Limitation of the Least Square Method in the Evaluation of Dimension of Fractal Brownian Motions
Abstract
With the standard deviation for the logarithm of the re-scaled range |F(t+τ)-F(t)| of simulated fractal Brownian motions F(t) given in a previous paper q14, the method of least squares is adopted to determine the slope, S, and intercept, I, of the log( |F(t+τ)-F(t)|) vs log(τ) plot to investigate the limitation of this procedure. It is found that the reduced 2 of the fitting decreases with the increase of the Hurst index, H (the expectation value of S), which may be attributed to the correlation among the re-scaled ranges. Similarly, it is found that the errors of the fitting parameters S and I are usually smaller than their corresponding standard deviations. These results show the limitation of using the simple least square method to determine the dimension of a fractal time series. Nevertheless, they may be used to reinterpret the fitting results of the least square method to determine the dimension of fractal Brownian motions more self-consistently. The currency exchange rate between Euro and Dollar is used as an example to demonstrate this procedure and a fractal dimension of 1.511 is obtained for spans greater than 30 transactions.