Spatio-temporal modeling of global ozone data using convolution

Abstract

Large spatial data are becoming more and more popular in environmental science and other related fields. Observations are often made over a substantial fraction of the surface of the Earth over a long period of time. It is necessary to model spatio-temporal random processes on the sphere which is challenging both conceptually and computationally. Convolution modeling method can be utilized to generate a random field with valid covariance structure on spheres. A latent dynamic process is defined on a grid covering the globe. The data vector is first projected onto the low-dimensional space spanned by those grids at each available time point. The resulting time series are fitted with seasonal ARIMA models. Forecasting is made by convolving the latent dynamic processes at all grid points using von Mises–Fisher kernel function. The procedure is illustrated by the total ozone data collected by Total Ozone Mapping Spectrometer during a 12-year period of time.

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11

References

  1. Barry, R. P., & Ver Hoef, J. M. (1996). Blackbox kriging: Spatial prediction without specifying variogram models. Journal of Agricultural, Biological, and Environmental Statistics, 1, 297–322.

    MathSciNet  Article  Google Scholar 

  2. Calder, C. A., Holloman, C., & Higdon, D. (2002). Exploring space-time structure in ozone concentration using a dynamic process convolution model. In C. Gatsonis, R. E. Kass, A. Carriquiry, A. Gelman, D. Higdon, D. Pauler, & I. Verdinelli (Eds.), Case Studies in Bayesian Statistics 6 (pp. 165–176). New York: Springer.

    Google Scholar 

  3. Castruccio, S., & Genton, M. G. (2015). Beyond axial symmetry: An improved class of models for global data. Stat, 3, 48–55.

    MathSciNet  Article  Google Scholar 

  4. Castruccio, S., & Genton, M. G. (2016). Compressing an ensemble with statistical models: An algorithm for global 3D spatio-temporal temperature. Technometrics, 58, 319–328.

    MathSciNet  Article  Google Scholar 

  5. Cressie, N. (1993). Statistics for spatial data. New York: Wiley.

    Google Scholar 

  6. Cressie, N., & Johannesson, G. (2008). Fixed rank kriging for very large spatial data sets. Journal of the Royal Statistical Society: Series B, 70, 209–226.

    MathSciNet  Article  Google Scholar 

  7. De Iaco, S., Palma, M., & Posa, D. (2019). Choosing suitable linear coregionalization models for spatio-temporal data. Stochastic Environmental Research and Risk Assessment, 33, 1419–1434.

    Article  Google Scholar 

  8. Gelfand, A. E., Diggle, P., Guttorp, P., & Fuentes, M. (2010). Handbook of spatial statistics. Boca Raton: CRC Press.

    Google Scholar 

  9. Gneiting, T. (2013). Strictly and non-strictly positive definite functions on spheres. Bernoulli, 19, 1327–1349.

    MathSciNet  Article  Google Scholar 

  10. Heaton, M. J., Datta, A., Finley, A. O., Furrer, R., Guinness, J., & Guhaniyogi, R. (2019). A case study competition among methods for analyzing large spatial data. Journal of Agricultural, Biological and Environmental Statistics, 24, 398–425.

    MathSciNet  Article  Google Scholar 

  11. Heaton, M. J., Katzfuss, M., Berrett, C., & Nychka, D. W. (2014). Constructing valid spatial processes on the sphere using kernel convolutions. Environmetrics, 25, 2–15.

    MathSciNet  Article  Google Scholar 

  12. Hegglin, M. I., Fahey, D. W., Mack, M., Montzka, S. A., & Nash, E. R. (2014). Twenty questions and answers about the ozone layer: 2014 update, scientific assessment of ozone depletion: 2014. Technical report, World Meteorological Organization, Geneva, Switzerland

  13. Higdon, D. (1998). A process-convolution approach to modelling temperatures in the North Atlantic Ocean. Environmental and Ecological Statistics, 5, 173–190.

    Article  Google Scholar 

  14. Higdon, D. (2002). Space and space time modeling using process convolutions. In A. C. In, V. Barnett, P. C. Chatwin, & A. H. El-Shaarawi (Eds.), Quantitative methods for current environmental issues (pp. 37–56). London: Springer.

