Difference correlation can distinguish deterministic chaos from 1/fα-type colored noise

Abstract

Distinguishing deterministic chaos from colored noise with the power-law spectra or random fractal sequences (fractional Brownian motions) is one of the important problems in chaotic time series analysis. In this paper, we describe a simple method for solving this problem, which seems easier than the other algorithms that have already been proposed. In order to show how well our procedure works, first we apply a nonlinear prediction to time series data, produced from both nonlinear dynamical systems and stochastic systems with the power-law spectra. Next, we evaluate the prediction performance by calculating two kinds of correlation coefficients between actual time series and predicted time series, which are called a conventional correlation coefficient and a difference correlation coefficient. The conventional correlation coefficient is a usual correlation coefficient between actual time series and predicted time series, and the difference correlation coefficient is between first-difference time series obtained from actual time series and predicted time series. When the one-step-ahead nonlinear prediction is applied to deterministic chaos without observational noise, not only conventional correlation coefficients but also difference correlation coefficients are very high values, namely, the coefficients take values almost 1.0 even if the number of data points is small. On the other hand, in the case of 1/${\mathrm{f}}^{\mathrm{\ensuremath{\alpha}}}$-type colored noise, although conventional correlation coefficients are relatively high values, difference correlation coefficients turn out to be low values, even though the scaling exponent of the power spectrum \ensuremath{\alpha} is large. This difference between conventional correlation and difference correlation can be a good criterion for distinguishing deterministic chaos from colored noise with the power-law spectra. Finally, several real time series data are analyzed in order to confirm the applicability of the proposed method.