The effect of time delay on Approximate & Sample Entropy calculations

Approximate and Sample Entropy are two widely used techniques to measure system complexity or regularity based on chosen parameters such as pattern length, m, and tolerance, r. In this paper, we investigate how different values of the time delay parameter, τ can be used in conjunction with standard values of m and r in the computation of Approximate and Sample Entropy. The results show that for time series generated by nonlinear dynamics that have long range correlation, a time delay equal to the first zero crossing or minimum of the autocorrelation function can provide additional information into the characteristics of the time series that may be useful in comparative analysis. With a unity delay, we demonstrate that Approximate and Sample Entropy are possibly measuring only the (linear) autocorrelation properties of the signal, and these are highly invariant under surrogate data generation methods. Hence when this occurs, the complexity measures of the surrogate and original data are not statistically different.