Distinguishing separate components in high-dimensional signals by using the modified embedding method and forecasting.

Abstract

The non-linear analysis of medium- and high-dimensional signals (HDS) encounters often difficulties due to the problem of the choice of the proper window width. This problem occurs due to the fact that the autocorrelation time (tau) of HDS reflecting the natural scale of the signal is similar or higher than the predictability of HDS and the predictability points to some degree to the maximal acceptable window width (W). The new approach to the embedding process is proposed in this paper that uses the constant W instead of the constant L. The lag is calculated by using the formula L = W/(m - 1). If L is non-integer, the cubic interpolation is performed to find the values of the signal between the given points. A forecasting analysis of two signals of the known structure and dimensional complexity is performed by employing the new approach. The analyzed signals consist of 3 and 4 Lorenz signals of different time scales, respectively. The used measure of the predictability is the predictive time t(p05) that is defined as the prediction time for which the correlation coefficient rho between the real and the predicted points equals 0.5. The relations t(p05)(W) possess the maxima that correspond to the autocorrelation times of the individual components. These maxima are observed approximately at W(i) = (0.6-1)tau(i), where i is the index of the given component of the signal. The saturation of predictability is observed for consecutive maxima in most cases at the embedding dimension equal to about m(i*) = 2 . d(k) + 1, where d(k) is the sum of the dimensional complexities of the components, whose autocorrelation time is equal or less than the W(i). Thus, it is clearly visible that only the components of the time scale equal or less than W(i) are demonstrated in the results of the analysis.