scoringrules.crps_2pexponential

scoringrules.crps_2pexponential(obs: ArrayLike, scale1: ArrayLike, scale2: ArrayLike, location: ArrayLike, *, backend: Backend = None) → ArrayLike

Compute the closed form of the CRPS for the two-piece exponential distribution.

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

\[\mathrm{CRPS}(F_{\sigma_{1}, \sigma_{2}, \mu}, y) = |y - \mu| + \frac{2\sigma_{\pm}^{2}}{\sigma_{1} + \sigma_{2}} \exp \left( - \frac{|y - \mu|}{\sigma_{\pm}} \right) - \frac{2\sigma_{\pm}^{2}}{\sigma_{1} + \sigma_{2}} + \frac{\sigma_{1}^{3} + \sigma_{2}^{3}}{2(\sigma_{1} + \sigma_{2})^2},\]

where \(F_{\sigma_{1}, \sigma_{2}, \mu}\) is the two-piece exponential distribution function with scale parameters \(\sigma_{1}, \sigma_{2} > 0\) and location parameter \(\mu\). The parameter \(\sigma_{\pm}\) is equal to \(\sigma_{1}\) if \(y < 0\) and \(\sigma_{2}\) if \(y \geq 0\).

Parameters:

obsarray_like

The observed values.

scale1array_like

First scale parameter of the forecast two-piece exponential distribution.

scale2array_like

Second scale parameter of the forecast two-piece exponential distribution.

locationarray_like

Location parameter of the forecast two-piece exponential distribution.

backendstr, optional

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

Returns:

crpsarray_like

The CRPS between 2pExp(sigma1, sigma2, location) and obs.

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_2pexponential(0.8, 3.0, 1.4, 0.0)
array(1.18038524)