The blessing of dimensionality for the analysis of climate data

Abstract

We give a simple description of the blessing of dimensionality with the main focus on the concentration phenomena. These phenomena imply that in high dimensions the lengths of independent random vectors from the same distribution have almost the same length and that independent vectors are almost orthogonal. In the climate and atmospheric sciences we rely increasingly on ensemble modelling and face the challenge of analysing large samples of long time series and spatially extended fields. We show how the properties of high dimensions allow us to obtain analytical results for e.g. correlations between sample members and the behaviour of the sample mean when the size of the sample grows. We find that the properties of high dimensionality with reasonable success can be applied to climate data. This is the case although most climate data show strong anisotropy and both spatial and temporal dependence, resulting in effective dimensions around 25–100.