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Approximate entropy (ApEn) has been widely used as an estimator of regularity in many scientific fields. It has proved to be a useful tool because of its ability to distinguish different system's dynamics when there is only available short-length noisy data. Incorrect parameter selection (embedding dimension m, threshold r and data length N) and the presence of noise in the signal can undermine the ApEn discrimination capacity. In this work we show that r max (ApEn(m, r max , N) = ApEn max ) can also be used as a feature to discern between dynamics. Moreover, the combined use of ApEn max and r max allows a better discrimination capacity to be accomplished, even in the presence of noise. We conducted our studies using real physiological time series and simulated signals corresponding to both low-and high-dimensional systems. When ApEn max is incapable of discerning between different dynamics because of the noise presence, our results suggest that r max provides additional information that can be useful for classification purposes. Based on cross-validation tests, we conclude that, for short length noisy signals, the joint use of ApEn max and r max can significantly decrease the misclassification rate of a linear classifier in comparison with their isolated use.

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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.