Skip to main content

Constrained energy variation for change point detection

Abstract

The problem of change point detection can be solved either by online methods, based on a discrepancy measure, or by offline methods. The former tries to detect the change points one by one with a sliding window and leads to a lower computational time but are more sensitive to noise. Conversely, offline methods consider the entire data to detect all the change points which make them more robust against the noise but at a price of higher computational cost. In this paper, we propose an operational search method that combines the benefits of both approaches with the double aim to get higher noise resistance while keeping a blazingly fast time. The search method slides over the edges of the signal to determines their state by considering a global constrained energy. Thanks to the calculus of variation, the computation of this energy is reduced to the estimation of the effective jump for each edge. We study the performance and accuracy of our energy variation method to detect the change points in synthetic and real-world examples. The results compare favorably against state of the art algorithm in terms of speed and accuracy.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11

Notes

  1. 1.

    A clique \(\mathcal {C}\) is a complete graph where each node \(X_i\) is connected to all the other nodes \(X_j\). For our case, we only have unary cliques \(X_i\) and binary cliques \((X_i, X_{i+1})\).

  2. 2.

    https://www.rdocumentation.org/packages/neuroblastoma.

References

  1. Adams, R. P., & MacKay, D. J. (2007). Bayesian online changepoint detection, arXiv preprint arXiv:0710.3742

  2. Aminikhanghahi, S., & Cook, D. J. (2017). A survey of methods for time series change point detection. Knowledge and information systems, 51, 339–367.

    Article  Google Scholar 

  3. Belcaid, A., & Douimi, M. (2020). A novel online change point detection using an approximate random blanket and the line process energy. International Journal on Artificial Intelligence Tools, 29, 2050018.

    Article  Google Scholar 

  4. Belcaid, A., Douimi, M., & Fihri, A. F. (2018). Recursive reconstruction of piecewise constant signals by minimization of an energy function. Inverse Problems & Imaging, 12, 903–920.

    MathSciNet  Article  Google Scholar 

  5. Bellman, R. (1958). On a routing problem. Quarterly of Applied Mathematics, 16, 87–90.

    MathSciNet  Article  Google Scholar 

  6. Bishop, C. M. (2006). Pattern recognition and machine learning. Springer.

  7. Blake, A., Kohli, P., & Rother, C. (2011). Markov random fields for vision and image processing. MIT Press.

  8. Blake, A., & Zisserman, A. (1987). Visual reconstruction. MIT Press.

  9. Boykov, Y., Veksler, O., & Zabih, R. (2001). Fast approximate energy minimization via graph cuts. IEEE Transactions on Pattern Analysis and Machine Intelligence, 23, 1222–1239.

    Article  Google Scholar 

  10. Celisse, A., Marot, G., Pierre-Jean, M., & Rigaill, G. (2018). New efficient algorithms for multiple change-point detection with reproducing kernels. Computational Statistics & Data Analysis, 128, 200–220.

    MathSciNet  Article  Google Scholar 

  11. Chen, J., & Gupta, A. K. (1997). Testing and locating variance changepoints with application to stock prices. Journal of the American Statistical association, 92, 739–747.

    MathSciNet  Article  Google Scholar 

  12. Chen, J., & Gupta, A. K. (2011). Parametric statistical change point analysis: With applications to genetics, medicine, and finance. Springer.

  13. Chu, H., Chung, C. K., Jeong, W., & Cho, K.-H. (2017). Predicting epileptic seizures from scalp EEG based on attractor state analysis. Computer Methods and Programs in Biomedicine, 143, 75–87.

    Article  Google Scholar 

  14. Clifford, P. (1990). Markov random fields in statistics, Disorder in physical systems: A volume in honour of John M. Hammersleyhttps://www.bibsonomy.org/bibtex/2a34c398322fa84ea2efa23b6109772a8/arnsholt (pp. 19–32).

  15. Cremers, D., Rousson, M., & Deriche, R. (2007). A review of statistical approaches to level set segmentation: Integrating color, texture, motion and shape. International Journal of Computer Vision, 72, 195–215.

    Article  Google Scholar 

  16. Davies, B. (2002). Integral transforms and their applications (Vol. 41). Springer.

  17. Fan, Z., Guan, L., et al. (2018). Approximate \(l_0\) -penalized estimation of piecewise-constant signals on graphs. The Annals of Statistics, 46, 3217–3245.

    MathSciNet  MATH  Google Scholar 

  18. Frecon, J., Pustelnik, N., Dobigeon, N., Wendt, H., & Abry, P. (2017). Bayesian selection for the \(l_2\)-Potts model regularization parameter: 1-d piecewise constant signal denoising. IEEE Transactions on Signal Processing, 65, 5215–5224.

