Time Series Analysis Using Fractal and Multifractal Methods

Abstract

The article presents a theoretical development of the application of mathematical methods for fractal and multifractal analysis in the study of non-linear, non-stationary time series. The research was conducted using Detrended Fluctuation Analysis (DFA) and MultiFractal Detrended Fluctuation Analysis (MFDFA). A characteristic feature of the application of these methods is that they allow for additional information to be obtained from the studied objects, providing an insight into the objects' nonlinear dynamic nature. In the present case, the subjects of the analyses are fluctuations in the physiological signals of the heartbeat. The data is obtained through specialized electronic devices (holters) that record intervals during a 24-hour heart rate registration period. The dynamic characteristics of the heart rate intervals, converted by electronic devices into a time series, show fractal and in some cases multifractal properties. Conducting studies on prolonged time series, which reflect the functioning of cardiac activity in both healthy subjects and those with cardiovascular changes, has made it possible to assess with a high level of confidence the degree of the disease or lack thereof. A suitable approach for these studies is the use of modern mathematical methods for fractal and multifractal analysis of the heart rate signals.