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

Rank histograms are a tool for evaluating ensemble forecasts. They are useful for determining the reliability of ensemble forecasts and for diagnosing errors in its mean and spread. Rank histograms are generated by repeatedly tallying the rank of the verification (usually, an observation) relative to values from an ensemble sorted from lowest to highest. However, an uncritical use of the rank histogram can lead to misinterpretations of the qualities of that ensemble. For example, a flat rank histogram, ususally taken as a sign of reliability, can still be generated from unreliable ensembles. Similarly, a U-shaped rank histogram, commonly understood as indicating a lack of variability in the ensemble, can also be a sign of conditional bias. It is also shown that flat rank histograms can be generated for some model variables if the variance of the ensemble is correctly specified, yet if covariances between model grid points are improperly specified, rank histograms for combinations of mo...

Summary. Probabilistic forecasts of continuous variables take the form of predictive densities or predictive cumulative distribution functions. We propose a diagnostic approach to the evaluation of predictive performance that is based on the paradigm of maximizing the sharpness of the predictive distributions subject to calibration. Calibration refers to the statistical consistency between the distributional forecasts and the observations and is a joint property of the predictions and the events that materialize. Sharpness refers to the concentration of the predictive distributions and is a property of the forecasts only. A simple theoretical framework allows us to distinguish between probabilistic calibration, exceedance calibration and marginal calibration. We propose and study tools for checking calibration and sharpness, among them the probability integral transform histogram, marginal calibration plots, the sharpness diagram and proper scoring rules. The diagnostic approach is illustrated by an assessment and ranking of probabilistic forecasts of wind speed at the Stateline wind energy centre in the US Pacific Northwest. In combination with cross-validation or in the time series context, our proposal provides very general, nonparametric alternatives to the use of information criteria for model diagnostics and model selection.

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Ensemble prediction systems typically show positive spread-error correlation, but they are subject to forecast bias and underdispersion, and therefore uncalibrated. This work proposes the use of ensemble model output statistics (EMOS), an easy to imple-ment post-processing technique that addresses both forecast bias and underdispersion and takes account of the spread-skill relationship. The technique is based on multiple lin-ear regression and 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 an optimal, bias-corrected weighted average of the ensemble member forecasts, with coefficients that are constrained to be nonnegative and associated with the member model skill. The EMOS predictive mean provides a highly accurate deterministic-style forecast. The EMOS predictive variance is a linear function of the ensemble spread. For fitting the EMOS coefficients, the method of minimum CRPS estimation is introduced.

Abstract An experiment is designed to evaluate and compare precipitation forecasts from a 5-member, 4-km grid-spacing (ENS4) and a 15-member, 20-km grid-spacing (ENS20) Weather Research and Forecasting (WRF) model ensemble, which cover a similar domain over the central United States. The ensemble forecasts are initialized at 2100 UTC on 23 different dates and cover forecast lead times up to 33 hours. Previous work has demonstrated that simulations using convection-allowing resolution (CAR; dx ~ 4-km) have a better representation of the spatial and temporal statistical properties of convective precipitation than coarser models using convective parameterizations. In addition, higher resolution should lead to greater ensemble spread as smaller scales of motion are resolved. Thus, CAR ensembles should provide more accurate and reliable probabilistic forecasts than parameterized-convection resolution (PCR) ensembles. Computation of various precipitation skill metrics for probabilistic and deterministic forecasts reveals that ENS4 generally provides more accurate precipitation forecasts than ENS20, with the differences tending to be statistically significant for precipitation thresholds above 0.25 inches at forecast lead times of 9 to 21 hours (0600 – 1800 UTC) for all accumulation intervals

We develop a method for assessing uncertainty about quantities of interest using urban simulation models. The method is called Bayesian melding, and extends a previous method developed for macrolevel deterministic simulation models to agent-based stochastic models. It encodes all the available information about model inputs and outputs in terms of prior probability distributions and likelihoods, and uses Bayes’s theorem to obtain the resulting posterior distribution of any quantity of interest that is a function of model inputs and/or outputs. It is Monte Carlo based, and quite easy to implement. We applied it to the projection of future household numbers by traffic activity zone in Eugene-Springfield, Oregon, using the UrbanSim model developed at the University of Washington. We compared it with a simpler method that uses repeated runs of the model with fixed estimated inputs. We found that the simple repeated runs method gave distributions of quantities of interest that were too narrow, while Bayesian melding gave well calibrated uncertainty statements.

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

Uncertainty in the initial condition is one of the factors that limits the utility of single-model-run predictions of even deterministic nonlinear systems. In practice, an ensemble of initial conditions is often used to generate forecasts with the dual aims of 1) estimating the reliability of the forecasts and 2) estimating the probability distribution of the future state of the system. Current rank histogram ensemble verification techniques can only evaluate scalars drawn from ensembles and associated verification; a new method is presented that allows verification in high-dimensional spaces, including those of the verifications for 10^6 dimensional numerical weather prediction forecasts.

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