Fractal dimensions of time sequences

We present a simple and efficient way for calculating the fractal dimension D of any time sequence sampled at a constant time interval. We calculated the error of a piecewise interpolation to N+1 points of the time sequence with respect to the next level of (2N+1)-point interpolation. This error was found to be proportional to the scale (i.e., 1/N) to the power of 1−D. A simple analysis showed that our method is equivalent to the inverse process of the method of random midpoint displacement widely used in generating fractal Brownian motion for a given D. The efficiency of our method makes the fractal dimension a practical tool in analyzing the abundant data in natural, economic, and social sciences.