Time series and dependent variables

A new method for analyzing time series which is designed to extract inherent deterministic dependencies in the series is presented, and quantities, ~j(e), based on conditional probabilities, whose magnitude is an indicator of the extent to which the kth element in a series is a deterministic function of the (k -j)th element to within a measurement uncertainty, are derived.

Abstract

We present a new method for analyzing time series which is designed to extract inherent deterministic dependencies in the series. The method is particularly suited to series with broad-band spectra such as chaotic series with or without noise. We derive quantities, [delta]j([var epsilon]), based on conditional probabilities, whose magnitude, roughly speaking, is an indicator of the extent to which the kth element in the series is a deterministic function of the (k - j)th element to within a measurement uncertainty, [var epsilon]. We apply our method to a number of deterministic time series generated by chaotic processes such as the tent, logistic and Henon maps, as well as to sequences of quasi-random numbers. In all cases the [delta]j correctly indicate the expected dependencies. We also show that the [delta]j are robust to the addition of substantial noise in a deterministic process. In addition, we derive a predictability index which is a measure of the extent to which a time series is predictable given some tolerance, [var epsilon]. Finally, we discuss the behavior of the [delta]i as [var epsilon] approaches zero.