Using scaling-region distributions to select embedding parameters

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