Relationship between logistic chaos and randomness using a matrix of probabilities and its application to the classification of time series: The line from chaos point to the reversed type of chaos is perpendicular to that from randomness's point to its reversed type

摘要

Chaos theory has been applied to analyze the dynamics of time series in many fields. Some primary questions that are addressed are "What distinguished chaos from randomness?" or "What is the relationship between chaos and randomness". In this paper, by converting the time series of {X} to the series of signs of {Δx} such that Δx = x - x, logistic chaos can be represented by a probability matrix. Applying the probability matrices of the logistic chaos and randomness gives the following geometric property: The line from the point of chaos characterized type is perpendicular to the line from the point of randomness to that of the reversed type. Although the probability matrix is a necessary condition of time series, it can be applied to introduce distances from the time series of data to the quasi-chaos or the quasi-randomness and to propose a classification method for time series.