Measuring the complexity of classification problems

We study a number of measures that characterize the difficulty of a classification problem. We compare a set of real world problems to random combinations of points in this measurement space and found that real problems contain structures that are significantly different from the random sets. Distribution of problems in this space reveals that there exist at least two independent factors affecting a problem's difficulty, and that they have notable joint effects. We suggest using this space to describe a classifier domain of competence. This can guide static and dynamic selection of classifiers for specific problems as well as sub-problems formed by confinement, projections, and transformations of the feature vectors.