Function-on-function linear quantile regression
Abstract
In this study, we propose a function-on-function linear quantile regression model that allows for more than one functional predictor to establish a more flexible and robust approach. The proposed model is first transformed into a finitedimensional space via the functional principal component analysis paradigm in the estimation phase. It is then approximated using the estimated functional principal component functions, and the estimated parameter of the quantile regression model is constructed based on the principal component scores. In addition, we propose a Bayesian information criterion to determine the optimum number of truncation constants used in the functional principal component decomposition. Moreover, a stepwise forward procedure and the Bayesian information criterion are used to determine the significant predictors for including in the model. We employ a nonparametric bootstrap procedure to construct prediction intervals for the response functions. The finite sample performance of the proposed method is evaluated via several Monte Carlo experiments and an empirical data example, and the results produced by the proposed method are compared with the ones from existing models.
Keyword : function-on-function regression, functional principal component analysis, median regression, quantile regression
How to Cite
Beyaztas, U., & Shang, H. L. (2022). Function-on-function linear quantile regression. Mathematical Modelling and Analysis, 27(2), 322–341. https://doi.org/10.3846/mma.2022.14664
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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G. Schwarz. Estimating the dimension of a model. The Annals of Statistics, 6(2):461–464, 1987. https://doi.org/10.1214/aos/1176344136
Q. Tang and L. Cheng. Partial functional linear quantile regression. Science China Mathematics, 57(12):2589–2608, 2014. https://doi.org/10.1007/s11425-014-4819-x
F. Yao, H.-G. Müller and J.-L. Wang. Functional linear regression analysis for longitudinal data. The Annals of Statistics, 33(6):2873–2903, 2005. https://doi.org/10.1214/009053605000000660
F. Yao, S. Sue-Chee and F. Wang. Regularized partially functional quantile regression. Journal of Multivariate Analysis, 156:39–56, 2017. https://doi.org/10.1016/j.jmva.2017.02.001
D. Yu, L. Kong and I. Mizera. Partial functional linear quantile regression for neuroimaging data analysis. Neurocomputing, 195:74–87, 2016. https://doi.org/10.1016/j.neucom.2015.08.116
U. Beyaztas and H.L. Shang. A comparison of parameter estimation in functionon-function regression. Communications in Statistics - Simulation and Computation, 0(0):1–31, 2020. https://doi.org/10.1080/03610918.2020.1746340
U. Beyaztas and H.L. Shang. On function-on-function regression: Partial least squares approach. Environmental and Ecological Statistics, 27(1):95–114, 2020. https://doi.org/10.1007/s10651-019-00436-1
H. Cardot, C. Crambes and P. Sarda. Quantile regression when the covariates are functions. Journal of Nonparametric Statistics, 17(7):841–856, 2005. https://doi.org/10.1080/10485250500303015
K. Chen and H-G. Mu¨ller. Conditional quantile analysis when covariates are functions, with application to growth data. Journal of the Royal Statistical Society: Series B, 74(1):67–89, 2012. https://doi.org/10.1111/j.1467-9868.2011.01008.x
J.-M. Chiou, Y.-F. Yang and Y.-T. Chen. Multivariate functional linear regression and prediction. Journal of Multivariate Analysis, 146:301–312, 2016. https://doi.org/10.1016/j.jmva.2015.10.003
Microsoft Corporation and S. Weston. doParallel: Foreach Parallel Adaptor for the ’parallel’ Package. R package version 1.0.1, 2020. Available from Internet: https://CRAN. R-project.org/package=doParallel
D. Şentürk and H.-G. Müller. Generalized varying coefficient models for longitudinal data. Biometrika, 95(3):653–666, 2008. https://doi.org/10.1093/biomet/asn006
A. Cuevas. A partial overview of the theory of statistics with functional data. Journal of Statistical Planning and Inference, 147:1–23, 2014. https://doi.org/10.1016/j.jspi.2013.04.002
F. Ferraty, A. Rabhi and P. Vieu. Conditional quantiles for dependent functional data with application to the climatic El Nino phenomenon. Sankhyā: The Indian Journal of Statistics, 67(2):378–398, 2005.
