## The Annals of Applied Statistics

### A stochastic space-time model for intermittent precipitation occurrences

#### Abstract

Modeling a precipitation field is challenging due to its intermittent and highly scale-dependent nature. Motivated by the features of high-frequency precipitation data from a network of rain gauges, we propose a threshold space-time $t$ random field (tRF) model for 15-minute precipitation occurrences. This model is constructed through a space-time Gaussian random field (GRF) with random scaling varying along time or space and time. It can be viewed as a generalization of the purely spatial tRF, and has a hierarchical representation that allows for Bayesian interpretation. Developing appropriate tools for evaluating precipitation models is a crucial part of the model-building process, and we focus on evaluating whether models can produce the observed conditional dry and rain probabilities given that some set of neighboring sites all have rain or all have no rain. These conditional probabilities show that the proposed space-time model has noticeable improvements in some characteristics of joint rainfall occurrences for the data we have considered.

#### Article information

Source
Ann. Appl. Stat., Volume 9, Number 4 (2015), 2110-2132.

Dates
Revised: August 2015
First available in Project Euclid: 28 January 2016

https://projecteuclid.org/euclid.aoas/1453994194

Digital Object Identifier
doi:10.1214/15-AOAS875

Mathematical Reviews number (MathSciNet)
MR3456368

Zentralblatt MATH identifier
06560824

#### Citation

Sun, Ying; Stein, Michael L. A stochastic space-time model for intermittent precipitation occurrences. Ann. Appl. Stat. 9 (2015), no. 4, 2110--2132. doi:10.1214/15-AOAS875. https://projecteuclid.org/euclid.aoas/1453994194

