A goodness of fit measure related to r² for model performance assessment

Abstract

Checking the predictive worth of an environmental model inevitably includes a goodness of fit metric to quantify the degree of matching to recorded data, thereby giving a measure of model performance. Considerable analysis and discussion have taken place over fit indices in hydrology, but a neglected aspect is the degree of communicability to other disciplines. It is suggested that a fit index is best communicated to colleagues via reference to models giving unbiased predictions, because unbiased environmental models are a desirable goal across disciplines. That is, broad recognition of a fit index is aided if it simplifies in the unbiased case to a familiar and logical expression. This does not hold for the Nash–Sutcliffe Efficiency E which reduces to the somewhat awkward unbiased expression E = 2 – 1/r², where r² is the coefficient of determination. A new goodness of fit index V is proposed for model validation as V = r²/(2-E), which simplifies to the easily communicated V = r4 in the unbiased case. The index is defined over the range 0 ≤ V ≤ 1, and it happens that V < E for larger values of E. Some synthetic and recorded data sets are used to illustrate characteristics of V in comparison to E