Determining the Fractal Dimension of a Time Series with a Neural Net

There are several methods for estimating the fractal dimension of a time series of data such as the box counting method and the correlation method [DeGrauwe, Dewachter and EmbrechtS, 1993] [peitgen and Saupe, 1988]. The application of these methods are often demanding in computing time and require expert interaction for interpreting the calculated fractal dimension. Artificial neural nets (ANN) offer a fast and elegant way to estimate the fractal dimension of a time series. A backpropagation net was trained to find the fractal dimension of a time series with encouraging results. Training patterns of time series with known fractal dimension were generated with the fractal interpolation method described by Barnsley [Barnsley, 1988]. Two artificial neural nets were trained with the backpropagation algorithm on the amplitude spectra of fractal signals. One ANN used local normalization, the other ANN used global normalization of the spectral components. The trained neural nets were then tested on entirely different fractal signals generated with the WeierstrassMandelbrot function [Mandelbrot, 1983]. Both ANN's could estimate the fractal dimension for other fractal time series generated with the fractal interpolation technique, but only the ANN with global normalization could correctly identify the fractal dimension of the Weierstrass-Mandelbrot based time series (within 10 percent error). Section 2 briefly reviews fractal interpolation. Section 3 summarizes how amplitude spectra can be generated from fractal interpolation graphs and explains the difference between local and global normalization of power spectra. Section 4 introduces the MandelbrotWeierstrass function, and provides details of the neural net architecture. The failure of the first ANN to correctly identify Weierstrass-Mandelbrot functions led to a deeper insight into the relationship between fractal time series and their amplitude spectra, and is discussed in Section 5.