# Self-affine scaling

Features that capture the properties of long-range correlations in time series.

*catch22* contains two features that capture the behaviour of two features based on fluctuation analysis, which aim to capture potential long-range correlations in time series, derived from the `SC_FluctAnal`

function in *hctsa*. Select one of the cards below to discover more information:

Here is a reference for understanding scaling methods: *Power spectrum and detrended fluctuation analysis: Application to daily temperatures* Talkner and Weber, *Phys. Rev. E* **62:** 150 (2000).

### Summary of the Method

The main steps of the method are as follows:

Compute a cumulative sum of the time series

Compute the level of fluctuation (e.g., root-mean-square deviations from local low-order trends) across windows corresponding to a given timescale. Different methods exist for detrending time-series windows at a given timescale, including (relevant to these two features):

Rescaled range analysis removes a line connecting the endpoints of each window and computes the range of the remaining points (Caccia et al., Physica A, 1997)

DFA fits a

*k*-order polynomial to each window and computes the residuals from this fit.

Looks for linear scaling in the log(timescale)–log(fluctuation) plot.

Some time series exhibit 'multifractal' scaling: there are different scaling rules at different ranges of timescales. The two *catch22* features fit two distinct scaling regimes and return the timescale at which the predicted change in scaling regime occurs.

Note that these features make quite strong assumptions about the data and can be unstable for time series that do not exhibit scaling, or exhibit strong but 'unifractal' scaling.

## 1. `rs_range`

`rs_range`

### What it does

uses rescaled range analysis described above to estimate points in logarithmic timescale–fluctuation space.

Time series (the best approximation to two distinct scaling regimes: the first scaling regime is indicated with red markers in the plots below) that change at a short timescale, like this AR(2) process, receive **low values** of this feature:

**Feature output: ****rs_range =**

**0.200**

**Feature output:**

**rs_range =**

**0.200**

## 2. `dfa`

`dfa`

### What it does

measures the same property as above, but from a timescale–fluctuation curve estimated using de-trended fluctuation analysis (DFA), using linear de-trending in each window, after down-sampling the time series by a factor of 2.

It displays qualitatively similar behaviour to`rs_range`

, e.g., with **low value **for this Duffing-van der Pol oscillator (which has a change in the scaling relationship at short timescales):

**Feature output: ****dfa =**

**0.204**

**Feature output:**

**dfa =**

**0.204**

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