Brazilian Journal of Probability and Statistics

Spatio-temporal dynamic model and parallelized ensemble Kalman filter for precipitation data

Luis Sánchez, Saba Infante, Victor Griffin, and Demetrio Rey

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


This paper presents a spatiotemporal dynamic model which allows Bayesian inference of precipitation states in some Venezuelan meteorological stations. One of the limitations that are reported in digital databases is the reliability of the records and the lack of information for certain days, weeks, or months. To complete the missing data, the Gibbs algorithm, a Markov Chain Monte Carlo (MCMC) procedure, was used. A feature of precipitation series is that their distribution contains discrete and continuous components implying complicated dynamics. A model is proposed based on a discrete representation of a stochastic integro-difference equation. Given the difficulty of obtaining explicit analytical expressions for the predictive posterior distribution, approximations were obtained using a sequential Monte Carlo algorithm called the parallelized ensemble Kalman filter. The proposed method permits the completion of the missing data in the series where required and secondly allows the splitting of a large database into smaller ones for separate evaluation and eventual combination of the individual results. The objective is to reduce the dimension and computational cost in order to obtain models that are able to describe the reality in real time. It was shown that the obtained models are able to predict spatially and temporally the states of rainfall for at least three to four days quickly, efficiently and accurately. Three methods of statistical validation were used to evaluate the performance of the model and showed no significant discrepancies. Speedup and efficiency factors were calculated to compare the speed of calculation using the parallelized ensemble Kalman filter algorithm with the speed of the sequential version. The improvement in speed for four pthread executions was greatest.

Article information

Braz. J. Probab. Stat., Volume 30, Number 4 (2016), 653-675.

Received: February 2014
Accepted: July 2015
First available in Project Euclid: 13 December 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Precipitation modeling missing data spatio temporal models stochastic integro-difference equation parallelized ensemble Kalman filter


Sánchez, Luis; Infante, Saba; Griffin, Victor; Rey, Demetrio. Spatio-temporal dynamic model and parallelized ensemble Kalman filter for precipitation data. Braz. J. Probab. Stat. 30 (2016), no. 4, 653--675. doi:10.1214/15-BJPS297.

