An adaptive sparse grid approach for time series predictions

Abstract

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 &#8477;^<em>d</em> and the states of the underlying, unknown dynamical system that determines the time series. The embedded data points are located on an <em>m</em>-dimensional manifold (or even fractal) called attractor of the time series. Takens’ theorem then states that an upper bound for the embedding dimension <em>d</em> can be given by <em>d</em> ≤ 2<em>m</em> + 1.<br /> The task of predicting future values thus becomes, together with an estimate on the manifold dimension m, a scattered data regression problem in <em>d</em> 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.<br /> 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.