Error Spread Score

The error spread score (Christensen et al., 2015) is given by:

\[ESS = \left(s^2 - e^2 - e \cdot s \cdot g\right)^2\]

where the mean \(m\), variance \(s^2\), and skewness \(g\) of the ensemble forecast of size \(F\) are computed as follows:

\[m = \frac{1}{F} \sum_{f=1}^{F} X_f, \quad s^2 = \frac{1}{F-1} \sum_{f=1}^{F} (X_f - m)^2, \quad g = \frac{F}{(F-1)(F-2)} \sum_{f=1}^{F} \left(\frac{X_f - m}{s}\right)^3\]

The error in the ensemble mean \(e\) is calculated as \(e = m - y\), where \(y\) is the observed value.

scoringrules.error_spread_score

error_spread_score(
    observations: ArrayLike,
    forecasts: Array,
    /,
    axis: int = -1,
    *,
    backend: Backend = None,
) -> Array

Compute the error-spread score (Christensen et al., 2015) for a finite ensemble.

Parameters:

Name	Type	Description	Default
observations	ArrayLike	The observed values.	required
forecasts	Array	The predicted forecast ensemble, where the ensemble dimension is by default represented by the last axis.	required
axis	int	The axis corresponding to the ensemble. Default is the last axis.	-1
backend	Backend	The name of the backend used for computations. Defaults to 'numba' if available, else 'numpy'.	None

Returns:

Type	Description
-Array	An array of error spread scores for each ensemble forecast, which should be averaged to get meaningful values.