Export citation


  • Amisigo, B. A. and van de Giesen, N. C. (2005). Using a spatio-temporal dynamic state-space model with the EM algorithm to patch gaps in daily river flow series. Hydrology and Earth System Sciences 9, 209–224.
  • Aram, P., Freestone, D., Dewar, M., Grayden, D., Kadirkamanathan, V. and Scerr, K. (2007). Estimation of integro-difference equation based spatio-temporal systems. Available at
  • Bakar, K. (2011). Bayesian Analysis of Daily Maximum Ozone Levels. Thesis for the degree of Doctor of Philosophy. Available at
  • Banerjee, S., Carlin, B. and Gelfand, A. (2004). Hierarchical Modeling and Analysis for Spatial Data. Monographs on Statistics and Applied Probability Chapman and Hall: New York.
  • Banerjee, S. and Fuentes, M. (2011). Bayesian modeling for large spatial data sets. Research report 2011-47. Division of Biostatistics, University of Minnesota. Wires Computational Statistics.
  • Berliner, L., Milliff, R. and Wikle, C. (2003). Bayesian hierarchical modeling of air–sea interaction. Journal of Geophysics 108, 3104.
  • Cameletti, M., Ignaccolo, R. and Bande, S. (2010). Comparing air quality statistical models. Techical Report 40, Graspa working paper (accepted for publication at environmetrics).
  • Cameletti, M., Lindgren, F., Simpson, D. and Rue, H. (2012). Spatio-temporal modeling of particulate matter concentration through the SPDE approach. 2, 109 - 131.
  • Calder, C., Berrett, C., Shi, T., Xiao, N. and Munroe, D. K. (2011). Modeling space time dynamics of aerosols using satellite data and atmospheric transport model output. Journal of Agricultural, Biological, and Environmental Statistics 16, 495–512.
  • Cocchi, D., Greco, F. and Trivisano, C. (2007). Hierarchical space–time modelling of pollution. Atmospheric Environment 41, 532–542.
  • Cressie, N. and Huang, H. (1999). Classes of nonseparable, spatio-temporal stationary covariance functions. Journal of the American Statistical Association 94, 1330–1340.
  • Cressie, N. and Wikle, C. (2011). Statistics for Spatio-Temporal Data. New York: Wiley.
  • Dewar, M. (2007). A framework for modelling dynamic spatiotemporal systems. Ph.D. Thesis.
  • Dewar, M., Scerri, K. and Kadirkamanathan (2011). Data-driven spatio-temporal modeling using the integro-difference equation. IEEE Transactions on Signal Processing 57, 1.
  • Evensen, G. (2009). Data Assimilation: The Ensemble Kalman Filter, 2nd ed. Berlin: Springer.
  • Evensen, G. (1994). Sequential data assimilation with a nonlinear quasi-geostrophic model using Monte Carlo methods to forecast error statistics. Journal of Geophysical Research 99, 10143–10162.
  • Evensen, G. (2003). The ensemble Kalman filter: Theoretical formulation and practical implementation. Ocean Dynamics 53, 343–367.
  • Evensen, G. and van Leeuwen, P. (1996). Assimilation of geosat altimeter data for the agulhas current using the ensemble Kalman filter with a quasi-geostrophic model. Monthly Weather Review 24, 85–96.
  • Fasso, A. and Cameletti, M. (2007). A general spatio-temporal model for environmental data. Technical Report No. 27, Graspa–The Italian Group of Environmental Statistics.
  • Fernández-Casal, R., González-Manteiga, W. and Febrero-Bande, M. (2003). Flexible spatio-temporal stationary variogram models. Statistics and Computing 13, 127–136.
  • Gneiting, T. (2002). Nonseparable, stationary covariance functions for space–time data. Journal of the American Statistical Association 97, 590–600.
  • Hernández, A., Guenni, L. and Sansó, B. (2009). Extreme limit distribution of truncated models for daily rainfall. Environmetrics 20, 962–980.
  • Hernández, A., Guenni, L. and Sansó, B. (2011). Características de la precipitación extrema en algunas localidades de Venezuela. Interciencia 36, 185–191.
  • Jones, R. H. and Zhang, Y. (1997). Models for continuous stationary space–time processes. In Modelling Longitudinal and Spatially Correlated Data (T. G. Gregoire, D. R. Brillinger, P. J. Diggle, E. Russek-Cohen, W. G. Warren and R. D. Wolfinger, eds.) 289–298. New York: Springer.
  • Kotecha, J. H. and Djuric, P. M. (1999). Gibbs sampling approach for generation of truncated multivariate Gaussian random variables. In Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP), 1757–1760. Los Alamitos: IEEE Computer Society.
