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Wang, Yafei; Kong, Linglong; Jiang, Bei; Zhou, Xingcai; Yu, Shimei; Zhang, Li; Heo, Giseon

Wavelet-based LASSO in functional linear quantile regression. (English) Zbl 07193772

J. Stat. Comput. Simulation 89, No. 6, 1111-1130 (2019).
Summary: In this paper, we develop an efficient wavelet-based regularized linear quantile regression framework for coefficient estimations, where the responses are scalars and the predictors include both scalars and function. The framework consists of two important parts: wavelet transformation and regularized linear quantile regression. Wavelet transform can be used to approximate functional data through representing it by finite wavelet coefficients and effectively capturing its local features. Quantile regression is robust for response outliers and heavy-tailed errors. In addition, comparing with other methods it provides a more complete picture of how responses change conditional on covariates. Meanwhile, regularization can remove small wavelet coefficients to achieve sparsity and efficiency. A novel algorithm, Alternating Direction Method of Multipliers (ADMM) is derived to solve the optimization problems. We conduct numerical studies to investigate the finite sample performance of our method and applied it on real data from ADHD studies.

Cited in 5 Documents

MSC:

62-XX Statistics

Keywords:

quantile regression; wavelets; functional data analysis; LASSO; ADMM; ADHD

Software:

quantreg; rwt; wavethresh; flare; RWT
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Full Text: DOI

References:

