scoringrules.crps_gev

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scoringrules.crps_gev#

scoringrules.crps_gev(obs: ArrayLike, shape: ArrayLike, location: ArrayLike = 0.0, scale: ArrayLike = 1.0, *, backend: Backend = None) → ArrayLike#

Compute the closed form of the CRPS for the generalised extreme value (GEV) distribution.

It is based on the following formulation from [1]:

\[\text{CRPS}(F_{\xi, \mu, \sigma}, y) = \sigma \cdot \text{CRPS}(F_{\xi}, \frac{y - \mu}{\sigma})\]

see Notes below for special cases.

Parameters:
obsarray_like

The observed values.

shapearray_like

Shape parameter of the forecast GEV distribution.

locationarray_like, optional

Location parameter of the forecast GEV distribution.

scalearray_like, optional

Scale parameter of the forecast GEV distribution.

backendstr, optional

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

Returns:
crpsarray_like

The CRPS between obs and GEV(shape, location, scale).

Notes

Special cases are handled as follows:

  • For \(\xi = 0\):

\[\text{CRPS}(F_{\xi}, y) = -y - 2\text{Ei}(\log F_{\xi}(y)) + C - \log 2\]

  • For \(\xi \neq 0\):

\[\text{CRPS}(F_{\xi}, y) = y(2F_{\xi}(y) - 1) - 2G_{\xi}(y) - \frac{1 - (2 - 2^{\xi}) \Gamma(1 - \xi)}{\xi}\]

where \(C\) is the Euler-Mascheroni constant, \(\text{Ei}\) is the exponential integral, and \(\Gamma\) is the gamma function. The GEV cumulative distribution function \(F_{\xi}\) and the auxiliary function \(G_{\xi}\) are defined as:

  • For \(\xi = 0\):

\[F_{\xi}(x) = \exp(-\exp(-x))\]

  • For \(\xi \neq 0\):

\[\begin{split}F_{\xi}(x) = \begin{cases} 0, & x \leq \frac{1}{\xi} \\ \exp(-(1 + \xi x)^{-1/\xi}), & x > \frac{1}{\xi} \end{cases}\end{split}\]
\[\begin{split}G_{\xi}(x) = \begin{cases} 0, & x \leq \frac{1}{\xi} \\ \frac{F_{\xi}(x)}{\xi} + \frac{\Gamma_u(1-\xi, -\log F_{\xi}(x))}{\xi}, & x > \frac{1}{\xi} \end{cases}\end{split}\]

References

[1]

Friederichs, P., & Thorarinsdottir, T. L. (2012). A comparison of parametric and non-parametric methods for forecasting extreme events. Environmetrics, 23(7), 595-611. https://doi.org/10.1002/env.2176

Examples

>>> import scoringrules as sr
>>> sr.crps_gev(0.3, 0.1)
0.2924712413052034
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