Incomplete classifications of a finite set of objects using Monotone Systems

The problem of incomplete classification is solved using a monotone system of a special kind. A classification method based on identification of the minimal cores of the monotone system is proposed. Existence conditions of a complete classification are given. The method is compared with previously published methods. Automatic classification methods, in addition to looking for complete classifications (i.e., partitioning of the initial set of objects), when classes are nonintersecting and form a covering of the entire set, also consider classifications with intersecting classes, with fuzzy classes, with macrostructure, etc. [1-3]. An independent group comprises incomplete classification methods, which identify a special class of “atypical” (background, special, or intermediate) objects and then assign the rest of the objects to nonintersecting classes [4-6]. Incomplete classifications are constructed when it is desirable to form classes comprising “strongly separated” subsets of objects, with all the background objects collected in a single class. This is archived either by two-step procedures, in which the first stage involving identification of the special objects is independent of the second stage involving classification proper, or by single-stage processing in which the classification functional is defined on the set of two-level classifications, thus complicating the discrete optimization problem. Moreover, both cases require specifying in advance the number of classes and the cardinality of the set of special objects, which leads to multi-alternative computations. Finally, most of the known algorithms are crudely approximate. In this paper, the sought incomplete classification is implicitly described by a separate estimate for each identified class of nonspecial objects, and this estimate should be extremal and equal on all classes of the sought classification. The proposed approach requires minimum prior information: we only need to know the measure of association of one object with a subset of objects. The number of classes and the number of objects identified, as special, is not fixed in advance. The proposed algorithm guarantees exact solution of the corresponding extremal problem.