The Korean Journal of Applied Statistics (응용통계연구)

The Korean Statistical Society (한국통계학회)
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An empirical evidence of inconsistency of the ℓ1 trend filtering in change point detection

ℓ1 추세필터의 변화점 식별에 있어서의 비일치성

  • Yu, Donghyeon (Department of Statistics, Inha University) ;
  • Lim, Johan (Department of Statistics, Seoul National University) ;
  • Son, Won (Department of Information Statistics, Dankook University)
  • 유동현 (인하대학교 통계학과) ;
  • 임요한 (서울대학교 통계학과) ;
  • 손원 (단국대학교 정보통계학과)
  • Received : 2022.01.26
  • Accepted : 2022.02.19
  • Published : 2022.06.30
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Abstract

The fused LASSO signal approximator (FLSA) can be applied to find change points from the data having piecewise constant mean structure. It is well-known that the FLSA is inconsistent in change points detection. This inconsistency is due to a total-variation denoising penalty of the FLSA. ℓ1 trend filter, one of the popular tools for finding an underlying trend from data, can be used to identify change points of piecewise linear trends. Since the ℓ1 trend filter applies the sum of absolute values of slope differences, it can be inconsistent for change points recovery as the FLSA. However, there are few studies on the inconsistency of the ℓ1 trend filtering. In this paper, we demonstrate the inconsistency of the ℓ1 trend filtering with a numerical study.

구간별 상수 구조를 가지는 관측값으로부터 변화점을 식별하기 위해 FLSA가 자주 사용되고 있다. FLSA는 총변동벌점을 이용하기 때문에 평균 수준이 단조성을 가지는 경우에는 변화점 식별에서의 일치성이 보장되지 않는다는 특징이 있다. ℓ1 추세필터는 오차제곱합과 기울기 차이에 대한 ℓ1 벌점의 합을 목적함수로 가지는 구간별 선형 구조 추정방법으로 구간별 선형 구조에서의 변화점을 식별하기 위해 활용할 수 있다. 한편, ℓ1 추세필터의 경우에도 총변동벌점을 이용하므로 FLSA와 마찬가지로 변화점 식별에 있어서 비일치성을 보일 것으로 예상할 수 있는데 이와 관련된 연구는 아직까지 많이 이루어져 있지 않다. 이 연구에서는 모의실험을 통해 구간별 선형 모형에서 변화점을 식별하기 위해 사용되는 ℓ1 추세필터의 비일치성에 대해 살펴본다.

Keywords

Acknowledgement

이 성과는 정부(과학기술정보통신부)의 재원으로 한국연구재단의 지원을 받아 수행된 연구임 (No. 2020R1F1A1A01048127(유동현), No. 2021R1A2C1010786(임요한), No. 2020R1F1A1A01051039(손원)).

References

  1. Friedman J, Hastie T, Hofling H, and Tibshirani R (2007). Pathwise coordinate optimization. The Annals of Applied Statistics, 1 (2), 302-332. https://doi.org/10.1214/07-AOAS131
  2. Fryzlewicz P (2014). Wild binary segmentation for multiple change-point detection. The Annals of Statistics, 42 (6), 2243-2281. https://doi.org/10.1214/14-AOS1245
  3. Hoefling H (2010). A path algorithm for the fused lasso signal approximator. Journal of Computational and Graphical Statistics, 19 (4), 984-1006. https://doi.org/10.1198/jcgs.2010.09208
  4. Hodrick RJ and Prescott EC (1997). Postwar US business cycles: an empirical investigation. Journal of Money, credit, and Banking, 29 (1), 1-16. https://doi.org/10.2307/2953682
  5. Kim SJ, Koh K, Boyd S, and Gorinevsky D (2009). ℓ1 trend filtering. SIAM Review, 51 (2), 339-360. https://doi.org/10.1137/070690274
  6. Olshen AB, Venkatraman ES, Lucito R, and Wigler M (2004). Circular binary segmentation for the analysis of array-based DNA copy number data. Biostatistics, 5 (4), 557-572. https://doi.org/10.1093/biostatistics/kxh008
  7. Pettitt AN (1979). A non-parametric approach to the change-point problem. Journal of the Royal Statistical Society: Series C (Applied Statistics), 28 (2), 126-135. https://doi.org/10.2307/2346729
  8. Rojas CR and Wahlberg B (2014). On change point detection using the fused lasso method. arXiv preprint arXiv:1401.5408.
  9. Son W and Lim J (2019). Modified path algorithm of fused lasso signal approximator for consistent recovery of change points. Jounal of Statistical Planning and Inference, 200, 223-238. https://doi.org/10.1016/j.jspi.2018.10.003
  10. 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, 67 (1), 91-108. https://doi.org/10.1111/j.1467-9868.2005.00490.x
  11. Yao YC (1988). Estimating the number of change-points via Schwarz' criterion. Statistics and Probability Letters, 6 (3), 181-189. https://doi.org/10.1016/0167-7152(88)90118-6