Methods for Mathematical Analysis of Simulated and Real Fractal Processes with Application in Cardiology

Abstract

In the article, a comparative analysis is performed regarding the accuracy parameter in determining the degree of self-similarity of fractal processes between the following methods: Variance-Time plot, Rescaled Range (R/S), Wavelet-based, Detrended Fluctuation Analysis (DFA) and Multifractal Detrended Fluctuation Analysis (MFDFA). To evaluate the methods, fractal processes based of Fractional Gaussian Noise were simulated and the dependence between the length of the simulated process and the degree of self-similarity was investigated by calculating the Hurst exponent (H > 0.5). It was found that the Wavelet-based, DFA and MFDFA methods, with a process length greater than 214 points, have a relative error of the Hurst exponent is less than 1%. A methodology for the Wavelet-based method related to determining the size of the scale and the wavelet algorithm was proposed, and it was investigated in terms of the exact determination of the Hurst exponent of two algorithms: Haar and Daubechies with different number of coefficients and different values of the scale. Based on the analysis, it was determined that the Daubechies algorithm with 10 coefficients and scale (i = 2, j = 10) has a relative error of less than 0.5%. The three most accurate methods are applied to the study of real cardiac signals of two groups of people: healthy and unhealthy (arrhythmia) subjects. The results of the statistical analysis, using the t-test, show that the proposed methods can distinguish the two studied groups and can be used for diagnostic purposes.

Keywords

• fractal process
• Hurst exponent
• multifractal process
• RR time series