  Scaling regions -- intervals on a graph where the dependent variable depends
linearly on the independent variable -- abound in dynamical systems, notably in
calculations of invariants like the correlation dimension or a Lyapunov
exponent. In these applications, scaling regions are generally selected by
hand, a process that is subjective and often challenging due to noise,
algorithmic effects, and confirmation bias. In this paper, we propose an
automated technique for extracting and characterizing such regions. Starting
with a two-dimensional plot -- e.g., the values of the correlation integral,
calculated using the Grassberger-Procaccia algorithm over a range of scales --
we create an ensemble of intervals by considering all possible combinations of
endpoints, generating a distribution of slopes from least-squares fits weighted
by the length of the fitting line and the inverse square of the fit error. The
mode of this distribution gives an estimate of the slope of the scaling region
(if it exists). The endpoints of the intervals that correspond to the mode
provide an estimate for the extent of that region. When there is no scaling
region, the distributions will be wide and the resulting error estimates for
the slope will be large. We demonstrate this method for computations of
dimension and Lyapunov exponent for several dynamical systems, and show that it
can be useful in selecting values for the parameters in time-delay
reconstructions.
