😎RAD

The Rescaled Auto-Density (RAD) is a noise-robust metric for inferring the distance to criticality (the DTC). It aims to perform well in settings where the noise level varies between time series.

📕 Original Paper

📙 B. Harris, L. Gollo, B.D. Fulcher. "Tracking the distance to criticality in systems with unknown noise", Physical Review X 14: 031021 (2024). https://doi.org/10.1103/PhysRevX.14.031021

❓ What is RAD?

In the original paper, we use a data-driven approach to develop theory on how to infer the DTC in the variable-noise setting. This theory relies on two key time-series properties: i) the distribution of values; and ii) the scale of fast fluctuations. Combining these two properties, such as by curve-fitting the distribution and solving the Fokker--Planck equation, partials-out the noise strength to give a noise-robust estimate of the shape of the potential function (and hence the control parameter). RAD implements these key algorithmic steps in a simplified way, efficiently estimating the DTC from data using elementary time-series operations. In RAD, the two algorithmic elements are captured by standard deviations above and below the median value (measuring the distribution) and the standard deviation of lag-1 differences (measuring fast fluctuations). Our RAD feature is given by:

$$f_\textrm{RAD} = \sigma(\Delta \mathbf{x}) \left[\frac{1}{\sigma(U)} - \frac{1}{\sigma(L)} \right]\,,$$

where $\mathbf{x}$ is the input time series, $\Delta$is the lag-1 difference operator, $\sigma(\cdot)$ is the standard deviation, and:

$$U = \{x_t: x_t \geq \tilde{x}\}\,,\\L = \{x_t: x_t < \tilde{x}\}\,,$$

where $\tilde{x}$ is the median value of $\mathbf{x}$.

🏃 Running RAD in Matlab, Julia, and Python

Code for implementing RAD is available in Matlab, Julia, and Python is collected in this RAD Github repository.

RAD is also available through existing toolboxes in both Matlab and Julia:

• Matlab: in hctsa (as of v1.08), as the CR_RAD()master function, and CR_RAD_1, CR_RAD_2, and CR_RAD_tau operations

• Julia: in the TimeseriesFeatures.jl package (StatsBase.jl extension), as the CR_RAD::Feature

⚠️ Usage Guide

RAD is best suited to univariate, regularly-sampled time-series data. The sampling period should be constant between time series, and if the time series are excessively smooth or over-sampled then the tau parameter should be set to a value greater than 1 (a suitable tau can often be inferred by inspecting the autocorrelation function of a time series).

RAD also includes an optional centering step (doAbs), which should be enabled if the time series do not represent radial values (e.g., enable doAbs if the distribution is approximately symmetric or 'two-sided').