The Korean Journal of Applied Statistics (응용통계연구)

The Korean Statistical Society (한국통계학회)
권호 표지
DOI QR코드

DOI QR Code

Detection of multiple change points using penalized least square methods: a comparative study between ℓ0 and ℓ1 penalty

벌점-최소제곱법을 이용한 다중 변화점 탐색

  • Son, Won (Department of Statistics, Seoul National University) ;
  • Lim, Johan (Department of Statistics, Seoul National University) ;
  • Yu, Donghyeon (Department of Statistics, Keimyung University)
  • Received : 2016.09.26
  • Accepted : 2016.09.30
  • Published : 2016.10.31
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, we numerically compare two penalized least square methods, the ${\ell}_0$-penalized method and the fused lasso regression (FLR, ${\ell}_1$ penalization), in finding multiple change points of a signal. We find that the ${\ell}_0$-penalized method performs better than the FLR, which produces many false detections in some cases as the theory tells. In addition, the computation of ${\ell}_0$-penalized method relies on dynamic programming and is as efficient as the FLR.

본 연구에서는 다중 변화점 탐색과 관련하여 최근 많은 관심을 받고 있는 ${\ell}_0$-벌점 최소제곱법과 fused-라쏘-회귀(fused lasso regression; FLR)방법을 모의 실험을 통하여 비교하였다. 모의 실험의 결과로 FLR방법은 비-변화점을 변화점으로 잘못 탐색하는 경향이 ${\ell}_0$-벌점 최소제곱법과 비교할 때 상대적으로 높게 나타났으며 ${\ell}_0$-벌점 최소제곱법이 전반적으로 FLR방법에 비하여 좋은 성능을 보였다. 더불어 ${\ell}_0$-벌점 최소제곱법은 동적프로그래밍을 통하여 FLR 방법과 유사하게 효율적인 계산이 가능하다.

Keywords

References

  1. Basseville, M. and Nikiforov, I. V. (1993). Detection of Abrupt Changes: Theory and Application (Vol. 104), Prentice Hall, Englewood Cliffs.
  2. Carlstein, E., Muller, H.-G., and Siegmund, D. (1994). Change-point Problems, Institute of Mathematical Statistics, California.
  3. Chen, J. and Gupta, A. K. (2001). On change point detection and estimation. Communications in Statistics-Simulation and Computation, 30, 665-697. https://doi.org/10.1081/SAC-100105085
  4. Csorgo, M. and Horvath, L. (1997). Limit Theorems in Change-Point Analysis, John Wiley & Sons, New York.
  5. Harchaoui, Z. and Levy-Leduc, C. (2010). Multiple change-point estimation with a total variation penalty. Journal of the American Statistical Association, 105, 1480-1493. https://doi.org/10.1198/jasa.2010.tm09181
  6. Jang, W., Lim, J., Lazar, N. A., Loh, J. M., and Yu, D. (2015). Some properties of generalized fused lasso and its applications to high dimensional data. Journal of the Korean Statistical Society, 44, 352-365. https://doi.org/10.1016/j.jkss.2014.10.002
  7. Johnson, N. A. (2013). A dynamic programming algorithm for the fused Lasso and $L_0$-segmentation. Journal of Computational and Graphical Statistics, 22, 246-260. https://doi.org/10.1080/10618600.2012.681238
  8. Kotz, S., Read, C. B., Balakrishnan, N., Vidakovic, B., and Johnson, N. L. (Eds.) (2006). Encyclopedia of Statistical Sciences (2nd ed.), John Wiley & Sons, NJ.
  9. Lim, E., Hahn, K. S., Lim, J., Kim, M., Park, J., and Yoon, J. (2012). Statistical properties of news coverage data. Communications for Statistical Applications and Methods, 19, 771-780. https://doi.org/10.5351/CKSS.2012.19.6.771
  10. Lin, K., Sharpnack, J., Rinaldo, A., and Tibshirani, R. J. (2016). Approximate recovery in changepoint Problems, from $\ell$2 estimation error rates, arXiv preprint, arXiv:1606.06746.
  11. Qian, J. and Jia, J. (2016). On stepwise pattern recovery of the fused Lasso. Computational Statistics and Data Analysis, 94, 221-237. https://doi.org/10.1016/j.csda.2015.08.013
  12. Rinaldo, A. (2009). Properties and refinements of the fused lasso. The Annals of Statistics, 37, 2922-2952. https://doi.org/10.1214/08-AOS665
  13. Rojas, C. R. and Wahlberg, B. (2015). How to monitor and mitigate stair-casing in L1 trend filtering, arXiv preprint, arXiv:1412.0607v1.
  14. Tibshirani, R., Saunders, M., Rosset, S., Zhu, J., and Knight, K. (2005). Sparsity and smoothness via the fused lasso. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 67, 91-108. https://doi.org/10.1111/j.1467-9868.2005.00490.x
  15. Tibshirani, R. and Wang, P. (2008). Spatial smoothing and hot spot detection for CGH data using the fused lasso. Biostatistics, 9, 18-29. https://doi.org/10.1093/biostatistics/kxm013
  16. Ye, G.-B. and Xie, X. (2011). Split Bregman method for large scale fused Lasso. Computational Statistics and Data Analysis, 55, 1552-1569. https://doi.org/10.1016/j.csda.2010.10.021
  17. Yu, D., Won, J., Lee, T., Lim, J., and Yoon, S. (2015a). High-dimensional fused lasso regression using majorization-minimization and parallel processing. Journal of Computational and Graphical Statistics, 24, 121-153. https://doi.org/10.1080/10618600.2013.878662
  18. Yu, D., Lee, S. J., Lee, W. J., Kim, S. C., Lim, J., and Kwon, S. W. (2015b). Classification of spectral data using fused lasso logistic regression. Chemometrics and Intelligent Laboratory Systems, 142, 70-77. https://doi.org/10.1016/j.chemolab.2015.01.006
  19. Zhao, P. and Yu, B. (2006). On model selection consistency of Lasso. Journal of Machine Learning Research, 7, 2541-2563.