Conditional entropy of nonlinear dynamical systems

Abstract

We present a method for detecting and quantifying relationships between two or more time series. It is based on the estimation of conditional entropies. In the case of continuous systems nonlinear effects have to be considered. For this case an appropriate conditional probability is calculated. Our method can be used for deciding, if one observable Y depends on another one X. If Y can be expressed as a continuous function f(X), the conditional entropy vanishes. Both dependences between different time series and between time shifted versions of a single one can be detected. In this paper we focus on temporal relationships in a single time series. In this case the observable Y is a time shifted version of X. The properties of the modified conditional entropy help to find optimal delay times in the reconstruction of nonlinear dynamics. Furthermore, we show the advantage of variable delay times.