Reconstructing state-space dynamics from scalar data using time-delay
embedding requires choosing values for the delay Ï„ and the dimension m.
Both parameters are critical to the success of the procedure and neither is
easy to formally validate. While embedding theorems do offer formal guidance
for these choices, in practice one has to resort to heuristics, such as the
average mutual information (AMI) method of Fraser & Swinney for Ï„ or the
false near neighbor (FNN) method of Kennel et al. for m. Best practice
suggests an iterative approach: one of these heuristics is used to make a good
first guess for the corresponding free parameter and then an "asymptotic
invariant" approach is then used to firm up its value by, e.g., computing the
correlation dimension or Lyapunov exponent for a range of values and looking
for convergence. This process can be subjective, as these computations often
involve finding, and fitting a line to, a scaling region in a plot: a process
that is generally done by eye and is not immune to confirmation bias. Moreover,
most of these heuristics do not provide confidence intervals, making it
difficult to say what "convergence" is. Here, we propose an approach that
automates the first step, removing the subjectivity, and formalizes the second,
offering a statistical test for convergence. Our approach rests upon a recently
developed method for automated scaling-region selection that includes
confidence intervals on the results. We demonstrate this methodology by
selecting values for the embedding dimension for several real and simulated
dynamical systems. We compare these results to those produced by FNN and
validate them against known results -- e.g., of the correlation dimension --
where these are available. We note that this method extends to any free
parameter in the theory or practice of delay reconstruction