Extraction of nonlinear dynamics from short and noisy time series

The aim of this paper is to define efficient strategies for the modeling and predicting of short and noisy time series with neural networks. Several complementary methods are tested on short series constructed from the Lorenz system which has been spoiled with various levels of measurement or dynamical noise. The best strategies are selected from the simulation results according to the level and the noise characteristics. In the presence of measurement noise we show that over-sizing of the embedding dimension of the learning set increases the powerness of neural network fits. In the case of dynamical noise spoiling, we found that generation of a new trajectory predicted with local operators, amplifying information of the original series, allows the usage of neural networks as in the case of measurement noise, and so, avoids over-fitting problems possible with very short series. The strategies applied to the real biological and astronomical data (whooping cough in Great Britain and Wolf sunspot numbers) revealed their deterministic skeletons showing chaotic attractors. (C) 2001 Elsevier Science Ltd. All rights reserved.