Abstract

We propose a novel method to detect and date structural breaks in the entire

distribution of functional data.

Theoretical guarantees are developed for our

procedure under fewer assumptions than in the existing work.

In particular,

we establish the asymptotic null distribution of the test statistic, which enables

us to test the null hypothesis at a certain significance level. Additionally, the

limiting distribution of the estimated structural break date is developed under two

situations of the break size: fixed and shrinking towards 0 at a specified rate. We

further propose a unified bootstrap procedure to construct a confidence interval

for the true structural break date for these two situations.

These theoretical

results are justified through comprehensive simulation studies in finite samples.

We apply the proposed method to Australian temperature data for detecting

structural beaks.

Key words and phrases: Change point analysis; Functional principal component analysis; Mean em- bedding; Reproducing kernel Hilbert space; Temperature data 1 1 Introduction In this article we consider structural break detection for functional data

Information

Preprint No.SS-2025-0404
Manuscript IDSS-2025-0404
Complete AuthorsPeijun Sang, Bing Li
Corresponding AuthorsPeijun Sang
Emailspeijun.sang@uwaterloo.ca

References

  1. Antoch, J., Hušková, M., and Veraverbeke, N. (1995). Change-point problem and bootstrap. Journaltitle of Nonparametric Statistics, 5(2):123– 144.
  2. Aston, J. A. and Kirch, C. (2012a). Detecting and estimating changes in dependent functional data. Journal of Multivariate Analysis, 109:204– 220.
  3. Aston, J. A. and Kirch, C. (2012b). Evaluating stationarity via changepoint alternatives with applications to fMRI data. Annals of Applied Statistics, 6(4):1906–1948.
  4. Aue, A., Gabrys, R., Horváth, L., and Kokoszka, P. (2009). Estimation of a change-point in the mean function of functional data. Journal of Multivariate Analysis, 100(10):2254–2269.
  5. Aue, A., Rice, G., and Sönmez, O. (2018). Detecting and dating structural breaks in functional data without dimension reduction. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 80(3):509– 529.
  6. Aue, A., Rice, G., and Sönmez, O. (2020). Structural break analysis for spectrum and trace of covariance operators. Environmetrics, 31(1):e2617.
  7. Berkes, I., Gabrys, R., Horváth, L., and Kokoszka, P. (2009). Detecting changes in the mean of functional observations. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 71(5):927–946.
  8. Chen, Y.-T., Chiou, J.-M., and Huang, T.-M. (2023). Greedy segmentation for a functional data sequence. Journal of the American Statistical Association, 118(542):959–971.
  9. Dette, H. and Kutta, T. (2021). Detecting structural breaks in eigensystems of functional time series. Electronic Journal of Statistics, 15(1):944–983.
  10. Fukumizu, K., Gretton, A., Sun, X., and Schölkopf, B. (2007). Kernel measures of conditional dependence. Advances in Neural Information Processing Systems, 20:489–496.
  11. Garreau, D., Jitkrittum, W., and Kanagawa, M. (2017). Large sample analysis of the median heuristic. arXiv preprint arXiv:1707.07269.
  12. Gretton, A., Borgwardt, K. M., Rasch, M. J., Schölkopf, B., and Smola,
  13. A. (2012). A kernel two-sample test. The Journal of Machine Learning Research, 13(1):723–773.
  14. Horváth, L. and Rice, G. (2024). Change Point Analysis for Time Series.
  15. Springer, New York.
  16. Horváth, L., Rice, G., and Zhao, Y. (2022). Change point analysis of covariance functions: A weighted cumulative sum approach. Journal of Multivariate Analysis, 189:104877.
  17. Hsing, T. and Eubank, R. (2015). Theoretical Foundations of Functional Data Analysis, with an Introduction to Linear Operators. John Wiley, Chichester.
  18. Li, B. and Song, J. (2017). Nonlinear sufficient dimension reduction for functional data. The Annals of Statistics, 45(3):1059–1095.
  19. Li, D., Li, R., and Shang, H. L. (2024). Detection and estimation of structural breaks in high-dimensional functional time series. The Annals of Statistics, 52(4):1716–1740.
  20. Muandet, K., Fukumizu, K., Sriperumbudur, B., and Schölkopf, B. (2017). Kernel mean embedding of distributions: A review and beyond. Foundations and Trends® in Machine Learning, 10(1-2):1–141.
  21. Rahimi, A. and Recht, B. (2007). Random features for large-scale kernel machines. Advances in Neural Information Processing Systems, 20.
  22. Ramsay, J. O. and Silverman, B. W. (2005). Functional Data Analysis 2nd edition. Springer, New York.
  23. Sang, P. and Li, B. (2026). Nonlinear function-on-function regression by RKHS. Journal of Machine Learning Research, 27(12):1–54.
  24. Sriperumbudur, B. K., Fukumizu, K., and Lanckriet, G. R. (2011). Universality, characteristic kernels and RKHS embedding of measures. Journal of Machine Learning Research, 12(7):2389–2410.
  25. Wang, J.-L., Chiou, J.-M., and Müller, H.-G. (2016). Functional data analysis. Annual Review of Statistics and Its Application, 3(1):257–295.
  26. Wynne, G. and Duncan, A. B. (2022). A kernel two-sample test for functional data. Journal of Machine Learning Research, 23(73):1–51. ---