Nonlinear Decomposition of Climate Data: a New Method for Reconstruction of Dynamical Modes

Modeling of multivariate time-series produced by complex systems requires efficient tools for compact data representation. In this report we consider this problem, relating to empirical modeling of climate, which implies an analysis of spatial-distributed time-series. The main goal is to establish the number of principal modes which have key contribution to data and actually governs the observed variability. Currently, the number of widely used methods based on PCA and factor analysis exists, which yield different data decompositions taking into consideration spatial/spatio-temporal correlations in observed dynamics: spatial empirical orthogonal functions, M-SSA, varimax rotation, empirical orthogonal teleconnections, and so on. However, the question about possibility of improving the decomposition by taking into account nonlinear couplings between variables often remains untouched.

In the report the method for construction of principal dynamic modes on the basis of low-dimensional nonlinear parametric representation of observed multivariate time-series is suggested. It is aimed to extracting the set of latent modes that both explains an essential part of variability, and obeys the simplest evolution law. Thus, this approach can be used for optimal reconstruction of the phase space for empirical prognostic modeling of observed dynamics. Criterion of evidence of the nonlinearity, which allows estimating optimal parameters of data representation (including the number of parameters in mode definition and the number of principal modes) is proposed.

The effectiveness of suggested method is firstly demonstrated on toy model example: two-dimensional strongly nonlinear problem of unknown dynamic mode retrieval from noisy data is considered. Next, the evidence of nonlinear couplings in SST space-distributed data covering the Globe is investigated by the proposed approach. It is demonstrated that the obtained principal modes capture more part of SST variability than principal components (PCs) constructed by EOF or spatio-temporal EOF expansions. In particular, the first nonlinear mode almost completely describes ENSO phenomenon by capturing three ENSO-related PCs obtained from EOF decomposition. The application of the approach to data-driven forecasting of ENSO dynamics is discussed.