Description:
A real valued, deterministic and stationary time series can be embedded in a sometimes high-dimensional real vector space. This leads to a one-to-one relationship between the embedded, time dependent vectors in Rd and the states of the underlying, unknown dynamical system that determines the time series. The embedded data points are located on an m-dimensional manifold (or even fractal) called attractor of the time series. Takens' theorem then states that an upper bound for the embedding dimension d can be given by d ≤ 2m + 1. The task of predicting future values thus becomes, together with an estimate on the manifold dimension m, a scattered data regression problem in d dimensions. In contrast to most of the common regression algorithms like support vector machines (SVMs) or neural networks, which follow a data-based approach, we employ in this paper a sparse grid-based discretization technique. This allows us to efficiently handle huge amounts of training data in moderate dimensions. Extensions of the basic method lead to space- and dimension-adaptive sparse grid algorithms. They become useful if the attractor is only located in a small part of the embedding space or if its dimension was chosen too large. We discuss the basic features of our sparse grid prediction method and give the results of numerical experiments for time series with both, synthetic data and real life data.