Fractal dimensions of time sequences

Abstract

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.