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Richard M. Chmielecki
and
Adrian E. Raftery

Abstract

Bayesian model averaging (BMA) is a statistical postprocessing technique that has been used in probabilistic weather forecasting to calibrate forecast ensembles and generate predictive probability density functions (PDFs) for weather quantities. The authors apply BMA to probabilistic visibility forecasting using a predictive PDF that is a mixture of discrete point mass and beta distribution components. Three approaches to developing predictive PDFs for visibility are developed, each using BMA to postprocess an ensemble of visibility forecasts. In the first approach, the ensemble is generated by a translation algorithm that converts predicted concentrations of hydrometeorological variables into visibility. The second approach augments the raw ensemble visibility forecasts with model forecasts of relative humidity and quantitative precipitation. In the third approach, the ensemble members are generated from relative humidity and precipitation alone. These methods are applied to 12-h ensemble forecasts from 2007 to 2008 and are tested against verifying observations recorded at Automated Surface Observing Stations in the Pacific Northwest. Each of the three methods produces predictive PDFs that are calibrated and sharp with respect to both climatology and the raw ensemble.

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Chris Fraley
,
Adrian E. Raftery
, and
Tilmann Gneiting

Abstract

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 be equal within each exchangeable group. With these adaptations, BMA can be applied to postprocess multimodel ensembles of any composition.

In experiments with surface temperature and quantitative precipitation forecasts from the University of Washington mesoscale ensemble and ensemble Kalman filter systems over the Pacific Northwest, the proposed extensions yield good results. The BMA method is robust to exchangeability assumptions, and the BMA postprocessed combined ensemble shows better verification results than any of the individual, raw, or BMA postprocessed ensemble systems. These results suggest that statistically postprocessed multimodel ensembles can outperform individual ensemble systems, even in cases in which one of the constituent systems is superior to the others.

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J. McLean Sloughter
,
Tilmann Gneiting
, and
Adrian E. Raftery

Abstract

Probabilistic forecasts of wind vectors are becoming critical as interest grows in wind as a clean and renewable source of energy, in addition to a wide range of other uses, from aviation to recreational boating. Unlike other common forecasting problems, which deal with univariate quantities, statistical approaches to wind vector forecasting must be based on bivariate distributions. The prevailing paradigm in weather forecasting is to issue deterministic forecasts based on numerical weather prediction models. Uncertainty can then be assessed through ensemble forecasts, where multiple estimates of the current state of the atmosphere are used to generate a collection of deterministic predictions. Ensemble forecasts are often uncalibrated, however, and Bayesian model averaging (BMA) is a statistical way of postprocessing these forecast ensembles to create calibrated predictive probability density functions (PDFs). It represents the predictive PDF as a weighted average of PDFs centered on the individual bias-corrected forecasts, where the weights reflect the forecasts’ relative contributions to predictive skill over a training period. In this paper the authors extend the BMA methodology to use bivariate distributions, enabling them to provide probabilistic forecasts of wind vectors. The BMA method is applied to 48-h-ahead forecasts of wind vectors over the North American Pacific Northwest in 2003 using the University of Washington mesoscale ensemble and is shown to provide better-calibrated probabilistic forecasts than the raw ensemble, which are also sharper than probabilistic forecasts derived from climatology.

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Veronica J. Berrocal
,
Adrian E. Raftery
, and
Tilmann Gneiting

Abstract

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 generated at minimal computational cost. The spatial BMA technique was applied to 48-h forecasts of surface temperature over the Pacific Northwest in 2004, using the University of Washington mesoscale ensemble. The spatial BMA ensemble generally outperformed the BMA and GOP ensembles and showed much better verification results than the raw ensemble, both at individual sites, for weather field forecasts, and for forecasts of composite quantities, such as average temperature in National Weather Service forecast zones and minimum temperature along the Interstate 90 Mountains to Sound Greenway.

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Laurence J. Wilson
,
Stephane Beauregard
,
Adrian E. Raftery
, and
Richard Verret

Abstract

Bayesian model averaging (BMA) has recently been proposed as a way of correcting underdispersion in ensemble forecasts. BMA is a standard statistical procedure for combining predictive distributions from different sources. The output of BMA is a probability density function (pdf), which is a weighted average of pdfs centered on the bias-corrected forecasts. The BMA weights reflect the relative contributions of the component models to the predictive skill over a training sample. The variance of the BMA pdf is made up of two components, the between-model variance, and the within-model error variance, both estimated from the training sample. This paper describes the results of experiments with BMA to calibrate surface temperature forecasts from the 16-member Canadian ensemble system. Using one year of ensemble forecasts, BMA was applied for different training periods ranging from 25 to 80 days. The method was trained on the most recent forecast period, then applied to the next day’s forecasts as an independent sample. This process was repeated through the year, and forecast quality was evaluated using rank histograms, the continuous rank probability score, and the continuous rank probability skill score. An examination of the BMA weights provided a useful comparative evaluation of the component models, both for the ensemble itself and for the ensemble augmented with the unperturbed control forecast and the higher-resolution deterministic forecast. Training periods around 40 days provided a good calibration of the ensemble dispersion. Both full regression and simple bias-correction methods worked well to correct the bias, except that the full regression failed to completely remove seasonal trend biases in spring and fall. Simple correction of the bias was sufficient to produce positive forecast skill out to 10 days with respect to climatology, which was improved by the BMA. The addition of the control forecast and the full-resolution model forecast to the ensemble produced modest improvement in the forecasts for ranges out to about 7 days. Finally, BMA produced significantly narrower 90% prediction intervals compared to a simple Gaussian bias correction, while achieving similar overall accuracy.

