Detailed information on the Mean Square Skill Score (MSSS) is available in attachment II.8. It is recommended users read the documentation in the attachment for more details.
The Mean Square Skill Score (MSSS) is applicable to deterministic forecasts only.
The MSSS is essentially the Mean Square Error (MSE) of the forecasts compared to the MSE of climatology for a station or grid point.
The MSE for a forecast at a grid point (or station) may be given by:
where x and f denote time series of observations and continuous deterministic forecasts.
The MSE for climatology is given by:
The Mean Square Skill Score is therefore given as:
For the domains (i.e., Tropics and northern and southern Extra-Tropics) over which the MSSS is calculated, it is recommended that an overall MSSS is provided. This is computed as:
where the weighting function, w, is unity for verifications at stations and for gridded data is equal to cos(Θ), where Θ is the latitude of each corresponding gridpoint.
MSSSj for forecasts fully cross validated can be expanded as:
where rfxj is the product moment correlation of the forecasts and observations at a point or station j.
The first three terms of the decomposition of MSSSj are related to phase errors (through the correlation), amplitude errors (through the ratio of the forecast to observed variances) and overall bias error respectively of the forecasts (see below). These three terms and MSSSj are a requirement of Level 2 and software is provided to calculate these.
An example of the MSSS score for Level 2 is shown below:
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