A generalisation of the signal-to-noise ratio using proper scoring rules

A generalised concept of the signal-to-noise ratio (or equivalently the ratio of predictable components, or RPC) is provided, based on proper scoring rules. This definition is the natural generalisation of the classical RPC, yet it allows one to define and analyse the signal-to-noise properties of any type of forecast that is amenable to scoring, thus drastically widening the applicability of these concepts. The methodology is illustrated for ensemble forecasts, scored using the continuous ranked probability score (CRPS), and for probability forecasts of a binary event, scored using the logarithmic score. Numerical examples are demonstrated using synthetic data with prescribed signal-to-noise ratios as well as seasonal ensemble hindcasts of the North Atlantic Oscillation (NAO) index. For the synthetic data, the RPC statistic as well as the scoring rule--based ones agree regarding which data sets exhibit anomalous signal-to-noise ratios, but exhibit different variance, indicating different statistical properties. For the NAO data, on the other hand, the results among the different statistics are more equivocal.