The Annals of Applied Statistics

A multi-resolution model for non-Gaussian random fields on a sphere with application to ionospheric electrostatic potentials

Minjie Fan, Debashis Paul, Thomas C. M. Lee, and Tomoko Matsuo

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


Gaussian random fields have been one of the most popular tools for analyzing spatial data. However, many geophysical and environmental processes often display non-Gaussian characteristics. In this paper, we propose a new class of spatial models for non-Gaussian random fields on a sphere based on a multi-resolution analysis. Using a special wavelet frame, named spherical needlets, as building blocks, the proposed model is constructed in the form of a sparse random effects model. The spatial localization of needlets, together with carefully chosen random coefficients, ensure the model to be non-Gaussian and isotropic. The model can also be expanded to include a spatially varying variance profile. The special formulation of the model enables us to develop efficient estimation and prediction procedures, in which an adaptive MCMC algorithm is used. We investigate the accuracy of parameter estimation of the proposed model, and compare its predictive performance with that of two Gaussian models by extensive numerical experiments. Practical utility of the proposed model is demonstrated through an application of the methodology to a data set of high-latitude ionospheric electrostatic potentials, generated from the LFM-MIX model of the magnetosphere-ionosphere system.

Article information

Ann. Appl. Stat. Volume 12, Number 1 (2018), 459-489.

Received: March 2017
Revised: September 2017
First available in Project Euclid: 9 March 2018

Permanent link to this document

Digital Object Identifier

Non-Gaussian random field multi-resolution analysis isotropic process on a sphere MCMC ionospheric electrostatic potential LFM-MIX model


Fan, Minjie; Paul, Debashis; Lee, Thomas C. M.; Matsuo, Tomoko. A multi-resolution model for non-Gaussian random fields on a sphere with application to ionospheric electrostatic potentials. Ann. Appl. Stat. 12 (2018), no. 1, 459--489. doi:10.1214/17-AOAS1104.

