分形布朗运动建模的误差估计
Error Assessment in Modeling with Fractal Brownian Motions
摘要：
为了用分形布朗运动（fBms）对给定的时间序列$F(t)$建模，有必要对相关量进行适当的误差评估。通常，分形维数$D$通过关系$D=2-H$从Hurst指数$H$导出，并且Hurst指数可以通过分析重新标度范围$\langle|F(t+τ)-F(t)|\rangle$对时间跨度$τ$的依赖性来评估。对于fBms，重新标度的范围的误差不仅取决于数据采样，而且由于长期记忆的存在而随$H$变化。如果不知道分形维数，则无法评估给定时间序列的此误差。我们进行了大量的数值模拟，探讨fBms的重新标度范围的误差，并发现$0<H<0.5$，$|F(t+τ)-F(t)|$可以被视为独立的时间跨度没有重叠; $0.5<H<1$，长期记忆使$|F(t+τ)-F(t)|$相关和近似方法来评估$\langle|F(t+τ)-F(t)|\rangle$的误差。误差和分形维数，然后可以确定自相一致的时间序列的建模与fBms。
英文摘要：
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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