# Linear Autocorrelation Function

These features capture properties of the linear autocorrelation function

**NB**: The name

`CO_f1ecac`

derives from an earlier version of *hctsa*(the current version of

*hctsa*names this feature as

`first1e_acf_tau`

).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).`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`

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.Last modified 10mo ago