Physica D 50 (1991) 95-116 North-Holland Time series and dependent variables Robert Savit and Matthew Green Physics Department, The University of Michigan, Ann Arbor, MI 48109, USA Received 19 October 1989 Revised manuscript received 8 January 1990 Accepted 4 August 1990 Communicated by R.M. Westervelt 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, ~j(e), 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 ) t h element to within a measurement uncertainty, e. We apply our method to a number of deterministic time series generated by chaotic processes such as the tent, logistic and H~non maps, as well as to sequences of quasi-random numbers. In all cases the 6j correctly indicate the expected dependencies. We also show that the ~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, e. Finally, we discuss the behavior of the 6i as e approaches zero. I. Introduction Time series which evidence characteristics of a broad-band spectrum are notoriously difficult to analyze. Traditional methods such as Fourier analysis or other linear transforms usually fail to offer many insights into the underlying structure of such a time series. Broad-band time series appear in many contexts including data signaling processes, biomedical applications, and economic systems and, unfortunately, are more the rule than the exception. Crudely speaking, the broad-band nature of such series may be due to "noise", to determinis- tic processes of a chaotic, or near-chaotic nature, or to a combination of both. One important ob- jective in the analysis of broad-band series is distinguishing among these alternatives. In this paper, we present a method for analyzing time series which allows one to distinguish between certain kinds of noise and certain deterministic processes. Our approach is based on the con- struction of conditional probabilities for the repe- tition of short sequential patterns of values in a time series. The conditional probabilities can be expressed in terms of the Grassberger-Procaccia correlation integrals [1] and contain information from the entire series. If the series is generated by a chaotic process, then our method samples all regions of the attractor. Using this method, we can determine, quantitatively, the extent to which a term in the time series is a function of previous terms in the series. As a by-product, we are able to determine, in the case of a chaotic system, the minimum embedding dimension necessary for a reasonable description of the dynamics. As we shall make clear below, this minimum embedding dimension generally depends on e, the tolerance or uncertainty with which one wishes to observe the dynamics. Finally, we introduce a predictabil- 0167-2789/91/$03.50 © 1991- Elsevier Science Publishers B.V, (North-Holland)