New Method for Estimating the Fractal Dimension of Discrete Temporal Signals

To measure the fractal dimension of time series, several methods and algorithms based on various coverings has been elaborated. In order to improve the complexity and the precision of the fractal dimension estimation of the discreet time series, we developed a simple method based on a multi-scale covering using the rectangle as a structuring element of covering. An optimization technique has been associated to this method in order to determine the optimal time interval Delta tau max through which the line log-log is fitted whose slope represents the fractal dimension. This optimization technique permitted to improve the precision and the time computing of the method. In order to measure the performance and the robustness of the proposed method, we applied it to fractal parametric signals whose theoretical fractal dimension is known, namely: the Weierstrass function (WF) and the fractional Brownian motion (FBM). Experimental results show that the proposed method presents a good precision since the estimation error averaged over 180 tests for the two types of signals is 3.7%.