Multi-level emulation of complex climate model responses to boundary forcing data

Abstract

Climate model components involve both high-dimensional input and output fields. It is desirable to efficiently generate spatio-temporal outputs of these models for applications in integrated assessment modelling or to assess the statistical relationship between such sets of inputs and outputs, for example, uncertainty analysis. However, the need for efficiency often compromises the fidelity of output through the use of low complexity models. Here, we develop a technique which combines statistical emulation with a dimensionality reduction technique to emulate a wide range of outputs from an atmospheric general circulation model, PLASIM, as functions of the boundary forcing prescribed by the ocean component of a lower complexity climate model, GENIE-1. Although accurate and detailed spatial information on atmospheric variables such as precipitation and wind speed is well beyond the capability of GENIE-1’s energy-moisture balance model of the atmosphere, this study demonstrates that the output of this model is useful in predicting PLASIM’s spatio-temporal fields through multi-level emulation. Meaningful information from the fast model, GENIE-1 was extracted by utilising the correlation between variables of the same type in the two models and between variables of different types in PLASIM. We present here the construction and validation of several PLASIM variable emulators and discuss their potential use in developing a hybrid model with statistical components.

Appendix: Gaussian process emulator

The climate model, $f(\cdot )$, can be viewed as a function of a set of inputs, ${\varvec{x}}=[x_1,\ldots ,x_d]$, where d is the number of perturbed model parameters. This number is commonly referred to as the number of dimensions of the emulator. The output of each model run is a scalar value y. Supposed we have n simulation runs, providing n realisations ${\varvec{y}}=[y_1=f({\varvec{x}}_1),\ldots ,y_n=f({\varvec{x}}_n)]$. These comprise the training set used to train an emulator.

First, the function $f(\cdot )$ is represented by a GP prior described by a mean function $m(\cdot )$ and a covariance function $V(\cdot ,\cdot )$

$$\begin{aligned} f(\cdot )|\varvec{\beta },\sigma ^2,\varvec{\theta } \sim {\mathcal {N}}(m(\cdot ),V(\cdot ,\cdot )). \end{aligned}$$

(12)

This GP is used as a prior for Bayesian inference. The prior does not depend on the training data but specifies the assumptions made about the function of interest. Then, the outputs from a selected number of simulations are incorporated, allowing us to update the prior to the posterior GP. This process is called training the GP model. Following (Kennedy and O’Hagan 2001), $m(\cdot )$ and $V(\cdot ,\cdot )$ are modelled hierarchically, meaning that they are parameterised in terms of hyperparameters. The mean function is given by:

$$\begin{aligned} m({\varvec{x}})= {\varvec{h}}^T({\varvec{x}})\varvec{\beta }, \end{aligned}$$

(13)

where ${\varvec{h}}({\varvec{x}})$ is a vector of known regression functions of the inputs, describing a class of shapes of the function $f(\cdot )$. $\varvec{\beta }$ is an unknown vector of coefficients. In the case of ordinary kriging, ${\varvec{h}}(\cdot )={\mathbf {1}}$, making $\varvec{\beta }$ the unknown overall mean. A variation of kriging, called universal kriging, uses a linear mean function:

$$\begin{aligned} {\varvec{h}}(\cdot )=({\varvec{1}},{\varvec{x}}^T), \end{aligned}$$

(14)

where ${\varvec{h}}({\varvec{x}})^T$ is a $(s\times 1)$ vector with $s=d+1$.

The covariance function is given by:

$$\begin{aligned} V({\varvec{x}},\varvec{x'})=\sigma ^2 { {\Psi }}({\varvec{x}},\varvec{x'}), \end{aligned}$$

(15)

in which $\sigma ^2$ is an unknown variance of the GP and ${\Psi }(\cdot ,\cdot )$ is the assumed correlation function:

$$\begin{aligned} { {\Psi }}({\mathbf {x}},\mathbf {x'})=\text {exp}\left[ -\sum _{j=1}^{d}10^{\theta _j}\left| {x_j}-{x'_j}\right| ^{p_j}\right] . \end{aligned}$$

(16)

The function ${ {\Psi }}$ represents the correlation between pairs of points, which is assumed to be stationary and continuous, that is, it only depends on the distance between the pair of inputs, (${\varvec{x}}-\varvec{x'}$). This power exponential form of covariance structure is a popular choice due to its flexibility.

