Direct test for determinism in a time series

A direct test for deterministic dynamics can be established by measurement of average directional vectors in a coarse-grained d-dimensional embedding of a time series. Theoretical analysis of the statistical properties of a random time series using the same embedding technique is possible by consideration of classical results concerning random walks in d dimensions. Examples are given to show the clear differences between deterministic dynamics, such as may be generated by chaotic systems, and stochastic dynamics.