Integral Fractional Ornstein-Uhlenbeck Process Model for Animal Movement

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.