scoringrules.crps_csg0

scoringrules.crps_csg0(obs: ArrayLike, shape: ArrayLike, rate: ArrayLike | None = None, *, scale: ArrayLike | None = None, shift: ArrayLike = 0.0, backend: Backend = None) → ArrayLike

Compute the closed form of the CRPS for the censored, shifted gamma distribution.

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

\[\mathrm{CRPS}\bigl(F^0_{\alpha,\beta,\delta},\,y\bigr) = (y+\delta)\,\bigl(2F_{\alpha,\beta}(y+\delta)-1\bigr) - \frac{\alpha}{\beta\pi}\,B\!\Bigl(\tfrac12,\alpha+\tfrac12\Bigr)\,\bigl(1 - F_{2\alpha,\beta}(2\delta)\bigr) + \frac{\alpha}{\beta}\,\Bigl(1 + 2\,F_{\alpha,\beta}(\delta)\,F_{\alpha+1,\beta}(\delta) - F_{\alpha,\beta}(\delta)^2 - 2\,F_{\alpha+1,\beta}(y+\delta)\Bigr) - \delta\,\bigl(F_{\alpha,\beta}(\delta)\bigr)^{2},\]

where \(F^0_{\alpha,\beta,\delta}\) is the censored, shifted gamma distribution function with shape parameter \(\alpha > 0\), rate parameter \(\beta > 0\) (equivalently, with scale parameter \(1/\beta\)) and shift parameter \(\delta > 0\).

Parameters:

obsarray_like

The observed values.

shapearray_like

Shape parameter of the forecast CSG distribution.

ratearray_like, optional

Rate parameter of the forecast CSG distribution. Either rate or scale must be provided.

scalearray_like, optional

Scale parameter of the forecast CSG distribution, where scale = 1 / rate. Either rate or scale must be provided.

shiftarray_like

Shift parameter of the forecast CSG 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 CSG(shape, rate, shift).

Raises:

ValueError

If both rate and scale are provided, or if neither is provided.

References

[1]

Scheuerer, M., & Hamill, T. M. (2015). Statistical postprocessing of ensemble precipitation forecasts by fitting censored, shifted gamma distributions. Monthly Weather Review, 143(11), 4578-4596. https://doi.org/10.1175/MWR-D-15-0061.1

Examples

>>> import scoringrules as sr
>>> sr.crps_csg0(0.7, shape=0.5, rate=2.0, shift=0.3)
0.5411044348806484