Searching for Relational Patterns in Data

We consider several basic classes of tolerance relations among objects. These (global) relations are defined from some predefined similarity measures on values of attributes. A tolerance relation in a given class of tolerance relations is optimal with respect to a given decision table A if it contains only pairs of objects with the same decision and the number of such pairs contained in the relation is maximal among all relations from the class. We present a method for (sub-)optimal tolerance relation learning from data (decision table). The presented method is based on rough set approach. We show that for some basic families of tolerance relations this problem can be transformed to a relative geometrical problem in a real affine space. Hence geometrical computations are becoming useful tools for solving the problem of global tolerance relation construction. The complexity of considered problems can be evaluated by the complexity of the corresponding geometrical problems. We propose some efficient heuristics searching for an approximation of optimal tolerance relations in considered families of tolerance relations. The global tolerance relations can be treated as patterns in the cartesian product of the object set. We show how to apply the relational patterns (global tolerance relations) in clustering and classification of objects.