    MathSciNet  Article  Google Scholar 

  19. Fryzlewicz, P., et al. (2014). Wild binary segmentation for multiple change-point detection. The Annals of Statistics, 42, 2243–2281.

    MathSciNet  Article  Google Scholar 

  20. Gelfand, I. M., Silverman, R. A., et al. (2000). Calculus of variations. Courier Corporation.

  21. Geman, S., & Geman, D. (1984). Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images. IEEE Transactions on Pattern Analysis and Machine Intelligence, 6, 721–741.

    Article  Google Scholar 

  22. Gretton, A., Borgwardt, K., Rasch, M., Schölkopf, B., & Smola, A. J. (2007). A kernel method for the two-sample-problem. In Advances in neural information processing systemshttps://proceedings.neurips.cc/paper/2006/file/e9fb2eda3d9c55a0d89c98d6c54b5b3e-Paper.pdf vol 19. (pp. 513–520).

  23. Harchaoui, Z., & Lévy-Leduc, C. (2010). Multiple change-point estimation with a total variation penalty. Journal of the American Statistical Association, 105, 1480–1493.

    MathSciNet  Article  Google Scholar 

  24. Harchaoui, Z., Vallet, F., Lung-Yut-Fong, A., & Cappé, O. (2009). A regularized kernel-based approach to unsupervised audio segmentation. In 2009 IEEE international conference on acoustics, speech and signal processing (pp. 1665–1668) IEEE.

  25. Hochba, D. S. (1997). Approximation algorithms for NP-hard problems. ACM Sigact News, 28, 40–52.

    Article  Google Scholar 

  26. Hocking, T. D., Schleiermacher, G., Janoueix-Lerosey, I., Boeva, V., Cappo, J., Delattre, O., et al. (2013). Learning smoothing models of copy number profiles using breakpoint annotations. BMC Bioinformatics, 14, 1–15.

    Article  Google Scholar 

  27. Hohm, K., Storath, M., & Weinmann, A. (2015). An algorithmic framework for Mumford–Shah regularization of inverse problems in imaging. Inverse Problems, 31, 115011.

    MathSciNet  Article  Google Scholar 

  28. Janoueix-Lerosey, I., Schleiermacher, G., Michels, E., Mosseri, V., Ribeiro, A., Lequin, D., et al. (2009). Overall genomic pattern is a predictor of outcome in neuroblastoma. Journal of Clinical Oncology, 27, 1026–1033.

    Article  Google Scholar 

  29. Kaplan, A., Röschke, J., Darkhovsky, B., & Fell, J. (2001). Macrostructural EEG characterization based on nonparametric change point segmentation: Application to sleep analysis. Journal of Neuroscience Methods, 106, 81–90.

    Article  Google Scholar 

  30. Keogh, E., Chu, S., Hart, D., & Pazzani, M. (2001). An online algorithm for segmenting time series. In Proceedings 2001 IEEE international conference on data mining (pp. 289–296) IEEE.

  31. Killick, R., Fearnhead, P., & Eckley, I. A. (2012). Optimal detection of changepoints with a linear computational cost. Journal of the American Statistical Association, 107, 1590–1598.

    MathSciNet  Article  Google Scholar 

  32. Komodakis, N., Tziritas, G., & Paragios, N. (2008). Performance vs computational efficiency for optimizing single and dynamic MRFs: Setting the state of the art with primal-dual strategies. Computer Vision and Image Understanding, 112, 14–29.

    Article  Google Scholar 

  33. Lai, W. R., Johnson, M. D., Kucherlapati, R., & Park, P. J. (2005). Comparative analysis of algorithms for identifying amplifications and deletions in array CGH data. Bioinformatics, 21, 3763–3770.

    Article  Google Scholar 

  34. Lavielle, M. (1998). Optimal segmentation of random processes. IEEE Transactions on Signal Processing, 46, 1365–1373.

    Article  Google Scholar 

  35. Lavielle, M. (2005). Using penalized contrasts for the change-point problem. Signal Processing, 85, 1501–1510.

    Article  Google Scholar 

  36. Lebarbier, É. (2005). Detecting multiple change-points in the mean of Gaussian process by model selection. Signal Processing, 85, 717–736.

    Article  Google Scholar 

  37. Lemenant, A. (2016). A selective review on Mumford–Shah minimizers. Bollettino dell’Unione Matematica Italiana, 9, 69–113.

