# Linear Autocorrelation Function

These features capture properties of the 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.