FRACTAL CORRELATION IN HETEROGENEOUS SYSTEMS

Noisy or irregular signals are usually thought of as being composed of or driven by random processes. However, some signals which appear noisy (such as fractal signals) are correlated, and the fractal dimension, D, provides a measure of the degree of correlation between elements of signals (densities, intensities of a property, voltages, etc.) over space or time. To arrive at this correlation, methods of analysis using the dispersion of a signal, such as the relative dispersion or the Hurst rescaled range analysis, give measures of the fractal dimension D and of the two-point correlation between adjacent segments of the signal. We have derived expressions for the two-point correlation over an extended range of observations of non-adjacent neighbors. The results allows estimation of the fractal dimension D from the autocorrelation function directly. We use the result as a test for self-similarity to determine whether or not the correlation fall-off is the same for different sized groupings of the data. This means that the estimate of the fractal dimension D and the correlation requires observations at one unit size to estimate the range-extended correlation behavior, with verification at one other unit size. The correlation coefficient between nth neighbor units is r(n) = 0.5[\n + 1\2H - 2\n\2H + \n - 1\2H], where the Hurst coefficient H = 2 - D for one-dimensional signals. Asymptotically, the ratio of successive values of r(n) is r(n)/r(n-1) = [n/(n - 1)]2H - 2. The extended range fractal correlation technique is therefore both efficient and relatively robust.