Modeling the trajectories of animals is challenging due to the complexity of their behaviors, the influence of unpredictable environmental factors, individual variability, and the lack of detailed data on their movements. Additionally, factors such as migration, hunting, reproduction, and social interactions add additional layers of complexity when attempting to accurately forecast their movements. In the literature, various models exits that aim to study animal telemetry, by modeling the velocity of the telemetry, the telemetry itself or both processes jointly through a Markovian process. In this work, we propose to model the velocity of each coordinate axis for animal telemetry data as a fractional Ornstein-Uhlenbeck (fOU) process. Then, the integral fOU process models position data in animal telemetry. Compared to traditional methods, the proposed model is flexible in modeling long-range memory. The Hurst parameter $H \in (0,1)$ is a crucial parameter in integral fOU process, as it determines the degree of dependence or long-range memory. The integral fOU process is nonstationary process. In addition, a higher Hurst parameter ($H > 0.5$) indicates a stronger memory, leading to trajectories with transient trends, while a lower Hurst parameter ($H < 0.5$) implies a weaker memory, resulting in trajectories with recurring trends. When H = 0.5, the process reduces to a standard integral Ornstein-Uhlenbeck process. We develop a fast simulation algorithm of telemetry trajectories using an approach via finite-dimensional distributions. We also develop a maximum likelihood method for parameter estimation and its performance is examined by simulation studies. Finally, we present a telemetry application of Fin Whales that disperse over the Gulf of Mexico.

翻译：动物运动轨迹建模因其行为复杂性、不可预测环境因素的影响、个体差异性以及运动详细数据的缺乏而极具挑战性。此外，迁徙、捕食、繁殖和社会互动等因素在试图准确预测其运动轨迹时又增加了额外的复杂性。文献中存在多种动物遥测模型，通过马尔可夫过程对遥测速度、遥测本身或两者联合进行建模。本研究提出将动物遥测数据各坐标轴的速度建模为分数阶Ornstein-Uhlenbeck（fOU）过程，进而利用积分fOU过程对动物遥测中的位置数据进行建模。与传统方法相比，该模型在长程记忆建模方面具有灵活性。Hurst参数 $H \in (0,1)$ 是积分fOU过程中的关键参数，它决定了依赖程度或长程记忆。积分fOU过程为非平稳过程。此外，较高的Hurst参数（$H > 0.5$）表示更强的记忆性，导致轨迹呈现瞬时趋势；较低的Hurst参数（$H < 0.5$）表示较弱的记忆性，导致轨迹呈现重复趋势。当H = 0.5时，该过程退化为标准积分Ornstein-Uhlenbeck过程。我们基于有限维分布方法开发了一种遥测轨迹快速模拟算法，并提出了参数估计的极大似然方法，通过模拟研究评估其性能。最后，我们展示了在墨西哥湾分布的鲱鲸遥测应用案例。