The Annals of Applied Statistics

A dynamic nonstationary spatio-temporal model for short term prediction of precipitation

Abstract

Precipitation is a complex physical process that varies in space and time. Predictions and interpolations at unobserved times and/or locations help to solve important problems in many areas. In this paper, we present a hierarchical Bayesian model for spatio-temporal data and apply it to obtain short term predictions of rainfall. The model incorporates physical knowledge about the underlying processes that determine rainfall, such as advection, diffusion and convection. It is based on a temporal autoregressive convolution with spatially colored and temporally white innovations. By linking the advection parameter of the convolution kernel to an external wind vector, the model is temporally nonstationary. Further, it allows for nonseparable and anisotropic covariance structures. With the help of the Voronoi tessellation, we construct a natural parametrization, that is, space as well as time resolution consistent, for data lying on irregular grid points. In the application, the statistical model combines forecasts of three other meteorological variables obtained from a numerical weather prediction model with past precipitation observations. The model is then used to predict three-hourly precipitation over 24 hours. It performs better than a separable, stationary and isotropic version, and it performs comparably to a deterministic numerical weather prediction model for precipitation and has the advantage that it quantifies prediction uncertainty.

Article information

Source
Ann. Appl. Stat., Volume 6, Number 4 (2012), 1452-1477.

Dates
First available in Project Euclid: 27 December 2012

https://projecteuclid.org/euclid.aoas/1356629047

Digital Object Identifier
doi:10.1214/12-AOAS564

Mathematical Reviews number (MathSciNet)
MR3058671

Zentralblatt MATH identifier
1257.62121

Citation

Sigrist, Fabio; Künsch, Hans R.; Stahel, Werner A. A dynamic nonstationary spatio-temporal model for short term prediction of precipitation. Ann. Appl. Stat. 6 (2012), no. 4, 1452--1477. doi:10.1214/12-AOAS564. https://projecteuclid.org/euclid.aoas/1356629047

