Fig. 3

Optimization of expected risk. Left side: Representation of the threshold optimization with respect to the expected risk \(\:\stackrel{\sim}{ER}\) using a diagram where the \(\:x\) axis represents the threshold variable \(\:s\) and the \(\:y\) axis the \(\:\stackrel{\sim}{ER}\left(s\right)\) function. The same artificial model was used as in Fig. 2, left side (i.e. with parameters \(\:{\sigma\:}_{FP}=\:{\sigma\:}_{FN}=0.3\)). The optimum threshold is the point where \(\:\stackrel{\sim}{ER}\left(s\right)\) reaches its minimum. Right side: \(\:\stackrel{\sim}{ER}\left(s\right)\) diagram for the same model with the weighted balanced accuracy metric (\(\:WBA\), see description below) metric overlaid in a color coding. Additionally, the contour lines of the metric are displayed. The optimization of \(\:WBA\) is equivalent to finding the optimum threshold for the expected risk \(\:\stackrel{\sim}{ER}\). In the representation on the right side, (local) optimization is equivalent to finding the points on the \(\:ROC\) curves which are tangent to the iso-contour lines of the function \(\:WBA\) (depicted by the dot). This tangent is shown in brown color (parallel to the iso-contour lines). The concrete \(\:\stackrel{\sim}{ER}\) values are directly shown on the iso-contour lines (colored lines, which are straight lines in this case). The diagonal line represents the symmetry line between positive and negative cases