## The Priestley Forecast Skill Score

 E. Linacre 1/'99

A convenient measure of the quality of a forecast of daily maximum temperature, for instance, is the Priestley Skill Score (PRSK), which involves a comparison of the variance of such temperatures at this time of year (V), with the mean square of the error in forecasting the temperature (E), thus

PRSK = 1 - E / V

The variance V is the arithmetic mean of the squared deviations from the average value. It is calculated as follows for this set of measured daily maximum temperatures Tx in ºC -

 Day 1 2 3 4 5 6 7 8 9 mean Tx 24 25 22 26 23 20 23 26 27 24, i.e. Tm Tx - Tm 0 1 -2 2 -1 -4 -1 2 3 - (ditto)2 0 1 4 4 1 16 1 4 9 4.4, i.e. V forecast 25 25 26 29 24 20 22 25 26 - errror 1 0 4 3 1 0 -1 -1 -1 - (ditto)2 1 0 16 9 1 0 1 1 1 3.3, i.e. E

This set of forecasts shows a forecasting skill of (1 - 3.3/4.4) i.e. 0.25. That positive number implies that the forecasts are better than chance estimates. However, the degree of skill is modest. If it were perfect, E would be zero and the PRSK would be unity.

It is also interesting to calculate the skill achieved by two other methods of forecasting. First, one might assume that the value for each day was always the average value (i.e. the ‘climatological forecast’). Then E by definition would equal V, and thus the skill works out as zero. Alternatively, the ‘persistence forecast’ involves assuming that tomorrow’s value will be the same as today’s proved to be. Therefore, the table above becomes the following -

 day 1 2 3 4 5 6 7 8 9 mean Tx 24 25 22 26 23 20 23 26 27 24, i.e. Tm Tx - Tm 0 1 -2 2 -1 -4 -1 2 3 - (ditto)2 0 1 4 4 1 16 1 4 9 4.4, i.e. V forecast - 24 25 22 26 23 20 23 26 - error - -1 3 -4 3 3 -3 -3 -1 - (ditto)2 1 1 9 16 9 9 9 9 1 8.0, i.e. E

In this case, the skill works out as minus 0.8, which is appalling. It would have been better to use a climatological forecast. However, this is not always the case.

 In summary, PRSK = 1 perfect forecast PRSK > 0 better than chance (has skill) PRSK = 0 climatology PRSK < 1 worse than climatology