The Annals of Applied Statistics

Identifying multiple changes for a functional data sequence with application to freeway traffic segmentation

Jeng-Min Chiou, Yu-Ting Chen, and Tailen Hsing

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


Motivated by the study of road segmentation partitioned by shifts in traffic conditions along a freeway, we introduce a two-stage procedure, Dynamic Segmentation and Backward Elimination (DSBE), for identifying multiple changes in the mean functions for a sequence of functional data. The Dynamic Segmentation procedure searches for all possible changepoints using the derived global optimality criterion coupled with the local strategy of at-most-one-changepoint by dividing the entire sequence into individual subsequences that are recursively adjusted until convergence. Then, the Backward Elimination procedure verifies these changepoints by iteratively testing the unlikely changes to ensure their significance until no more changepoints can be removed. By combining the local strategy with the global optimal changepoint criterion, the DSBE algorithm is conceptually simple and easy to implement and performs better than the binary segmentation-based approach at detecting small multiple changes. The consistency property of the changepoint estimators and the convergence of the algorithm are proved. We apply DSBE to detect changes in traffic streams through real freeway traffic data. The practical performance of DSBE is also investigated through intensive simulation studies for various scenarios.

Article information

Ann. Appl. Stat., Volume 13, Number 3 (2019), 1430-1463.

Received: May 2018
Revised: October 2018
First available in Project Euclid: 17 October 2019

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Changepoint analysis covariance operator functional principal component projection segmentation


Chiou, Jeng-Min; Chen, Yu-Ting; Hsing, Tailen. Identifying multiple changes for a functional data sequence with application to freeway traffic segmentation. Ann. Appl. Stat. 13 (2019), no. 3, 1430--1463. doi:10.1214/19-AOAS1242.

