Prediction of functional data with spatial dependence: a penalized approach

  • M. Carmen Aguilera-Morillo
  • María Durbán
  • Ana M. Aguilera
Original Paper


This paper is focus on spatial functional variables whose observations are a set of spatially correlated sample curves obtained as realizations of a spatio-temporal stochastic process. In this context, as alternative to other geostatistical techniques (kriging, kernel smoothing, among others), a new method to predict the curves of temporal evolution of the process at unsampled locations and also the surfaces of geographical evolution of the variable at unobserved time points is proposed. In order to test the good performance of the proposed method, two simulation studies and an application with real climatological data have been carried out. Finally, the results were compared with ordinary functional kriging.


Spatial functional data Spatial correlation P-spline penalty Functional regression 


  1. Aguilera AM, Aguilera-Morillo MC (2013) Comparative study of different B-spline approaches for functional data. Math Comput Model 58:1568–1579CrossRefGoogle Scholar
  2. Aguilera AM, Aguilera-Morillo MC (2013) Penalized PCA approaches for B-spline expansions of smooth functional data. Appl Math Comput 219:7805–7819Google Scholar
  3. Aguilera-Morillo MC, Aguilera AM, Escabias M, Valderrama MJ (2013) Penalized spline approaches for functional logit regression. Test 22:251–277CrossRefGoogle Scholar
  4. Caballero W, Giraldo R, Mateu J (2013) A universal kriging approach for spatial functional data. Stoch Environ Res Risk Assess 27:1553–1563CrossRefGoogle Scholar
  5. Chiou JM, Müller HG, Wang JL (2004) Functional response models. Stat Sin 14:659–677Google Scholar
  6. Delicado P, Giraldo R, Comas C, Mateu J (2009) Statistics for spatial functional data: some recent contributions. Environmetrics 21:224–239CrossRefGoogle Scholar
  7. Dubrule O (1984) Comparing kriging and splines. Comput Geosci 10(2–3):327–338CrossRefGoogle Scholar
  8. Eilers PHC, Marx B (1996) Flexible smoothing with B-splines and penalties. Stat Sci 11:89–121CrossRefGoogle Scholar
  9. Eilers PHC, Currie I, Durban M (2006) Fast and compact smoothing on large multidimensional grids. Comput Stat Data Anal 50:61–76CrossRefGoogle Scholar
  10. Escabias M, Aguilera AM, Valderrama MJ (2005) Modeling environmental data by functional principal component logistic regression. Environmetrics 16:95–107CrossRefGoogle Scholar
  11. Faraway JJ (1997) Regression analysis for a functional response. Technometrics 39:254–261CrossRefGoogle Scholar
  12. Fernandez-Pascual RM, Espejo R, Ruiz-Medina MD (2015) Moment and Bayesian wavelet regression from spatially correlated functional data. Stoch Environ Res Risk Assess. doi:10.1007/s00477-015-1130-5
  13. Ferraty F, Vieu P (2006) Nonparametric functional data analysis. Springer, New YorkGoogle Scholar
  14. Giraldo R (2010) Geostatistical analysis of functional data. PhD Thesis, Universitat Politècnica de Catalunya, CatalunyaGoogle Scholar
  15. Giraldo R, Delicado P, Mateu J (2010) Continuous time-varying kriging for spatial prediction of functional data: an environmental application. J Agric Biol Environ Stat 15:66–82CrossRefGoogle Scholar
  16. Giraldo R, Delicado P, Mateu J (2011) Ordinary kriging for function-valued spatial data. Environ Ecol Stat 18:411–426CrossRefGoogle Scholar
  17. Giraldo R, Mateu J, Delicado P (2012) geofd: an R package for function-valued geostatistical prediction. Rev Colomb Estad 35:385–407Google Scholar
  18. Goulard M, Voltz M (1993) Geostatistical interpolation of curves: a case study in soil science. Springer, Dordrecht, pp 805–816Google Scholar
