Most classification methods provide either a prediction of group membership
or an assessment of class membership probability. In the case of two-group
classification the predicted probability can be described as "risk" of
belonging to a special group. When the required output is a set of ordinal risk
categories, a discretization of the continuous risk prediction is achieved by
two common methods: dividing the output of an "optimal" classification model
into adjacent intervals that correspond to the desired risk categories, or
constructing a set of models that describe the conditional risk function at
specific points (quantile regression). By defining a new error measure for the
distribution of risk onto intervals we are able identify lower bounds on the
accuracy of these methods, showing sub-optimality both in their distribution of
risk and in the efficiency of their resulting partition into intervals. Using
the framework of discriminant analysis, we combine the two approaches by adding
a new form of constraint to the existing discriminant analysis optimization
problem and by introducing a penalty function to avoid degenerate solutions.
Finally we show an example using linear discriminant analysis as a reference.