## Probabilistic Quantitative Precipitation Forecasting using a Two-Stage Spatial Model (2008)

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### BibTeX

@MISC{Berrocal08probabilisticquantitative,

author = {Veronica J. Berrocal and Adrian E. Raftery and Tilmann Gneiting},

title = {Probabilistic Quantitative Precipitation Forecasting using a Two-Stage Spatial Model},

year = {2008}

}

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### Abstract

Multidisciplinary University Research Initiative (MURI) program administered by the Office of

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Citation Context ...recipitation accumulation have been proposed in the literature. Stidd (1973), Bell (1987), Bardossy and Plate (1992), Hutchinson (1995), and Sansò and Guenni (1999, 2000, 2004) adapted a Tobit model (=-=Tobin 1958-=-; Chib 1992) to precipitation accumulation, working with a latent Gaussian process that relates to precipitation via a power transformation and a truncation. The power truncated normal (PTN) model off... |

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Citation Context ... probability score (11) is a proper scoring rule for a probabilistic forecast of a scalar quantity; for a deterministic forecast, it reduces to the absolute error. The Brier score or quadratic score (=-=Brier 1950-=-) for a probability forecast of a binary event is defined as bs(f, o) = (f − o) 2 , where f is the forecast probability for the event and o equals 1 if the event occurs and 0 otherwise. As the represe... |

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Citation Context ... of the predictive distributions, we use verification rank histograms (Anderson 1996; Talagrand et al. 1997; Hamill and Colucci 1997; Hamill 2001) and probability integral transform (PIT) histograms (=-=Diebold et al. 1998-=-; Gneiting et al. 2007). Verification rank histograms are used for ensemble forecasts when the number of members m is small. For each forecast case, the rank of the verifying observation is tallied wi... |

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Citation Context ...hods have been developed to statistically postprocess numerical predictions of precipitation occurrence and produce quantitative probabilistic precipitation forecasts. They include linear regression (=-=Glahn and Lowry 1972-=-; Bermowitz 1975; Antolik 2000), quantile regression (Bremnes 2004; Friederichs and Hense 2007), logistic regression (Applequist et al. 2002; Hamill et al. 2004), neural networks (Koizumi 1999; Ramire... |

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Citation Context ...patial grid is of size approximately 10,000, so simulation from the required multivariate normal distributions is not a straightforward task. For doing this, we use the circulant embedding technique (=-=Wood and Chan 1994-=-; Dietrich and Newsam 1997; Gneiting et al. 2006) as implemented in the R package RandomFields (Schlather 2001). This is a very fast technique that can readily be used in real time. For verification p... |

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Citation Context ...n using a single latent spatial process. To assess the calibration of the predictive distributions, we use verification rank histograms (Anderson 1996; Talagrand et al. 1997; Hamill and Colucci 1997; =-=Hamill 2001-=-) and probability integral transform (PIT) histograms (Diebold et al. 1998; Gneiting et al. 2007). Verification rank histograms are used for ensemble forecasts when the number of members m is small. F... |

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Citation Context ...precipitation accumulation using a single latent spatial process. To assess the calibration of the predictive distributions, we use verification rank histograms (Anderson 1996; Talagrand et al. 1997; =-=Hamill and Colucci 1997-=-; Hamill 2001) and probability integral transform (PIT) histograms (Diebold et al. 1998; Gneiting et al. 2007). Verification rank histograms are used for ensemble forecasts when the number of members ... |

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Citation Context ...ommodate precipitation occurrence and precipitation accumulation using a single latent spatial process. To assess the calibration of the predictive distributions, we use verification rank histograms (=-=Anderson 1996-=-; Talagrand et al. 1997; Hamill and Colucci 1997; Hamill 2001) and probability integral transform (PIT) histograms (Diebold et al. 1998; Gneiting et al. 2007). Verification rank histograms are used fo... |

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Citation Context ... accumulation have been proposed in the literature. Stidd (1973), Bell (1987), Bardossy and Plate (1992), Hutchinson (1995), and Sansò and Guenni (1999, 2000, 2004) adapted a Tobit model (Tobin 1958; =-=Chib 1992-=-) to precipitation accumulation, working with a latent Gaussian process that relates to precipitation via a power transformation and a truncation. The power truncated normal (PTN) model offers a unifi... |

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Citation Context ...ze approximately 10,000, so simulation from the required multivariate normal distributions is not a straightforward task. For doing this, we use the circulant embedding technique (Wood and Chan 1994; =-=Dietrich and Newsam 1997-=-; Gneiting et al. 2006) as implemented in the R package RandomFields (Schlather 2001). This is a very fast technique that can readily be used in real time. For verification purposes, we need statistic... |

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Citation Context ...itation occurrence and precipitation accumulation using a single latent spatial process. To assess the calibration of the predictive distributions, we use verification rank histograms (Anderson 1996; =-=Talagrand et al. 1997-=-; Hamill and Colucci 1997; Hamill 2001) and probability integral transform (PIT) histograms (Diebold et al. 1998; Gneiting et al. 2007). Verification rank histograms are used for ensemble forecasts wh... |

