scoringrules.gksmv_ensemble

scoringrules.gksmv_ensemble(obs: Array, fct: Array, m_axis: int = -2, v_axis: int = -1, *, ens_w: Array = None, estimator: str = 'nrg', backend: Backend = None) → Array

Compute the multivariate Gaussian Kernel Score (GKS) for a finite ensemble.

The GKS is the kernel score associated with the Gaussian kernel

\[k(x_{1}, x_{2}) = \exp \left(- \frac{ \| x_{1} - x_{2} \| ^{2}}{2} \right),\]

where :math:` | cdot |` is the euclidean norm.

Given an ensemble forecast \(F_{ens}\) comprised of multivariate members \(\mathbf{x}_{1}, \dots, \mathbf{x}_{M}\), the GKS is

\[\text{GKS}(F_{ens}, y)= - \frac{1}{M} \sum_{m=1}^{M} k(\mathbf{x}_{m}, \mathbf{y}) + \frac{1}{2 M^{2}} \sum_{m=1}^{M} \sum_{j=1}^{M} k(\mathbf{x}_{m}, \mathbf{x}_{j}) + \frac{1}{2}k(y, y)\]

If the fair estimator is to be used, then \(M^{2}\) in the second component of the right-hand-side is replaced with \(M(M - 1)\).

Parameters:

obsarray_like

The observed values, where the variables dimension is by default the last axis.

fctarray_like

The predicted forecast ensemble, where the ensemble dimension is by default represented by the second last axis and the variables dimension by the last axis.

m_axisint

The axis corresponding to the ensemble dimension on the forecasts array. Defaults to -2.

v_axisint

The axis corresponding to the variables dimension on the forecasts array (or the observations array with an extra dimension on m_axis). Defaults to -1.

ens_warray_like

Weights assigned to the ensemble members. Array with one less dimension than fct (without the v_axis dimension). Default is equal weighting. Weights are normalised so that they sum to one across the ensemble members.

estimatorstr

Indicates the estimator to be used.

backendstr

The name of the backend used for computations. Defaults to ‘numba’ if available, else ‘numpy’.

Returns:

scorearray_like

The GKS between the forecast ensemble and obs.