Predicting physical variables in time-delay embedding.

Abstract

In time-delay reconstruction of chaotic attractors we can accurately predict the short-term future behavior of the observed variable x(t)=x(n)=x(${\mathit{t}}_{0}$+${\mathrm{\ensuremath{\tau}}}_{\mathit{s}}$n) without prior knowledge of the equations of motion by building local or global models in the state space. In many cases we also want to predict variables other than the one which is observed and require methods for determining models to predict these variables in the same space. We present a method which takes measurements of two variables x(n) and z(n) and builds models for the determination of z(n) in the phase-space made out of the x(n) and its time lags. Similarly we show that one may produce models for x(n) in the z(n) space, except where special symmetries prevent this, such as in the familiar Lorenz model. Our algorithm involves building local polynomial models in the reconstructed phase space of the observed variable of low order (linear or quadratic) which approximate the function z(n)=F(x(n)) where x(n) is a vector constructed from a sequence of values of observed variables in a time delay fashion. We train the models on a partial data set of measured values of both x(n) and z(n) and then predict the z(n) in a recovery set of observations of x(n) alone. In all of our analyses we assume that the observed data alone are available to us and that we possess no knowledge of the dynamical equations. We test this method on the numerically generated data set from the Lorenz model and also on a number of experimental data sets from electronic circuits.