Reconstruction expansion as a geometry-based framework for choosing proper delay times

The quality of attractor reconstruction using the method of delays is known to be sensitive to the delay parameter, τ. Here we develop a new, computationally efficient approach to choosing τ that quantifies reconstruction expansion from the identity line of the embedding space. We show that reconstruction expansion is related to the concept of reconstruction signal strength and that increased expansion corresponds to diminished effects of measurement error. Thus, reconstruction expansion represents a simple, geometrical framework for choosing τ. Furthermore, we describe the role of dynamical error in attractor expansion and argue that algorithms for determining τ should be considered as attempts at estimating an upper bound to the optimal delay.