Search for the optimal metric in classification problems with ordinal features

A method for finding the optimal distance function for the classification problem with two classes in which the objects are specified by vectors of their ordinal features is proposed. An optimal distance function is sought by the minimization of the weighted difference of the average intraclass and interclass distances. It is assumed that a specific distance function is given for each feature, which is defined on the Cartesian product of the set of integer numbers in the range from $0$ to $N-1$ and takes values from $0$ to $M$. Distance functions satisfy modified metric properties. The number of admissible distance functions is calculated, which enables one to significantly reduce the complexity of the problem. To verify the appropriateness of metric optimization and to perform experiments, the nearest neighbor algorithm is used.