Error Assessment in Modeling with Fractal Brownian Motions

To model a given time series $F(t)$ with fractal Brownian motions (fBms), it is necessary to have appropriate error assessment for related quantities. Usually the fractal dimension $D$ is derived from the Hurst exponent $H$ via the relation $D=2-H$, and the Hurst exponent can be evaluated by analyzing the dependence of the rescaled range $\langle|F(t+τ)-F(t)|\rangle$ on the time span $τ$. For fBms, the error of the rescaled range not only depends on data sampling but also varies with $H$ due to the presence of long term memory. This error for a given time series then can not be assessed without knowing the fractal dimension. We carry out extensive numerical simulations to explore the error of rescaled range of fBms and find that for $0<H<0.5$, $|F(t+τ)-F(t)|$ can be treated as independent for time spans without overlap; for $0.5<H<1$, the long term memory makes $|F(t+τ)-F(t)|$ correlated and an approximate method is given to evaluate the error of $\langle|F(t+τ)-F(t)|\rangle$. The error and fractal dimension can then be determined self-consistently in the modeling of a time series with fBms.

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