Export citation


  • Aston, J. A. D. and Kirch, C. (2012). Detecting and estimating changes in dependent functional data. J. Multivariate Anal. 109 204–220.
  • Aue, A., Gabrys, R., Horváth, L. and Kokoszka, P. (2009). Estimation of a change-point in the mean function of functional data. J. Multivariate Anal. 100 2254–2269.
  • Benko, M., Härdle, W. and Kneip, A. (2009). Common functional principal components. Ann. Statist. 37 1–34.
  • Berkes, I., Gabrys, R., Horváth, L. and Kokoszka, P. (2009). Detecting changes in the mean of functional observations. J. R. Stat. Soc. Ser. B. Stat. Methodol. 71 927–946.
  • Bosq, D. (2000). Linear Processes in Function Spaces: Theory and Applications. Lecture Notes in Statistics 149. Springer, New York.
  • Braun, J. V., Braun, R. K. and Müller, H.-G. (2000). Multiple changepoint fitting via quasilikelihood, with application to DNA sequence segmentation. Biometrika 87 301–314.
  • Chiou, J.-M., Chen, Y.-T. and Hsing, T. (2019). Supplement to “Identifying multiple changes for a functional data sequence with application to freeway traffic segmentation.” DOI:10.1214/19-AOAS1242SUPP.
  • Chiou, J.-M., Chen, Y.-T. and Yang, Y.-F. (2014). Multivariate functional principal component analysis: A normalization approach. Statist. Sinica 24 1571–1596.
  • Ciuperca, G. (2011). A general criterion to determine the number of change-points. Statist. Probab. Lett. 81 1267–1275.
  • Ferraty, F. and Vieu, P. (2006). Nonparametric Functional Data Analysis: Theory and Practice. Springer Series in Statistics. Springer, New York.
  • Fremdt, S., Steinebach, J. G., Horváth, L. and Kokoszka, P. (2013). Testing the equality of covariance operators in functional samples. Scand. J. Stat. 40 138–152.
  • Frick, K., Munk, A. and Sieling, H. (2014). Multiscale change point inference. J. R. Stat. Soc. Ser. B. Stat. Methodol. 76 495–580.
  • Fryzlewicz, P. (2014). Wild binary segmentation for multiple change-point detection. Ann. Statist. 42 2243–2281.
  • Gromenko, O., Kokoszka, P. and Reimherr, M. (2017). Detection of change in the spatiotemporal mean function. J. R. Stat. Soc. Ser. B. Stat. Methodol. 79 29–50.
  • Happ, C. and Greven, S. (2018). Multivariate functional principal component analysis for data observed on different (dimensional) domains. J. Amer. Statist. Assoc. 113 649–659.
  • Harchaoui, Z. and Lévy-Leduc, C. (2010). Multiple change-point estimation with a total variation penalty. J. Amer. Statist. Assoc. 105 1480–1493.
  • Hörmann, S. and Kokoszka, P. (2010). Weakly dependent functional data. Ann. Statist. 38 1845–1884.
  • Horváth, L. and Kokoszka, P. (2012). Inference for Functional Data with Applications. Springer Series in Statistics. Springer, New York.
  • Hsing, T. and Eubank, R. (2015). Theoretical Foundations of Functional Data Analysis, with an Introduction to Linear Operators. Wiley Series in Probability and Statistics. Wiley, Chichester.
  • Hu, S. R. and Wang, C. M. (2008). Vehicle detector deployment strategies for the estimation of network origin-destination demands using link traffic counts. IEEE Transactions on Intelligent Transportation Systems 9 283–300.
  • Killick, R., Fearnhead, P. and Eckley, I. A. (2012). Optimal detection of changepoints with a linear computational cost. J. Amer. Statist. Assoc. 107 1590–1598.
  • May, A. D. (1990). Traffic Flow Fundamentals. Prentice Hall, Englewood Cliffs, NJ.
  • Newey, W. K. (1991). Uniform convergence in probability and stochastic equicontinuity. Econometrica 59 1161–1167.
  • Niu, Y. S., Hao, N. and Zhang, H. (2016). Multiple change-point detection: A selective overview. Statist. Sci. 31 611–623.
  • Niu, Y. S. and Zhang, H. (2012). The screening and ranking algorithm to detect DNA copy number variations. Ann. Appl. Stat. 6 1306–1326.
  • Olshen, A. B., Venkatraman, E., Lucito, R. and Wigler, M. (2004). Circular binary segmentation for the analysis of array-based DNA copy number data. Biostatistics 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.
  • Panaretos, V. M., Kraus, D. and Maddocks, J. H. (2010). Second-order comparison of Gaussian random functions and the geometry of DNA minicircles. J. Amer. Statist. Assoc. 105 670–682.
  • Pötscher, B. M. and Prucha, I. R. (1994). Generic uniform convergence and equicontinuity concepts for random functions: An exploration of the basic structure. J. Econometrics 60 23–63.
  • Ramsay, J. O. and Silverman, B. W. (2005). Functional Data Analysis, 2nd ed. Springer Series in Statistics. Springer, New York.
  • Transportation Research Board (2010). HCM 2010: Highway Capacity Manual. TRB, Washington, DC.
  • van der Vaart, A. W. and Wellner, J. A. (1996). Weak Convergence and Empirical Processes: With Applications to Statistics. Springer Series in Statistics. Springer, New York.
  • Wang, J.-L., Chiou, J.-M. and Müller, H.-G. (2016). Functional data analysis. Annual Review of Statistics and Its Application 3 257–295.
  • Yao, Y.-C. (1988). Estimating the number of change-points via Schwarz’ criterion. Statist. Probab. Lett. 6 181–189.
  • Zhang, J.-T. (2014). Analysis of Variance for Functional Data. Monographs on Statistics and Applied Probability 127. CRC Press, Boca Raton, FL.

Supplemental materials

  • Additional simulation results. We provide additional simulation results of Section 5.4 for DSBE, BScust and CBSec with the sample size $N=100$ in comparison with those with $N=200$.