Linear autocorrelation structure

Features quantifying linear autocorrelation structure (from the autocorrelation function or power spectrum)

CO_f1ecac (first1e_acf_tau)

NB: The name CO_f1ecac derives from an earlier version of hctsa (the current version of hctsa names this feature as first1e_acf_tau).

What it does

The first1e_acf_tau feature in catch22 computes the first 1/e crossing of the autocorrelation function of the time series. In hctsa, this can be computed as CO_FirstCrossing(x_z,'ac',1/exp(1),'discrete').
This feature measures the first time lag at which the autocorrelation function drops below 1/e (= 0.3679).

What it measures

first1e_acf_tau captures the approximate scale of autocorrelation in a time series. This can be thought of as the number of steps into the future at which a value of the time series at the current point and that future point remain substantially (>1/e) correlated. For a continuous-time system, this statistic is high when the sampling rate is high relative to the timescale of the dynamics.
  • For uncorrelated noise, like the Poisson-distributed series shown below, the autocorrelation function drops to ~0 immediately, and we obtain the minimum value of this statistic: first1e_acf_tau = 1.
  • For processes with a greater level of autocorrelation, the autocorrelation function decays more slowly, and we can obtain a larger value of this feature. Take this series simulated from a Chirikov map, which has first1e_acf_tau = 6
  • We obtain even larger values for even more slowly varying time series, like this ODE, measured at a very high sampling rate, yielding first1e_acf_tau = 17
  • Financial series (and many non-stationary stochastic processes) are highly autocorrelated, like this series for which first1e_acf_tau = 176

CO_FirstMin_ac (firstMin_acf)

This feature is named CO_FirstMin_ac in catch22 and matches the hctsa feature named firstMin_acf.
Similar to the 1/e crossing feature above, firstMin_acf computes the first minimum of the autocorrelation function. It exhibits similar behavior.


This feature returns the first peak in the autocorrelation function satisfying a set of conditions (after detrending the time series using a single-knot cubic regression spline).
It is based on a method by Wang et al. (2007) (described in their paper: "Structure-based Statistical Features and Multivariate Time Series Clustering" Link).
Broadly, it gives high values to slowly-varying time series like this slow (on the timescale of
Δt\Delta t
) oscillator (feature value of 62):
And lower values to this fast (on the timescale of
Δt\Delta t
) map (the Gingerbread map) (feature value = 4):


This feature computes the relative power in the lowest 20% of frequencies (relative to the sampling rate of the data) [the output area_5_1 from the hctsa code SP_Summaries(x_z,'welch','rect',[],false)].
It gives high values to time series with lots of power in low frequencies, and low values to time series that have most of their power in higher frequencies.
The area under the power spectrum is estimated in linear space, where the power spectral density is estimated using Welch's method (with a rectangular window).
Here's an example of a slow-varying stocahstic process with a very high value for this feature, 0.987, reflecting 98.7% of power is this low-frequency band (relevant portion of the power spectrum shaded red below):
This Lozi map has a low value of 0.03 (3% of power is in the red low-frequency band):


Like the previous feature, this one is also extracted from the power spectrum (estimated using a Welch's method with a rectangular window). But this time, it returns the frequency,
, at which the amount of power in frequencies low and higher than
is the same: the "centroid".
It gives high values to time series that have their power in high frequencies, like this audio of an animal sound (centroid point shown with a red circle), 2.82:
And it gives low values to slower-varying time series like this snippet of an electrocardiogram recording from a patient with congestive heart failure. 0.15: