Venkatraman, E. S. and Olshen, A. (2007). A faster circular binary segmentation algorithm for the analysis of array CGH data. Bioinformatics 23 657–663.
  • Vostrikova, L. (1981). Detecting ‘disorder’ in multidimensional random processes. Sov. Math., Dokl. 24 55–59.
  • Wang, Y. (1995). Jump and sharp cusp detection by wavelets. Biometrika 82 385–397.
  • Wang, H., Zhang, D. and Shin, K. (2004). Change-point monitoring for the detection of DoS attacks. IEEE Trans. Dependable Secure Comput. 1 193–208.
  • Wu, Y. (2008). Simultaneous change point analysis and variable selection in a regression problem. J. Multivariate Anal. 99 2154–2171.
  • Yao, Y.-C. (1988). Estimating the number of change-points via Schwarz’ criterion. Statist. Probab. Lett. 6 181–189.
  • Yao, Y.-C. and Au, S. T. (1989). Least-squares estimation of a step function. Sankhyā Ser. A 51 370–381.
  • Zhang, N. R. and Siegmund, D. O. (2007). A modified Bayes information criterion with applications to the analysis of comparative genomic hybridization data. Biometrics 63 22–32.

Supplemental materials

  • Supplement to “Tail-greedy bottom-up data decompositions and fast multiple change-point detection”. Extension of the TGUH methodology to dependent non-Gaussian data; refinements to post-processing; study of the accuracy of TGUH in estimating change-point locations; additional figure for the analysis of Section 4.3.