Sikora Grzegorz, Höll Marc, Gajda Janusz, Kantz Holger, Chechkin Aleksei, Wyłomańska Agnieszka
Faculty of Pure and Applied Mathematics, Hugo Steinhaus Center, Wroclaw University of Science and Technology, 50-370 Wroclaw, Poland.
Department of Physics, Institute of Nanotechnology and Advanced Materials, Bar-Ilan University, Ramat-Gan, 5290002 Israel.
Phys Rev E. 2020 Mar;101(3-1):032114. doi: 10.1103/PhysRevE.101.032114.
Detrended fluctuation analysis (DFA) is one of the most widely used tools for the detection of long-range dependence in time series. Although DFA has found many interesting applications and has been shown to be one of the best performing detrending methods, its probabilistic foundations are still unclear. In this paper, we study probabilistic properties of DFA for Gaussian processes. Our main attention is paid to the distribution of the squared error sum of the detrended process. We use a probabilistic approach to derive general formulas for the expected value and the variance of the squared fluctuation function of DFA for Gaussian processes. We also get analytical results for the expected value of the squared fluctuation function for particular examples of Gaussian processes, such as Gaussian white noise, fractional Gaussian noise, ordinary Brownian motion, and fractional Brownian motion. Our analytical formulas are supported by numerical simulations. The results obtained can serve as a starting point for analyzing the statistical properties of DFA-based estimators for the fluctuation function and long-memory parameter.
去趋势波动分析（DFA）是检测时间序列中长程相关性最广泛使用的工具之一。尽管DFA已发现许多有趣的应用，并且已被证明是性能最佳的去趋势方法之一，但其概率基础仍不明确。在本文中，我们研究高斯过程的DFA的概率性质。我们主要关注去趋势过程的平方误差和的分布。我们使用概率方法推导高斯过程DFA的平方波动函数的期望值和方差的通用公式。对于高斯过程的特定示例，如高斯白噪声、分数高斯噪声、普通布朗运动和分数布朗运动，我们也得到了平方波动函数期望值的解析结果。我们的解析公式得到了数值模拟的支持。所获得的结果可作为分析基于DFA的波动函数和长记忆参数估计器统计性质的起点。
