Elastic trend filtering
-
Juyoung Jeong
and
Sangwoon Yun
Abstract
Trend filtering aims to estimate underlying trends in time series data, which is necessary to investigate data in a variety of disciplines. We propose a new method called elastic trend filtering. The proposed method combines ℓ 2 and ℓ 1 norm penalties to exploit the benefits and strengths of Hodrick–Prescott and ℓ 1 trend filterings. We apply the alternating direction method of multipliers for its efficient computation and numerical experiments show the soundness and efficiency of the proposed method. We further apply the proposed method to graph cases for potential applications and suggest a trend filtering for its variance estimate.
Funding source: National Research Foundation of Korea
Award Identifier / Grant number: NRF-2016R1A5A1008055
Award Identifier / Grant number: NRF-2021R1C1C2008350
Award Identifier / Grant number: NRF-2019R1F1A1047134, 2022R1A2C1010537
Award Identifier / Grant number: NRF-2019R1F1A1057051, 2022R1A2C1011503
-
Author contribution: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.
-
Research funding: This work was supported by the National Research Foundation of Korea NRF-2016R1A5A1008055. The first author was supported by the National Research Foundation of Korea NRF-2021R1C1C2008350. The second author was supported by the National Research Foundation of KoreaNRF-2019R1F1A1047134, 2022R1A2C1010537. The third author was supported by the National Research Foundation of Korea NRF-2019R1F1A1057051, 2022R1A2C1011503.
-
Conflict of interest statement: The authors declare no conflicts of interest regarding this article.
References
[1] S.-J. Kim, K. Koh, S. Boyd, and D. Gorinevsky, “ℓ1 trend filtering,” SIAM Rev., vol. 51, no. 2, pp. 339–360, 2009. https://doi.org/10.1137/070690274.Search in Google Scholar
[2] R. J. Hodrick and E. C. Prescott, “Postwar U.S. business cycles: an empirical investigation,” J. Money Credit Bank., vol. 29, no. 1, p. 1, 1997. https://doi.org/10.2307/2953682.Search in Google Scholar
[3] R. J. Tibshirani, Divided Differences, Falling Factorials, and Discrete Splines: Another Look at Trend Filtering and Related Problems, 2020. arXiv:2003.03886.Search in Google Scholar
[4] Y. M. Jung, T. Jeong, and S. Yun, “Non-convex TV denoising corrupted by impulse noise,” Inverse Probl. Imaging, vol. 11, no. 4, pp. 689–702, 2017.10.3934/ipi.2017032Search in Google Scholar
[5] Y. M. Jung, J. J. Whang, and S. Yun, “Sparse probabilistic k-means,” Appl. Math. Comput., vol. 382, p. 125328, 2020. https://doi.org/10.1016/j.amc.2020.125328.Search in Google Scholar
[6] L. Ziegelmeier, M. Kirby, and C. Peterson, “Stratifying high-dimensional data based on proximity to the convex hull boundary,” SIAM Rev., vol. 59, no. 2, pp. 346–365, 2017. https://doi.org/10.1137/15m1047921.Search in Google Scholar
[7] H. Zou and T. Hastie, “Regularization and variable selection via the elastic net,” J. Roy. Stat. Soc. B, vol. 67, pp. 301–320, 2005. https://doi.org/10.1111/j.1467-9868.2005.00503.x.Search in Google Scholar
[8] Y-X. Wang, S. James, A. J. Smola, and R. J. Tibshirani, “Trend filtering on graphs,” J. Mach. Learn. Res., vol. 17, no. 105, pp. 1–41, 2016.10.1007/s10994-015-5539-3Search in Google Scholar
[9] Centers for Disease Control and Prevention, United States Covid-19 Cases and Deaths by State over Time, 2020. Available at: https://data.cdc.gov/Case-Surveillance/United-States-COVID-19-Cases-and-Deaths-by-State-o/9mfq-cb36.Search in Google Scholar
[10] D. E. Giles, “Constructing confidence bands for the hodrick–prescott filter,” Appl. Econ. Lett., vol. 20, no. 5, pp. 480–484, 2013. https://doi.org/10.1080/13504851.2012.714057.Search in Google Scholar
[11] I. Gutman and W. Xiao, “Generalized inverse of the laplacian matrix and some applications,” Bulletin: Classe des sciences mathematiques et natturalles, vol. 129, no. 29, pp. 15–23, 2004. https://doi.org/10.2298/bmat0429015g.Search in Google Scholar
[12] S. A. Gershgorin, “Uber die abgrenzung der eigenwerte einer matrix,” Bulletin de l’Académie des Sciences de l’URSS. Classe des sciences mathématiques et na, vol. 6, pp. 749–754, 1931.Search in Google Scholar
[13] Texas Demographic Center, Population Estimates by County, San Antonio, 2019. Available at: https://demographics.texas.gov/Data/TPEPP/Estimates/.Search in Google Scholar
© 2022 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- Original Research Articles
