The Annals of Applied Statistics

Evaluation of the cooling trend in the ionosphere using functional regression with incomplete curves

Oleksandr Gromenko, Piotr Kokoszka, and Jan Sojka

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We develop a statistical framework to test the hypothesis of the existence of an ionospheric cooling trend related to the global warming hypothesis; both are attributed to the same driver, namely the increased concentration of greenhouse gases. However, the study of a temporal trend in the ionosphere is easier because there are fewer covariates to be taken into account. The hypothesis that a cooling trend in the ionosphere exists has been an important focus of space physics research for over two decades. A central difficulty in reaching broadly agreed—on conclusions has been the absence of data with sufficiently long temporal and sufficiently broad spatial coverage. Complete time series of data that cover several decades exist only in a few separated (industrialized) regions. The space physics community has struggled to combine the information contained in these data, and often contradictory conclusions have been reported based on the analyses relying on one or a few locations. We present a statistical analysis that uses all data, even those with incomplete temporal coverage. It is based on a new functional regression approach that can handle spatially indexed curves whose temporal domain depends on location and may contain gaps. The test statistic combines spatial and temporal dependence in the data and is approximately normally distributed. We conclude that a statistically significant cooling trend exists in the Northern Hemisphere. This confirms the hypothesis put forward in the space physics community over two decades ago.

Article information

Ann. Appl. Stat., Volume 11, Number 2 (2017), 898-918.

Received: March 2014
Revised: January 2017
First available in Project Euclid: 20 July 2017

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Cooling trend functional regression incomplete time series ionosphere solar activity spatial averaging spatio-temporal modeling


Gromenko, Oleksandr; Kokoszka, Piotr; Sojka, Jan. Evaluation of the cooling trend in the ionosphere using functional regression with incomplete curves. Ann. Appl. Stat. 11 (2017), no. 2, 898--918. doi:10.1214/17-AOAS1022.

