scoringrules.crps_gpd

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

Compute the closed form of the CRPS for the generalised pareto distribution (GPD).

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

\[\mathrm{CRPS}(F_{M, \xi}, y) = |y| - \frac{2 (1 - M)}{1 - \xi} \left( 1 - (1 - F_{\xi}(y))^{1 - \xi} \right) + \frac{(1 - M)^{2}}{2 - \xi},\]

\[\mathrm{CRPS}(F_{M, \xi, \mu, \sigma}, y) = \sigma \mathrm{CRPS} \left( F_{M, \xi}, \frac{y - \mu}{\sigma} \right),\]

where \(F_{M, \xi, \mu, \sigma}\) is the GPD distribution function with shape parameter \(\xi < 1\), location parameter \(\mu\), scale parameter \(\sigma > 0\), and point mass \(M \in [0, 1]\) at the lower boundary. \(F_{M, \xi} = F_{M, \xi, 0, 1}\).

Parameters:

obsarray_like

The observed values.

shapearray_like

Shape parameter of the forecast GPD distribution.

locationarray_like

Location parameter of the forecast GPD distribution.

scalearray_like

Scale parameter of the forecast GPD distribution.

massarray_like

Mass parameter at the lower boundary of the forecast GPD 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 GPD(shape, location, scale, mass).

References

[1]

Jordan, A., Krüger, F., & Lerch, S. (2019). Evaluating Probabilistic Forecasts with scoringRules. Journal of Statistical Software, 90(12), 1-37. https://doi.org/10.18637/jss.v090.i12

Examples

>>> import scoringrules as sr
>>> sr.crps_gpd(0.3, 0.9)
0.6849331901197213