#### 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.
• Aiyyer, A. R. and Thorncroft, T. (2006). Climatology of vertical wind shear in the tropical Atlantic. J. Climate 19 2969–2983.
• Akaike, H. (1973). Information theory and an extension of the maximum likelihood principle. In Second International Symposium on Information Theory (Tsahkadsor, 1971) (B. N. Petrov and F. Csake, eds.) 267–281. Akadémiai Kiadó, Budapest.
• Bárdossy, A. and Plate, E. J. (1992). Space-time model for daily rainfall using atmospheric circulation patterns. Water Resour. Res. 28 1247–1259.
• Bell, T. L. (1987). A space-time stochastic model of rainfall for satellite remote-sensing studies. J. Geophys. Res. 92 9631–9643.
• Bell, T. L. and Kundu, P. K. (1996). A study of the sampling error in satellite rainfall estimates using optimal averaging of data and a stochastic model. J. Climate 9 1251–1268.
• Bell, T. L. and Kundu, P. K. (2003). Comparing satellite rainfall estimates with rain gauge data: Optimal strategies suggested by a spectral model. J. Geophys. Res. 108 4121.
• 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.
• Cowpertwait, P. S. P. (1994). A generalized point process model for rainfall. Proc. Roy. Soc. London Ser. A 447 23–37.
• Cox, D. R. and Isham, V. (1988). A simple spatial-temporal model of rainfall. Proc. Roy. Soc. London Ser. A 415 317–328.
• Glasbey, C. A. and Nevison, I. M. (1997). Rainfall modelling using a latent Gaussian variable. In Modelling Longitudinal and Spatially Correlated Data: Methods, Applications, and Future Directions (Gregoire, T. G., Brillinger, D. R., Diggle, P. J., Russek-Cohen, E., Warren, W. G. and Wolfinger, R. D., eds.) 233–242. Lecture Notes in Statistics 122. Springer, New York.
• Helgason, H., Pipiras, V. and Abry, P. (2011). Fast and exact synthesis of stationary multivariate Gaussian time series using circulant embedding. Signal Process. 91 1123–1133.
• Hernández, A., Guenni, L. and Sansó, B. (2009). Extreme limit distribution of truncated models for daily rainfall. Environmetrics 20 962–980.
• Hughes, J. P. and Guttorp, P. (1999). A non-homogeneous hidden Markov model for precipitation occurrence. Appl. Stat. 48 15–30.
• Katz, R. W. (1977). Precipitation as a chain-dependent process. J. Appl. Meteorol. 16 671–676.
• Katz, R. W. (1996). Use of conditional stochastic models to generate climate change scenarios. Clim. Change 32 237–255.
• Kleiber, W., Katz, R. W. and Rajagopalan, B. (2012). Daily spatiotemporal precipitation simulation using latent and transformed Gaussian processes. Water Resour. Res. 48 W01523.
• Kundu, P. K. and Siddani, R. K. (2007). A new class of probability distributions for describing the spatial statistics of area-averaged rainfall. J. Geophys. Res. D 18113 112.
• Kundu, P. K. and Siddani, R. K. (2011). Scale dependence of spatiotemporal intermittence of rain. Water Resour. Res. 47 318–340.
• Le Cam, L. (1961). A stochastic description of precipitation. In Proc. 4th Berkeley Sympos. Math. Statist. and Prob., Vol. III (J. Newman, ed.) 165–186. Univ. California Press, Berkeley, CA.
• López-Pintado, S. and Romo, J. (2009). On the concept of depth for functional data. J. Amer. Statist. Assoc. 104 718–734.
• Maraun, D. et al. (2010). Precipitation downscaling under climate change: Recent developments to bridge the gap between dynamical models and the end user. Rev. Geophys. 48 3003.
• Marsan, D., Schertzer, D. and Lovejoy, S. (1996). Causal space-time multi-fractal processes: Predictability and forecasting of rain fields. J. Geophys. Res. 101 26333–26346.
• Over, T. M. and Gupta, V. K. (1996). A space-time theory of mesoscale rainfall using random cascades. J. Geophys. Res. 101 26319–26331.
• R Core Team (2013). R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria. Available at http://www.R-project.org/.
• Richardson, C. W. (1981). Stochastic simulation of daily precipitation, temperature, and solar radiation. Water Resour. Res. 17 182–190.
• Richardson, C. W. and Wright, D. A. (1984). WGEN: A model for generating daily weather variables. USDA, ARS-8, NTIS, Springfield, VA.
• Rodriguez-Iturbe, I., Cox, D. R. and Isham, V. (1987). Some models for rainfall based on stochastic point processes. Proc. Roy. Soc. London Ser. A 410 269–288.
• Rodriguez-Iturbe, I., Cox, D. R. and Isham, V. (1988). A point process model for rainfall: Further developments. Proc. Roy. Soc. London Ser. A 417 283–298.
• Røislien, J. and Omre, H. (2006). T-distributed random fields: A parametric model for heavy-tailed well-log data. Math. Geol. 38 821–849.
• Sansó, B. and Guenni, L. (1999). Venezuelan rainfall data analysis using a Bayesian space-time model. J. R. Stat. Soc. Ser. C Appl. Stat. 48 345–362.
• Sigrist, F., Künsch, H. R. and Stahel, W. A. (2012). A dynamic nonstationary spatio-temporal model for short term prediction of precipitation. Ann. Appl. Stat. 6 1452–1477.
• Stein, M. L. (1992). Prediction and inference for truncated spatial data. J. Comput. Graph. Statist. 1 91–110.
• Stein, M. L. (2005). Statistical methods for regular monitoring data. J. R. Stat. Soc. Ser. B Stat. Methodol. 67 667–687.
• Stein, M. L. (2009). Spatial interpolation of high-frequency monitoring data. Ann. Appl. Stat. 3 272–291.
• Sun, Y. and Genton, M. G. (2011). Functional boxplots. J. Comput. Graph. Statist. 20 316–334.
• Sun, Y. and Genton, M. G. (2012). Adjusted functional boxplots for spatio-temporal data visualization and outlier detection. Environmetrics 23 54–64.
• Sun, Y., Genton, M. G. and Nychka, D. (2012). Exact fast computation of band depth for large functional datasets: How quickly can one million curves be ranked? Stat 1 68–74.
• Sun, Y., Bowman, K. P., Genton, M. G. and Tokay, A. (2015). A Matérn model of the spatial covariance structure of point rain rates. Stoch. Environ. Res. Risk Assess. 29 411–416.
• Tokay, A., Bashor, P. G. and McDowell, V. L. (2010). Comparison of rain gauge measurements in the mid-Atlantic region. J. Hydrometeorol. 11 553–565.
• Waymire, E. D., Gupta, V. K. and Rodríguez-Iturbe, I. (1984). Spectral theory of rainfall intensity at the meso-$\beta$ scale. Water Resour. Res. 20 1453–1465.
• Wilks, D. S. (2010). Use of stochastic weather generators for precipitation downscaling. Wiley Interdiscip. Rev.: Clim. Change 1 898–907.
• Wood, A. T. A. and Chan, G. (1994). Simulation of stationary Gaussian processes in $[0,1]^{d}$. J. Comput. Graph. Statist. 3 409–432.
• Zheng, X. and Katz, R. W. (2008). Simulation of spatial dependence in daily rainfall using multisite generators. Water Resour. Res. 44 W09403.
• Zheng, X., Renwick, J. and Clark, A. (2010). Simulation of multisite precipitation using an extended chain-dependent process. Water Resour. Res. 46 W01504.