Figure A3: Maps of ClimateBench target variables for each
baseline model and the target NorESM values under the test ssp245scenario averaged between 2050-2100.
A.1 Gaussian process models
specifications
The GP models kernel k have the same form for all four climate
response variables
k = \(k_{CO2}\ +\ k_{CH4}\ +\ k_{\text{BC}}\ +\ k_{SO2}\)
where \(k_{CO2}\) and \(k_{CH4}\) are kernels that respectively take as
inputs CO2 and CH4 emissions. \(k_{\text{BC}}\) and \(k_{SO2}\) are
kernels that take as inputs the 5 principal components of BC and SO2
emission maps respectively, each principal component being rescaled by
an independent length scale term. We choose the Matérn-1.5 class of
kernel,
\(k_{X}(x,\ x^{\prime})\ =(1+\sqrt{3}\ d(x,x^{\prime}))exp(-\sqrt{3}\ d(x,x^{\prime}))\ \),
where \(X\) is a general notation for CO2, CH4, BC or SO2, and\(d(x,x^{\prime})\) is a distance between inputs typically given by
\(d(x,x^{\prime})\ =\ \sum_{i}{}|x_{i}-x_{i}^{\prime}|/l_{i}\).
\(l_{i}\) is a length scale associated to the \(i^{\text{th}}\)coordinate \(x_{i}\). Global CO2 and CH4 emissions are scalar inputs,
hence the corresponding distances only involve one length scale
parameter. The principal components decompositions of BC and SO2
emission maps both have 5 coordinates, hence we set each principal
component to be a different coordinate with its own length scale
parameter. The Matérn-1.5 kernel guarantees that the corresponding GP
lies in a space of continuous functions, hence providing regularity to
the climate response predictions. We refer the reader to Rasmussen and
Williams, 2005, Chapter 4 for more details on the Matérn kernel. Each
kernel is multiplied by a variance term \({\sigma_{X}}^{2}\) , which
rescales the kernel in the above sum and allows to balance relative
features importance. Variances and length scales are tuned during the
optimization step.