Informative Symbolic Representations as a Way to Qualitatively Analyze Time Series

Abstract

A qualitative analysis refers to the identification of the qualitative characteristics of a time series, such as its periodicity, final periodicity or near periodicity, convergence to the limit state, randomness, etc. For such an analysis, methods of symbolic dynamics are used, with the time series being represented as a sequence of symbols from a finite alphabet. Such a representation of the time series corresponds to the division of the state space of the process into a finite number of parts, each of which is denoted by a separate symbol. Thus, the symbolic representation of the time series shows in which part of the state space the process is at each moment. The choice of the cardinality of the alphabet of the symbols used affects the dependence between the laboriousness of obtaining the results of the analysis of the time series and their accuracy. To choose the cardinality of the alphabet, it was proposed to use the method of analysis of the entropy of words obtained using a sliding window of variable length. Two approaches that differ in the criterion of splitting are considered: coding of the time series levels based on dividing the range of its values, and coding trends, that is, changes in these values. A method for calculating the entropy function is described. Decision rules are proposed for obtaining various qualitative characteristics of symbolic representations of time series based on the values of the entropy function.