References

• Ailliot, P., Thompson, C. and Thomson, P. (2009). Space–time modelling of precipitation by using a hidden Markov model and censored Gaussian distributions. J. R. Stat. Soc. Ser. C. Appl. Stat. 58 405–426.
• Allcroft, D. J. and Glasbey, C. A. (2003). A latent Gaussian Markov random-field model for spatiotemporal rainfall disaggregation. J. R. Stat. Soc. Ser. C. Appl. Stat. 52 487–498.
• Banerjee, S., Gelfand, A. E., Finley, A. O. and Sang, H. (2008). Gaussian predictive process models for large spatial data sets. J. R. Stat. Soc. Ser. B Stat. Methodol. 70 825–848.
• Bardossy, A. and Plate, E. J. (1992). Space–time model for daily rainfall using atmospheric circulation patterns. Water Resources Research 28 1247–1259.
• Bell, T. (1987). A space–time stochastic model of rainfall for satellite remote-sensing studies. Journal of Geophysical Research 92 9631–9643.
• Bellone, E., Hughes, J. P. and Guttorp, P. (2000). A hidden Markov model for downscaling synoptic atmospheric patterns to precipitation amounts. Climate Research 15 1–12.
• Berger, J. O., De Oliveira, V. and Sansó, B. (2001). Objective Bayesian analysis of spatially correlated data. J. Amer. Statist. Assoc. 96 1361–1374.
• Berrocal, V. J., Raftery, A. E. and Gneiting, T. (2008). Probabilistic quantitative precipitation field forecasting using a two-stage spatial model. Ann. Appl. Stat. 2 1170–1193.
• Brown, P. E., Kåresen, K. F., Roberts, G. O. and Tonellato, S. (2000). Blur-generated non-separable space–time models. J. R. Stat. Soc. Ser. B Stat. Methodol. 62 847–860.
• Brown, P. E., Diggle, P. J., Lord, M. E. and Young, P. C. (2001). Space–time calibration of radar rainfall data. J. R. Stat. Soc. Ser. C. Appl. Stat. 50 221–241.
• Carter, C. K. and Kohn, R. (1994). On Gibbs sampling for state space models. Biometrika 81 541–553.
• Charles, S., Bates, B. and Hughes, J. (1999). A spatiotemporal model for downscaling precipitation occurrence and amounts. Journal of Geophysical Research 104 31657–31669.
• Chib, S. and Greenberg, E. (1995). Understanding the Metropolis–Hastings algorithm. Amer. Statist. 49 327–335.
• Coe, R. and Stern, R. (1982). Fitting models to daily rainfall data. Journal of Applied Meteorology 21 1024–1031.
• Cox, D. R. and Isham, V. (1988). A simple spatial–temporal model of rainfall. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 415 317–328.
• Cressie, N. A. C. (1993). Statistics for Spatial Data, 2nd ed. Wiley, New York.
• Cressie, N. and Huang, H.-C. (1999). Classes of nonseparable, spatio-temporal stationary covariance functions. J. Amer. Statist. Assoc. 94 1330–1340.
• Cressie, N. and Wikle, C. K. (2011). Statistics for Spatio-Temporal Data. Wiley, Hoboken, NJ.
• Diggle, P. J., Tawn, J. A. and Moyeed, R. A. (1998). Model-based geostatistics. J. R. Stat. Soc. Ser. C. Appl. Stat. 47 299–350. With discussion and a reply by the authors.
• Fowler, H. J., Kilsby, C. G., O’Connell, P. E. and Burton, A. (2005). A weather-type conditioned multi-site stochastic rainfall model for the generation of scenarios of climatic variability and change. Journal of Hydrology 308 50–66.
• Frühwirth-Schnatter, S. (1994). Data augmentation and dynamic linear models. J. Time Series Anal. 15 183–202.
• Fuentes, M., Reich, B. and Lee, G. (2008). Spatial–temporal mesoscale modeling of rainfall intensity using gage and radar data. Ann. Appl. Stat. 2 1148–1169.
• Gelfand, A. E., Banerjee, S. and Gamerman, D. (2005). Spatial process modelling for univariate and multivariate dynamic spatial data. Environmetrics 16 465–479.
• Gelfand, A. E. and Smith, A. F. M. (1990). Sampling-based approaches to calculating marginal densities. J. Amer. Statist. Assoc. 85 398–409.
• Gelfand, A. E., Diggle, P. J., Fuentes, M. and Guttorp, P., eds. (2010). Handbook of Spatial Statistics. CRC Press, Boca Raton, FL.
• Gneiting, T. (2002). Nonseparable, stationary covariance functions for space–time data. J. Amer. Statist. Assoc. 97 590–600.
• Gneiting, T. (2011). Making and evaluating point forecasts. J. Amer. Statist. Assoc. 106 746–762.