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Laurence J. Wilson
,
Stéphane Beauregard
,
Adrian E. Raftery
, and
Richard Verret
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Tilmann Gneiting
,
Adrian E. Raftery
,
Anton H. Westveld III
, and
Tom Goldman

Abstract

Ensemble prediction systems typically show positive spread-error correlation, but they are subject to forecast bias and dispersion errors, and are therefore uncalibrated. This work proposes the use of ensemble model output statistics (EMOS), an easy-to-implement postprocessing technique that addresses both forecast bias and underdispersion and takes into account the spread-skill relationship. The technique is based on multiple linear regression and is akin to the superensemble approach that has traditionally been used for deterministic-style forecasts. The EMOS technique yields probabilistic forecasts that take the form of Gaussian predictive probability density functions (PDFs) for continuous weather variables and can be applied to gridded model output. The EMOS predictive mean is a bias-corrected weighted average of the ensemble member forecasts, with coefficients that can be interpreted in terms of the relative contributions of the member models to the ensemble, and provides a highly competitive deterministic-style forecast. The EMOS predictive variance is a linear function of the ensemble variance. For fitting the EMOS coefficients, the method of minimum continuous ranked probability score (CRPS) estimation is introduced. This technique finds the coefficient values that optimize the CRPS for the training data. The EMOS technique was applied to 48-h forecasts of sea level pressure and surface temperature over the North American Pacific Northwest in spring 2000, using the University of Washington mesoscale ensemble. When compared to the bias-corrected ensemble, deterministic-style EMOS forecasts of sea level pressure had root-mean-square error 9% less and mean absolute error 7% less. The EMOS predictive PDFs were sharp, and much better calibrated than the raw ensemble or the bias-corrected ensemble.

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Adrian E. Raftery
,
Tilmann Gneiting
,
Fadoua Balabdaoui
, and
Michael Polakowski

Abstract

Ensembles used for probabilistic weather forecasting often exhibit a spread-error correlation, but they tend to be underdispersive. This paper proposes a statistical method for postprocessing ensembles based on Bayesian model averaging (BMA), which is a standard method for combining predictive distributions from different sources. The BMA predictive probability density function (PDF) of any quantity of interest is a weighted average of PDFs centered on the individual bias-corrected forecasts, where the weights are equal to posterior probabilities of the models generating the forecasts and reflect the models' relative contributions to predictive skill over the training period. The BMA weights can be used to assess the usefulness of ensemble members, and this can be used as a basis for selecting ensemble members; this can be useful given the cost of running large ensembles. The BMA PDF can be represented as an unweighted ensemble of any desired size, by simulating from the BMA predictive distribution.

The BMA predictive variance can be decomposed into two components, one corresponding to the between-forecast variability, and the second to the within-forecast variability. Predictive PDFs or intervals based solely on the ensemble spread incorporate the first component but not the second. Thus BMA provides a theoretical explanation of the tendency of ensembles to exhibit a spread-error correlation but yet be underdispersive.

The method was applied to 48-h forecasts of surface temperature in the Pacific Northwest in January–June 2000 using the University of Washington fifth-generation Pennsylvania State University–NCAR Mesoscale Model (MM5) ensemble. The predictive PDFs were much better calibrated than the raw ensemble, and the BMA forecasts were sharp in that 90% BMA prediction intervals were 66% shorter on average than those produced by sample climatology. As a by-product, BMA yields a deterministic point forecast, and this had root-mean-square errors 7% lower than the best of the ensemble members and 8% lower than the ensemble mean. Similar results were obtained for forecasts of sea level pressure. Simulation experiments show that BMA performs reasonably well when the underlying ensemble is calibrated, or even overdispersed.

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Hannah M. Director
,
Adrian E. Raftery
, and
Cecilia M. Bitz

Abstract

A new method, called contour shifting, is proposed for correcting the bias in forecasts of contours such as sea ice concentration above certain thresholds. Retrospective comparisons of observations and dynamical model forecasts are used to build a statistical spatiotemporal model of how predicted contours typically differ from observed contours. Forecasted contours from a dynamical model are then adjusted to correct for expected errors in their location. The statistical model changes over time to reflect the changing error patterns that result from reducing sea ice cover in the satellite era in both models and observations. For an evaluation period from 2001 to 2013, these bias-corrected forecasts are on average more accurate than the unadjusted dynamical model forecasts for all forecast months in the year at four different lead times. The total area, which is incorrectly categorized as containing sea ice or not, is reduced by 3.3 × 105 km2 (or 21.3%) on average. The root-mean-square error of forecasts of total sea ice area is also reduced for all lead times.

Open access
William Kleiber
,
Adrian E. Raftery
,
Jeffrey Baars
,
Tilmann Gneiting
,
Clifford F. Mass
, and
Eric Grimit

Abstract

The authors introduce two ways to produce locally calibrated grid-based probabilistic forecasts of temperature. Both start from the Global Bayesian model averaging (Global BMA) statistical postprocessing method, which has constant predictive bias and variance across the domain, and modify it to make it local. The first local method, geostatistical model averaging (GMA), computes the predictive bias and variance at observation stations and interpolates them using a geostatistical model. The second approach, Local BMA, estimates the parameters of BMA at a grid point from stations that are close to the grid point and similar to it in elevation and land use. The results of these two methods applied to the eight-member University of Washington Mesoscale Ensemble (UWME) are given for the 2006 calendar year. GMA was calibrated and sharper than Global BMA, with prediction intervals that were 8% narrower than Global BMA on average. Examples using sparse and dense training networks of stations are shown. The sparse network experiment illustrates the ability of GMA to draw information from the entire training network. The performance of Local BMA was not statistically different from Global BMA in the dense network experiment, and was superior to both GMA and Global BMA in areas with sufficient nearby training data.

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