Normalized coefficients of prediction accuracy for comparative forecast verification and modeling

Abstract

The error coefficients (Formula presented.), (Formula presented.), and the linear model measures r and R2 are used in several disciplines to quantify the agreement between predicted and observed numerical data. Typical applications include forecast verification and model calibration. However, these coefficients have major drawbacks: Whereas the error coefficients are not comparable between data with different scaling, the measures from linear models are insensitive to additive or multiplicative biases. Here, we present a new categorization of the various normalizations and other modifications proposed in the literature to overcome these problems. We then propose a novel and simple idea to normalize (Formula presented.), and (Formula presented.) in analogy to the construction of r: We divide each error coefficient by its maximum that is possible when considering the given sets of predictions and observations separately. Unlike existing normalizations of error coefficients or skill scores, our new normalized coefficients treat observations and predictions symmetrically and do not rely on past data. As a result, they are not subject to artificial bias or spurious accuracy due to a long-term trend in the data. After discussing properties of the new coefficients and illustrating them with simple artificial data, we demonstrate their advantages with real data from atmospheric science.