Export citation


  • Andrews, D. F. and Mallows, C. L. (1974). Scale mixtures of normal distributions. J. Roy. Statist. Soc. Ser. B 36 99–102.
  • Andrieu, C. and Thoms, J. (2008). A tutorial on adaptive MCMC. Stat. Comput. 18 343–373.
  • Atkinson, K. and Han, W. (2012). Spherical Harmonics and Approximations on the Unit Sphere: An Introduction. Lecture Notes in Math. 2044. Springer, Heidelberg.
  • Baldi, P., Kerkyacharian, G., Marinucci, D. and Picard, D. (2009). Asymptotics for spherical needlets. Ann. Statist. 37 1150–1171.
  • Barndorff-Nielsen, O. (1979). Models for non-Gaussian variation, with applications to turbulence. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 368 501–520.
  • Berg, J., Natarajan, A., Mann, J. and Patton, E. G. (2016). Gaussian vs non-Gaussian turbulence: Impact on wind turbine loads. Wind Energy 19 1975–1989.
  • Bhattacharya, A., Chakraborty, A. and Mallick, B. K. (2016). Fast sampling with Gaussian scale mixture priors in high-dimensional regression. Biometrika 103 985–991.
  • Chu, J.-H., Clyde, M. A. and Liang, F. (2009). Bayesian function estimation using continuous wavelet dictionaries. Statist. Sinica 19 1419–1438.
  • Codrescu, M. V., Fuller-Rowell, T. J. and Foster, J. C. (1995). On the importance of E-field variability for Joule heating in the high-latitude thermosphere. Geophys. Res. Lett. 22 2393–2396.
  • Codrescu, M. V., Fuller-Rowell, T. J., Foster, J. C., Holt, J. M. and Cariglia, S. J. (2000). Electric field variability associated with the Millstone Hill electric field model. J. Geophys. Res. 105 5265–5273.
  • Cousins, E. D. P., Matsuo, T. and Richmond, A. D. (2013a). SuperDARN assimilative mapping. J. Geophys. Res. 118 7954–7962.
  • Cousins, E. D. P., Matsuo, T. and Richmond, A. D. (2013b). Mesoscale and large-scale variability in high-latitude ionospheric convection: Dominant modes and spatial/temporal coherence. J. Geophys. Res. 118 7895–7904.
  • Cousins, E. D. P. and Shepherd, S. G. (2012). Statistical characteristics of small-scale spatial and temporal electric field variability in the high-latitude ionosphere. J. Geophys. Res. 117 A03317.
  • Cressie, N. (1993). Statistics for Spatial Data. Wiley, New York, NY.
  • Cressie, N. and Johannesson, G. (2008). Fixed rank kriging for very large spatial data sets. J. R. Stat. Soc. Ser. B. Stat. Methodol. 70 209–226.
  • Cressie, N., Shi, T. and Kang, E. L. (2010). Fixed rank filtering for spatio-temporal data. J. Comput. Graph. Statist. 19 724–745. With supplementary material available online.
  • Dai, F. and Xu, Y. (2013). Approximation Theory and Harmonic Analysis on Spheres and Balls. Springer, New York, NY.
  • De Oliveira, V., Kedem, B. and Short, D. A. (1997). Bayesian prediction of transformed Gaussian random fields. J. Amer. Statist. Assoc. 92 1422–1433.
  • Fan, M. (2015). A note on spherical needlets. Preprint. Available at arXiv:1508.05406.
  • Fan, M., Paul, D., Lee, C. M. T. and Matsuo, T. (2017). Modeling tangential vector fields on a sphere. J. Amer. Statist. Assoc. To appear.
  • Fan, M., Paul, D., Lee, C. M. T. and Matsuo, T. (2018). Supplement to “A multi-resolution model for non-Gaussian random fields on a sphere with application to ionospheric electrostatic potentials.” DOI:10.1214/17-AOAS1104SUPP.
  • Gelman, A., Roberts, G. O. and Gilks, W. R. (1996). Efficient Metropolis jumping rules. Bayesian Stat. 5 599–608.
  • Genton, M. G. and Kleiber, W. (2015). Cross-covariance functions for multivariate geostatistics. Statist. Sci. 30 147–163.
  • Gneiting, T. (2013). Strictly and non-strictly positive definite functions on spheres. Bernoulli 19 1327–1349.
  • Gneiting, T., Balabdaoui, F. and Raftery, A. E. (2007). Probabilistic forecasts, calibration and sharpness. J. R. Stat. Soc. Ser. B. Stat. Methodol. 69 243–268.
  • Gneiting, T. and Raftery, A. E. (2007). Strictly proper scoring rules, prediction, and estimation. J. Amer. Statist. Assoc. 102 359–378.
  • Gneiting, T. and Ranjan, R. (2011). Comparing density forecasts using threshold-and quantile-weighted scoring rules. J. Bus. Econom. Statist. 29 411–422.
  • Górski, K. M., Hivon, E., Banday, A. J., Wandelt, B. D., Hansen, F. K., Reinecke, M. and Bartelmann, M. (2005). HEALPix: A framework for high-resolution discretization and fast analysis of data distributed on the sphere. Astrophys. J. 622 759.
  • Guinness, J. and Fuentes, M. (2016). Isotropic covariance functions on spheres: Some properties and modeling considerations. J. Multivariate Anal. 143 143–152.
  • Heaton, M. J., Kleiber, W., Sain, S. R. and Wiltberger, M. (2015). Emulating and calibrating the multiple-fidelity Lyon–Fedder–Mobarry magnetosphere-ionosphere coupled computer model. J. R. Stat. Soc. Ser. C. Appl. Stat. 64 93–113.
  • Hunsucker, R. D. and Hargreaves, J. K. (2007). The High-Latitude Ionosphere and Its Effects on Radio Propagation. Cambridge Univ. Press, Cambridge.
