Linear Autocorrelation Function

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.

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CO_f1ecac (first1e_acf_tau)

What it does

What it measures

CO_FirstMin_ac (firstMin_acf)