A correlation function for choosing time delays in phase portrait reconstructions

A new method is proposed for the selection of the time delay used in creating phase portrait reconstructions of dynamical systems. Using the singular values computed from a singular value decomposition (SVD) of the embedded flow, we compute a quantity which we term the singular value fraction (SVF) as a function of the chosen lag time Δ, and denoted as f_sv (Δ). For a given value of k, it gives the fractional power in the first k singular values. We seek to choose Δ so that folding of the attractor is minimised, but the attractor volume is maximised, and we thus choose Δ to be at the first minimum of f_sv. Equivalently we use a window of size Δ_w, where Δ_w is apprximately the first major maximum of f_sv. It is shown that this method compares favourably with some other extant methods.