Reconstruction of the dynamics of noisy multivariate time series

We generalize the multi step prediction cost function for an unbiased reconstruction of the dynamics for noisy time series recently proposed for the case of scalar time series (which requires a reconstruction in delay space) to the case of multivariate time series data. 1 Given a time series with irregular time dependence the reconstruction of the underlying deterministic dynamics ooers probably the most charming possibility to proof that the data source is a nonlinear, deterministic process. The nonlinear modelling of data has been an extensive subject of research over the last years 1]. The most common situation found is the case of a given scalar time series as a result of measurements of a single variable, from which the state space has to be reconstructed by delay coordinates. This restriction to a scalar signal is mostly caused by experimental constraints. The so reconstructed phase space is equivalent to the original space as is guaranteed by the Takens' delay embedding theorem provided its dimension is large enough 2]. Modelling of a system in the reconstructed delay space faces certain problems. A large number of papers have been published on the question of the optimal delay time and embedding dimension for the modelling 4, 5]. Theoretically, all embeddings are equivalent, independent of the delay time or the embedding dimension as soon as it exceeds twice the fractal dimension of the attractor. But when dealing with a nite number of data points, this is no longer true. From a bad choice of the delay time severe problems for the reconstruction arise, and in extreme cases a state space reconstruction is of no value for practical considerations anymore. In the presence of noise the situation becomes even worse. There exists a principal limitation in state space reconstruction from time series in the presence of noise 3]. Since the diiculties are related to the reconstruction problems, it is generally preferable to use multivariate data, if available, rather than extracting the same information from a time delayed copy of a single measurement. Besides the reason that in many experiments only a single observable is recorded, the simplicity and intuitiveness of scalar embeddings is another reason for many people to use only one scalar observable, even if there are more components measured. We want to point out one huge drawback of time delay embedding, which occurs even in the case of optimal time delay and embedding dimension: …