• 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., Genton, M. G. and Guttorp, P. (2007). Geostatistical space–time models, stationarity, separability and full symmetry. In Statistical Methods for Spatio-Temporal Systems (B. Finkenstädt, L. Held and V. Isham, eds.). Monographs on Statistics and Applied Probability 107 151–175. Chapman & Hall/CRC, Boca Raton.
• Gneiting, T. and Raftery, A. E. (2007). Strictly proper scoring rules, prediction, and estimation. J. Amer. Statist. Assoc. 102 359–378.
• Gneiting, T., Larson, K., Westrick, K., Genton, M. G. and Aldrich, E. (2006). Calibrated probabilistic forecasting at the stateline wind energy center: The regime-switching space–time method. J. Amer. Statist. Assoc. 101 968–979.
• Hastings, W. K. (1970). Monte Carlo sampling methods using Markov chains and their applications. Biometrika 57 97–109.
• Hernández, A., Guenni, L. and Sansó, B. (2009). Extreme limit distribution of truncated models for daily rainfall. Environmetrics 20 962–980.
• Huang, H.-C. and Hsu, N.-J. (2004). Modeling transport effects on ground-level ozone using a non-stationary space–time model. Environmetrics 15 251–268.
• Hughes, J. and Guttorp, P. (1994). A class of stochastic models for relating synoptic atmospheric patterns to regional hydrologic phenomena. Water Resources Research 30 1535–1546.
• Hughes, J., Guttorp, P. and Charles, S. P. (1999). A non-homogeneous hidden Markov model for precipitation occurrence. J. R. Stat. Soc. Ser. C. Appl. Stat. 48 15–30.
• Hutchinson, M. (1995). Stochastic space–time weather models from ground-based data. Agricultural and Forest Meteorology 73 237–264.
• Isham, V. and Cox, D. R. (1994). Stochastic models of precipitation. In Statistics for the Environment, Vol. 2 (V. Barnett and K. F. Turkmann, eds.). Wiley, Chichester.
• Jones, R. H. and Zhang, Y. (1997). Models for continuous stationary space–time processes. In Statistical Methods for Spatio-Temporal Systems (T. G. Gregoire, D. R. Brillinger, P. J. Diggle, E. Russek-Cohen, W. G. Warren and R. D. Wolfinger, eds.). Lecture Notes in Statistics 122 289–298. Springer, New York.
• Kober, K., Craig, G. C., Keil, C. and Dörnbrack, A. (2012). Blending a probabilistic nowcasting method with a high-resolution numerical weather prediction ensemble for convective precipitation forecasts. Quarterly Journal of the Royal Meteorological Society 138 755–768.
• Künsch, H. R. (2001). State space and hidden Markov models. In Complex Stochastic Systems (Eindhoven, 1999). Monographs on Statistics and Applied Probability 87 109–173. Chapman & Hall/CRC, Boca Raton, FL.
• Kyriakidis, P. C. and Journel, A. G. (1999). Geostatistical space–time models: A review. Math. Geol. 31 651–684.
• Le Cam, L. (1961). A stochastic description of precipitation. In Proc. 4th Berkeley Sympos. Math. Statist. and Prob., Vol. III 165–186. Univ. California Press, Berkeley, CA.
• 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. With discussion and a reply by the authors.
• Little, M. A., McSharry, P. E. and Taylor, J. W. (2009). Generalized linear models for site-specific density forecasting of U.K. daily rainfall. Monthly Weather Review 137 1029–1045.
• Ma, C. (2003). Families of spatio-temporal stationary covariance models. J. Statist. Plann. Inference 116 489–501.
• Makhnin, O. V. and McAllister, D. L. (2009). Stochastic precipitation generation based on a multivariate autoregression model. Journal of Hydrometeorology 10 1397–1413.
• Mardia, K. V. and Watkins, A. J. (1989). On multimodality of the likelihood in the spatial linear model. Biometrika 76 289–295.
• Mason, J. (1986). Numerical weather prediction. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 407 51–60.
• Matheson, J. E. and Winkler, R. L. (1976). Scoring rules for continuous probability distributions. Manag. Sci. 22 1087–1096.
• Metropolis, N., Rosenbluth, A. W., Rosenbluth, M. N., Teller, A. H. and Teller, E. (1953). Equation of state calculations by fast computing machines. J. Chem. Phys. 21 1087–1092.
• Okabe, A., Boots, B., Sugihara, K. and Chiu, S. N. (2000). Spatial Tessellations: Concepts and Applications of Voronoi Diagrams, 2nd ed. Wiley, Chichester.
