Computational Statistics

, Volume 30, Issue 3, pp 647–671 | Cite as

Partial linear modelling with multi-functional covariates

  • Germán Aneiros
  • Philippe VieuEmail author
Original Paper


This paper takes part on the current literature on semi-parametric regression modelling for statistical samples composed of multi-functional data. A new kind of partially linear model (so-called MFPLR model) is proposed. It allows for more than one functional covariate, for incorporating as well continuous and discrete effects of functional variables and for modelling these effects as well in a nonparametric as in a linear way. Based on the continuous specificity of functional data, a new method is proposed for variable selection (so-called PVS method). In addition, from this procedure, new estimates of the various parameters involved in the partial linear model are constructed. A simulation study illustrates the finite sample size behavior of the PVS procedure for selecting the influential variables. Through some real data analysis, it is shown how the method is reaching the three main objectives of any semi-parametric procedure. Firstly, the flexibility of the nonparametric component of the model allows to get nice predictive behavior; secondly, the linear component of the model allows to get interpretable outputs; thirdly, the low computational cost insures an easy applicability. Even if the intent is to be used in multi-functional problems, it will briefly discuss how it can also be used in uni-functional problems as a boosting tool for improving prediction power. Finally, note that the main feature of this paper is of applied nature but some basic asymptotics are also stated in a final “Appendix”.


Semi-parametrics Functional data analysis Multi-functional covariates Partial linear model Variable selection 

JEL Classification




The authors wish to express their great gratitude to the Editors and the Reviewers who have provided very interesting comments. Their suggestions were of great help when revising this work and will certainly increase its impact. In particular the reviewing procedure has been the opportunity to have highly interesting cross exchanges with one Referee about the interest and the meaning of our model (see Remark 2), which have greatly contributed to improve this work, at least by pointing our some challenging issues for the future. The research of G. Aneiros was partially supported by Grants MTM2011-22392 and CN2012/130 from Spanish Ministerio de Economía y Competitividad and Xunta Galicia, respectively.


