Choosing Time Delays
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The embedding theorem [Man81, Tak81, CSY91] is silent on the choice of time delay to use in constructing d-dimensional data vectors. Indeed, it allows any time delay, except certain multiples of the precise period of a periodic signal [CSY91], as long as one has an infinite amount of infinitely accurate data. Since none of us will ever acquire an infinite amount of infinitely accurate data and still have time to compute anything else, this means one must turn to some principle other than the same geometric construction used in the unfolding of the attractor to establish a workable value for the time delay T. So that absent another theorem—and such a theorem is so far absent—we must find some prescription for choosing T. To put it another way, we have no hope of finding the very best or optimal time delay without giving some additional rationale for that optimality. Any such rationale will, of course, determine some T, but such optimality is tautological. At best it simply confirms the prescription one establishes for its realization.
KeywordsMutual Information Chaotic System Lorenz System Average Mutual Information Lorenz Attractor
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