Automatic estimation of attractor invariants

Abstract

The invariants of an attractor have been the most used resource to characterize a nonlinear dynamics. Their estimation is a challenging endeavor in short-time series and/or in presence of noise. In this article we present two new coarse-grained estimators for the correlation dimension and for the correlation entropy. They can be easily estimated from the calculation of two U-correlation integrals. We have also developed an algorithm that is able to automatically obtain these invariants and the noise level in order to process large data sets. This method has been statistically tested through simulations in low-dimensional systems. The results show that it is robust in presence of noise and short data lengths. In comparison with similar approaches, our algorithm outperforms the estimation of the correlation entropy

Invariant Measures Based on the U‐Correlation Integral: An Application to the Study of Human Voice

“…Such functions are useful because they allow us to estimate the invariants and to visually confirm a scaling regime. In [16], we have presented three coarse-grained functions:…”

“…As an advantage, they can be calculated in an automated manner from two U-correlation integrals. Moreover, we have proposed an automated algorithm to estimate , 2 , and from these coarse-grained estimators [16].…”

“…This choice is critical since, in practical application, the coarse-grained estimators strongly vary as a function of the scale. To avoid subjective judgements, we have proposed an algorithm (see Algorithm 1) for the automated estimation of , 2 , and based on the coarse-grained estimators (ℎ), (ℎ), and (ℎ), respectively [16]. This algorithm selects the scaling range where those invariants should be estimated based on the next criteria: (i) the coarse-grained estimator must be constant for a range of scale values and (ii) the value of the invariant should converge as the embedding dimension increases.…”