- One-dimensional optimal system and similarity transformations for the 3 + 1 Kudryashov–Sinelshchikov equation
- Hopf bifurcations in a network of FitzHugh–Nagumo biological neurons
- Gravity modulation effect on weakly nonlinear thermal convection in a fluid layer bounded by rigid boundaries
- A new numerical formulation for the generalized time-fractional Benjamin Bona Mohany Burgers’ equation
- Chebyshev spectral method for solving a class of local and nonlocal elliptic boundary value problems
- Adaptive extragradient methods for solving variational inequalities in real Hilbert spaces
- Elastic trend filtering
- A new approach to representations of homothetic motions in Lorentz space
- Numerical simulation of variable-order fractional differential equation of nonlinear Lane–Emden type appearing in astrophysics
- Stability analysis and numerical simulations of the fractional COVID-19 pandemic model
- Numerical solutions of the Bagley–Torvik equation by using generalized functions with fractional powers of Laguerre polynomials
- N-fold Darboux transformation and exact solutions for the nonlocal Fokas–Lenells equation on the vanishing and plane wave backgrounds
- Closed form soliton solutions to the space-time fractional foam drainage equation and coupled mKdV evolution equations
- Master-slave synchronization in the Duffing-van der Pol and Φ6 Duffing oscillators
- Singularity analysis and analytic solutions for the Benney–Gjevik equations
- Trajectory controllability of nonlinear fractional Langevin systems
- Similarity transformations for modified shallow water equations with density dependence on the average temperature
- Non-equidistant partition predictor–corrector method for fractional differential equations with exponential memory
- Dynamical analysis of fractional-order of IVGTT glucose–insulin interaction
- Dynamic response of Mathieu–Duffing oscillator with Caputo derivative
- Non-Newtonian fluid flow having fluid–particle interaction through a porous zone in a channel with permeable walls
- A class of computationally efficient Newton-like methods with frozen inverse operator for nonlinear systems
Articles in the same Issue
- Frontmatter
- Original Research Articles
- One-dimensional optimal system and similarity transformations for the 3 + 1 Kudryashov–Sinelshchikov equation
- Hopf bifurcations in a network of FitzHugh–Nagumo biological neurons
- Gravity modulation effect on weakly nonlinear thermal convection in a fluid layer bounded by rigid boundaries
- A new numerical formulation for the generalized time-fractional Benjamin Bona Mohany Burgers’ equation
- Chebyshev spectral method for solving a class of local and nonlocal elliptic boundary value problems
- Adaptive extragradient methods for solving variational inequalities in real Hilbert spaces
- Elastic trend filtering
- A new approach to representations of homothetic motions in Lorentz space
- Numerical simulation of variable-order fractional differential equation of nonlinear Lane–Emden type appearing in astrophysics
- Stability analysis and numerical simulations of the fractional COVID-19 pandemic model
- Numerical solutions of the Bagley–Torvik equation by using generalized functions with fractional powers of Laguerre polynomials
- N-fold Darboux transformation and exact solutions for the nonlocal Fokas–Lenells equation on the vanishing and plane wave backgrounds
- Closed form soliton solutions to the space-time fractional foam drainage equation and coupled mKdV evolution equations
- Master-slave synchronization in the Duffing-van der Pol and Φ6 Duffing oscillators
- Singularity analysis and analytic solutions for the Benney–Gjevik equations
- Trajectory controllability of nonlinear fractional Langevin systems
- Similarity transformations for modified shallow water equations with density dependence on the average temperature
- Non-equidistant partition predictor–corrector method for fractional differential equations with exponential memory
- Dynamical analysis of fractional-order of IVGTT glucose–insulin interaction
- Dynamic response of Mathieu–Duffing oscillator with Caputo derivative
- Non-Newtonian fluid flow having fluid–particle interaction through a porous zone in a channel with permeable walls
- A class of computationally efficient Newton-like methods with frozen inverse operator for nonlinear systems