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Citation Context ...stage spatial yes yes no yes yes at several sites simultaneously. In our assessment, we are guided by the principle of maximizing the sharpness of the predictive distributions subject to calibration (=-=Gneiting et al. 2007-=-). In other words, we aim at predictive distributions that are as concentrated as possible while being statistically consistent with the observations. To provide summary measures of predictive perform... |

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Citation Context ... at individual sites as a function of the length M in days. To assess the quality of the predictive distributions for daily precipitation accumulation, we use the continuous ranked probability score (=-=Matheson and Winkler 1976-=-; Gneiting and Raftery 2007), which is a strictly proper scoring rule for the evaluation of probabilistic forecasts of a univariate quantity. It is negatively oriented, that is, the smaller the better... |

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Citation Context ...tributions (Krzysztofowicz and Maranzano 2006), and two-stage models in which a Gamma density is employed to model precipitation accumulation (Wilks 1990; Hamill and Colucci 1998; Wilson et al. 1999; =-=Sloughter et al. 2007-=-). Common to all these methods is the underlying assumption that forecast errors at different locations are spatially independent. While this is not necessarily true, assuming conditional spatial inde... |

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Citation Context ... a prediction horizon of 48 hours. We compare our method to competing prediction techniques, including an ensemble of NWP forecasts (University of Washington mesoscale ensemble; Grimit and Mass 2002; =-=Eckel and Mass 2005-=-), the Bayesian model averaging technique of Sloughter et al. (2007) and versions of the power truncated normal model of Bardossy and Plate (1992). In Section 4 we review other statistical postprocess... |

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Citation Context ... shortcoming in weather-related decision-making (National Research Council 2006). One approach to incorporate uncertainty information into weather forecasting is via ensembles of numerical forecasts (=-=Palmer 2002-=-; Gneiting and Raftery 2005). While this is a major advance, the use of statistical postprocessing techniques for numerical forecasts remains essential. Several methods have been developed to statisti... |

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Citation Context ...not a straightforward task. For doing this, we use the circulant embedding technique (Wood and Chan 1994; Dietrich and Newsam 1997; Gneiting et al. 2006) as implemented in the R package RandomFields (=-=Schlather 2001-=-). This is a very fast technique that can readily be used in real time. For verification purposes, we need statistical forecast ensembles at observation sites, as opposed to the gridded forecasts in F... |

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Citation Context ...lly postprocess numerical predictions of precipitation occurrence and produce quantitative probabilistic precipitation forecasts. They include linear regression (Glahn and Lowry 1972; Bermowitz 1975; =-=Antolik 2000-=-), quantile regression (Bremnes 2004; Friederichs and Hense 2007), logistic regression (Applequist et al. 2002; Hamill et al. 2004), neural networks (Koizumi 1999; Ramirez et al. 2005), binning techni... |

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Citation Context ...s of precipitation occurrence and produce quantitative probabilistic precipitation forecasts. They include linear regression (Glahn and Lowry 1972; Bermowitz 1975; Antolik 2000), quantile regression (=-=Bremnes 2004-=-; Friederichs and Hense 2007), logistic regression (Applequist et al. 2002; Hamill et al. 2004), neural networks (Koizumi 1999; Ramirez et al. 2005), binning techniques (Gahrs et al. 2003; Yussouf and... |

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Citation Context ...ccumulation are modeled as stationary and isotropic Gaussian processes with an exponential correlation function. More general covariance structures such as the Matérn covariance function (Stein 1999; =-=Guttorp and Gneiting 2006-=-) could be employed. In fitting the model, we used a sliding training window consisting of the recent past. This is a simple adaptive approach that yields good results. A different approach would be t... |

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Citation Context ... overall steady improvement in the quality of numerical weather predictions, forecasts of precipitation accumulation are still not as accurate and reliable as those of other meteorological variables (=-=Applequist et al. 2002-=-; Stensrud and Yussouf 2007). Furthermore, quantitative precipitation forecasts obtained from a single NWP model are deterministic, and thus, do not convey any information about the degree of uncertai... |

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Citation Context ...ent in the quality of numerical weather predictions, forecasts of precipitation accumulation are still not as accurate and reliable as those of other meteorological variables (Applequist et al. 2002; =-=Stensrud and Yussouf 2007-=-). Furthermore, quantitative precipitation forecasts obtained from a single NWP model are deterministic, and thus, do not convey any information about the degree of uncertainty in the prediction, whic... |

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Citation Context ...ence first, and then model nonzero precipitation accumulation conditionally on its occurrence. Common choices for the distribution of nonzero precipitation accumulation include exponential densities (=-=Todorovic and Woolhiser 1975-=-), mixtures of exponentials (Woolhiser and Pegram 1979; Foufoula-Georgiou and Lettenmaier 1987) and gamma densities (Stern and Coe 1984; Wilks 1989; Hamill and Colucci 1998; Wilson et al. 1999; Slough... |

