Combining single-value streamflow forecasts – A review and guidelines for selecting techniques

Choosing an appropriate method for combining single-value forecasts should depend on characteristics of the individual forecasts being combined and their relationships with each other. This study attempts to develop a guideline to choose effective combining techniques by using analytical derivations and/or hydrological experiments. The two most popular combining techniques, Simple Average (SA) and Weighted Average (WA), are compared from theoretical angles. The standard deviation of the combined forecast error is quantified as a function of the ratio of the standard deviation and the correlation coefficient between the two constituent forecast errors. Following the theoretical study, empirical research for eight combining methods including SA and WA methods was conducted to confirm the theoretical findings of this study and to verify results from other research carried out. The results of the empirical experiments are summarized to confirm the effects of the eight combining methods. The major findings include that: (1) SA yields reasonable results for any combination of forecasts when information of constituent forecasts is absent, (2) one cannot expect combining technique to yield significant improvement when two constituent forecasts are highly correlated, (3) the Regression and ANN combining methods can remove the effects of bias in the constituent forecasts and yield unbiased combining forecasts, and (4) when the constituent forecasts have nonstationary errors, a time-varying-weight combining method yields better results than the constant-weight methods in most cases. Based on these theoretical findings and empirical results, a guideline for combining methods is provided. The guideline suggests appropriate methods for combining single-value streamflow forecasts by considering bias and nonstationarity of the errors in the individual forecasts; the ratio of the error variance of any two forecasts and cross-correlation among the forecasts.