scoringrules.crps_mixnorm

scoringrules.crps_mixnorm(obs: ArrayLike, m: ArrayLike, s: ArrayLike, w: ArrayLike = None, m_axis: ArrayLike = -1, *, backend: Backend = None) → ArrayLike

Compute the closed form of the CRPS for a mixture of normal distributions.

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

\[\mathrm{CRPS}(F, y) = \sum_{i=1}^{M} w_{i} A(y - \mu_{i}, \sigma_{i}^{2}) - \frac{1}{2} \sum_{i=1}^{M} \sum_{j=1}^{M} w_{i} w_{j} A(\mu_{i} - \mu_{j}, \sigma_{i}^{2} + \sigma_{j}^{2}),\]

where \(F(x) = \sum_{i=1}^{M} w_{i} \Phi \left( \frac{x - \mu_{i}}{\sigma_{i}} \right)\), and \(A(\mu, \sigma^{2}) = \mu (2 \Phi(\frac{\mu}{\sigma}) - 1) + 2\sigma \phi(\frac{\mu}{\sigma}).\)

Parameters:

obsarray_like

The observed values.

m: array_like

Means of the component normal distributions.

s: array_like

Standard deviations of the component normal distributions.

w: array_like

Non-negative weights assigned to each component.

m_axisint

The axis corresponding to the mixture components. Default is the last axis.

backendstr, optional

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

Returns:

crpsarray_like

The CRPS between MixNormal(m, s) and obs.

References

[1]

Grimit, E.P., Gneiting, T., Berrocal, V.J. and Johnson, N.A. (2006), The continuous ranked probability score for circular variables and its application to mesoscale forecast ensemble verification. Q.J.R. Meteorol. Soc., 132: 2925-2942. https://doi.org/10.1256/qj.05.235

Examples

>>> import scoringrules as sr
>>> sr.crps_mixnorm(0.0, [0.1, -0.3, 1.0], [0.4, 2.1, 0.7], [0.1, 0.2, 0.7])
0.46806866729387275