A graphical analysis of cost-sensitive regression problems

Several efforts have been done to bring ROC analysis beyond (binary) classification, especially in regression. However, the mapping and possibilities of these proposals do not correspond to what we expect from the analysis of operating conditions, dominance, hybrid methods, etc. In this paper we present a new representation of regression models in the so-called regression ROC (RROC) space. The basic idea is to represent over-estimation on the xaxis and under-estimation on the y-axis. The curves are just drawn by adjusting a shift, a constant that is added (or subtracted) to the predictions, and plays a similar role as a threshold in classification. From here, we develop the notions of optimal operating condition, convexity, dominance, and explore several evaluation metrics that can be shown graphically, such as the area over the RROC curve (AOC). In particular, we show a novel and significant result, the AOC is equal to the error variance (multiplied by a factor which does not depend on the model). The derivation of RROC curves with non-constant shifts and soft regression models, and the relation with cost plots is also discussed.