Inferring resource selection and utilization distributions from irregular and error-prone animal tracking data

Technical Summary: Inferring Resource Selection and Utilization Distributions from Irregular and Error-Prone Animal Tracking Data

Problem Statement
Understanding animal distribution and informing conservation strategies relies on accurately quantifying habitat selection and deriving utilization distributions (UDs) from telemetry data. Traditional methods, such as Resource Selection Functions (RSFs) and Step-Selection Functions (SSFs), often assume regular sampling intervals and negligible measurement error. These assumptions are frequently violated in marine systems where data are collected via Argos or Fastloc GPS systems, which are characterized by irregular sampling intervals and significant location uncertainty.

Current practice typically employs a "two-step" approach: first, filtering raw data to correct for measurement error and impute missing values (often using state-space models like correlated random walks), and second, fitting habitat selection models to the corrected locations. This decoupled procedure fails to propagate uncertainty from the filtering stage into the ecological inference, potentially yielding biased estimates of habitat selection parameters and inaccurate UDs. Furthermore, standard habitat-driven Langevin diffusion models, which offer a continuous-time framework for linking local movement decisions to broad-scale space use, have historically struggled to incorporate measurement error when habitat covariates depend on unobserved true locations.

Methodology
The authors propose a unified, single-stage framework that integrates a habitat-driven Langevin diffusion model with a state-space formulation to handle both measurement error and temporal irregularity simultaneously.

1. Langevin Movement Model: The animal's movement is modeled as an underdamped Langevin diffusion process. The stationary distribution of the location process, $\pi$, is defined as a Resource Selection Function (RSF) dependent on spatial covariates. This ensures that the long-term Utilization Distribution (UD) is strictly proportional to the RSF. The dynamics are governed by stochastic differential equations (SDEs) involving a friction term, Brownian noise, and an environmental drift driven by the gradient of the log-potential surface ($\nabla \log \pi$).
2. Measurement Error Model: The observed locations are treated as noisy realizations of the true latent locations, modeled via a Gaussian observation process. For Argos data, the error covariance is derived from provided error ellipses; for Fastloc GPS, a circular error assumption is used.
3. Inference via Laplace Approximation and TMB: The core challenge is that the habitat covariates are functions of the unobserved true locations (latent variables). To address this, the authors utilize the Template Model Builder (TMB) framework. They employ the Laplace approximation to integrate over the high-dimensional space of latent true locations and velocities. Crucially, this implementation extracts covariate values and calculates gradients from raster data during the optimization process, allowing the model to account for covariate uncertainty dependent on latent variables. This enables efficient maximum likelihood estimation (MLE) using automatic differentiation.
4. Spatial Constraints (Narwhal Case Study): In the application to narwhal data, the authors incorporate spatial constraints directly into the likelihood via penalty terms. A penalty is applied to the squared distance of latent locations from water to prevent the model from estimating locations on land, while additional penalties on speed and error scaling parameters ensure biological realism.

Key Contributions

• Unified Framework: The paper presents the first TMB implementation of a state-space model capable of handling covariates that depend on latent variables. This allows for the simultaneous estimation of movement parameters, habitat selection coefficients, and true trajectories without decoupling error correction from ecological inference.
• Computational Efficiency: By combining the Laplace approximation with automatic differentiation in TMB, the method achieves fast and scalable maximum likelihood inference, making it feasible to fit complex models to large datasets (e.g., 25,000 observations per simulation run) where sampling-based approaches (like MCMC) would be computationally prohibitive.
• Open Source Implementation: The authors provide an open-source R package (langevinSSM) for simulating and fitting the model, facilitating reproducibility and adoption by the movement ecology community.

Results

• Simulation Study: The Langevin SSM consistently outperformed the traditional two-step method.
  • Bias Reduction: The two-step method systematically underestimated habitat selection strength, with estimates biased toward zero even under low measurement error. In contrast, the Langevin SSM recovered the sign and magnitude of selection parameters with substantially less bias, even under 50% measurement error and high levels of missing data.
  • Coverage: The two-step method failed to provide valid confidence intervals (0% coverage) for selection parameters under moderate to high error, whereas the Langevin SSM maintained coverage between 85% and 97%.
  • Trajectory Recovery: The Langevin SSM recovered trajectories nearly identical to the true simulated paths, while the two-step method retained significant noise.
  • Utilization Distribution: The Langevin SSM preserved the true UD (high Bhattacharyya's affinity), whereas the two-step method produced distorted UDs.
• Narwhal Case Study: Applied to narwhal telemetry data in Qikiqtaaluk, Canada:
  • Habitat Selection: The unified approach identified a strong preference for deep waters (estimated coefficient $\hat{\beta}_1 = -3.1$), whereas the two-step method yielded a much weaker, attenuated signal ($\hat{\beta}_1 = -1.1$).
  • Parameter Stability: The two-step method produced implausibly high estimates for speed and persistence parameters (e.g., speed estimates reaching 522,420 km/h), indicating instability. The Langevin SSM produced biologically plausible estimates consistent with known narwhal behavior.
  • Spatial Correction: The unified model successfully pushed a substantial number of raw "land" locations into the water via its penalty structure, whereas the two-step method retained all erroneous land points.

Significance and Claims
The authors claim that their approach addresses long-standing challenges in movement ecology regarding measurement error and temporal irregularity. By unifying the filtering and inference stages, the method prevents the attenuation bias inherent in two-step procedures, leading to more accurate estimates of habitat selection and utilization distributions. The paper asserts that this framework is applicable across a wide range of taxa and environments, particularly in marine systems where data quality is often compromised. The authors note that while the method assumes a linear relationship between covariates and selection (and stationarity), it provides a computationally efficient, flexible foundation for future extensions, such as time-varying covariates or more complex discretization schemes to handle large observation gaps. The study emphasizes that jointly estimating movement and habitat selection is inherently superior to decoupling them, as habitat selection fundamentally influences movement dynamics.