Short-term predictions by statistical methods in regions of varying dynamical error growth in a chaotic system

In a nonlinear, chaotic dynamical system, there are typically regions in which an infinitesimal error grows and regions in which it decays. If the observer does not know the evolution law, recourse is taken to non-dynamical methods, which use the past values of the observables to fit an approximate evolution law. This fitting can be local, based on past values in the neighborhood of the present value as in the case of Farmer-Sidorowich (FS) technique, or it can be global, based on all past values, as in the case of Artificial Neural Networks (ANN). Short-term predictions are then made using the approximate local or global mapping so obtained. In this study, the dependence of statistical prediction errors on dynamical error growth rates is explored using the Lorenz-63 model. The regions of dynamical error growth and error decay are identified by the bred vector growth rates or by the eigenvalues of the symmetric Jacobian matrix. The prediction errors by the FS and ANN techniques in these two regions are compared. It is found that the prediction errors by statistical methods do not depend on the dynamical error growth rate. This suggests that errors using statistical methods are independent of the dynamical situation and the statistical methods may be potentially advantageous over dynamical methods in regions of low dynamical predictability.