scoringrules.crps_gtct

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scoringrules.crps_gtct#

scoringrules.crps_gtct(obs: ArrayLike, df: ArrayLike, location: ArrayLike = 0.0, scale: ArrayLike = 1.0, lower: ArrayLike = -inf, upper: ArrayLike = inf, lmass: ArrayLike = 0.0, umass: ArrayLike = 0.0, *, backend: Backend = None) → ArrayLike#

Compute the closed form of the CRPS for the generalised truncated and censored t distribution.

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

\[\mathrm{CRPS}(F_{l, L, \nu}^{u, U}, y) = |y - z| + uU^{2} - lL^{2} + \left( \frac{1 - L - U}{F_{\nu}(u) - F_{\nu}(l)} \right) z \left( 2 F_{\nu}(z) - \frac{(1 - 2L) F_{\nu}(u) + (1 - 2U) F_{\nu}(l)}{1 - L - U} \right) - \left( \frac{1 - L - U}{F_{\nu}(u) - F_{\nu}(l)} \right) \left( 2 G_{\nu}(z) - 2 G_{\nu}(u)U - 2 G_{\nu}(l)L \right) - \left( \frac{1 - L - U}{F_{\nu}(u) - F_{\nu}(l)} \right)^{2} \bar{B}_{\nu} \left( H_{\nu}(u) - H_{\nu}(l) \right),\]
\[\mathrm{CRPS}(F_{l, L, \nu, \mu, \sigma}^{u, U}, y) = \sigma \mathrm{CRPS}(F_{(l - \mu)/\sigma, L, \nu}^{(u - \mu)/\sigma, U}, \frac{y - \mu}{\sigma}),\]
\[G_{\nu}(x) = - \left( \frac{\nu + x^{2}}{\nu - 1} \right) f_{\nu}(x),\]
\[H_{\nu}(x) = \frac{1}{2} + \frac{1}{2} \mathrm{sgn}(x) I \left( \frac{1}{2}, \nu - \frac{1}{2}, \frac{x^{2}}{\nu + x^{2}} \right),\]
\[\bar{B}_{\nu} = \left( \frac{2 \sqrt{\nu}}{\nu - 1} \right) \frac{B(\frac{1}{2}, \nu - \frac{1}{2})}{B(\frac{1}{2}, \frac{\nu}{2})^{2}},\]

where \(F_{\nu}\) is the CDF of the standard t distribution with \(\nu > 1\) degrees of freedom, distribution, \(F_{l, L, \nu, \mu, \sigma}^{u, U}\) is the CDF of the t distribution truncated below at \(l\) and above at \(u\), with point masses \(L, U > 0\) at the lower and upper boundaries, respectively, and degrees of freedom, location and scale parameters \(\nu > 1\), \(\mu\) and \(\sigma > 0\). \(F_{l, L, \nu}^{u, U} = F_{l, L, \nu, 0, 1}^{u, U}\).

Parameters:
obsarray_like

The observed values.

dfarray_like

Degrees of freedom parameter of the forecast distribution.

locationarray_like

Location parameter of the forecast distribution.

scalearray_like

Scale parameter of the forecast distribution.

lowerarray_like

Lower boundary of the truncated forecast distribution.

upperarray_like

Upper boundary of the truncated forecast distribution.

lmassarray_like

Point mass assigned to the lower boundary of the forecast distribution.

umassarray_like

Point mass assigned to the upper boundary of the forecast distribution.

Returns:
crpsarray_like

The CRPS between gtct(df, location, scale, lower, upper, lmass, umass) 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_gtct(0.0, 2.0, 0.1, 0.4, -1.0, 1.0, 0.1, 0.1)
0.13997789333289662
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