Cross-validation analysis of bias models in Bayesian multi-model projections of climate

Abstract

Climate change projections are commonly based on multi-model ensembles of climate simulations. In this paper we consider the choice of bias models in Bayesian multimodel predictions. Buser et al. (Clim Res 44(2–3):227–241, 2010a) introduced a hybrid bias model which combines commonly used constant bias and constant relation bias assumptions. The hybrid model includes a weighting parameter which balances these bias models. In this study, we use a cross-validation approach to study which bias model or bias parameter leads to, in a specific sense, optimal climate change projections. The analysis is carried out for summer and winter season means of 2 m-temperatures spatially averaged over the IPCC SREX regions, using 19 model runs from the CMIP5 data set. The cross-validation approach is applied to calculate optimal bias parameters (in the specific sense) for projecting the temperature change from the control period (1961–2005) to the scenario period (2046–2090). The results are compared to the results of the Buser et al. (Clim Res 44(2–3):227–241, 2010a) method which includes the bias parameter as one of the unknown parameters to be estimated from the data.

Appendix

1.1 Continuous ranked probability score

Let x be a scalar variable (e.g. 2 m-temperature). Suppose that a probability function forecast of x is given by p(x) and we have also an observation $x^{\text {obs}}$ of the x. The Continuous Ranked Probability Score (CRPS) (Stanski et al. 1989; Hersbach 2000; Candille and Talagrand 2005; Grimit et al. 2006) is defined as

$$\begin{aligned} \mathrm {CRPS}=\int _{-\infty }^\infty \left[ P^{\text {pred}}(x)-P^{\text {obs}}(x)\right] ^2{\,\mathrm d}x \end{aligned}$$

(14)

where $P^{\text {pred}}(x)=\int _{-\infty }^x p(x'){\,\mathrm d}x'$ is the cumulative distribution function of p(x) and $P^{\text {obs}}$ is the cumulative distribution function for the observation:

$$\begin{aligned} P^{\text {obs}}(x)= & {} \left\{ \begin{array}{cc} 0,&{} \text {if }x<x^{\text {obs}}\\ 1,&{}\text {if }x\ge x^{\text {obs}}\end{array}\right. =H(x-x^{\text {obs}}) . \end{aligned}$$

where H is the Heaviside function ($H(x)=1$ if $x\ge 0$ and 0 otherwise).

In this paper, the “distance” between the predictions of the future temperatures $Y_{0,t}$ (when $\ell$’th model is taken as the “truth” in the cross-validation) and the actual future temperatures $Y_{\ell ,t}$ is measured using the CRPS. The (total) CRPS score is calculated as the mean of CRPSs calculated for every year $t=1,\ldots ,T$. By Eq. (13), the prediction distribution is a (weighted) sum of Gaussian distributions and the CRPS for such Gaussian mixture model can be calculated in closed form using an expression given in Grimit et al. (2006).