    MathSciNet  Article  Google Scholar 

  38. Li, S. Z. (2009). Markov random field modeling in image analysis. Springer.

  39. Little, M. A., & Jones, N. S. (2011). Generalized methods and solvers for noise removal from piecewise constant signals. ii. New methods. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 467, 3115–3140.

    Article  Google Scholar 

  40. Malladi, R., Kalamangalam, G. P., & Aazhang, B. (2013). Online bayesian change point detection algorithms for segmentation of epileptic activity. In 2013 Asilomar conference on signals, systems and computers (pp. 1833–1837) IEEE.

  41. Nikolova, M. (1999). Markovian reconstruction using a GNC approach. IEEE Transactions on Image Processing, 8, 1204–1220.

    Article  Google Scholar 

  42. Nikolova, M., Ng, M. K., Zhang, S., & Ching, W.-K. (2008). Efficient reconstruction of piecewise constant images using nonsmooth nonconvex minimization. SIAM Journal on Imaging Sciences, 1, 2–25.

    MathSciNet  Article  Google Scholar 

  43. Olshen, A. B., Venkatraman, E., Lucito, R., & Wigler, M. (2004). Circular binary segmentation for the analysis of array-based DNA copy number data. Biostatistics, 5, 557–572.

    Article  Google Scholar 

  44. Pallara, L. A. -N. F. -D., Ambrosio, L., & Fusco, N. (2000). Functions of bounded variation and free discontinuity problems. Oxford University Press.

  45. Pinkel, D., Segraves, R., Sudar, D., Clark, S., Poole, I., Kowbel, D., et al. (1998). High resolution analysis of DNA copy number variation using comparative genomic hybridization to microarrays. Nature Genetics, 20, 207.

    Article  Google Scholar 

  46. Rosskopf, J., Paul-Yuan, K., Plenio, M. B., & Michaelis, J. (2016). Energy-based scheme for reconstruction of piecewise constant signals observed in the movement of molecular machines. Physical Review E, 94, 022421.

    Article  Google Scholar 

  47. Saatçi, Y., Turner, R. D., & Rasmussen, C. E. (2010). Gaussian process change point models. In ICML (pp. 927–934).

  48. Schleiermacher, G., Janoueix-Lerosey, I., Ribeiro, A., Klijanienko, J., Couturier, J., Pierron, G., et al. (2010). Accumulation of segmental alterations determines progression in neuroblastoma. Journal of Clinical Oncology, 28, 3122–3130.

    Article  Google Scholar 

  49. Sen, A., & Srivastava, M. S. (1975). On tests for detecting change in mean. The Annals of Statistics, 3, 98–108.

    MathSciNet  Article  Google Scholar 

  50. Shumway, R. H., & Stoffer, D. S. (2017). Time series analysis and its applications: with R examples. Springer.

  51. Staudacher, M., Telser, S., Amann, A., Hinterhuber, H., & Ritsch-Marte, M. (2005). A new method for change-point detection developed for on-line analysis of the heart beat variability during sleep. Physica A: Statistical Mechanics and Its Applications, 349, 582–596.

    Article  Google Scholar 

  52. Storath, M., Weinmann, A., & Demaret, L. (2014). Jump-sparse and sparse recovery using Potts functionals. IEEE Transactions on Signal Processing, 62, 3654–3666.

    MathSciNet  Article  Google Scholar 

  53. Tartakovsky, A., Nikiforov, I., & Basseville, M. (2014). Sequential analysis: Hypothesis testing and changepoint detection. Chapman and Hall/CRC.

  54. Truong, C., Oudre, L., & Vayatis, N. (2019). Selective review of offline change point detection methods. Signal Processing, 167, 107299.

    Article  Google Scholar 

  55. Weinmann, A., Storath, M., & Demaret, L. (2015). The \(\text{ l}^{1}\)-Potts functional for robust jump-sparse reconstruction. SIAM Journal on Numerical Analysis, 53, 644–673.

    MathSciNet  Article  Google Scholar 

  56. Wu, D., Faria, A. V., Younes, L., Ross, C. A., Mori, S., & Miller, M. I. (2018). Whole-brain segmentation and change-point analysis of anatomical brain mri—Application in premanifest Huntington’s disease. JoVE (Journal of Visualized Experiments), 136, e57256.

    Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to A. Belcaid.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Belcaid, A., Belkbir, H. Constrained energy variation for change point detection. Multidim Syst Sign Process (2021). https://doi.org/10.1007/s11045-021-00785-w

Download citation

Keywords

  • Change point detection
  • Segmentation
  • Energy variation
  • MRF networks