Multidimensional Integral Fractional Ornstein--Uhlenbeck Process with an Application to Animal Movement

Abstract

Fractional Ornstein--Uhlenbeck (fOU) processes model temporal dependence and memory, including long-range dependence, while retaining the classical Ornstein--Uhlenbeck process as a special case. We extend the integral fractional Ornstein--Uhlenbeck (ifOU) process to a multidimensional setting for animal telemetry. Longitude, Latitude, and Altitude velocities are represented by coordinate-specific fOU processes driven by a multivariate fractional Brownian motion, allowing each coordinate to retain its own damping, scale, and Hurst parameters. We establish covariance validity, characterize the admissible cross-correlation region, and derive the asymptotic behavior of cross-covariances and separated increments. We develop procedures for finite-dimensional simulation, Gaussian likelihood inference, and conditional velocity reconstruction. Replicated simulations examine estimation of cross-coordinate correlations, while joint estimation of the complete parameter vector is illustrated for one three-dimensional trajectory. The proposed model is applied to telemetry records from five common noctule bats migrating in Germany, including three trajectories with Altitude measurements.