Grzesiek Aleksandra, Gajda Janusz, Thapa Samudrajit, Wyłomańska Agnieszka
Faculty of Pure and Applied Mathematics, Hugo Steinhaus Center, Wroclaw University of Science and Technology, Wyspianskiego 27, 50-370 Wroclaw, Poland.
Faculty of Economic Sciences, University of Warsaw, Długa 44/50, 00-241 Warsaw, Poland.
Chaos. 2024 Apr 1;34(4). doi: 10.1063/5.0201436.
Fractional Brownian motion (FBM) is a canonical model for describing dynamics in various complex systems. It is characterized by the Hurst exponent, which is responsible for the correlation between FBM increments, its self-similarity property, and anomalous diffusion behavior. However, recent research indicates that the classical model may be insufficient in describing experimental observations when the anomalous diffusion exponent varies from trajectory to trajectory. As a result, modifications of the classical FBM have been considered in the literature, with a natural extension being the FBM with a random Hurst exponent. In this paper, we discuss the problem of distinguishing between two models: (i) FBM with the constant Hurst exponent and (ii) FBM with random Hurst exponent, by analyzing the probabilistic properties of statistics represented by the quadratic forms. These statistics have recently found application in Gaussian processes and have proven to serve as efficient tools for hypothesis testing. Here, we examine two statistics-the sample autocovariance function and the empirical anomaly measure-utilizing the correlation properties of the considered models. Based on these statistics, we introduce a testing procedure to differentiate between the two models. We present analytical and simulation results considering the two-point and beta distributions as exemplary distributions of the random Hurst exponent. Finally, to demonstrate the utility of the presented methodology, we analyze real-world datasets from the financial market and single particle tracking experiment in biological gels.
分数布朗运动（FBM）是描述各种复杂系统中动力学的典型模型。它由赫斯特指数表征，该指数决定了FBM增量之间的相关性、其自相似性以及反常扩散行为。然而，最近的研究表明，当反常扩散指数在不同轨迹间变化时，经典模型可能不足以描述实验观测结果。因此，文献中考虑了对经典FBM的修正，一种自然的扩展是具有随机赫斯特指数的FBM。在本文中，我们通过分析二次型表示的统计量的概率性质，讨论区分两种模型的问题：（i）具有恒定赫斯特指数的FBM和（ii）具有随机赫斯特指数的FBM。这些统计量最近在高斯过程中得到应用，并已被证明是假设检验的有效工具。在这里，我们利用所考虑模型的相关性质研究两个统计量——样本自协方差函数和经验异常测度。基于这些统计量，我们引入一种检验程序来区分这两种模型。我们给出了将两点分布和贝塔分布作为随机赫斯特指数的示例性分布的分析和模拟结果。最后，为了证明所提出方法的实用性，我们分析了来自金融市场和生物凝胶中单粒子追踪实验的真实数据集。