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Citation Context ...tion occurrence and produce quantitative probabilistic precipitation forecasts. They include linear regression (Glahn and Lowry 1972; Bermowitz 1975; Antolik 2000), quantile regression (Bremnes 2004; =-=Friederichs and Hense 2007-=-), logistic regression (Applequist et al. 2002; Hamill et al. 2004), neural networks (Koizumi 1999; Ramirez et al. 2005), binning techniques (Gahrs et al. 2003; Yussouf and Stensrud 2006), hierarchica... |

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Citation Context ...ccurrence. Common choices for the distribution of nonzero precipitation accumulation include exponential densities (Todorovic and Woolhiser 1975), mixtures of exponentials (Woolhiser and Pegram 1979; =-=Foufoula-Georgiou and Lettenmaier 1987-=-) and gamma densities (Stern and Coe 1984; Wilks 1989; Hamill and Colucci 1998; Wilson et al. 1999; Sloughter 3NWP forecast of precipitation accumulation valid January 5, 2004 4$ 44 46 48 50 !130 !1$... |

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Citation Context ...ion include exponential densities (Todorovic and Woolhiser 1975), mixtures of exponentials (Woolhiser and Pegram 1979; Foufoula-Georgiou and Lettenmaier 1987) and gamma densities (Stern and Coe 1984; =-=Wilks 1989-=-; Hamill and Colucci 1998; Wilson et al. 1999; Sloughter 3NWP forecast of precipitation accumulation valid January 5, 2004 4$ 44 46 48 50 !130 !1$5 !1$0 !115 0 $0 40 60 80 100 (a) NWP forecast of pre... |

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Citation Context ...tion conditionally on its occurrence. Common choices for the distribution of nonzero precipitation accumulation include exponential densities (Todorovic and Woolhiser 1975), mixtures of exponentials (=-=Woolhiser and Pegram 1979-=-; Foufoula-Georgiou and Lettenmaier 1987) and gamma densities (Stern and Coe 1984; Wilks 1989; Hamill and Colucci 1998; Wilson et al. 1999; Sloughter 3NWP forecast of precipitation accumulation valid... |

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Citation Context ...y 1972; Bermowitz 1975; Antolik 2000), quantile regression (Bremnes 2004; Friederichs and Hense 2007), logistic regression (Applequist et al. 2002; Hamill et al. 2004), neural networks (Koizumi 1999; =-=Ramirez et al. 2005-=-), binning techniques (Gahrs et al. 2003; Yussouf and Stensrud 2006), hierarchical models based on climatic prior distributions (Krzysztofowicz and Maranzano 2006), and two-stage models in which a Gam... |

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Citation Context ...is negatively oriented, that is, the smaller the better. To assess calibration for ensemble forecasts of multivariate weather quantities, we use j=1 19the minimum spanning tree (MST) rank histogram (=-=Smith and Hansen 2004-=-; Wilks 2004). If the ensemble has m members, the MST rank is found by tallying the length of the MST that connects the m ensemble members within the combined set of the m + 1 lengths of the ensemble-... |

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Citation Context ...ed to statistically postprocess numerical predictions of precipitation occurrence and produce quantitative probabilistic precipitation forecasts. They include linear regression (Glahn and Lowry 1972; =-=Bermowitz 1975-=-; Antolik 2000), quantile regression (Bremnes 2004; Friederichs and Hense 2007), logistic regression (Applequist et al. 2002; Hamill et al. 2004), neural networks (Koizumi 1999; Ramirez et al. 2005), ... |

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Citation Context ...in weather-related decision-making (National Research Council 2006). One approach to incorporate uncertainty information into weather forecasting is via ensembles of numerical forecasts (Palmer 2002; =-=Gneiting and Raftery 2005-=-). While this is a major advance, the use of statistical postprocessing techniques for numerical forecasts remains essential. Several methods have been developed to statistically postprocess numerical... |

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Citation Context ...Glahn and Lowry 1972; Bermowitz 1975; Antolik 2000), quantile regression (Bremnes 2004; Friederichs and Hense 2007), logistic regression (Applequist et al. 2002; Hamill et al. 2004), neural networks (=-=Koizumi 1999-=-; Ramirez et al. 2005), binning techniques (Gahrs et al. 2003; Yussouf and Stensrud 2006), hierarchical models based on climatic prior distributions (Krzysztofowicz and Maranzano 2006), and two-stage ... |

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Citation Context ...cipitation accumulation include exponential densities (Todorovic and Woolhiser 1975), mixtures of exponentials (Woolhiser and Pegram 1979; Foufoula-Georgiou and Lettenmaier 1987) and gamma densities (=-=Stern and Coe 1984-=-; Wilks 1989; Hamill and Colucci 1998; Wilson et al. 1999; Sloughter 3NWP forecast of precipitation accumulation valid January 5, 2004 4$ 44 46 48 50 !130 !1$5 !1$0 !115 0 $0 40 60 80 100 (a) NWP for... |