On the Upper Bounds of the Real-Valued Predictions

Predictions are fundamental in science as they allow to test and falsify theories. Predictions are ubiquitous in bioinformatics and also help when no first principles are available. Predictions can be distinguished between classifications (when we associate a label to a given input) or regression (when a real value is assigned). Different scores are used to assess the performance of regression predictors; the most widely adopted include the mean square error, the Pearson correlation (rho), and the coefficient of determination (or R-2). The common conception related to the last 2 indices is that the theoretical upper bound is 1; however, their upper bounds depend both on the experimental uncertainty and the distribution of target variables. A narrow distribution of the target variable may induce a low upper bound. The knowledge of the theoretical upper bounds also has 2 practical applications: (1) comparing different predictors tested on different data sets may lead to wrong ranking and (2) performances higher than the theoretical upper bounds indicate overtraining and improper usage of the learning data sets. Here, we derive the upper bound for the coefficient of determination showing that it is lower than that of the square of the Pearson correlation. We provide analytical equations for both indices that can be used to evaluate the upper bound of the predictions when the experimental uncertainty and the target distribution are available. Our considerations are general and applicable to all regression predictors.