The Annals of Applied Statistics

Estimation and testing for spatially indexed curves with application to ionospheric and magnetic field trends

Oleksandr Gromenko, Piotr Kokoszka, Lie Zhu, and Jan Sojka

Full-text: Open access


We develop methodology for the estimation of the functional mean and the functional principal components when the functions form a spatial process. The data consist of curves $X(\mathbf{s}_{k};t)$, $t\in[0,T]$, observed at spatial locations $\mathbf{s}_{1},\mathbf{s}_{2},\ldots,\mathbf{s}_{N}$. We propose several methods, and evaluate them by means of a simulation study. Next, we develop a significance test for the correlation of two such functional spatial fields. After validating the finite sample performance of this test by means of a simulation study, we apply it to determine if there is correlation between long-term trends in the so-called critical ionospheric frequency and decadal changes in the direction of the internal magnetic field of the Earth. The test provides conclusive evidence for correlation, thus solving a long-standing space physics conjecture. This conclusion is not apparent if the spatial dependence of the curves is neglected.

Article information

Ann. Appl. Stat. Volume 6, Number 2 (2012), 669-696.

First available in Project Euclid: 11 June 2012

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Gromenko, Oleksandr; Kokoszka, Piotr; Zhu, Lie; Sojka, Jan. Estimation and testing for spatially indexed curves with application to ionospheric and magnetic field trends. Ann. Appl. Stat. 6 (2012), no. 2, 669--696. doi:10.1214/11-AOAS524.

Export citation


  • Bel, L., Bar-Hen, A., Petit, R. and Cheddadi, R. (2011). Spatio-temporal functional regression on paleoecological data. J. Appl. Stat. 38 695–704.
  • Cnossen, I. and Richmond, A. D. (2008). Modelling the effects of changes in the Earth’s magnetic field from 1957 to 1997 on the ionospheric hmF2 and foF2 parameters. Journal of Atmospheric and Solar-Terrestrial Physics 70 1512–1524.
  • Delicado, P., Giraldo, R., Comas, C. and Mateu, J. (2010). Statistics for spatial functional data: Some recent contributions. Environmetrics 21 224–239.
  • Finkenstädt, B., Held, L. and Isham, V., eds. (2007). Statistical Methods for Spatio-Temporal Systems. Monographs on Statistics and Applied Probability 107. Chapman & Hall/CRC, Boca Raton, FL.
  • Gelfand, A. E., Diggle, P. J., Fuentes, M. and Guttorp, P., eds. (2010). Handbook of Spatial Statistics. CRC Press, Boca Raton, FL.
  • Giraldo, R., Delicado, P. and Mateu, J. (2011a). Ordinary kriging for function-valued spatial data. Environ. Ecol. Stat. 18 411–426.
  • Giraldo, R., Delicado, P. and Mateu, J. (2011b). A generalization of cokriging and multivariable spatial prediction for functional data. Technical report, Univ. Politécnica de Catalunya, Barcelona.
  • Gneiting, T. (2011). Making and evaluating point forecasts. J. Amer. Statist. Assoc. 106 746–762.
  • Hörmann, S. and Kokoszka, P. (2012). Consistency of the mean and the principal components of spatially distributed functional data. Bernoulli. To appear.
  • Horváth, L. and Kokoszka, P. (2012). Inference for Functional Data with Applications. Springer, Berlin.
  • Kivelson, M. G. and Russell, C. T., eds. (1997). Introduction to Space Physics. Cambridge Univ. Press, Cambridge.
  • Kokoszka, P., Maslova, I., Sojka, J. and Zhu, L. (2008). Testing for lack of dependence in the functional linear model. Canad. J. Statist. 36 207–222.
  • Laštovička, J. (2009). Global pattern of trends in the upper atmosphere and ionosphere: Recent progress. Journal of Atmospheric and Solar-Terrestrial. Physics 71 1514–1528.
  • Laštovička, J., Akmaev, R. A., Beig, G., Bremer, J., Emmert, J. T., Jacobi, C., Jarvis, J. M., Nedoluha, G., Portnyagin, Y. I. and Ulich, T. (2008). Emerging pattern of global change in the upper atmosphere and ionosphere. Annales Geophysicae 26 1255–1268.
  • Maslova, I., Kokoszka, P., Sojka, J. and Zhu, L. (2009). Removal of nonconstant daily variation by means of wavelet and functional data analysis. Journal of Geophysical Research 114 A03202.
  • Maslova, I., Kokoszka, P., Sojka, J. and Zhu, L. (2010a). Estimation of Sq variation by means of multiresolution and principal component analyses. Journal of Atmospheric and Solar-Terrestial Physics 72 625–632.
  • Maslova, I., Kokoszka, P., Sojka, J. and Zhu, L. (2010b). Statistical significance testing for the association of magnetometer records at high-, mid- and low latitudes during substorm days. Planetary and Space Science 58 437–445.
  • Mikhailov, A. V. and Marin, D. (2001). An interpretation of the foF2 and hmF2 long-term trends in the framework of the geomagnetic control concept. Annales Geophysicae 19 733–748.
  • Nerini, D., Monestiez, P. and Manté, C. (2010). Cokriging for spatial functional data. J. Multivariate Anal. 101 409–418.
  • Percival, D. B. and Walden, A. T. (2000). Wavelet Methods for Time Series Analysis. Cambridge Series in Statistical and Probabilistic Mathematics 4. Cambridge Univ. Press, Cambridge.
  • Qian, L., Burns, A. G., Solomon, S. C. and Roble, R. G. (2009). The effect of carbon dioxide cooling on trends in the F2-layer ionosphere. Journal of Atmospheric and Solar-Terrestrial Physics 71 1592–1601.
  • Ramsay, J., Hooker, G. and Graves, S. (2009). Functional Data Analysis with R and MATLAB. Springer, New York.
  • Rishbeth, H. (1990). A greenhouse effect in the ionosphere? Planetary and Space Science 38 945–948.
  • Roble, R. G. and Dickinson, R. E. (1989). How will changes in carbon dioxide and methane modify the mean structure of the mesosphere and thermosphere? Geophysical Research Letters 16 1441–1444.
  • Schabenberger, O. and Gotway, C. A. (2005). Statistical Methods for Spatial Data Analysis. Chapman & Hall/CRC, Boca Raton, FL.
  • Székely, G. J., Rizzo, M. L. and Bakirov, N. K. (2007). Measuring and testing dependence by correlation of distances. Ann. Statist. 35 2769–2794.
  • Székely, G. J. and Rizzo, M. L. (2009). Brownian distance covariance. Ann. Appl. Stat. 3 1236–1265.
  • Ulich, T., Clilverd, M. A. and Rishbeth, H. (2003). Determining long-term change in the ionosphere. Eos, Transactions American Geophysical Union 84 581–585.
  • Wackernagel, H. (2003). Multivariate Geostatistics, 3rd ed. Springer, New York.
  • Yamanishi, Y. and Tanaka, Y. (2003). Geographically weighted functional multiple regression analysis: A numerical investigation. J. Japanese Soc. Comput. Statist. 15 307–317.