Both p and $\theta$ can be estimated for each dimension. For simplicity and to reduce computational cost, $p=2$ is assumed for all dimensions. An independent value of $\theta$ is obtained for each dimension by maximising the likelihood of ${\varvec{y}}$.

The specified GP is used as a prior for Bayesian inference and is parameterised in terms of the hyperparameters $\varvec{\beta }$, $\sigma ^2$, $\varvec{\theta }$ and p. Given that the prior is Gaussian, by analytically marginalising $\beta$ and $\sigma ^2$, the marginal likelihood of the observed outputs at n training points, ${\varvec{y}}$, given $\theta$ and p can then be computed (estimated by maximising the likelihood of ${\varvec{y}}$). A more detailed description of the derivations and formulations can be found in Mardia and Marshall (1984).

Prior beliefs about the model behaviour are combined with observations from training points to produce a posterior distribution for the model. Having obtained estimates for $\varvec{\theta }$ and p, the posterior distribution found can be used to make predictions about the model’s outputs at unsampled inputs. The predictive distribution is a Student’s t-distribution, with $n-s$ degrees of freedom

$$\begin{aligned} p(f({\varvec{x}})|{\mathbf {y}},\theta ) \sim t_{n-s}(m_1({\varvec{x}}),V_1({\varvec{x}},\varvec{x'})), \end{aligned}$$

(17)

with

$$\begin{aligned} m_1({\varvec{x}})={\varvec{h}}^T({\varvec{x}})\varvec{{\hat{\beta }}}+{\varvec{T}}({\varvec{x}}){\mathbf {A}}^{-1}({\varvec{y}}-{\varvec{H}}\varvec{{\hat{\beta }}}), \end{aligned}$$

(18)

and

$$\begin{aligned} V_1({\varvec{x}},{\varvec{x}}')={\hat{\sigma }}^2 [{\Psi }({\varvec{x}},\varvec{x'})-{\varvec{T}}({\varvec{x}})^T{\mathbf {A}}^{-1}{\varvec{T}}(\varvec{x'})+{\mathbf {P}}({\varvec{x}})({\mathbf {H}}^T{\mathbf {A}}^{-1}{\mathbf {H}})^{-1}{\mathbf {P}}(\varvec{x'})^T], \end{aligned}$$

(19)

where ${\mathbf {H}}$ is the regression matrix of the design points, ${\mathbf {H}}= {\varvec{h}}({\varvec{x}})^T$, and ${\mathbf {A}}$ is the design points correlation matrix, ${\mathbf {A}}= {\Psi }({\varvec{x}},{\varvec{x}}')$; ${\varvec{t}}({\varvec{x}})$ is the correlation vector between ${\varvec{x}}$ and the training set, i.e. $({\varvec{T}}({\varvec{x}}))_i = {\Psi }({\varvec{x}},{\varvec{x}}_i)$ and ${\mathbf {P}}({\varvec{x}})={\varvec{h}}({\mathbf {x}})^T-{\varvec{T}}({\varvec{x}}){\mathbf {A}}^{-1}{\mathbf {H}}$. The estimated values of $\sigma ^2$ and $\varvec{\beta }$ are indicated as ${\hat{\sigma }}^2$ and $\varvec{{\hat{\beta }}}$, respectively:

$$\begin{aligned} \varvec{{\hat{\beta }}} = ({\mathbf {H}}^{\mathbf {T}}{\mathbf {A}}^{-1}{\mathbf {H}})^{-1}{\mathbf {H}}^T{\mathbf {A}}^{-1}{\varvec{y}} \end{aligned}$$

(20)

and

$$\begin{aligned} {\hat{\sigma }}^2 = \frac{{\varvec{y}}^T({\mathbf {A}}^{-1}-{\mathbf {A}}^{-1}{\mathbf {H}}({\mathbf {H}}^T{\mathbf {A}}^{-1}{\mathbf {H}})^{-1}{\mathbf {H}}^T{\mathbf {A}}^{-1}){\varvec{y}}}{n-q-2}. \end{aligned}$$

(21)

A full description of the derivation of the posterior distribution is available in Rasmussen and Williams (2006).