• Paciorek, C. J. and Schervish, M. J. (2006). Spatial modelling using a new class of nonstationary covariance functions. Environmetrics 17 483–506.
• Rue, H. and Held, L. (2005). Gaussian Markov Random Fields: Theory and Applications. Monographs on Statistics and Applied Probability 104. Chapman & Hall/CRC, Boca Raton, FL.
• Sansó, B. and Guenni, L. (1999a). Venezuelan rainfall data analysed by using a Bayesian space–time model. J. R. Stat. Soc. Ser. C. Appl. Stat. 48 345–362.
• Sansó, B. and Guenni, L. (1999b). A stochastic model for tropical rainfall at a single location. Journal of Hydrology 214 64–73.
• Sansó, B. and Guenni, L. (2000). A nonstationary multisite model for rainfall. J. Amer. Statist. Assoc. 95 1089–1100.
• Sansó, B. and Guenni, L. (2004). A Bayesian approach to compare observed rainfall data to deterministic simulations. Environmetrics 15 597–612.
• Sigrist, F., Künsch, H. R. and Stahel, W. A. (2012). An SPDE based spatio-temporal model for large data sets with an application to postprocessing precipitation forecasts. Preprint. Available at http://arxiv.org/abs/1204.6118.
• Sloughter, J. M., Raftery, A. E., Gneiting, T. and Fraley, C. (2007). Probabilistic quantitative precipitation forecasting using Bayesian model averaging. Monthly Weather Review 135 3209–3220.
• Smith, A. F. M. and Roberts, G. O. (1993). Bayesian computation via the Gibbs sampler and related Markov chain Monte Carlo methods. J. R. Stat. Soc. Ser. B Stat. Methodol. 55 3–23.
• Sølna, K. and Switzer, P. (1996). Time trend estimation for a geographic region. J. Amer. Statist. Assoc. 91 577–589.
• Stehlik, J. and Bardossy, A. (2002). Multivariate stochastic downscaling model for generating daily precipitation series based on atmospheric circulation. Journal of Hydrology 256 120–141.
• Stein, M. (1990). Uniform asymptotic optimality of linear predictions of a random field using an incorrect second-order structure. Ann. Statist. 18 850–872.
• Stein, M. L. (2005). Space–time covariance functions. J. Amer. Statist. Assoc. 100 310–321.
• Stern, R. D. and Coe, R. (1984). A model fitting analysis of daily rainfall data. J. Roy. Statist. Soc. Ser. A 147 1–34.
• Stidd, C. K. (1973). Estimating the precipitation climate. Water Resources Research 9 1235–1241.
• Tobin, J. (1958). Estimation of relationships for limited dependent variables. Econometrica 26 24–36.
• Voronoi, G. (1908). Nouvelles applications des paramètres continus à la théorie des formes quadratiques. Deuxième mémoire. Recherches sur les parallélloèdres primitifs. Journal Für die Reine und Angewandte Mathematik (Crelles Journal) 1908 198–287.
• Warnes, J. J. and Ripley, B. D. (1987). Problems with likelihood estimation of covariance functions of spatial Gaussian processes. Biometrika 74 640–642.
• Waymire, E., Gupta, V. K. and Rodriguez-Iturbe, I. (1984). A spectral theory of rainfall intensity at the meso-$\beta$ scale. Water Resources Research 20 1453–1465.
• West, M. and Harrison, J. (1997). Bayesian Forecasting and Dynamic Models, 2nd ed. Springer, New York.
• Wikle, C. K. and Cressie, N. (1999). A dimension-reduced approach to space–time Kalman filtering. Biometrika 86 815–829.
• Wikle, C. K. and Hooten, M. B. (2010). A general science-based framework for dynamical spatio-temporal models. TEST 19 417–451.
• Wilks, D. (1990). Maximum likelihood estimation for the gamma distribution using data containing zeros. Journal of Climate 3 1495–1501.
• Wilks, D. (1998). Multisite generalization of a daily stochastic precipitation generation model. Journal of Hydrology 210 178–191.
• Wilks, D. (1999). Multisite downscaling of daily precipitation with a stochastic weather generator. Climate Research 11 125–136.
• Xu, K., Wikle, C. K. and Fox, N. I. (2005). A kernel-based spatio-temporal dynamical model for nowcasting weather radar reflectivities. J. Amer. Statist. Assoc. 100 1133–1144.
• Zhang, H. (2004). Inconsistent estimation and asymptotically equal interpolations in model-based geostatistics. J. Amer. Statist. Assoc. 99 250–261.