Improved estimators of correlation and R2 for skewed hydrologic data

The coefficient of determination R-2 and Pearson correlation coefficient rho = R are standard metrics in hydrology for the evaluation of the goodness of fit between model simulations and observations, and as measures of the degree of dependence of one variable upon another. We show that the standard product moment estimator of rho, termed r, while well-behaved for bivariate normal data, is upward biased and highly variable for bivariate non-normal data. We introduce three alternative estimators of rho which are nearly unbiased and exhibit much less variability than r for non-normal data. We also document remarkable upward bias and tremendous increases in variability associated with r using both synthetic data and daily streamflow simulations from 905 calibrated rainfall-runoff models. We show that estimators of rho = R accounting for skewness are needed for daily streamflow series because they exhibit high variability and skewness compared to, for example, monthly/annual series, where r should perform well.