Co-kriging is an extension to this technique, which is applicable when a fast approximation of the primary simulator is available. For this method to work, the primary simulator and its approximation need to be correlated and contain information about one another.

When only a small number of expensive runs are available, it has been shown that by combining these with cheaper runs from a simplified code, an emulator of the expensive model can be built at a lower cost (Forrester et al. 2007).

We make a simplification that the expensive and cheap models, $f_e$ and $f_c$ respectively, can be represented by GP emulators with the same value of p. The cheap model is first emulated and then linked to the expensive one using the single multiplier approach:

$$\begin{aligned} f_e({\varvec{x}})=\rho f_c({\varvec{x}})+f_d({\varvec{x}}). \end{aligned}$$

(22)

The right-hand side of the equation consists of a cheap GP, $f_c$, multiplied by a scaling factor $\rho$ and a separate GP, $f_d$, modelling the stochastic residual of the expensive model (Kennedy and O’Hagan 2000; Forrester et al. 2007). Together these two terms describe the emulator of the expensive model. This approximation is chosen for its simplicity as well as the assumption that the main difference between the two models is largely a matter of scale. This assumption is made based on the fact that both EMBM and PLASIM are driven by the boundary conditions specified by GENIE-1’s ocean. They essentially share similar inputs but have the ability to respond differently.

Two sets of training points are required for the construction of a co-kriging emulator; a cheap set ${\varvec{y}}_c=f_c({\varvec{x}}_c)$, which finely samples the input space, and a small, sparse set ${\varvec{y}}_e=f_e({\varvec{x}}_e)$ of expensive points. When the number of PLASIM training points is small, such that a kriging emulator cannot be built with high accuracy, co-kriging employing a large additional number of training points from GENIE-1’s EMBM can be used instead. The number of points required depends on the size of the problem as well as the smoothness of the function being emulated. A general rule of thumb for the number of training points for kriging is 10 times the number of parameters (Loeppky et al. 2009). The inputs at which the expensive training set is obtained, ${\varvec{x}}_e$, is a subset of the cheap set, ${\varvec{x}}_c$. These expensive points are chosen using an exchange algorithm described by Cook and Nachtsheim (1980).

The covariance matrix for co-kriging, ${\Psi }_{ck}$, can be written in block form as

$$\begin{aligned} {\Psi }_{ck}= \begin{pmatrix} \sigma ^2_c{\mathbf {A}}_c({\varvec{x}}_c) &{} \quad \rho \sigma _c^2 {\mathbf {A}}_c({\varvec{x}}_c,{\varvec{x}}_e) \\ \rho \sigma _c^2 {\mathbf {A}}_c({\varvec{x}}_e,{\varvec{x}}_c) &{} \quad \rho \sigma _c^2 {\mathbf {A}}_c({\varvec{x}}_e) + \sigma _e^2{\mathbf {A}}_d({\varvec{x}}_e) \\ \end{pmatrix}, \end{aligned}$$

(23)

with ${\mathbf {A}}_c= {\Psi }({\varvec{x}},\varvec{x'};\varvec{\theta }_c)$ and ${\mathbf {A}}_d= {\Psi }({\varvec{x}},\varvec{x'};\varvec{\theta }_d)$. This covariance matrix encompasses the correlation between cheap points (${\mathbf {A}}_c({\varvec{x}}_c)$), expensive points (${\mathbf {A}}_c({\varvec{x}}_e)$ and ${\mathbf {A}}_d({\varvec{x}}_e)$) and the cross-correlation between the cheap and expensive points (${\mathbf {A}}_c({\varvec{x}}_c,{\varvec{x}}_e)$ and ${\mathbf {A}}_c({\varvec{x}}_e,{\varvec{x}}_c)$). Details on the formulation and derivation of this equation can be found in Kennedy and O’Hagan (2000) and Forrester et al. (2007).