Determining embedding dimension for phase-space reconstruction using a geometrical construction

We examine the issue of determining an acceptable minimum embedding dimension by looking at the behavior of near neighbors under changes in the embedding dimension from d{→}d+1. When the number of nearest neighbors arising through projection is zero in dimension dE, the attractor has been unfolded in this dimension. The precise determination of dE is clouded by ``noise,'' and we examine the manner in which noise changes the determination of dE. Our criterion also indicates the error one makes by choosing an embedding dimension smaller than dE. This knowledge may be useful in the practical analysis of observed time series.