Bassingthwaighte J B, Raymond G M
Center for Bioengineering, University of Washington, Seattle 98195, USA.
Ann Biomed Eng. 1995 Jul-Aug;23(4):491-505. doi: 10.1007/BF02584449.
Fractal signals can be characterized by their fractal dimension plus some measure of their variance at a given level of resolution. The Hurst exponent, H, is < 0.5 for rough anticorrelated series, > 0.5 for positively correlated series, and = 0.5 for random, white noise series. Several methods are available: dispersional analysis, Hurst rescaled range analysis, autocorrelation measures, and power special analysis. Short data sets are notoriously difficult to characterize; research to define the limitations of the various methods is incomplete. This numerical study of fractional Brownian noise focuses on determining the limitations of the dispersional analysis method, in particular, assessing the effects of signal length and of added noise on the estimate of the Hurst coefficient, H, (which ranges from 0 to 1 and is 2 - D, where D is the fractal dimension). There are three general conclusions: (i) pure fractal signals of length greater than 256 points give estimates of H that are biased but have standard deviations less than 0.1; (ii) the estimates of H tend to be biased toward H = 0.5 at both high H (> 0.8) and low H (< 0.5), and biases are greater for short time series than for long; and (iii) the addition of Gaussian noise (H = 0.5) degrades the signals: for those with negative correlation (H < 0.5) the degradation is great, the noise has only mild degrading effects on signals with H > 0.6, and the method is particularly robust for signals with high H and long series, where even 100% noise added has only a few percent effect on the estimate of H. Dispersional analysis can be regarded as a strong method for characterizing biological or natural time series, which generally show long-range positive correlation.
分形信号可以通过其分形维数以及在给定分辨率水平下的某种方差度量来表征。赫斯特指数H，对于粗糙的反相关序列小于0.5，对于正相关序列大于0.5，对于随机白噪声序列等于0.5。有几种方法可用：离散分析、赫斯特重标极差分析、自相关度量和功率谱分析。众所周知，短数据集很难进行表征；确定各种方法局限性的研究并不完整。这项关于分数布朗噪声的数值研究重点在于确定离散分析方法的局限性，特别是评估信号长度和添加噪声对赫斯特系数H估计值的影响（H的范围是0到1，且H = 2 - D，其中D是分形维数）。有三个总体结论：（i）长度大于256个点的纯分形信号给出的H估计值有偏差，但标准差小于0.1；（ii）在高H（> 0.8）和低H（< 0.5）时，H的估计值往往偏向于H = 0.5，并且短时间序列的偏差比长时间序列的更大；（iii）添加高斯噪声（H = 0.5）会使信号退化：对于负相关（H < 0.5）的信号，退化程度很大，噪声对H > 0.6的信号只有轻微的退化影响，并且该方法对于高H和长序列的信号特别稳健，在这种情况下，即使添加100%的噪声对H估计值的影响也只有几个百分点。离散分析可被视为表征生物或自然时间序列的一种有力方法，这些序列通常呈现长程正相关。
