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 $⟨|F(t+\ddot{I}„)-F(t)|⟩$ of simulated fractal Brownian motions $F(t)$ given in a previous paper \cite{q14}, the method of least squares is adopted to determine the slope, $S$, and intercept, $I$, of the log$(⟨|F(t+\ddot{I}„)-F(t)|⟩)$ vs $\log (\ddot{\mathrm{I}}„)$ plot to investigate the limitation of this procedure. It is found that the reduced $\ddot{I}{‡}^2$ 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.

7 pages,23 figures, submitted to Multiscale Modeling and Simulation Journal. Data Analysis, Statistics and Probability (physics.data-an), High Energy Astrophysical Phenomena (astro-ph.HE)