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  • Bakar, K. and Sahu, S. (2015). spTimer: Spatio-temporal Bayesian modeling using R. J. Stat. Softw. 63 1–32.
  • Bremer, J., Damboldt, T., Mielich, J. and Suessmann, P. (2012). Comparing long-term trends in the ionospheric F2—Region with two different methods. Journal of Atmospheric and Solar-Terrestrial Physics 77 174–185.
  • Crainiceanu, C. M., Staicu, A.-M., Ray, S. and Punjabi, N. (2012). Bootstrap-based inference on the difference in the means of two correlated functional processes. Stat. Med. 31 3223–3240.
  • Cressie, N. and Hawkins, D. M. (1980). Robust estimation of the variogram. I. J. Int. Assoc. Math. Geol. 12 115–125.
  • Cressie, N. and Wikle, C. K. (2011). Statistics for Spatio-Temporal Data. Wiley, Hoboken, NJ.
  • Damboldt, T. and Suessmann, P. (2012). Consolidated Database of worldwide measured monthly medians of ionospheric characteristics foF2 and M(3000)F2. INAG Bulletin on the Web, INAG-73. Available at
  • Delicado, P., Giraldo, R., Comas, C. and Mateu, J. (2010). Statistics for spatial functional data: Some recent contributions. Environmetrics 21 224–239.
  • Gelfand, A. E., Diggle, P., Guttorp, P. and Fuentes, M., eds. (2010). Handbook of Spatial Statistics. Chapman & Hall/CRC, Boca Raton, FL.
  • Genton, M. G. (1998). Highly robust variogram estimation. Math. Geol. 30 213–221.
  • Genton, M. G. (2007). Separable approximations of space–time covariance matrices. Environmetrics 18 681–695.
  • Giraldo, R., Delicado, P. and Mateu, J. (2010). Continuous time-varying kriging for spatial prediction of functional data: An environmental application. J. Agric. Biol. Environ. Stat. 15 66–82.
  • Giraldo, R., Delicado, P. and Mateu, J. (2011). Ordinary kriging for function-valued spatial data. Environ. Ecol. Stat. 18 411–426.
  • Giraldo, R., Delicado, P. and Mateu, J. (2012). Hierarchical clustering of spatially correlated functional data. Stat. Neerl. 66 403–421.
  • Gromenko, O. and Kokoszka, P. (2012). Testing the equality of mean functions of ionospheric critical frequency curves. J. R. Stat. Soc. Ser. C. Appl. Stat. 61 715–731.
  • Gromenko, O. and Kokoszka, P. (2013). Nonparametric inference in small data sets of spatially indexed curves with application to ionospheric trend determination. Comput. Statist. Data Anal. 59 82–94.
  • 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. Ann. Appl. Stat. 6 669–696.
  • Haas, T. C. (1995). Local prediction of a spatio-temporal process with an application to wet sulfate deposition. J. Amer. Statist. Assoc. 90 1189–1199.
  • Hoff, P. D. (2011). Separable covariance arrays via the Tucker product, with applications to multivariate relational data. Bayesian Anal. 6 179–196.
  • Horváth, L. and Kokoszka, P. (2012). Inference for Functional Data with Applications. Springer, New York.
  • Jiang, H. and Serban, N. (2012). Clustering random curves under spatial interdependence with application to service accessibility. Technometrics 54 108–119.
  • Kelly, M. C. (2009). The Earth’s Ionosphere, 2nd ed. Academic Press, San Diego, CA.
  • Kraus, D. (2015). Components and completion of partially observed functional data. J. R. Stat. Soc. Ser. B. Stat. Methodol. 77 777–801.
  • Lastovicka, J., Solomon, S. C. and Qian, L. (2012). Trends in the neutral and ionized atmosphere. Reviews of Space Physics 168 113–145.
  • Lastovicka, J., Mikhailov, A. V., Ulich, T., Bremer, J., Elias, A., Ortiz de Adler, N., Jara, V., Abbarca del Rio, R., Foppiano, A., Ovalle, E. and Danilov, A. (2006). Long term trends in foF2: A comparison of various methods. Journal of Atmospheric and Solar-Terrestrial Physics 68 1854–1870.
  • Liebl, D. (2013). Modeling and forecasting electricity spot prices: A functional data perspective. Ann. Appl. Stat. 7 1562–1592.
  • Luttinen, J. and Ilin, A. (2009). Variational Gaussian-process factor analysis for modeling spatio-temporal data. In Advances in Neural Information Processing Systems 22 (Y. Bengio, D. Schuurmans, J. D. Lafferty, C. K. I. Williams and A. Culotta, eds.) 1177–1185. Curran Associates, Red Hook, NY.
  • Luttinen, J. and Ilin, A. (2012). Efficient Gaussian process inference for short-scale spatio–temporal modeling. In Proceedings of the International Conference on Artificial Intelligence and Statistics 22. JMLR W&CP.
  • Mielich, J. and Bremer, J. (2013). Long-term trends in the ionospheric F2 region with different solar activity indices. Annals of Geophysics 31 291–303.
  • Nerini, D., Monestiez, P. and Manté, C. (2010). Cokriging for spatial functional data. J. Multivariate Anal. 101 409–418.
  • Paul, D. and Peng, J. (2011). Principal components analysis for sparsely observed correlated functional data using a kernel smoothing approach. Electron. J. Stat. 5 1960–2003.
  • 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.
  • Secchi, P., Vantini, S. and Vitelli, V. (2011). A clustering algorithm for spatially dependent functional data. Procedia Environmental Sciences 7 176–181.
  • Secchi, P., Vantini, S. and Vitelli, V. (2012). Bagging Voronoi classifiers for clustering spatial functional data. International Journal of Applied Earth Observation and Geoinformation 22 53–64.
  • Sherman, M. (2011). Spatial Statistics and Spatio-Temporal Data: Covariance Functions and Directional Properties. Wiley, Chichester.
  • Staicu, A.-M., Crainiceanu, C. and Carroll, R. J. (2010). Fast methods for spatially correlated multilevel functional data. Biostatistics 11 177–194.
  • Staicu, A.-M., Crainiceanu, C. M., Reich, D. S. and Ruppert, D. (2012). Modeling functional data with spatially heterogeneous shape characteristics. Biometrics 68 331–343.
  • Stein, M. L. (1999). Interpolation of Spatial Data: Some Theory for Kriging. Springer, New York.
  • Stein, M. L. (2005). Space–time covariance functions. J. Amer. Statist. Assoc. 100 310–321.
  • Sun, Y., Li, B. and Genton, M. G. (2012). Geostatistics for large datasets. In Advances and Challenges in Space–Time Modelling of Natural Events (E. Porcu, J. M. Montero and M. Schlather, eds.) 3 55–77. Springer, Berlin.
  • Thébault, E., Finlay, C. C., Beggan, C. D., Alken, P., Aubert, J., Barrois, O., Bertrand, F., Bondar, T., Boness, A., Brocco, L., Canet, E., Chambodut, A., Chulliat, A., Coïsson, P., Civet, F., Du, A., Fournier, A., Fratter, I., Gillet, N., Hamilton, B., Hamoudi, M., Hulot, G., Jager, T., Korte, M., Kuang, W., Lalanne, X., Langlais, B., Léger, J.-M., Lesur, V., Lowes, F. J., Macmillan, S., Mandea, M., Manoj, C., Maus, S., Olsen, N., Petrov, V., Ridley, V., Rother, M., Sabaka, T. J., Saturnino, D., Schachtschneider, R., Sirol, O., Tangborn, A., Thomson, A., Tøffner-Clausen, L., Vigneron, P., Wardinski, I. and Zvereva, T. (2015). International geomagnetic reference field: The 12th generation. Earth, Planets and Space 67 1–19.
  • Ulich, T., Clilverd, M. A. and Rishbeth, H. (2003). Determining long-term change in the ionosphere. Eos, Transactions American Geophysical Union 84 581–585.
  • Yang, J., Zhu, H., Choi, T. and Cox, D. D. (2016). Smoothing and mean-covariance estimation of functional data with a Bayesian hierarchical model. Bayesian Anal. 11 649–670.
  • Yao, F., Müller, H.-G. and Wang, J.-L. (2005). Functional data analysis for sparse longitudinal data. J. Amer. Statist. Assoc. 100 577–590.

Supplemental materials

  • Supplementary material: Evaluation of the cooling trend in the ionosphere using functional regression with incomplete curves. The Supplementary Material contains the code and data used in this paper.