A Partition-induced Hereditary Metric for Equivalence Classes - Theory and Application

Measuring the distance between equivalence classes has its theoretical and practical merit, in particular, in the aspect of rough sets or the application on information systems. The typical metric for measuring the distance between partitions is the Hausdorff metric. Another candidate is the minimal matching metric which matches the pairwise minimal distance between the compartments. However, both methods need to involve or imbed Jaccard metric, which is essentially a static metric and less informative, since it scales the distance between 0 and 1. In this article, we devise a third metric which is defined inductively by some non-negative real functions. This mechanism enables its flexibility in applying metrics in real problems and delve deeper into the structures. We then apply this hereditary metric on two occasions: one with simulated data regarding algorithms and the other with real data regarding ontology population process. This metric per se is suitable for categorising procedures, methods, or other attributes