Towards automated extraction and characterization of scaling regions in dynamical systems

TL;DR

This paper introduces an automated method for identifying and characterizing scaling regions in dynamical systems, reducing subjectivity and improving accuracy in calculations like correlation dimension and Lyapunov exponents.

Contribution

The authors develop a novel ensemble-based approach to automatically detect and estimate the extent of scaling regions in dynamical system plots, enhancing objectivity and robustness.

Findings

Effective in identifying scaling regions in various dynamical systems

Reduces subjectivity in parameter selection for analysis

Provides reliable estimates of scaling region slopes and extents

Abstract

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.