CRPS decomposition

[sbfnk]

For the WIS we have a nice decomposition into overprediction, underprediction and dispersion. We could do the same for CRPS, either via the WIS with all quantile levels between 0 and 1 or, for samples via the median forecast $m$ (thanks @jbracher):

• dispersion as $D=\mathrm{CRPS}(F, m)$
• overprediction as $O=0$ if $y\geq m$ and $O=\mathrm{CRPS}(F, y) - D$ if $y < m$
• underprediction as $U=0$ if $y\leq m$ and $U=\mathrm{CRPS}(F, y) - D$ if $y > m$

[seabbs]

Great idea. Doing it via the median seems quite appealing.