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With the standard deviation for the logarithm of the re-scaled range $\langle|F(t+\tau)-F(t)|\rangle$ of simulated fractal Brownian motions $F(t)$ given ina previous paper \cite{q14}, the method of least squares is adopted todetermine the slope, $S$, and intercept, $I$, of the log$(\langle|F(t+\tau)-F(t)|\rangle)$ vs $\rm{log}(\tau)$ plot to investigate thelimitation of this procedure. It is found that the reduced $\chi^2$ of thefitting decreases with the increase of the Hurst index, $H$ (the expectationvalue of $S$), which may be attributed to the correlation among the re-scaledranges. Similarly, it is found that the errors of the fitting parameters $S$and $I$ are usually smaller than their corresponding standard deviations. Theseresults show the limitation of using the simple least square method todetermine the dimension of a fractal time series. Nevertheless, they may beused to reinterpret the fitting results of the least square method to determinethe dimension of fractal Brownian motions more self-consistently. The currencyexchange rate between Euro and Dollar is used as an example to demonstrate thisprocedure and a fractal dimension of 1.511 is obtained for spans greater than30 transactions.

Limitation of the Least Square Method in the Evaluation of Dimension of Fractal Brownian Motions