scoringrules.crps_ensemble

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scoringrules.crps_ensemble#

scoringrules.crps_ensemble(obs: ArrayLike, fct: Array, m_axis: int = -1, *, ens_w: Array = None, estimator: str = 'qd', sorted_ensemble: bool = False, backend: Backend = None) → Array#

Estimate the Continuous Ranked Probability Score (CRPS) for a finite ensemble.

For a forecast \(F\) and observation \(y\), the CRPS is formally defined as:

\[\begin{split}\mathrm{CRPS}(F, y) &= \int_{-\infty}^{\infty} (F(x) - \mathbf{1}\{y \le x\})^{2} dx \\ &= \mathbb{E} | X - y | - \frac{1}{2} \mathbb{E} | X - X^{\prime} |,\end{split}\]

where \(X, X^{\prime} \sim F\) are independent. When \(F\) is the empirical distribution function of an ensemble forecast \(x_{1}, \dots, x_{M}\), the CRPS can be estimated in several ways. Currently, scoringrules supports several alternatives for estimator:

  • the energy estimator ("nrg"),

  • the fair estimator ("fair"),

  • the probability weighted moment estimator ("pwm"),

  • the approximate kernel representation estimator ("akr").

  • the AKR with circular permutation estimator ("akr_circperm").

  • the integral estimator ("int").

  • the quantile decomposition estimator ("qd").

Parameters:
obsarray_like

The observed values.

fctarray_like, shape (…, m)

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

m_axisint

The axis corresponding to the ensemble. Default is the last axis.

ens_warray, shape (…, m)

Weights assigned to the ensemble members. Array with the same shape as fct. Default is equal weighting. Weights are normalised so that they sum to one across the ensemble members.

estimatorstr

Indicates the CRPS estimator to be used.

sorted_ensemblebool

Boolean indicating whether the ensemble members are already in ascending order. Default is False.

backendstr, optional

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

Returns:
crpsarray_like

The CRPS between the forecast ensemble and obs.

See also

twcrps_ensemble, owcrps_ensemble, vrcrps_ensemble

Weighted variants of the CRPS.

crps_quantile

CRPS for quantile forecasts.

es_ensemble

The multivariate equivalent of the CRPS.

Notes

Ensemble forecasts

Information on the different estimators available for the CRPS.

References

[1]

Matheson JE, Winkler RL (1976). “Scoring rules for continuous probability distributions.” Management Science, 22(10), 1087-1096. doi:10.1287/mnsc.22.10.1087.

[2]

Gneiting T, Raftery AE (2007). “Strictly proper scoring rules, prediction, and estimation.” Journal of the American Statistical Association, 102(477), 359-378. doi:10.1198/016214506000001437.

Examples

>>> import numpy as np
>>> import scoringrules as sr
>>> rng = np.random.default_rng(123)
>>> obs = rng.normal(size=3)
>>> pred = rng.normal(size=(3, 10))
>>> sr.crps_ensemble(obs, pred)
array([0.69605316, 0.32865417, 0.39048665])
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