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 cross-validation 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.

the Joint Ensemble Forecasting System (JEFS) under subcontract S06-47225 from the University

Forecast ensembles typically show a spread-skill 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 flow-dependent 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

Summary. We discuss tools for the evaluation of probabilistic forecasts and the critique of statistical models for ordered discrete data. Our proposals include a non-randomized 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 age-period-cohort models for larynx cancer counts in Germany.

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. S06-47225 with the University Corporation for Atmospheric Research (UCAR), as well as grants No. ATM-0724721 and No. DMS-0706745. 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 bias-corrected 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

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 post-processing 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 state-of-the-art circular-circular 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 bias-corrected ensemble member forecasts. We apply these techniques to 48-hour 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

Road maintenance is one of the main problems Departments of Transportation face during winter time. Anti-icing, i.e. applying chemicals to the road to prevent ice formation, is often used to keep the roads free of ice. Given the preventive nature of anti-icing, accurate predictions of road ice are needed. Currently, anti-icing decisions are usually based on deterministic weather forecasts. However the costs of the two kinds of error are highly asymmetric because the cost of a road closure due to ice is much greater than that of taking anti-icing measures. As a result, probabilistic forecasts are needed to optimize decisionmaking. We propose two methods for forecasting the probability of ice formation. Starting with deterministic numerical weather predictions, they produce a joint predictive probability distribution of temperature and precipitation. This then yields the probability of ice formation, defined here as the occurrence of precipitation when the temperature is below freezing. In the first method, temperature and precipitation at different spatial locations are treated as conditionally independent given the numerical weather predictions. In the second method, spatial dependence between forecast errors at different locations is modeled. The model

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

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.

As wind energy penetration continues to grow, there is a critical need for probabilistic forecasts of wind resources. In addition, there are many other societally relevant uses for forecasts of wind speed, ranging from aviation to ship routing and recreational boating. Over the past two decades, ensembles of numerical weather prediction (NWP) models have been developed, in which multiple estimates of the current state of the atmosphere are used to generate a collection of deterministic forecasts. However, even state-of-the-art ensemble systems are uncalibrated and biased. Here we propose a novel way of statistically post-processing NWP ensembles for wind speed using heteroskedastic censored (Tobit) regression, where location and spread derive from the ensemble forecast. The resulting ensemble model output statistics (EMOS) method is applied to 48-hour ahead forecasts of maximum wind speed over the North American Pacific Northwest in 2003 using the University of Washington Mesoscale Ensemble. The statistically post-processed EMOS density forecasts turn out to be calibrated and sharp, and result in substantial improvement over the unprocessed NWP ensemble or climatological reference forecasts.