Reducing noise in discretized time series.

We show that applying a noise-reduction algorithm to a discretized time series increases its average error, compared to the original series. We find that adding external noise comparable to the discretization step before noise reduction limits the increase of the average error and improves the estimation of Lyapunov exponents.

Multiscale analysis of complex systems.

We calculate block information versus size profiles for two-symbol strings generated by several dynamical processes: random, periodic, regular language, and substitutive. The profiles provide a good diagnostic of the complexity of the strings.

Finite statistical complexity for sofic systems.

We propose a measure of complexity for symbolic sequences, which is based on conditional probabilities, and captures computational aspects of complexity without the explicit construction of minimal deterministic finite automata (DFA). Moreover, if the sequence is obtained from a dynamical system through a suitable encoding and its equations of motion are known, we show how to estimate the regions of phase space that correspond to computational states with statistically equivalent futures (causal states).

Observation of structure in the Lorenz map.

Through massive numerical integration of the Lorenz system, we are able to discern structure in its Poincare map. We are also able to estimate its capacity dimension; our result is consistent with previous measurements of the correlation dimension of the Lorenz attractor. (c) 1999 American Institute of Physics.