F. Ferraty and P. Vieu. Nonparametric Functional Data Analysis. Springer, New York, 2006. https://doi.org/10.1007/0-387-36620-2
P. Hall and J.L. Horowitz. Methodology and convergence rates for functional linear regression. The Annals of Statistics, 35(1):70–91, 2007. https://doi.org/10.1214/009053606000000957
J. Harezlak, B.A. Coull, N.M. Laird, S.R. Magari and D.C. Christiani. Penalized solutions to functional regression problems. Computational Statistics and Data Analysiss, 51(10):4911–4925, 2007. https://doi.org/10.1016/j.csda.2006.09.034
L. Horváth and P. Kokoszka. Inference for Functional Data with Applications. Springer, New York, 2012. https://doi.org/10.1007/978-1-4614-3655-3
T. Hsing and R. Eubank. Theoretical Foundations of Functional Data Analysis, with an Introduction to Linear Operators. John Wiley & Sons, Chennai, India, 2015. https://doi.org/10.1002/9781118762547
H. Hullait, D.S. Leslie, N.G. Pavlidis and S. King. Robust function-on-function regression. Technometrics, 63(3):396–409, 2021. https://doi.org/10.1080/00401706.2020.1802350
A.E. Ivanescu, A.M. Staicu, F. Scheipl and S. Greven. Penalized functionon-function regression. Computational Statistics, 30(2):539–568, 2015. https://doi.org/10.1007/s00180-014-0548-4
R. Koenker. Quantile Regression. Cambridge University Press, Cambridge, 2005. https://doi.org/10.1017/CBO9780511754098
R. Koenker and Jr. G. Bassett. Regression quantiles. Econometrica, 46(1):33–50, 1978. https://doi.org/10.2307/1913643
R. Koenker. quantreg: Quantile Regression. R package version 5.85. 2021. Available from Internet: https://CRAN.R-project.org/package=quantreg
P. Kokoszka and M. Reimherr. Introduction to Functional Data Analysis. CRC Press, Boca Raton, 2017. https://doi.org/10.1201/9781315117416
E.R. Lee, H. Noh and B.U. Park. Model selection via Bayesian information criterion for quantile regression models. Journal of the American Statistical Association: Theory and Methods, 109(505):216–229, 2014. https://doi.org/10.1080/01621459.2013.836975
Y. Liu, M. Li and J.S. Morris. Function-on-scalar quantile regression with application to mass spectrometry proteomics data. The Annals of Applied Statistics, 114(2):521–541, 2020. https://doi.org/10.1214/19-AOAS1319
R. Luo and X. Qi. Interaction model and model selection for function-on-function regression. Journal of Computational and Graphical Statistics, 28(2):309–322, 2019. https://doi.org/10.1080/10618600.2018.1514310
H. Ma, T. Li, H. Zhu and Z. Zhu. Quantile regression for functional partially linear model in ultra-high dimensions. Computational Statistics and Data Analysis, 129:135–147, 2019. https://doi.org/10.1016/j.csda.2018.06.005
H. Matsui, S. Kawano and S. Konishi. Regularized functional regression modeling for functional response and predictors. Journal of Math-for-Industry, 1(A3):17– 25, 2009.
Microsoft and S. Weston. foreach: Provides Foreach Looping Construct. R package version 1.5.1. 2020. Available from Internet: https://CRAN.R-project.org/package=foreach
H.G. Müller, J.M. Chiou and X. Leng. Inferring gene expression dynamics via functional regression analysis. BMC Bioinformatics, 9(60):1–20, 2008. https://doi.org/10.1186/1471-2105-9-60
J. O. Ramsay, S. Graves and G. Hooker. fda: Functional Data Analysis. R package version 5.1.9. 2020. Available from Internet: https://CRAN.R-project.org/package=fda
J.O. Ramsay and C.J. Dalzell. Some tools for functional data analysis. Journal of the Royal Statistical Society, Series B, 53(3):539–572, 1991. https://doi.org/10.1111/j.2517-6161.1991.tb01844.x
J.O. Ramsay and B.W. Silverman. Applied Functional Data Analysis 2nd edition. Springer, New York, 2002. https://doi.org/10.1007/b98886
J.O. Ramsay and B.W. Silverman. Functional Data Analysis. Springer, New York, 3 edition, 2006. https://doi.org/10.1007/b98888
G. Schwarz. Estimating the dimension of a model. The Annals of Statistics, 6(2):461–464, 1987. https://doi.org/10.1214/aos/1176344136
Q. Tang and L. Cheng. Partial functional linear quantile regression. Science China Mathematics, 57(12):2589–2608, 2014. https://doi.org/10.1007/s11425-014-4819-x
F. Yao, H.-G. Müller and J.-L. Wang. Functional linear regression analysis for longitudinal data. The Annals of Statistics, 33(6):2873–2903, 2005. https://doi.org/10.1214/009053605000000660
F. Yao, S. Sue-Chee and F. Wang. Regularized partially functional quantile regression. Journal of Multivariate Analysis, 156:39–56, 2017. https://doi.org/10.1016/j.jmva.2017.02.001
D. Yu, L. Kong and I. Mizera. Partial functional linear quantile regression for neuroimaging data analysis. Neurocomputing, 195:74–87, 2016. https://doi.org/10.1016/j.neucom.2015.08.116