Singular-value decomposition and embedding dimension

Data from dynamical experiments are often studied with use of results due to Shaw et al. and to Takens, which generate points in a space of relatively high dimension by embedding measurements which are typically one dimensional. A number of questions arise from this, the most obvious being how should one choose the dimension of the embedding space. In this paper we show that a method which seems promising at first sight, estimating the rank of the matrix of embedded data, is unfortunately not useful in general. Previous encouraging results have almost certainly been due to numerical problems which can, in part, be avoided by a careful application of singular-value decomposition. We show that this process does not give useful dynamical information, though it is often useful in noise control.