Fractal aspects of classification problem modeling

Abstract

The focus of this work is a mathematical model of the classification problem, which is based on the concepts of the theory of fractal sets and features of modeling mental processes. Within the framework of this approach, classification activities can be viewed from two points of view. On the one hand, it is working with the empirical representation of classification objects, on the other hand, it is a mental reproduction of the task of constructing classification partitions. The realization of these two types of activity occurs in the process of constructing a fractal set generated by a randomized system of iterated functions. The mathematical model of the classification problem is presented in the form of two spaces – metric, associated with the phenomenological component, and ultrametric, reflecting the cognitive side of the problem solution. As the analysis of the results of solving classification problems shows, fractal models well meet the requirements for the formulation of the problem and the algorithmic features of its solution. The very specificity of the problem statement is reflected in the need to take into account such characteristics as the isolation of individual objects, the ability to establish the similarity /difference of objects, the compactness of the feature space, etc. These properties are characteristic of the ultrametric space created during the solution. The relationship of these two spaces is carried out by modeling the fractal structure. The paper shows exactly how the use of the fractal approach in solving classification problems is associated with the construction of ultrametric spaces. It is characteristic that these ultrametric spaces are an integral part of the algorithm for solving the problem itself. This algorithmic component of solving the classification problem is directly related to cognitive processes and interpreted as a model of processes inherent in mental activity.