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# 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](https://github.com/DynamicsAndNeuralSystems/RAD).

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').


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