IDEAS home Printed from https://ideas.repec.org/a/eee/jmvana/v105y2012i1p1-17.html
   My bibliography  Save this article

Sparse estimation in functional linear regression

Author

Listed:
  • Lee, Eun Ryung
  • Park, Byeong U.

Abstract

As a useful tool in functional data analysis, the functional linear regression model has become increasingly common and been studied extensively in recent years. In this paper, we consider a sparse functional linear regression model which is generated by a finite number of basis functions in an expansion of the coefficient function. In this model, we do not specify how many and which basis functions enter the model, thus it is not like a typical parametric model where predictor variables are pre-specified. We study a general framework that gives various procedures which are successful in identifying the basis functions that enter the model, and also estimating the resulting regression coefficients in one-step. We adopt the idea of variable selection in the linear regression setting where one adds a weighted L1 penalty to the traditional least squares criterion. We show that the procedures in our general framework are consistent in the sense of selecting the model correctly, and that they enjoy the oracle property, meaning that the resulting estimators of the coefficient function have asymptotically the same properties as the oracle estimator which uses knowledge of the underlying model. We investigate and compare several methods within our general framework, via a simulation study. Also, we apply the methods to the Canadian weather data.

Suggested Citation

  • Lee, Eun Ryung & Park, Byeong U., 2012. "Sparse estimation in functional linear regression," Journal of Multivariate Analysis, Elsevier, vol. 105(1), pages 1-17.
    • Handle: RePEc:eee:jmvana:v:105:y:2012:i:1:p:1-17
    DOI: 10.1016/j.jmva.2011.08.005
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0047259X11001643
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.jmva.2011.08.005?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Hansheng Wang & Bo Li & Chenlei Leng, 2009. "Shrinkage tuning parameter selection with a diverging number of parameters," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 71(3), pages 671-683, June.
    2. Peter Hall & You‐Jun Yang, 2010. "Ordering and selecting components in multivariate or functional data linear prediction," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 72(1), pages 93-110, January.
    3. Zou, Hui, 2006. "The Adaptive Lasso and Its Oracle Properties," Journal of the American Statistical Association, American Statistical Association, vol. 101, pages 1418-1429, December.
    4. Cardot, Hervé & Johannes, Jan, 2010. "Thresholding projection estimators in functional linear models," Journal of Multivariate Analysis, Elsevier, vol. 101(2), pages 395-408, February.
    5. Yuhong Yang, 2005. "Can the strengths of AIC and BIC be shared? A conflict between model indentification and regression estimation," Biometrika, Biometrika Trust, vol. 92(4), pages 937-950, December.
    6. Cardot, Hervé & Ferraty, Frédéric & Sarda, Pascal, 1999. "Functional linear model," Statistics & Probability Letters, Elsevier, vol. 45(1), pages 11-22, October.
    7. Cardot, Herve & Johannes, Jan, 2010. "Thresholding projection estimators in functional linear models," LIDAM Reprints ISBA 2010034, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    8. James, Gareth M. & Silverman, Bernard W., 2005. "Functional Adaptive Model Estimation," Journal of the American Statistical Association, American Statistical Association, vol. 100, pages 565-576, June.
    9. Wang, Hansheng & Leng, Chenlei, 2007. "Unified LASSO Estimation by Least Squares Approximation," Journal of the American Statistical Association, American Statistical Association, vol. 102, pages 1039-1048, September.
    10. Gareth M. James, 2002. "Generalized linear models with functional predictors," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 64(3), pages 411-432, August.
    11. Fan J. & Li R., 2001. "Variable Selection via Nonconcave Penalized Likelihood and its Oracle Properties," Journal of the American Statistical Association, American Statistical Association, vol. 96, pages 1348-1360, December.
    12. Hansheng Wang & Runze Li & Chih-Ling Tsai, 2007. "Tuning parameter selectors for the smoothly clipped absolute deviation method," Biometrika, Biometrika Trust, vol. 94(3), pages 553-568.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Belli, Edoardo, 2022. "Smoothly adaptively centered ridge estimator," Journal of Multivariate Analysis, Elsevier, vol. 189(C).
    2. Mousavi, Seyed Nourollah & Sørensen, Helle, 2017. "Multinomial functional regression with wavelets and LASSO penalization," Econometrics and Statistics, Elsevier, vol. 1(C), pages 150-166.
    3. A. Delaigle & P. Hall, 2016. "Approximating fragmented functional data by segments of Markov chains," Biometrika, Biometrika Trust, vol. 103(4), pages 779-799.
    4. Brunel, Élodie & Mas, André & Roche, Angelina, 2016. "Non-asymptotic adaptive prediction in functional linear models," Journal of Multivariate Analysis, Elsevier, vol. 143(C), pages 208-232.
