Skill

The accuracy of weather forecasts relative to the accuracy of forecasts based on a reference approach, such as random chance, climatology, or persistence.

Skill is often measured using a skill score (SS), which can be computed using the following general formula:

$$\mathrm {SS} _{x}={\frac {X_{\mathrm {test} }-X_{\mathrm {ref} }}{X_{\mathrm {perfect} }-X_{\mathrm {ref} }}}$$

where the subscript x refers to the accuracy measure for which a skill score is being computed, such as mean square error (MSE), X_test is the value of x for the test/experiment data, X_ref is the value of x for the reference forecast, and X_perfect is the value of x for a perfect forecast. A common usage of this general function is the Brier skill score (BSS), derived from the Brier score using the following derivation:

$$\mathrm {BSS} ={\frac {\mathrm {BS} -\mathrm {BS} _{\mathrm {climo} }}{\mathrm {BS} _{\mathrm {perfect} }-\mathrm {BS} _{\mathrm {climo} }}}=1-{\frac {\mathrm {BS} }{\mathrm {BS} _{\mathrm {climo} }}}$$

since BS_perfect = 0. A climatological value, such as the mean observation, is commonly used as the reference forecast (see Murphy 1988 for details).

Just about any accuracy measure can be converted from its formulaic definition or common use into a skill score. See Wilks (2011), section 8.1.4, for details.

References:

Wilks, D. S., 2011: Statistical Methods in the Atmospheric Sciences. Academic Press, 676 pp.

Murphy, A. H., 1988: Skill scores based on the mean square error and their relationships to the correlation coefficient. Mon. Wea. Rev., 116, 2417–2424, https://doi.org/10.1175/1520-0493(1988)116<2417:SSBOTM>2.0.CO;2.