  1. Aneiros G, Cao R, Vilar-Fernández JM, Muñoz-San-Roque A (2013) Functional prediction for the residual demand in electricity spot markets. IEEE Trans Power Syst 28(4):4201–4208CrossRefGoogle Scholar
  2. Aneiros G, Ferraty F, Vieu P (2014) Variable selection in partial linear regression with functional covariate. Statistics. doi: 10.1080/02331888.2014.998675
  3. Aneiros-Pérez G, Vieu P (2006) Semi-functional partial linear regression. Stat Probab Lett 76(11):1102–1110zbMATHCrossRefGoogle Scholar
  4. Aneiros-Pérez G, Vieu P (2011) Automatic estimation procedure in partial linear model with functional data. Stat Pap 52(4):751–771zbMATHCrossRefGoogle Scholar
  5. Aneiros-Pérez G, Vieu P (2013) Testing linearity in semi-parametric functional data analysis. Comput Stat 28(2):413–434zbMATHCrossRefGoogle Scholar
  6. Aneiros G, Vieu P (2014) Variable selection in infinite-dimensional problems. Stat Probab Lett 94:12–20zbMATHMathSciNetCrossRefGoogle Scholar
  7. Chen D, Hall P, Müller HG (2011) Single and multiple index functional regression models with nonparametric link. Ann Stat 39(3):1720–1747zbMATHCrossRefGoogle Scholar
  8. Cuevas A (2014) A partial overview pof the theory of statistics with functional data. J Stat Plann Inference 147:1–23zbMATHMathSciNetCrossRefGoogle Scholar
  9. Du J, Zhang Z, Sun Z (2013) Variable selection for partially linear varying coefficient quantile regression model. Int J Biomath 6(3):14MathSciNetGoogle Scholar
  10. Fan J, Li R (2001) Variable selection via nonconcave penalized likelihood and its oracle properties. J Am Stat Assoc 96:1348–1360zbMATHMathSciNetCrossRefGoogle Scholar
  11. Fan J, Peng H (2004) Nonconcave penalized likelihood with a diverging number of parameters. Ann Stat 32:928–961zbMATHMathSciNetCrossRefGoogle Scholar
  12. Ferraty F, Goia A, Salinelli E, Vieu P (2013) Functional projection pursuit regression. Test 22:293–320zbMATHMathSciNetCrossRefGoogle Scholar
  13. Ferraty F, Hall P, Vieu P (2010) Most-predictive design points for functional data predictors. Biometrika 97:807–824zbMATHMathSciNetCrossRefGoogle Scholar
  14. Ferraty F, Laksaci A, Tadj A, Vieu P (2010) Rate of uniform consistency for nonparametric estimates with functional variables. J Stat Plann Inference 140:335–352zbMATHMathSciNetCrossRefGoogle Scholar
  15. Ferraty F, Park J, Vieu P (2011) Estimation of a functional single index model. In: Recent advances in functional data analysis and related topics, Contrib Stat Springer, Heidelberg, pp 111–116Google Scholar
  16. Ferraty F, Vieu P (2006) Nonparametric functional data analysis: theory and practice. Springer, New YorkGoogle Scholar
  17. Gertheiss J, Maity A, Staicu AM (2013) Variable selection in generalized functional linear models. Stat 2:86–101CrossRefGoogle Scholar
  18. Goia A, Vieu P (2014) Some advances on semi-parametric functional data modelling. In: Contributions in infinite-dimensional statistics and related topics, Esculapio, BolognaGoogle Scholar
  19. Goia A, Vieu P (2014) A partitioned single functional index model. Comput Stat. doi: 10.1007/s00180-014-0530-1
  20. Goldsmith J, Bobb J, Crainiceanu C, Caffo B, Reich D (2011) Penalized functional regression. J Comput Graph Stat 20:830851MathSciNetCrossRefGoogle Scholar
  21. Guo J, Tang M, Tian M, Zhu K (2013) Variable selection in high-dimensional partially linear additive models for composite quantile regression. Comput Stat Data Anal 65:56–67MathSciNetCrossRefGoogle Scholar
  22. Härdle W, Liang H, Gao J (2000) Partially linear models. Physica-Verlag, HeidelbergzbMATHCrossRefGoogle Scholar
  23. Härdle W, Liang H (2007) Statistical methods for biostatistics and related fields. Springer, Berlin, pp 87–103zbMATHCrossRefGoogle Scholar
  24. Hong Z, Hu Y, Lian H (2013) Variable selection for high-dimensional varying coefficient partially linear models via nonconcave penalty. Metrika 76(7):887–908zbMATHMathSciNetCrossRefGoogle Scholar
  25. Hu Y, Lian H (2013) Variable selection in a partially linear proportional hazards model with a diverging dimensionality. Stat Probab Lett 83(1):61–69zbMATHMathSciNetCrossRefGoogle Scholar
  26. Huang J, Xie H (2007) Asymptotic oracle properties of SCAD-penalized least squared estimators. Asymptotics: particles, processes and inverse problems. In: IMS Lecture Notes-Monograph Series. 55, pp 149–166Google Scholar
  27. Hunter DR, Li RA (2005) Variable selection using MM algorithms. Ann Stat 33(4):1617–1642zbMATHMathSciNetCrossRefGoogle Scholar
  28. Kneip A, Poss D, Sarda P. Functional linear regression with points of impact. (Preprint)Google Scholar
  29. Lian H (2011) Functional partial linear model. J Nonparametr Stat 23(1):115–128zbMATHMathSciNetCrossRefGoogle Scholar
  30. Liang H, Härdle W, Carroll RJ (1999) Estimation in a semiparametric partially linear errors-in-variables model. Ann Stat 27(5):1519–1535zbMATHCrossRefGoogle Scholar
  31. Maity A, Huang JZ (2012) Partially linear varying coefficient models stratified by a functional covariate. Stat Probab Lett 82(10):1807–1814zbMATHMathSciNetCrossRefGoogle Scholar
  32. McKeague IW, Sen B (2010) Fractals with point impact in functional linear regression. Ann Stat 38:2559–2586zbMATHMathSciNetCrossRefGoogle Scholar
  33. Ni X, Zhang HH, Zhang D (2009) Automatic model selection for partially linear models. J Multivar Anal 100:2100–2111zbMATHMathSciNetCrossRefGoogle Scholar
  34. Pateiro-López B, González-Manteiga W (2006) Multivariate partially linear models. Stat Probab Lett 76:1543–1549zbMATHCrossRefGoogle Scholar
  35. Rachdi M, Vieu P (2007) Nonparametric regression for functional data: automatic smoothing parameter selection. J Stat Plann Inference 137(9):2784–2801zbMATHMathSciNetCrossRefGoogle Scholar
  36. Robinson PM (1988) Root-n-consistent semiparametric regression. Econometrica 56(4):931–954zbMATHMathSciNetCrossRefGoogle Scholar
  37. Speckman P (1988) Kernel smoothing in partial linear models. J R Stat Soc Ser B 50(3):413–436zbMATHMathSciNetGoogle Scholar
  38. Wang H, Zou G, Wan A (2013) Adaptive LASSO for varying-coefficient partially linear measurement error models. J Stat Plann Inference 143(1):40–54zbMATHMathSciNetCrossRefGoogle Scholar
  39. Xia Y, Härdle W (2006) Semi-parametric estimation of partially linear single-index models. J Multivar Anal 97(5):1162–1184zbMATHCrossRefGoogle Scholar
  40. Xie H, Huang J (2009) SCAD-penalized regression in high-dimensional partially linear models. Ann Stat 37(2):673–696zbMATHMathSciNetCrossRefGoogle Scholar
  41. Zhang J, Wang T, Zhu L, Liang H (2012) A dimension reduction based approach for estimating and variable selection in partially linear single-index models with high-dimensional covariates. Electron J Stat 6:2235–2273zbMATHMathSciNetCrossRefGoogle Scholar
  42. Zhang R, Zhao W, Liu J (2013) Robust estimation and variable selection for semiparametric partially linear varying coefficient model based on modal regression. J Nonparametr Stat 25(2):523–544zbMATHMathSciNetCrossRefGoogle Scholar
  43. Zou H, Li R (2008) One-step sparse estimates in nonconcave penalized likelihood models. Ann Stat 36(4):1509–1533zbMATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Departamento de MatemáticasUniversidad de A CoruñaA CoruñaSpain
  2. 2.Institut de MathématiquesUniversité Paul SabatierToulouseFrance

Personalised recommendations