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39
Stochastic Partial Differential Equations. Birkhauser
 Stoch. Dyn
, 1996
"... In this paper we develop a white noise framework for the study of stochastic partial differential equations driven by a dparameter (pure jump) Lévy white noise. As an example we use this theory to solve the stochastic Poisson equation with respect to Lévy white noise for any dimension d. The soluti ..."
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Cited by 111 (3 self)
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In this paper we develop a white noise framework for the study of stochastic partial differential equations driven by a dparameter (pure jump) Lévy white noise. As an example we use this theory to solve the stochastic Poisson equation with respect to Lévy white noise for any dimension d. The solution is a stochastic distribution process given explicitly. We also show that if d≤3, then this solution can be represented as a classical random field in L 2 (µ), where µ is the probability law of the Lévy process. The starting point of our theory is a chaos expansion in terms of generalized Charlier polynomials. Based on this expansion we define Kondratiev spaces and the Lévy Hermite transform. 1. Introduction. White
Bayesian Modeling of Uncertainty in Ensembles of Climate Models
, 2008
"... Projections of future climate change caused by increasing greenhouse gases depend critically on numerical climate models coupling the ocean and atmosphere (GCMs). However, different models differ substantially in their projections, which raises the question of how the different models can best be co ..."
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Cited by 35 (7 self)
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Projections of future climate change caused by increasing greenhouse gases depend critically on numerical climate models coupling the ocean and atmosphere (GCMs). However, different models differ substantially in their projections, which raises the question of how the different models can best be combined into a probability distribution of future climate change. For this analysis, we have collected both current and future projected mean temperatures produced by nine climate models for 22 regions of the earth. We also have estimates of current mean temperatures from actual observations, together with standard errors, that can be used to calibrate the climate models. We propose a Bayesian analysis that allows us to combine the different climate models into a posterior distribution of future temperature increase, for each of the 22 regions, while allowing for the different climate models to have different variances. Two versions of the analysis are proposed, a univariate analysis in which each region is analyzed separately, and a multivariate analysis in which the 22 regions are combined into an overall statistical model. A crossvalidation approach is proposed to confirm the reasonableness of our Bayesian predictive distributions. The results of this analysis allow for a quantification of the uncertainty of climate model projections as a Bayesian posterior distribution, substantially extending previous approaches to uncertainty in climate models.
Predictive model assessment for count data
, 2007
"... Summary. We discuss tools for the evaluation of probabilistic forecasts and the critique of statistical models for ordered discrete data. Our proposals include a nonrandomized version of the probability integral transform, marginal calibration diagrams and proper scoring rules, such as the predicti ..."
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Cited by 20 (1 self)
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Summary. We discuss tools for the evaluation of probabilistic forecasts and the critique of statistical models for ordered discrete data. Our proposals include a nonrandomized version of the probability integral transform, marginal calibration diagrams and proper scoring rules, such as the predictive deviance. In case studies, we critique count regression models for patent data, and assess the predictive performance of Bayesian ageperiodcohort models for larynx cancer counts in Germany.
Probabilistic Wind Speed Forecasting using Ensembles and Bayesian Model Averaging
, 2008
"... the Joint Ensemble Forecasting System (JEFS) under subcontract S0647225 from the University ..."
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Cited by 17 (9 self)
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the Joint Ensemble Forecasting System (JEFS) under subcontract S0647225 from the University
Combining spatial statistical and ensemble information in probabilistic weather forecasts
 Monthly Weather Review
, 2007
"... Forecast ensembles typically show a spreadskill relationship, but they are also often underdispersive, and therefore uncalibrated. Bayesian model averaging (BMA) is a statistical postprocessing method for forecast ensembles that generates calibrated probabilistic forecast products for weather quant ..."
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Cited by 9 (7 self)
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Forecast ensembles typically show a spreadskill relationship, but they are also often underdispersive, and therefore uncalibrated. Bayesian model averaging (BMA) is a statistical postprocessing method for forecast ensembles that generates calibrated probabilistic forecast products for weather quantities at individual sites. This paper introduces the Spatial BMA technique, which combines BMA and the geostatistical output perturbation (GOP) method, and extends BMA to generate calibrated probabilistic forecasts of whole weather fields simultaneously, rather than just weather events at individual locations. At any site individually, Spatial BMA reduces to the original BMA technique. The Spatial BMA method provides statistical ensembles of weather field forecasts that take the spatial structure of observed fields into account and honor the flowdependent information contained in the dynamical ensemble. The members of the Spatial BMA ensemble are obtained by dressing the weather field forecasts from the dynamical ensemble with simulated spatially correlated error fields, in proportions that correspond to the BMA weights for the member models in the dynamical ensemble. Statistical ensembles of any size can be
Calibrating MultiModel Forecast Ensembles with Exchangeable and Missing Members using Bayesian Model Averaging ∗
, 2009
"... Sloughter for sharing their insights and providing data. This research was sponsored by the National Science Foundation under Joint Ensemble Forecasting System (JEFS) subaward No. S0647225 with the University Corporation for Atmospheric Research (UCAR), as well as grants No. ATM0724721 and No. DMS ..."