    5. Floriello, Davide & Vitelli, Valeria, 2017. "Sparse clustering of functional data," Journal of Multivariate Analysis, Elsevier, vol. 154(C), pages 1-18.
    6. Fraiman, Ricardo & Gimenez, Yanina & Svarc, Marcela, 2016. "Feature selection for functional data," Journal of Multivariate Analysis, Elsevier, vol. 146(C), pages 191-208.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Fei Jin & Lung-fei Lee, 2018. "Lasso Maximum Likelihood Estimation of Parametric Models with Singular Information Matrices," Econometrics, MDPI, vol. 6(1), pages 1-24, February.
    2. Jin, Fei & Lee, Lung-fei, 2018. "Irregular N2SLS and LASSO estimation of the matrix exponential spatial specification model," Journal of Econometrics, Elsevier, vol. 206(2), pages 336-358.
    3. Fan, Rui & Lee, Ji Hyung & Shin, Youngki, 2023. "Predictive quantile regression with mixed roots and increasing dimensions: The ALQR approach," Journal of Econometrics, Elsevier, vol. 237(2).
    4. Ping Zeng & Yongyue Wei & Yang Zhao & Jin Liu & Liya Liu & Ruyang Zhang & Jianwei Gou & Shuiping Huang & Feng Chen, 2014. "Variable selection approach for zero-inflated count data via adaptive lasso," Journal of Applied Statistics, Taylor & Francis Journals, vol. 41(4), pages 879-894, April.
    5. Hao, Meiling & Lin, Yunyuan & Zhao, Xingqiu, 2016. "A relative error-based approach for variable selection," Computational Statistics & Data Analysis, Elsevier, vol. 103(C), pages 250-262.
    6. Yanxin Wang & Qibin Fan & Li Zhu, 2018. "Variable selection and estimation using a continuous approximation to the $$L_0$$ L 0 penalty," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 70(1), pages 191-214, February.
    7. Pötscher, Benedikt M., 2007. "Confidence Sets Based on Sparse Estimators Are Necessarily Large," MPRA Paper 5677, University Library of Munich, Germany.
    8. repec:hum:wpaper:sfb649dp2016-047 is not listed on IDEAS
    9. Caner, Mehmet & Fan, Qingliang, 2015. "Hybrid generalized empirical likelihood estimators: Instrument selection with adaptive lasso," Journal of Econometrics, Elsevier, vol. 187(1), pages 256-274.
    10. Guang Cheng & Hao Zhang & Zuofeng Shang, 2015. "Sparse and efficient estimation for partial spline models with increasing dimension," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 67(1), pages 93-127, February.
    11. Sakyajit Bhattacharya & Paul McNicholas, 2014. "A LASSO-penalized BIC for mixture model selection," Advances in Data Analysis and Classification, Springer;German Classification Society - Gesellschaft für Klassifikation (GfKl);Japanese Classification Society (JCS);Classification and Data Analysis Group of the Italian Statistical Society (CLADAG);International Federation of Classification Societies (IFCS), vol. 8(1), pages 45-61, March.
    12. Ramon I. Garcia & Joseph G. Ibrahim & Hongtu Zhu, 2010. "Variable Selection in the Cox Regression Model with Covariates Missing at Random," Biometrics, The International Biometric Society, vol. 66(1), pages 97-104, March.
    13. Dengke Xu & Zhongzhan Zhang & Liucang Wu, 2014. "Variable selection in high-dimensional double generalized linear models," Statistical Papers, Springer, vol. 55(2), pages 327-347, May.
    14. Xingwei Tong & Xin He & Liuquan Sun & Jianguo Sun, 2009. "Variable Selection for Panel Count Data via Non‐Concave Penalized Estimating Function," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 36(4), pages 620-635, December.
    15. Joel L. Horowitz, 2015. "Variable selection and estimation in high-dimensional models," CeMMAP working papers 35/15, Institute for Fiscal Studies.
    16. Joel L. Horowitz & Lars Nesheim, 2018. "Using penalized likelihood to select parameters in a random coefficients multinomial logit model," CeMMAP working papers CWP29/18, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    17. Hirose, Kei & Tateishi, Shohei & Konishi, Sadanori, 2013. "Tuning parameter selection in sparse regression modeling," Computational Statistics & Data Analysis, Elsevier, vol. 59(C), pages 28-40.
    18. Li-Ping Zhu & Lin-Yi Qian & Jin-Guan Lin, 2011. "Variable selection in a class of single-index models," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 63(6), pages 1277-1293, December.
    19. Heng Lian & Peng Lai & Hua Liang, 2013. "Partially Linear Structure Selection in Cox Models with Varying Coefficients," Biometrics, The International Biometric Society, vol. 69(2), pages 348-357, June.
    20. Lian, Heng, 2014. "Semiparametric Bayesian information criterion for model selection in ultra-high dimensional additive models," Journal of Multivariate Analysis, Elsevier, vol. 123(C), pages 304-310.
    21. Guo-Liang Tian & Mingqiu Wang & Lixin Song, 2014. "Variable selection in the high-dimensional continuous generalized linear model with current status data," Journal of Applied Statistics, Taylor & Francis Journals, vol. 41(3), pages 467-483, March.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:jmvana:v:105:y:2012:i:1:p:1-17. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/wps/find/journaldescription.cws_home/622892/description#description .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.