    Google Scholar 

  15. Higdon, D., Swall, J., & Kern, J. (1999). Non-stationary spatial modeling. Bayesian statistics 6 (pp. 761–768). Oxford: Oxford University Press.

    Google Scholar 

  16. Huang, C., Zhang, H., & Robeson, S. M. (2011). On the validity of commonly used covariance and variogram functions on the sphere. Mathematical Geosciences, 43, 721–733.

    MathSciNet  Article  Google Scholar 

  17. Hyndman, R. J., & Khandakar, Y. (2008). Automatic time series forecasting: The forecast package for R. Journal of Statistical Software, 26, 1–22.

    Google Scholar 

  18. Jeong, J., & Jun, M. (2015). A class of Matérn-like covariance functions for smooth processes on a sphere. Spatial Statistics, 11, 1–18.

    MathSciNet  Article  Google Scholar 

  19. Jeong, J., Jun, M., & Genton, M. G. (2017). Spherical process models for global spatial statistics. Statistical Science, 32, 501–513.

    MathSciNet  Article  Google Scholar 

  20. Jones, R. H. (1962). Stochastic processes on a sphere. The Annals of Mathematical Statistics, 34, 213–218.

    MathSciNet  Article  Google Scholar 

  21. Jun, M., & Stein, M. L. (2007). An approach to producing space-time covariance functions on spheres. Technometrics, 49, 468–479.

    MathSciNet  Article  Google Scholar 

  22. Jun, M., & Stein, M. L. (2008). Nonstationary covariance models for global data. The Annals of Applied Statistics, 2, 1271–1289.

    MathSciNet  Article  Google Scholar 

  23. Kern, J. (2000). Bayesian process-convolution approaches to specifying spatial dependence structure. PhD thesis, Duke University

  24. Li, Y., & Zhu, Z. (2016). Modeling nonstationary covariance function with convolution on spheres. Computational Statistics and Data Analysis, 104, 233–246.

    MathSciNet  Article  Google Scholar 

  25. NASA Goddard Ozone & Air Quality. https://ozoneaq.gsfc.nasa.gov/data/toms/. Accessed 30 May 2016.

  26. Porcu, E., Alegria, A., & Furrer, R. (2018). Modeling temporally evolving and spatially globally dependent data. International Statistical Review, 86, 344–377.

    MathSciNet  Article  Google Scholar 

  27. Porcu, E., Bevilacqua, M., & Genton, M. G. (2016). Spatio-temporal covariance and cross-covariance functions of the great circle distance on a sphere. Journal of the American Statistical Association, 111, 888–898.

    MathSciNet  Article  Google Scholar 

  28. Ripley, B. D. (2005). Spatial statistics. New York: Wiley.

    Google Scholar 

  29. Rodrigues, A., & Diggle, P. J. (2010). A class of convolution-based models for spatio-temporal processes with non-separable covariance structure. Scandinavian Journal of Statistics, 37, 553–567.

    MathSciNet  Article  Google Scholar 

  30. Sansó, B., Schmidt, A. M., & Nobre, A. A. (2008). Bayesian spatio-temporal models based on discrete convolutions. The Canadian Journal of Statistics, 36, 239–258.

    MathSciNet  Article  Google Scholar 

  31. Shumway, R. H., & Stoffer, D. S. (2017). Time series analysis and its applications: With R examples (4th ed.). Berlin: Springer.

    Google Scholar 

  32. Stein, M. L. (1999). Interpolation of spatial data: Some theory for kriging. New York: Springer.

    Google Scholar 

  33. White, P., & Porcu, E. (2019). Towards a complete picture of covariance functions on spheres cross time, Electronic Journal of Statistics, 13, 2566–2594.

    MathSciNet  Article  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Zhengyuan Zhu.

Ethics declarations

Conflict of interest

On behalf of all authors, Zhengyuan Zhu states that there is no conflict of interest.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Li, Y., Zhu, Z. Spatio-temporal modeling of global ozone data using convolution. Jpn J Stat Data Sci 3, 153–166 (2020). https://doi.org/10.1007/s42081-019-00069-5

Download citation

Keywords

  • Spatio-temporal data
  • Kernel function
  • Convolution
  • Total ozone