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+1. When the number of nearest neighbors arising through projection is zero in dimension d₄, the attractor has been unfolded in this dimension. The precise determination of d₄ is clouded by ``noise, '' and we examine the manner in which noise changes the determination of d₄. Our criterion also indicates the error one makes by choosing an embedding dimension smaller than d₄. This knowledge may be useful in the practical analysis of observed time series.