Electronic Journal of Statistics

A Universal Kriging predictor for spatially dependent functional data of a Hilbert Space

Alessandra Menafoglio, Piercesare Secchi, and Matilde Dalla Rosa

Full-text: Open access


We address the problem of predicting spatially dependent functional data belonging to a Hilbert space, with a Functional Data Analysis approach. Having defined new global measures of spatial variability for functional random processes, we derive a Universal Kriging predictor for functional data. Consistently with the new established theoretical results, we develop a two-step procedure for predicting georeferenced functional data: first model selection and estimation of the spatial mean (drift), then Universal Kriging prediction on the basis of the identified model. The proposed methodology is applied to daily mean temperatures curves recorded in the Maritimes Provinces of Canada.

Article information

Electron. J. Statist., Volume 7 (2013), 2209-2240.

Received: October 2012
First available in Project Euclid: 19 September 2013

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62H11: Directional data; spatial statistics
Secondary: 62M20: Prediction [See also 60G25]; filtering [See also 60G35, 93E10, 93E11] 62M30: Spatial processes

Functional data analysis Sobolev metrics spatial prediction variogram


Menafoglio, Alessandra; Secchi, Piercesare; Dalla Rosa, Matilde. A Universal Kriging predictor for spatially dependent functional data of a Hilbert Space. Electron. J. Statist. 7 (2013), 2209--2240. doi:10.1214/13-EJS843. https://projecteuclid.org/euclid.ejs/1379596770

Export citation


  • Armstrong, M. and Diamond, P. (1984). Testing variogram for positive-definiteness., Mathematical geology 16(4), 407–421.
  • Arnold, L. (2003)., Random Dynamical Systems (Second ed.). Springer.
  • Bosq, D. (2000)., Linear Processes in Function Spaces. Springer, New York.
  • Caballero, W., Giraldo, R., and Mateu, J. (2013). A universal kriging approach for spatial functional data., Stochastic Environmental Research and Risk Assessment 27, 1553–1563.
  • Chilès, J. P. and Delfiner, P. (1999)., Geostatistics: Modeling Spatial Uncertainty. John Wiley & Sons, New York.
  • Cressie, N. (1993)., Statistics for Spatial data. John Wiley & Sons, New York.
  • Curriero, F. (2006). On the use of non-euclidean distance measures in geostatistics., Mathematical Geology 38, 907–926.
  • Delicado, P., Giraldo, R., Comas, C., and Mateu, J. (2010). Statistics for spatial functional data., Environmetrics 21(3–4), 224–239.
  • Dunford, N. and Schwartz, J. (1958)., Linear Operators I. Interscience, New York.
  • Efron, B. and Tibshirani, R. (1993)., An Introduction to the Bootstrap. Chapman & Hall/CRC.
  • Ferraty, F., Keilegom, I. V., and Vieu, F. (2010). On the validity of the bootstrap in non-parametric functional regression., Scandinavian Journal of Statistics 37(2), 286–306.
  • Ferraty, F. and Vieu, P. (2006)., Nonparametric Functional Data Analysis: Theory and Practice. Springer, New York.
  • Giraldo, R. (2009)., Geostatistical analysis of functional data. Ph. D. thesis, Universitat Politècnica da Catalunya, Barcellona.
  • Giraldo, R., Delicado, P., and Mateu, J. (2008a). Geostatistics for functional data: An ordinary kriging approach. Technical report. Universitat Politècnica de Catalunya, http://hdl.handle.net/2117/1099.
  • Giraldo, R., Delicado, P., and Mateu, J. (2008b)., Point-wise kriging for spatial prediction of functional data, Chapter 22, pp. 135–142. Functional and Operatorial Statistics. Proceedings of the First International Workshop on Functional and Operatorial Statistics. Springer, Toulouse, France.
  • Giraldo, R., Delicado, P., and Mateu, J. (2010a). Continuous time-varying kriging for spatial prediction of functional data: An environmental application., Journal of Agricultural, Biological, and Environmental Statistics 15(1), 66–82.
  • Giraldo, R., Delicado, P., and Mateu, J. (2010b)., Geofd: A package for prediction for functional data. Ph. D. thesis, Universitat Politecnica de Catalunya.
  • Giraldo, R., Delicado, P., and Mateu, J. (2011). Ordinary kriging for function-valued spatial data., Environmental and Ecological Statistics. 18(3), 411–426.
  • Goulard, M. and Voltz, M. (1993). Geostatistical interpolation of curves: A case study in soil science. In A. Soares (Ed.), Geostatistics Tróia ‘92, Volume 2, pp. 805–816. Kluwer Academic, Dordrecht.
  • Gromenko, O., Kokoszka, P., Zhu, L., and Sojka, J. (2012). Estimation and testing for spatially indexed curves with application to ionospheric and magnetic field trends., Annals of Applied Statistics 6(2), 669–696.
  • Hörmann, S. and Kokoszka, P. (2011). Consistency of the mean and the principal components of spatially distributed functional data. In F. Ferraty (Ed.), Recent Advances in Functional Data Analysis and Related Topics, Contributions to Statistics, pp. 169–175. Physica-Verlag HD.
  • Horváth, L. and Kokoszka, P. (2012)., Inference for Functional Data with Applications. Springer Series in Statistics. Springer.
  • Huang, C., Zhang, H., and Robeson, S. M. (2011). On the validity of commonly used covariance and variogram functions on the sphere., Mathematical Geosciences 43(6), 721–733.
  • Mahalanobis, P. C. (1936). On the generalised distance in statistics., Proceedings National Institute of Science, India 2(1), 49–55.
  • Menafoglio, A., Dalla Rosa, M., and Secchi, P. (2013). Supplement to “A Universal Kriging predictor for spatially dependent functional data of a Hilbert Space”. DOI:, 10.1214/00-EJS843SUPP.
  • Monestiez, P. and Nerini, D. (2008). A cokriging method for spatial functional data with applications in oceanology. In S. Dabo-Niang and F. Ferraty (Eds.), Functional and Operatorial Statistics, Chapter 36, pp. 237–242. Springer.
  • Natural Resources of Canada website, (2012).
  • Nerini, D., Monestiez, P., and Manté, C. (2010). Cokriging for spatial functional data., Journal of Multivariate Analysis 101(2), 409–418.
  • Ramsay, J. and Silverman, B. (2005)., Functional data analysis (Second ed.). Springer, New York.
  • Reyes, A., Giraldo, R., and Mateu, J. (2012). Residual kriging for functional data. Application to the spatial prediction of salinity curves. Reporte interno de investigación no. 20, Universidad Nacional de, Colombia.
  • Shumway, R. H. and Dean, W. (1968). Best linear unbiased estimation for multivariate stationary processes., Technometrics 10(3), 523– 534.
  • Stanley, D. (2002)., Canada’s Maritime provinces (First ed.). Lonely Planet Pubblications, Marybirnong.
  • Yamanishi, Y. and Tanaka, Y. (2003). Geographically weighted functional multiple regression analysis: A numerical investigation., Journal of Japanese Society of Computational Statistics (15), 307–317.
  • Zhu, K. (2007)., Operator Theory in Function Spaces (Second ed.). American Mathematical Society.

Supplemental materials