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

TL;DR

This paper investigates the limitations of using the least squares method to estimate the fractal dimension of fractal Brownian motions, revealing biases and proposing reinterpretation strategies for more accurate results.

Contribution

It identifies the correlation-induced biases in least squares fitting of fractal Brownian motions and suggests a more self-consistent reinterpretation approach.

Findings

Reduced chi-squared decreases with higher Hurst index

Errors in slope and intercept are smaller than their standard deviations

Fractal dimension of 1.511 obtained for Euro-Dollar exchange rate

Abstract

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 in a previous paper \cite{q14}, the method of least squares is adopted to determine the slope, $S$, and intercept, $I$, of the log$(\langle |F(t+\tau)-F(t)|\rangle)$ vs $\rm{log}(\tau)$ plot to investigate the limitation of this procedure. It is found that the reduced $\chi^2$ 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.