We present a novel Bayesian statistical approach to computing model weights in climate change projection ensembles in order to create probabilistic projections. The weight of each climate model is obtained by weighting the current day observed data under the posterior distribution admitted under competing climate models. We use a linear model to describe the model output and observations. The approach accounts for uncertainty in model bias, trend and internal variability, including error in the observations used. Our framework is general, requires very little problem-specific input, and works well with default priors. We carry out cross-validation checks that confirm that the method produces the correct coverage.

Regional climate models (RCMs) are powerful tools to produce regional climate
projections
(

Along with these ensemble modelling studies, methods for extracting
probabilistic projections have followed
(

Several authors have shown that in many regions, future changes are
positively correlated with present-day internal variability in the models:
see

In this article, we propose a new method to obtain model weights using raw
model output, so the method better accounts for model output uncertainty. Our
framework allows us to compute weights efficiently, simultaneously penalising
for model bias, deviations in trend and model internal variability. One of
the main advantages of the current approach is that improper and vague
priors for the model parameters can be used, which makes implementation of
the method much more straightforward. In the

Below the Bayesian methodology developed is described followed by a Markov
chain Monte Carlo (MCMC) method to obtain solutions for the posterior
distributions. The technique is then applied to a regional climate model
ensemble and compared with results found in previous work
(

Pictorial representation of the weight distribution on

In this section, we introduce the Bayesian methodology for weighting model
output based on current day observations. The framework we describe below is
not limited to any particular distributional form, although the analysis
presented is based on the univariate normal distribution. We have also
implemented the same procedure using the asymmetric Laplace distribution for
median regression to obtain robust estimators for our analyses, but we have
excluded them from presentation as the procedure produced similar results to
that of the normal error assumption (indicating no major violations from
normality). We suppose that current day observations are
denoted as

The parameters

New South Wales planning regions, the ACT and the state of Victoria.

Results for the CC region of south-eastern Australia, in the DJF
season. Top row: weights

Results for the FW region of south-eastern Australia, in the DJF
season. Top row: weights

Results for the CWO region of south-eastern Australia, in the MAM
season. Top row: weights

Posterior predictive projections of DJF temperature change in
2060–2079 compared to 1990–2009 for regions in south-eastern Australia.
Black lines correspond to

Bootstrapped weighted projections of DJF temperature change in
2060–2079 compared to 1990–2009 for regions in south-eastern Australia.
Black lines correspond to

We would like to weight the models based on the similarity of output

Finally, we define the weight for each model

The ensemble models can now be combined into a single posterior model, using
the weights

In order to understand this weight, we suppose for the moment that the data

It is worth noting that even if we specify non-informative priors in
Eq. (

The procedure for the calculation of weights is designed to be applicable
regardless of the distributional forms chosen to model the data. In most
cases, the posterior distributions

In addition to obtaining simulations from the posteriors of the

Finally, the predictive distribution for the future climate

Here we consider the same data as

Here we average the temperatures over south-eastern Australian regions that
include New South Wales (NSW) planning regions, ACT, and Victoria; see
Fig.

In addition to computing weights of the form in Eq. (

Cross validation of weighted projections of DJF temperature change
in 2060–2079 compared to 1990–2009 for region CC in south-eastern
Australia. Black lines correspond to

Mean squared error and 95 % coverage probabilities for the three sets of weights.

Figure

The weighted fits are shown in the last two plots in the bottom row of
Fig.

For seasons JJA and MAM, weights

The corresponding posterior predictive distribution of projections of change
in temperature for season DJF over the different regions in south-eastern
Australia are plotted in Fig.

With probability

Simulate a predictive temperature series

Compute current model estimate

Compute the mean of the differences between future prediction

We present the results for season DJF in Fig.

The incident of bimodality or multimodality is reduced in our approach
compared to

In order to assess the ensemble pdf, we performed a series of
cross-validation checks. For each region at a given season, we have 12
current model outputs and 12 future model outputs. We select 1 of the models,

Table

In this article we have introduced a new framework for computing Bayesian model weights. Our framework is novel, and requires minimal expert knowledge of model parameters. The fact that we do not require subjective expert prior knowledge makes the method more robust, since prior elicitation can sometimes be difficult, and different priors can lead to different conclusions.

We provided two alternative weight specifications under the same framework to aid interpretation of our weighting. One of the weights favours models with intercept terms that are close to the observation intercept. This weight does not penalise for trend deviations very well. An alternative weight which does not penalise for the intercept term can capture trend in the model very well. Both alternatives have deficiencies, and our proposed weight is a combination of the two. However, there are other potential avenues to explore with these alternative weights. For instance, for the weights based on trend and internal variability, it can be seen that the weighted model can capture trend extremely well but fails to account for bias, but applying some kind of post hoc bias correction may be a fruitful direction to pursue.

We validated our approach using cross validation, and showed that our posterior predictive distributions obtained correct empirical coverages, which is a desired property to possess, and provides us with some confidence in our approach. Our posterior predictive distributions also provided narrower confidence intervals than previous approaches. Finally, our model weighting framework is not restricted to data from univariate normal distributions, or linear models. This approach could be extended to handle dependent Gaussian data via a multivariate normal distribution, as well as non-linear and non-normal models.

Code and data for the analyses carried out in this article are available in the Supplement.

The authors declare that they have no conflict of interest.

This work was supported by the National Research Foundation of Korea Grant funded by the Korean Government (MEST) (NRF-2009-0093069).Edited by: James Annan Reviewed by: Hans R. Künsch and one anonymous referee