[1] Fass L.Imaging and cancer: a review. Mol Oncol. 2008;2(2):115-152. [Crossref], [PubMed], [Web of Science ®], [Google Scholar]
[2] Friston KJ.Modalities, modes, and models in functional neuroimaging. Science. 2009;326(5951):399-403. [Crossref], [PubMed], [Web of Science ®], [Google Scholar]
[3] Zhao Y, Ogden RT, Reiss PT.Wavelet-based LASSO in functional linear regression. J Comput Graph Stat. 2012;21(3):600-617. [Taylor & Francis Online], [Web of Science ®], [Google Scholar]
[4] Yu D, Kong L, Mizera I.Partial functional linear quantile regression for neuroimaging data analysis. Neurocomputing. 2016;195:74-87. [Crossref], [Web of Science ®], [Google Scholar]
[5] Koenker R, Bassett G.Regression quantiles. Econometrica. 1978;46:33-50. [Crossref], [Web of Science ®], [Google Scholar] · Zbl 0373.62038
[6] Koenker R. Quantile regression. Cambridge: Cambridge University Press; 2005. [Crossref], [Google Scholar] · Zbl 1111.62037
[7] Müller H, Stadtmüller U.Generalized functional linear models. Ann Stat. 2005;33:774-805. [Crossref], [Web of Science ®], [Google Scholar] · Zbl 1068.62048
[8] Müller H, Yao F.Functional additive models. J Am Stat Assoc. 2008;103:1534-1544. [Taylor & Francis Online], [Web of Science ®], [Google Scholar] · Zbl 1286.62040
[9] Delaigle A, Hall P, Apanasovich TV.Weighted least squares methods for prediction in the functional linear model. Electron J Stat. 2009;3:865-885. [Crossref], [Web of Science ®], [Google Scholar] · Zbl 1326.62151
[10] Marx BD, Eilers PHC.Generalized linear regression for sampled signals or curves: a p-spline approach. Technometrics. 1999;41:1-13. [Taylor & Francis Online], [Web of Science ®], [Google Scholar]
[11] Cardot H, Ferraty F, Sarda P.Functional linear model. Stat Probab Lett. 1999;45:11-22. [Crossref], [Web of Science ®], [Google Scholar] · Zbl 0962.62081
[12] Reiss PT, Ogden RT.Functional principal component regression and functional partial least squares. J Am Stat Assoc. 2007;102:984-996. [Taylor & Francis Online], [Web of Science ®], [Google Scholar] · Zbl 1469.62237
[13] Delaigle A, Hall P.Methodology and theory for partial least squares applied to functional data. Ann Stat. 2012;40:322-352. [Crossref], [Web of Science ®], [Google Scholar] · Zbl 1246.62084
[14] Apenteng OO, Ismail NA.The impact of the wavelet propagation distribution on SEIRS modeling with delay. PLoS ONE. 2014;9(6):e98288. [Crossref], [PubMed], [Web of Science ®], [Google Scholar]
[15] Härdle W, Kerkyacharian G, Picard D, et al. Wavelets: approximation and statistical applications. New York (NY): Springer-Verlag; 1998. [Crossref], [Google Scholar] · Zbl 0899.62002
[16] Vidakovic B. Statistical modeling by wavelets. New York (NY): Wiley; 1999. [Crossref], [Google Scholar] · Zbl 0924.62032
[17] Tibshirani R.Regression shrinkage and selection via the LASSO. J R Stat Soc B. 1996;58:267-288. [Google Scholar] · Zbl 0850.62538
[18] Wang X, Nan B, Zhu J, et al. Regularized 3d functional regression for brain image data via haar wavelets. Ann Appl Stat. 2014;8(2):1045-1064. [Crossref], [PubMed], [Web of Science ®], [Google Scholar] · Zbl 1454.62415
[19] Zhao W, Zhang R, Liu J.Sparse group variable selection based on quantile hierarchical LASSO. J Appl Stat. 2014;41(8):1658-1677. [Taylor & Francis Online], [Web of Science ®], [Google Scholar] · Zbl 1514.62979
[20] Zou H.The adaptive LASSO and its oracle properties. J Am Stat Assoc. 2006;101(476):1418-1429. [Taylor & Francis Online], [Web of Science ®], [Google Scholar] · Zbl 1171.62326
[21] Fan J, Li R.Variable selection via nonconcave penalized likelihood and its oracle properties. J Am Stat Assoc. 2001;96(456):1348-1360. [Taylor & Francis Online], [Web of Science ®], [Google Scholar] · Zbl 1073.62547
[22] Zhang CH, et al. Nearly unbiased variable selection under minimax concave penalty. Ann Stat. 2010;38(2):894-942. [Crossref], [Web of Science ®], [Google Scholar] · Zbl 1183.62120
[23] Zou H, Yuan M.Composite quantile regression and the oracle model selection theory. Ann Stat. 2008;36:1108-1126. [Crossref], [Web of Science ®], [Google Scholar] · Zbl 1360.62394
[24] Kai B, Li R, Zou H.Local composite quantile regression smoothing: an efficient and safe alternative to local polynomial regression. J R Stat Soc B. 2010;72(1):49-69. [Crossref], [Google Scholar] · Zbl 1411.62101
[25] Fan J, Lv J.A selective overview of variable selection in high dimensional feature space. Stat Sin. 2010;20(1):101. [PubMed], [Web of Science ®], [Google Scholar] · Zbl 1180.62080
[26] Bradic J, Fan J, Wang W.Penalized composite quasi-likelihood for ultrahigh dimensional variable selection. J R Stat Soc B. 2011;73(3):325-349. [Crossref], [Google Scholar] · Zbl 1411.62181
[27] Koenker R, Park BJ.An interior point algorithm for nonlinear quantile regression. J Econom. 1996;71(1):265-283. [Crossref], [Web of Science ®], [Google Scholar] · Zbl 0855.62030
[28] Gabay D, Mercier B.A dual algorithm for the solution of nonlinear variational problems via finite element approximation. Comput Math Appl. 1976;2(1):17-40. [Crossref], [Google Scholar] · Zbl 0352.65034
[29] Boyd S, Parikh N, Chu E, et al. Distributed optimization and statistical learning via the alternating direction method of multipliers. Found Trends Mach Learn. 2011;3(1):1-122. [Crossref], [Google Scholar] · Zbl 1229.90122
[30] Hestenes MR.Multiplier and gradient methods. J Optim Theory Appl. 1969;4(5):303-320. [Crossref], [Google Scholar] · Zbl 0174.20705
[31] Lin Z, M C, Ma Y. The augmented Lagrange multiplier method for exact recovery of corrupted low-rank matrices; 2013. ArXiv:1009.5055. [Google Scholar]
[32] Li X, Zhao T, Yuan X, et al. The flare package for high dimensional linear regression and precision matrix estimation in r. J Mach Learn Res. 2015;16(1):553-557. [PubMed], [Google Scholar] · Zbl 1337.62007
[33] Roebuck P. group RUD. rwt: Rice wavelet toolbox wrapper; 2014. R package version 1.0.0; Available from: http://CRAN.R-project.org/package=rwt. [Google Scholar]
[34] Nason G. wavethresh: Wavelets statistics and transforms.; 2013. R package version 4.6.6; Available from: http://CRAN.R-project.org/package=wavethresh. [Google Scholar]
[35] Koenker R. quantreg: Quantile regression; 2015. R package version 5.11; Available from: http://CRAN.R-project.org/package=quantreg. [Google Scholar]
[36] Campbell SB.Handbook of developmental psychopathology. In: Sameroff A.J., Lewis M., Miller S.M., editors. Attention-deficit/hyperactivity disorder. In. Boston: Springer; 2000. p. 383-401. [Google Scholar]
[37] Tzourio-Mazoyer N, Landeau B, Papathanassiou D, et al. Automated anatomical labeling of activations in SPM using a macroscopic anatomical parcellation of the MNI MRI single-subject brain. Neuroimage. 2002;15(1):273-289. [Crossref], [PubMed], [Web of Science ®], [Google Scholar]
[38] Tang Q, Kong L.Quantile regression in functional linear semiparametric model. Statistics. 2017;51(6):1342-1358. [Taylor & Francis Online], [Web of Science ®], [Google Scholar] · Zbl 1387.62054
[39] Yuan M, Lin Y.Model selection and estimation in regression with grouped variables. J R Stat Soc B. 2006;68(1):49-67. [Crossref], [Google Scholar] · Zbl 1141.62030
[40] Simon N, Friedman J, Hastie T, et al. A sparse-group LASSO. J Comput Graph Stat. 2013;22(2):231-245. [Taylor & Francis Online], [Web of Science ®], [Google Scholar]
[41] Mallat S. A wavelet tour of signal processing. Burlington: Elsevier; 2009. [Google Scholar] · Zbl 1170.94003
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