The partition of N-dimensional space, using shells

Abstract

In the field of pattern recognition there are problems where a decision can be made upon a set of training patterns. A training pattern represents a particular known case of the problem under investigation. Here it is represented as a point in the n-dimensional space Rⁿ, where each coordinate denotes a particular parameter of an observation or measurement. Each pattern point belongs to a single answer, which is the solution of this case. In this way, solved examples of the problem under investigation are represented by points of training patterns in the n-dimensional space. The question arises how to make the decision in order to find the answer for a given unknown pattern, i.e. a case without an answer, upon a set of training patterns used as basic knowledge.

By the proposed method, shells are used to divide the n-dimensional space into regions where points of training patterns are located. Each shell has the shape of an n-dimensional rectangle and covers pattern points of the same answer. The partition of the n-dimensional space is achieved first in the adaptation phase, where a single shell belongs to the same answer as the pattern points it covers. The coverage of the space obtained after the adaptation phase is then improved in the following self-adaptation phase. Here, pairs of shells belonging to the same answer are merged into substitute shells. Thus, the number of shells is reduced without damage to the obtained quality of the coverage of the space Rⁿ. Upon such a coverage, a decision can be made for a given pattern by searching a shell which covers its pattern point. The answer belonging to this shell is also the answer for that pattern.

The efficiency of this model has been satisfactorily demonstrated in two medical fields: prognosis of acute pancreatitis and in the diagnosis of disseminated cancer of unknown origin.