Appraisal of the generalized likelihood uncertainty estimation (GLUE) method

[1] Recent research documents that the widely accepted generalized likelihood uncertainty estimation (GLUE) method for describing forecasting precision and the impact of parameter uncertainty in rainfall/runoff watershed models fails to achieve the intended purpose when used with an informal likelihood measure. In particular, GLUE generally fails to produce intervals that capture the precision of estimated parameters, and the difference between predictions and future observations. This paper illustrates these problems with GLUE using a simple linear rainfall/runoff model so that model calibration is a linear regression problem for which exact expressions for prediction precision and parameter uncertainty are well known and understood. The simple regression example enables us to clearly and simply illustrate GLUE deficiencies. Beven and others have suggested that the choice of the likelihood measure used in a GLUE computation is subjective and may be selected to reflect the goals of the modeler. If an arbitrary likelihood is adopted that does not reasonably reflect the sampling distribution of the model errors, then GLUE generates arbitrary results without statistical validity that should not be used in scientific work. The traditional subjective likelihood measures that have been used with GLUE also fail to reflect the nonnormality, heteroscedasticity, and serial correlation among the residual errors generally found in real problems, and hence are poor metrics for even simple sensitivity analyses and model calibration. Most previous applications of GLUE only produce uncertainty intervals for the average model prediction, which by construction should not be expected to include future observations with the prescribed probability. We show how the GLUE methodology when properly implemented with a statistically valid likelihood function can provide prediction intervals for future observations which will agree with widely accepted and statistically valid analyses.