Nonlinear Empirical Decomposition of Space-Time Distributed Data

Abstract:

This report is devoted to the problem of an efficient expansion of multivariate time series, suitable for construction of principal components capturing major part of variability. We present an algorithm for nonlinear transformation of the basis of observational variables, which yields a set of hidden modes of the system. It is based on the nonlinear generalization of standard empirical orthogonal functions (EOFs) decomposition, i.e. the observed dataset is projected onto nonlinear curves which represent nonlinear transformations from latent principal component (PC) time series to the space of observational variables. Both the parameters of the transformation and the time series of PCs are found simultaneously by using Bayesian approach. As a prior restriction we use the simplest evolution operator to provide smoothness of the PC’s time series. Finally, Bayesian evidence technique is used to complete the task of correct choice of both the smoothness and the degree of nonlinearity of the transformation.
The capabilities of the method are demonstrated on the number of model examples: the applications of the proposed expansion to time series generated by distributed dynamical systems of different complexity are analyzed. The comparison of the expansion with traditional linear decompositions such as EOF or MSSA as well as some existing nonlinear decompositions is presented and discussed. The efficiency of obtained modes for construction of low-dimensional phase space capturing the key dynamical properties of analyzed system is investigated.

This research was supported by the Government of the Russian Federation (Agreement No. 14.Z50.31.0033 with Institute of Applied Physics RAS)