  19. Harville DA (1997) Matrix algebra from a statistician’s perspective. Springer, New YorkCrossRefGoogle Scholar
  20. Horvath L, Kokoszka P (2012) Inference for functional data with applications. Springer, New YorkCrossRefGoogle Scholar
  21. Hsing T, Eubank R (2015) Theoretical foundations of functional data analysis with an introduction to linear operators. Wiley, ChichesterCrossRefGoogle Scholar
  22. Ignaccolo R, Mateu J, Giraldo R (2014) Kriging with external drift for functional data for air quality monitoring. Stoch Environ Res Risk Assess 28:1171–1186CrossRefGoogle Scholar
  23. Kaufman CG, Sain SR (2010) Bayesian functional ANOVA modeling using Gaussian process prior distributions. Bayesian Anal 5(1):123–150CrossRefGoogle Scholar
  24. Laslett GM (1994) Kriging and splines: an empirical comparison of their predictive performance in some applications. J Am Stat Assoc 89(426):391–400CrossRefGoogle Scholar
  25. Lee DJ, Durban M (2009) Smooth-car mixed models for spatial count data. Comput Stat Data Anal 53:2968–2977CrossRefGoogle Scholar
  26. Lee DJ, Durban M (2011) Pspline ANOVA type interaction models for spatio temporal smoothing. Stat Model 11:49–69CrossRefGoogle Scholar
  27. Marx BD, Eilers PHC (1999) Generalized linear regression on sampled signals and curves. A P-spline approach. Technometrics 41:1–13CrossRefGoogle Scholar
  28. Menafoglio A, Secchi P, Dalla Rosa M (2013) A universal kriging predictor for spatially dependent functional data of a Hilbert Space. Electron J Stat 7:2209–2240CrossRefGoogle Scholar
  29. Ramsay JO, Silverman BW (1997) Functional data analysis, 1st edn. Springer, New YorkCrossRefGoogle Scholar
  30. Rao CR, Rao MB (1998) Matrix algebra and its applications to statistics and econometrics. World Scientific Publishing Co., Pte. Ltd., SingaporeCrossRefGoogle Scholar
  31. Reiss PT, Huang L, Mennes M (2010) Fast function-on-scalar regression with penalized basis expansions. Int J Biostat 6:1–28CrossRefGoogle Scholar
  32. Ruiz-Medina MD, Espejo RM (2012) Spatial autoregressive functional plug-in prediction of ocean surface temperature. Stoch Environ Res Risk Assess 26:335–344CrossRefGoogle Scholar
  33. Ruiz-Medina MD, Espejo RM, Ugarte MD, Militino AF (2014) Functional time series analysis of spatiotemporal epidemiological data. Stoch Environ Res Risk Assess 28:943–954CrossRefGoogle Scholar
  34. Ruppert D (2002) Selecting the number of knots for penalized splines. J Comput Graph Stat 11:735–757CrossRefGoogle Scholar
  35. Sangalli LM, Ramsay JO, Ramsay TO (2013) Spatial spline regression models. J R Stat Soc B 75:1–23CrossRefGoogle Scholar
  36. Shi JQ, Choi T (2011) Gaussian process regression analysis for functional data. CRC Press, Chapman and Hall, Boca RatonGoogle Scholar
  37. Sigrist F, Kuensch HR, Stahel WA (2015) spate: an R package for spatio-temporal modeling with a stochastic advection–diffusion process. J Stat Softw 63:1–23CrossRefGoogle Scholar
  38. Ugarte MD, Goicoa T, Militino AF, Durban M (2009) Spline smoothing in small area trend estimation and smoothing. Comput Stat Data Anal 53:3616–3629CrossRefGoogle Scholar
  39. Yakowitz SJ, Szidarovsky F (1985) A comparison of kriging with non-parametric regression methods. J Multivar Anal 16:21–53CrossRefGoogle Scholar
  40. Zhang J-T (2013) Analysis of variance for functional data. CRC Press, Chapman and HallGoogle Scholar
  41. Zhang J-T, Chen J (2007) Statistical inference for functional data. Ann Stat 35(3):1052–1079CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • M. Carmen Aguilera-Morillo
    • 1
  • María Durbán
    • 1
  • Ana M. Aguilera
    • 2
  1. 1.University Carlos III de MadridLeganésSpain
  2. 2.University of GranadaGranadaSpain

Personalised recommendations