  • Jones, R. H. (1963). Stochastic processes on a sphere. Ann. Math. Stat. 34 213–218.
  • Jun, M. and Stein, M. L. (2008). Nonstationary covariance models for global data. Ann. Appl. Stat. 2 1271–1289.
  • Kleiber, W., Sain, S. R., Heaton, M. J., Wiltberger, M., Reese, C. S., Bingham, D. (2013). Parameter tuning for a multi-fidelity dynamical model of the magnetosphere. Ann. Appl. Stat. 7 1286–1310.
  • Kleiber, W., Hendershott, B., Sain, S. R. and Wiltberger, M. (2016). Feature-based validation of the Lyon–Fedder–Mobarry magnetohydrodynamical model. J. Geophys. Res. 121 1192-1200.
  • Lindgren, F., Rue, H. and Lindström, J. (2011). An explicit link between Gaussian fields and Gaussian Markov random fields: The stochastic partial differential equation approach. J. R. Stat. Soc. Ser. B. Stat. Methodol. 73 423–498.
  • Lyon, J. G., Fedder, J. A. and Mobarry, C. M. (2004). The Lyon–Fedder–Mobarry (LFM) global MHD magnetospheric simulation code. J. Atmos. Sol.-Terr. Phys. 66 1333–1350.
  • Marinucci, D. and Peccati, G. (2011). Random Fields on the Sphere: Representation, Limit Theorems and Cosmological Applications. Cambridge Univ. Press, Cambridge.
  • Matsuo, T. and Richmond, A. D. (2008). Effects of high-latitude ionospheric electric field variability on global thermospheric Joule heating and mechanical energy transfer rate. J. Geophys. Res. 113 A07309.
  • Matsuo, T., Richmond, A. D. and Hensel, K. (2003). High-latitude ionospheric electric field variability and electric potential derived from DE-2 plasma drift measurements: Dependence on IMF and dipole tilt. J. Geophys. Res. 108 SIA 1-1–SIA 1-15.
  • Matsuo, T., Richmond, A. D. and Nychka, D. W. (2002). Modes of high-latitude electric field variability derived from DE-2 measurements: Empirical Orthogonal Function (EOF) analysis. Geophys. Res. Lett. 29 11-1–11-4.
  • Narcowich, F. J., Petrushev, P. and Ward, J. D. (2006). Localized tight frames on spheres. SIAM J. Math. Anal. 38 574–594.
  • Palacios, M. B. and Steel, M. F. J. (2006). Non-Gaussian Bayesian geostatistical modeling. J. Amer. Statist. Assoc. 101 604–618.
  • Palmroth, M., Janhunen, P., Pulkkinen, T. I., Aksnes, A., Lu, G., Østgaard, N., Watermann, J., Reeves, G. D. and Germany, G. A. (2005). Assessment of ionospheric Joule heating by GUMICS-4 MHD simulation, AMIE, and satellite-based statistics: Towards a synthesis. Ann. Geophysicae 23 2051–2068.
  • Perron, M. and Sura, P. (2013). Climatology of non-Gaussian atmospheric statistics. J. Climate 26 1063–1083.
  • Røislien, J. and Omre, H. (2006). T-distributed random fields: A parametric model for heavy-tailed well-log data. Math. Geol. 38 821–849.
  • Ruohoniemi, J. M. and Baker, K. B. (1998). Large-scale imaging of high-latitude convection with Super Dual Auroral Radar Network HF radar observations. J. Geophys. Res. 103 20797–20811.
  • Schoenberg, I. J. (1942). Positive definite functions on spheres. Duke Math. J. 9 96–108.
  • Stein, M. L. (1988). Asymptotically efficient prediction of a random field with a misspecified covariance function. Ann. Statist. 16 55–63.
  • Stein, M. L. (2007). Spatial variation of total column ozone on a global scale. Ann. Appl. Stat. 1 191–210.
  • Terdik, G. (2015). Angular spectra for non-Gaussian isotropic fields. Braz. J. Probab. Stat. 29 833–865.
  • Wallin, J. and Bolin, D. (2015). Geostatistical modelling using non-Gaussian Matérn fields. Scand. J. Stat. 42 872–890.
  • Weimer, D. R. (1995). Models of high-latitude electric potentials derived with a least error fit of spherical harmonic coefficients. J. Geophys. Res. 100 19595–19607.
  • West, M. (1987). On scale mixtures of normal distributions. Biometrika 74 646–648.
  • Wiltberger, M., Rigler, E. J., Merkin, V. and Lyon, J. G. (2017). Structure of high latitude currents in magnetosphere-ionosphere models. Space Sci. Rev. 206 575–598.
  • Wolfe, P. J., Godsill, S. J. and Ng, W.-J. (2004). Bayesian variable selection and regularization for time-frequency surface estimation. J. R. Stat. Soc. Ser. B. Stat. Methodol. 66 575–589.
  • Womersley, R. S. (2015). Efficient spherical designs with good geometric properties. Available at
  • Xu, G. and Genton, M. G. (2017). Tukey $g$-and-$h$ random fields. J. Amer. Statist. Assoc. 112 1236–1249.
  • Zhang, B., Lotko, W., Wiltberger, M. J., Brambles, O. J. and Damiano, P. A. (2011). A statistical study of magnetosphere–ionosphere coupling in the Lyon–Fedder–Mobarry global MHD model. J. Atmos. Sol.-Terr. Phys. 73 686–702.

Supplemental materials

  • Supplement to “A multi-resolution model for non-Gaussian random fields on a sphere with application to ionospheric electrostatic potentials”. This supplement provides additional figures and details of the numerical experiments and application.