  • Lasinio, J., Sahu, S. and Mardia, K. (2007). Modeling rainfall data using a Bayesian Kriged–Kalman model. In Bayesian Statistics and Its Applocations (S. K. Upadhya, U. Singh and D. K. Dey, eds.). London: Anshan Ltd.
  • Li, Y. and Ghosh, S. (2013). Efficient sampling methods for truncated multivariate normal and Student-t distributions subject to linear inequality constraints. Technical reports 2649, NC State Department of Statistics.
  • Ma, C. (2003). Families of spatio-temporal stationary covariance models. Journal of statistical planning and inference. To appear.
  • Majda, A. and Harlim, J. (2012). Filtering Complex Turbulent Systems. Cambridge: Cambridge University Press.
  • Robert, C. (1995). Simulation of truncated normal variables. Statistics and Computing 5, 121–125.
  • Sánchez, L. and Infante, S. (2013). Reconstruction of chaotic dynamic systems using non-linear filters. Chilean Journal of Statistic 4, 1–19.
  • Sánchez, L. Infante, S. Marcano, J. and Griffin, V. (2015). Polinomial Chaos based on the parallelized ensamble Kalman filter to estimate precipitation states Statistics, Optimization and Information Computing 1, 79–95.
  • Sahu, S. (2011). Hierarchical Bayesian models for space–time air pollution data. In Handbook of Statistics–Time Series Analysis, Methods and Applications (C. Rao, ed.). Handbook of Statistics 30. Holland: Elsevier Publishers.
  • Sahu, S. K., Yip, S. and Holland, D. M. (2011). A fast Bayesian method for updating and forecasting hourly ozone levels. Environmental and Ecological Statistics 18, 185–207.
  • Sansó, B. and Guenni, L. (1999a). Stochastic model for tropical rainfall at a single location. Journal of Hydrology 214, 64–73.
  • Sansó, B. and Guenni, L. (1999b). Venezuelan rainfall data analysis using a Bayesian space–time model. Journal of the Royal Statistical Society Series C Applied Statistics 48, 345–362.
  • Sansó, B. and Guenni, L. (2000). A non-stationary multi-site model for rainfall. Journal of the American Statistical Association 95, 1089–1100.
  • Scerri, K., Dewar, M. and Kadirkamanathan, V. (2009). Estimation and model selection for an IDE-based spatio-temporal model. IEEE Transactions on Signal Processing 57, 482–492.
  • Scerri, K., Dewar, M., Parham, A., Freestone, D., Kadirkamanathan, V. and Grayden, D. (2011). Balanced reduction of an IDE-based spatio-temporal model computation tools. The second international on computational logics, algebras, programming, tools, and benchmarking, 7–122.
  • Sigrist, F., Künsch, H. and Stahel, W. (2012). A dynamic nonstationary spatio-temporal model for short term prediction of precipitation. Annals of Applied Statistics 6, 1452–1477.
  • Stein, M. (2005). Space–time covariance functions. Journal of the American Statistical Association 100, 310–321.
  • Stroud, J., Stein, M., Lesht, B., Schwar, D. and Beletsky, D. (2010). An ensemble Kalman filter and smoother for satellite data assimilation. Journal of the American Statistical Association 105, 978–990.
  • West, M. and Harrinson, J. (1997). Bayesian Forescasting and Dynamic Models, 2nd ed. New York: Springer.
  • Wilhelm, S. (2013). Truncated multivariate normal and student t distribution. Available at
  • Wilkinson, B. and Allen, M. (2005). Parallel Programming: Techniques and Application Using Networked Workstations and Parallel Computers, 2nd ed. New York: Prentice-Hall.
  • Wikle, C. K. and Holan, S. H. (2011). Polynomial nonlinear spatio-temporal integro-difference equation models. Journal of Time Series Analysis 32, 339–350. doi:10.1111//j.1467-9892.2011.00729.x.
  • Wikle, C. and Hooten, M. (2006). Hierarchical Bayesian spatio temporal models for population spread. In Applications of Computational Statistics in the Environmental Sciences: Hierarchical Bayes and MCMC Methods (J. S. Clark and A. Gelfand, eds.). London: Oxford University Press.
  • Wikle, C. (2002). A kernel-based spectral model for non-Gaussian spatio-temporal processes. Statistical modelling: An international journal 2, 299–314.
  • Wikle, C., Milliff, R., Nychka, D. and Berliner, L. (2001). Spatiotemporal hierarchical Bayesian modeling: Tropical ocean surface winds. Journal of the American Statistical Association 96, 382–397.
  • Xu, K., Wikle, C. K. and Fox, N. I. (2005). A kernel-based spatio-temporal dynamical model for nowcasting weather radar reflectivities. Journal of the American Statistical Association 100, 1133–1144.