Noise-assisted estimation of attractor invariants

In this article, the noise-assisted correlation integral (NCI) is proposed. The purpose of the NCI is to estimate the invariants of a dynamical system, namely the correlation dimension (D), the correlation entropy (K_{2}), and the noise level (σ). This correlation integral is induced by using random noise in a modified version of the correlation algorithm, i.e., the noise-assisted correlation algorithm. We demonstrate how the correlation integral by Grassberger et al. and the Gaussian kernel correlation integral (GCI) by Diks can be thought of as special cases of the NCI. A third particular case is the U-correlation integral proposed herein, from which we derived coarse-grained estimators of the correlation dimension (D_{m}^{U}), the correlation entropy (K_{m}^{U}), and the noise level (σ_{m}^{U}). Using time series from the Henon map and the Mackey-Glass system, we analyze the behavior of these estimators under different noise conditions and data lengths. The results show that the estimators D_{m}^{U} and σ_{m}^{U} behave in a similar manner to those based on the GCI. However, for the calculation of K_{2}, the estimator K_{m}^{U} outperforms its GCI-based counterpart. On the basis of the behavior of these estimators, we have proposed an automatic algorithm to find D,K_{2}, and σ from a given time series. The results show that by using this approach, we are able to achieve statistically reliable estimations of those invariants.