Symbol sequence statistics in noisy chaotic signal reconstruction

A method is discussed for reconstructing chaotic systems from noisy signals using a symbolic approach. The state space of the dynamical system is partitioned into subregions and a symbol is assigned to each subregion. Consequently, an orbit in a continuous state space is converted into a long symbol string. The probabilities of occurrence for different symbol sequences constitute the symbol sequence statistics. The symbol sequence statistics are easily measured from the signal output and are used as the target for reconstruction (i.e., for assessing the goodness of fit of proposed models). Reliable reconstructions were achieved given a noisy chaotic signal, provided the general class of the model of the underlying dynamics is known. Both observational and dynamical noise were considered, and they were not limited to small amplitudes. Substantial noise produces a strong bias in the symbol sequence statistics, but such bias can be tracked and effectively eliminated by including the noise characteristics in the model. This is demonstrated by the robust reconstruction of the Hénon and Ikeda maps even when the signal to noise ratio is 1. Applications of this method include extracting control parameters for nonlinear dynamical systems and nonlinear model evaluation from experimental data.