Kiyani K H, Chapman S C, Watkins N W
Department of Physics, Centre for Fusion, Space and Astrophysics, University of Warwick, Gibbet Hill Road, Coventry CV4 7AL, United Kingdom.
Phys Rev E Stat Nonlin Soft Matter Phys. 2009 Mar;79(3 Pt 2):036109. doi: 10.1103/PhysRevE.79.036109. Epub 2009 Mar 17.
The accurate estimation of scaling exponents is central in the observational study of scale-invariant phenomena. Natural systems unavoidably provide observations over restricted intervals; consequently, a stationary stochastic process (time series) can yield anomalous time variation in the scaling exponents, suggestive of nonstationarity. The variance in the estimates of scaling exponents computed from an interval of N observations is known for finite variance processes to vary as approximately 1N as N-->infinity for certain statistical estimators; however, the convergence to this behavior will depend on the details of the process, and may be slow. We study the variation in the scaling of second-order moments of the time-series increments with N for a variety of synthetic and "real world" time series, and we find that in particular for heavy tailed processes, for realizable N , one is far from this approximately 1N limiting behavior. We propose a semiempirical estimate for the minimum N needed to make a meaningful estimate of the scaling exponents for model stochastic processes and compare these with some "real world" time series.
在对尺度不变现象的观测研究中，准确估计标度指数至关重要。自然系统不可避免地会在有限区间内提供观测数据；因此，一个平稳随机过程（时间序列）可能会在标度指数中产生异常的时间变化，这暗示着非平稳性。对于有限方差过程，由N次观测的区间计算出的标度指数估计值的方差，对于某些统计估计量而言，当N趋于无穷大时，其变化近似为1/N；然而，收敛到这种行为将取决于过程的细节，并且可能会很缓慢。我们研究了各种合成和“真实世界”时间序列中，时间序列增量的二阶矩的标度随N的变化情况，并且发现，特别是对于重尾过程，对于可实现的N，其远未达到这种近似1/N的极限行为。我们针对模型随机过程提出了一个半经验估计，用于估计对标度指数进行有意义估计所需的最小N，并将这些估计与一些“真实世界”时间序列进行比较。
