Astronomical Observatory, Jagiellonian University, Orla 171, PL-30-244 Kraków, Poland.
Phys Rev E. 2019 Dec;100(6-1):062144. doi: 10.1103/PhysRevE.100.062144.
Closed-form expressions, parametrized by the Hurst exponent H and the length n of a time series, are derived for paths of fractional Brownian motion (fBm) and fractional Gaussian noise (fGn) in the A-T plane, composed of the fraction of turning points T and the Abbe value A. The exact formula for A_{fBm} is expressed via Riemann ζ and Hurwitz ζ functions. A very accurate approximation, yielding a simple exponential form, is obtained. Finite-size effects, introduced by the deviation of fGn's variance from unity, and asymptotic cases are discussed. Expressions for T for fBm, fGn, and differentiated fGn are also presented. The same methodology, valid for any Gaussian process, is applied to autoregressive moving average processes, for which regions of availability of the A-T plane are derived and given in analytic form. Locations in the A-T plane of some real-world examples as well as generated data are discussed for illustration.
导出了分数布朗运动（fBm）和分数高斯噪声（fGn）在 A-T 平面中的路径的闭式表达式，参数为赫斯特指数 H 和时间序列的长度 n，A-T 平面由转折点分数 T 和阿贝值 A 组成。fBm 的 A_{fBm}的精确公式表示为 Riemann ζ 和 Hurwitz ζ 函数。得到了一个非常精确的近似，呈现出简单的指数形式。讨论了有限尺寸效应，由 fGn 的方差偏离 1 引起的和渐近情况。还给出了 fBm、fGn 和差分 fGn 的 T 表达式。对于任何高斯过程，相同的方法都适用于自回归移动平均过程，为其推导出了 A-T 平面的可用区域，并以解析形式给出。讨论了一些现实世界示例以及生成数据在 A-T 平面中的位置，以作说明。
