Skill Scores Based on the Mean Square Error and Their Relationships to the Correlation Coefficient

Several skill scores are defined, based on the mean-square-error measure of accuracy and alternative climatological standards of reference. Decompositions of these skill scores are formulated, each of which is shown to possess terms involving 1) the coefficient of correlation between the forecasts and observations, 2) a measure of the nonsystematic (i.e., conditional) bias in the forecast, and 3) a measure of the systematic (i.e., unconditional) bias in the forecasts. Depending on the choice of standard of reference, a particular decomposition may also contain terms relating to the degree of association between the reference forecasts and the observations. These decompositions yield analytical relationships between the respective skill scores and the correlation coefficient, document fundamental deficiencies in the correlation coefficient as a measure of performance, and provide additional insight into basic characteristics of forecasting performance. Samples of operational precipitation probability and minimum temperature forecasts are used to investigate the typical magnitudes of the terms in the decompositions. Some implications of the results for the practice of forecast verification are discussed.