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Cited by 8 (4 self)
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Sloughter for sharing their insights and providing data. This research was sponsored by the National Science Foundation under Joint Ensemble Forecasting System (JEFS) subaward No. S0647225 with the University Corporation for Atmospheric Research (UCAR), as well as grants No. ATM0724721 and No. DMS0706745. Bayesian model averaging (BMA) is a statistical postprocessing technique that generates calibrated and sharp predictive probability density functions (PDFs) from forecast ensembles. It represents the predictive PDF as a weighted average of PDFs centered on the biascorrected ensemble members, where the weights reflect the relative skill of the individual members over a training period. This work adapts the BMA approach to situations that arise frequently in practice, namely, when one or more of the member forecasts are exchangeable, and when there are missing ensemble members. Exchangeable members differ in random perturbations only, such as the members of bred ensembles, singular vector ensembles, or ensemble Kalman filter systems. Accounting for exchangeability simplifies the BMA approach, in that the BMA weights and the parameters of the component PDFs can be assumed to
Combining Probability Forecasts
, 2008
"... Linear pooling is by the far the most popular method for combining probability forecasts. However, any nontrivial weighted average of two or more distinct, calibrated probability forecasts is necessarily uncalibrated and lacks sharpness. In view of this, linear pooling requires recalibration, even i ..."
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Cited by 5 (0 self)
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Linear pooling is by the far the most popular method for combining probability forecasts. However, any nontrivial weighted average of two or more distinct, calibrated probability forecasts is necessarily uncalibrated and lacks sharpness. In view of this, linear pooling requires recalibration, even in the ideal case in which the individual forecasts are calibrated. Toward this end, we propose a beta transformed linear opinion pool (BLP) for the aggregation of probability forecasts from distinct, calibrated or uncalibrated sources. The BLP method fits an optimal nonlinearly recalibrated forecast combination, by compositing a beta transform and the traditional linear opinion pool. The technique is illustrated in a simulation example and in a case study on statistical and National Weather Service probability of precipitation forecasts.
Probabilistic forecasts of wind speed: Ensemble model output statistics by using heteroscedastic censored regression
 Journal of the Royal Statistical Society Series A
, 2010
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PROBABILISTIC QUANTITATIVE PRECIPITATION FIELD FORECASTING USING A TWOSTAGE SPATIAL MODEL 1
, 2008
"... Shortrange forecasts of precipitation fields are needed in a wealth of agricultural, hydrological, ecological and other applications. Forecasts from numerical weather prediction models are often biased and do not provide uncertainty information. Here we present a postprocessing technique for such n ..."
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Cited by 4 (3 self)
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Shortrange forecasts of precipitation fields are needed in a wealth of agricultural, hydrological, ecological and other applications. Forecasts from numerical weather prediction models are often biased and do not provide uncertainty information. Here we present a postprocessing technique for such numerical forecasts that produces correlated probabilistic forecasts of precipitation accumulation at multiple sites simultaneously. The statistical model is a spatial version of a twostage model that represents the distribution of precipitation by a mixture of a point mass at zero and a Gamma density for the continuous distribution of precipitation accumulation. Spatial correlation is captured by assuming that two Gaussian processes drive precipitation occurrence and precipitation amount, respectively. The first process is latent and drives precipitation occurrence via a threshold. The second process explains the spatial correlation in precipitation accumulation. It is related to precipitation via a sitespecific transformation function, so as to retain the marginal rightskewed distribution of precipitation while modeling spatial dependence. Both processes take into account the information contained in the numerical weather forecast and are modeled as stationary isotropic spatial processes with an exponential correlation function. The twostage spatial model was applied to 48hourahead forecasts of daily precipitation accumulation over the Pacific Northwest
Bias Correction and Bayesian Model Averaging for Ensemble Forecasts of Surface Wind Direction
"... Wind direction is an angular variable, as opposed to weather quantities such as temperature, quantitative precipitation or wind speed, which are linear variables. Consequently, traditional model output statistics and ensemble postprocessing methods become ineffective, or do not apply at all. We pro ..."
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Cited by 3 (2 self)
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Wind direction is an angular variable, as opposed to weather quantities such as temperature, quantitative precipitation or wind speed, which are linear variables. Consequently, traditional model output statistics and ensemble postprocessing methods become ineffective, or do not apply at all. We propose an effective bias correction technique for wind direction forecasts from numerical weather prediction models, which is based on a stateoftheart circularcircular regression approach. To calibrate forecast ensembles, a Bayesian model averaging scheme for directional variables is introduced, where the component distributions are von Mises densities centered at the individually biascorrected ensemble member forecasts. We apply these techniques to 48hour forecasts of surface wind direction over the Pacific Northwest, using the University of Washington Mesoscale Ensemble, where they yield consistent improvements in forecast performance. 1