THE ROLE AND DEFINITION OF THE FRACTAL CONSTANT OF CYCLIC TIME SERIES

Abstract

Many natural and anthropogenic processes around us are cyclic in nature. The sequence of such cycles mainly occurs in time. During their existence, the sources that generate them are influenced in different ways by existing factors. The action of these factors leads to differences in characteristics between cycles. Many data within a cycle are chaotic in nature. In such cases, the following fractal indicators are used to describe the data: fractal dimension, Hirst index, fractal constant. These indicators are directly related to the statistics of the cycle data. This relationship is at the heart of the RS analysis method, namely the cumulative swing and standard deviation of cycle data. If the first two indicators are used quite often, then the fractal constant is mostly ignored or its presence is only noted. From the point of view of the mathematical description of the laws of nature, for example in physics, biology, ecology, the constant always characterizes the nature, features, structure, and material of the source within the framework of these laws. From the point of view of the system approach, the stay of the system in one or another cycle may correspond to the states of the system. In such cases, the fractal constant characterizes these states. The purpose of this study is to determine the role of the fractal constant as an indicator of the state of the source in these cycles. For this, two sequences of data were used. The first of them is the monthly consumption of electricity at an industrial facility, and the second is the certification of a group of professional training participants, when each of them is presented with the same task. The results of the study are presented in tables and presented in diagrams. The tables show indicators by cycles: data volume, arithmetic mean (used in calculating the cumulative range), standard deviation, cumulative range, fractal dimension, Gerst exponent (exponential) and fractal constant. Differences in fractal constants are shown in the corresponding diagrams. These two examples already show the value of the fractal constant as a characteristic of the nature of the source. When using the RS-analysis method to determine the fractal indicators of the data sequence, it is worth paying attention to the fact that the ratio of the cumulative range to the standard deviation is an exponential function, for which the fractal constant is no less important than the Hurst exponent or the volume of data.