Detecting spatial patterns with the cumulant function – Part 1: The theory

Abstract. In climate studies, detecting spatial patterns that largely deviate from the sample mean still remains a statistical challenge. Although a Principal Component Analysis (PCA), or equivalently a Empirical Orthogonal Functions (EOF) decomposition, is often applied for this purpose, it provides meaningful results only if the underlying multivariate distribution is Gaussian. Indeed, PCA is based on optimizing second order moments, and the covariance matrix captures the full dependence structure of multivariate Gaussian vectors. Whenever the application at hand can not satisfy this normality hypothesis (e.g. precipitation data), alternatives and/or improvements to PCA have to be developed and studied.

To go beyond this second order statistics constraint, that limits the applicability of the PCA, we take advantage of the cumulant function that can produce higher order moments information. The cumulant function, well-known in the statistical literature, allows us to propose a new, simple and fast procedure to identify spatial patterns for non-Gaussian data. Our algorithm consists in maximizing the cumulant function. Three families of multivariate random vectors, for which explicit computations are obtained, are implemented to illustrate our approach. In addition, we show that our algorithm corresponds to selecting the directions along which projected data display the largest spread over the marginal probability density tails.