Naming info: This feature matches the hctsa feature called FC_LocalSimple_mean3_stderr. It computes the stderr output from running the code FC_LocalSimple(x_z,'mean',3) in hctsa.
forecast_error returns the a measure of error from using the mean of the previous 3 values of the time series to predict the next value. The error statistic used here is the standard deviation of the residuals from the full set of simple 1-step forecasts. Because the input time series is z-scored (standard deviation of 1), the residuals should have a standard deviation less than 1 if this forecasting method is doing something (at least minimally) useful.
Time series that are easy to predict (i.e., time series for which the mean of the 3 previous time steps are a good prediction of the current value) will get low values of this feature.
Here's an example of a predator-prey system (black) for which the dynamics are varying on a timescale of 3 steps, such that mean-3 predictions (blue) are very poor forecasts. For this time series, the residuals have a higher variance than the original time series.
This ODE system, by contrast is very highly sampled, such that the time series is a similar value in 3 steps, and the forecasts (blue) are very close to the original time series, yielding residuals with a very low standard deviation (0.03).