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

2026-08-04

AI Summary

This study addresses the challenge of modeling three-dimensional animal movement trajectories—comprising longitude, latitude, and vertical speed—that exhibit memory effects and long-range dependence. The authors propose a novel multidimensional integrated fractional Ornstein–Uhlenbeck (ifOU) process driven by multivariate fractional Brownian motion, extending the ifOU framework to three dimensions for the first time. The model incorporates coordinate-specific damping, scaling, and Hurst parameters, rigorously characterizes the admissible region for cross-correlation coefficients, and derives asymptotic properties of its cross-covariance and increment separation. By integrating a Gaussian process framework with finite-dimensional simulation and conditional velocity reconstruction, the method successfully recovers full three-dimensional movement parameters—including altitude—from migration trajectories of five noctule bats, accurately capturing inter-coordinate dependence structures.

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.

Problem

• animal movement
• fractional Ornstein--Uhlenbeck process
• multidimensional modeling
• telemetry data
• long-range dependence

Innovation

• multidimensional fractional Ornstein–Uhlenbeck
• animal telemetry
• multivariate fractional Brownian motion
• cross-coordinate correlation
• Gaussian likelihood inference