Fractal Time Series Analysis by Using Entropy and Hurst Exponent

The paper discusses the analysis of simulated and real fractal time series by applying approximate entropy (ApEn) and sample entropy (SampEn) to determine the complexity and predictability of the time series. The simulated fractal time series are based on the Fractal Gaussian Noise with different values of the Hurst exponent and different lengths. To study the accuracy of the simulated time series, the method of detrended fluctuation analysis was applied, which determines the value of the Hurst exponent. The results obtained on the basis of relative standard error (RSE) show that the simulated time series have a high degree of accuracy, as the RSE is less than 2% for all input values of the Hurst exponent. Studies show that ApEn and SampEn are methods that are very sensitive to input parameters: subseries length, tolerance value and data length. Due to the fact that there is no consensus on the choice of values for these parameters, in the article they are studied on simulated fractal time series with lengths of 600-3000 points. The values of ApEn, SampEn and the value of the Hurst exponent were found to be negatively correlated, and the predictability was positively related to the Hurst exponent. Both entropy-based methods and the Hurst exponent were used to study the dynamics of the behavior of real-time series related to heart rate intervals (RR interval series) to extract useful information related to the influence of the age of the subjects. The results show that the values of ApEn and SampEn are higher for young subjects than for adults, therefore the RR time series of young subjects is more complex than the time series of adult subjects. The value of the Hurst exponent of young subjects was found to be lower than that of adult subjects, therefore the time series of adult subjects have a higher degree of self-similarity. The results reported in this study may be useful in analysing cardiac data to extract useful information in clinical decision making.