A Reparametrization Approach for Dynamic Space-Time Models

  • Hyeyoung LeeEmail author
  • Sujit K. Ghosh


Researchers in diverse areas such as environmental and health sciences are increasingly working with data collected across space and time. The space-time processes that are generally used in practice are often complicated in the sense that the auto-dependence structure across space and time is non-trivial, often non-separable and non-stationary in space and time. Moreover, the dimension of such data sets across both space and time can be very large leading to computational difficulties due to numerical instabilities. Hence, space-time modeling is a challenging task and in particular parameter estimation based on complex models can be problematic due to the curse of dimensionality. We propose a novel reparametrization approach to fit dynamic space-time models which allows the use of a very general form for the spatial covariance function. Our modeling contribution is to present an unconstrained reparametrization method for a covariance function within dynamic space-time models. A major benefit of the proposed unconstrained reparametrization method is that we are able to implement the modeling of a very high dimensional covariance matrix that automatically maintains the positive definiteness constraint. We demonstrate the applicability of our proposed reparametrized dynamic space-time models for a large data set of total nitrate concentrations.

AMS Subject Classification

62H11 62F15 65C60 


Computational efficiency Dynamic models Reparametrization Spatial models 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Banerjee, S., Gamerman, D., Gelfand, A. E., 2005. Spatial process modelling for univariate and multivariate dynamic spatial data. Environmetrics, 16, 465–479.MathSciNetCrossRefGoogle Scholar
  2. Chen, Z., Dunson, D. B., 2003. Random effects selection in linear mixed models. Biometrics, 59, 762–769.MathSciNetCrossRefGoogle Scholar
  3. Cressie, N. A. C., Huang, H.-C., 1999. Classes of nonseparable, spatiotemporal stationary covariance functions. Journal of the American Statistical Association, 94, 1330–1340.MathSciNetCrossRefGoogle Scholar
  4. Daniels, M. J., Pourahmadi M., 2002. Bayesian analysis of covariance matrices and dynamic models for longitudinal data. Biometrika, 89, 553–566.MathSciNetCrossRefGoogle Scholar
  5. Daniels, M. J., Zhao, Y., 2003. Modeling the random effects covariance matrix in longitudinal data. Statistics in Medicine, 22, 1631–1647.CrossRefGoogle Scholar
  6. Fuentes, M., Raftery, A. E., 2005. Model evaluation and spatial interpolation by Bayesian combination of observations with outputs from numerical models. Biometrics, 66, 36–45.MathSciNetCrossRefGoogle Scholar
  7. Gelfand, A. E., Ghosh, S. K., Knight, J. R., Sirmans, C. F., 1998. Spatio-temporal modeling of residential sales data. Journal of Business & Economic Statistics, 16, 312–321.Google Scholar
  8. Gneiting, T., 2002. Nonseparable, stationary covariance functions for space-time data. Journal of the American Statistical Association, 97, 590–600.MathSciNetCrossRefGoogle Scholar
  9. Huang, H.-C., Cressie, N. A. C., 1996. Spatio-temporal prediction of snow water equivalent using the Kalman filter. Computational Statistics and Data Analysis, 22, 159–175.MathSciNetCrossRefGoogle Scholar
  10. Huang, H.-C., Hsu, N.-J., 2004. Modeling transport effects on ground-level ozone using a non-stationary space-time model. Environmetrics, 15, 251–268.CrossRefGoogle Scholar
  11. Huerta, G., Sanso, B., Stroud, J. R., 2004. A spatio-temporal model for Mexico city ozone levels. Journal of the Royal Statistical Society, Series C, 53, 231–248.MathSciNetCrossRefGoogle Scholar
  12. Kyriakidis, P. C., Journel, A. G., 1999. Geostatistical space-time models: a review. Mathematical Geology, 31 (6), 651–684.MathSciNetCrossRefGoogle Scholar
  13. Mardia, K. V., Goodall, C., Redfern, E. J., Alonso, F. J., 1998. The kriged Kalman filter (with discussion). Test, 7, 217–285.MathSciNetCrossRefGoogle Scholar
  14. National Research Council, 2004. Air Quality Management in the United States, The National Academic Press.Google Scholar
  15. Pole, A., West M., Harrison, P. J., 1994. Applied Bayesian forecasting and times series analysis, Chapman and Hall: New York.CrossRefGoogle Scholar
  16. Pourahmadi M., 1999. Joint mean-covariance models with applications to longitudinal data: unconstrained parameterisation. Biometrika, 86, 677–690.MathSciNetCrossRefGoogle Scholar
  17. Sansó, B., Guenni, L., 1999. Venezuelan rainfall data analyzed using a Bayesian space-time model. Applied Statistics, 48, 345–362.zbMATHGoogle Scholar
  18. Stein, M. L., 2003. Space-time covariance functions. Journal of the American Statistical Association, 100, 310–321.MathSciNetCrossRefGoogle Scholar
  19. Stroud, J. R., Müller, P., Sanso, B., 2001. Dynamic models for spatio-temporal data. Journal of Royal Statistical Society, Series B, 63, 673–689.MathSciNetCrossRefGoogle Scholar
  20. Tonellato, S., 1997. Bayesian dynamic linear models for spatial time series, Technical report (Rapporto di riceria 5/1997), Dipartimento di Statistica, Universita CaFoscari di Venezia, Venice, Italy.Google Scholar
  21. West M., Harrison, P. J., 1997. Bayesian Forecasting and Dynamic Models, Springer: New York, 2nd ed.zbMATHGoogle Scholar
  22. Wikle, C., Berliner, M., Cressie, N., 1999. Hierarchical Bayesian space-time models. Environmental and Ecological Statistics, 5, 117–154.CrossRefGoogle Scholar
  23. Wikle, C., Cressie, N., 1999. A dimension reduced approach to space-time kalman filtering. Biometrika, 86, 815–829.MathSciNetCrossRefGoogle Scholar
  24. Xu, K., Wikle, C., 2005. Estimation of Parameterized Spatio-Temporal Dynamic Models. Ecological and Environmental Statistics, to appear.Google Scholar

Copyright information

© Grace Scientific Publishing 2008

Authors and Affiliations

  1. 1.Korea Institute of Patent InformationSeoulKorea
  2. 2.Department of Statistics at North Carolina N]State UniversityRaleighUSA

Personalised recommendations