Dependent variables in broad band continuous time series

Abstract

In this paper we continue our development of new methods for the analysis of broad band time series by deriving quantities which are able to indicate deterministic dependence of an element in one time series on elements in other time series. These methods are very broadly applicable and are particularly well suited to the study of continuous time series, in which the value of the function may depend on derivatives of the function itself, or on other quantities. We apply our methods to a number of mathematical examples including the Lorentz equation, the Henon-Heiles equations, the forced Brusselator and the Mackey-Glass equation. We show that our methods are very successful at indicating deterministic dependencies in these systems, even if the time series are highly chaotic. Statistical aspects of our procedure are discussed, as are a number of interesting and surprising epistomological implications.