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Sparse estimation in functional linear regression

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  • 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
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    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.
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    Cited by:

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

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