The Effect of Noise and Nonlinear Noise Reduction Methods on the Fractal Dimension of Chaotic Time Series

Abstract

The fractal dimension (FD) of a signal is a useful measure for characterizing its complexity. The real signals are contaminated with noise, which leads to reduced efficiency of the fractal analysis. This paper investigates the effect of noise on the FD computation and the compares the fractal dimensions obtained from noise-reduced signals. To this aim, the FD of different continuous and discrete chaotic time series is computed in the case of noise-free and noisy signals, using Katz, Higuchi, and Leibovich and Toth algorithms. Also for further investigations, the result of Lyapunov and Hurst analysis is presented. It is observed that in the continuous-time systems, the fractal dimensions obtained from all algorithms have significant changes in the presence of noise. While for the discrete-time signals, the Katz and the Higuchi dimensions have more robustness to noise. Furthermore, the Lyapunov exponent is decreased, and the Hurst exponent is increased. Then, the noisy signals are filtered with four nonlinear noise-reduction methods. The fractal dimensions are calculated from each denoised signal, and the performances of the noise-reduction methods are compared. The results show that the Katz and the Higuchi methods have less percentage change in calculating the dimension of original and filtered signals than the Leibovich and Toth method. The analysis is also done on the real heart rate signals to expand the results to real processes.