Abstract
Movement for many animal species is constrained in space by barriers such as rivers, shorelines, or impassable cliffs. We develop an approach for modeling animal movement constrained in space by considering a class of constrained stochastic processes, reflected stochastic differential equations. Our approach generalizes existing methods for modeling unconstrained animal movement. We present methods for simulation and inference based on augmenting the constrained movement path with a latent unconstrained path and illustrate this augmentation with a simulation example and an analysis of telemetry data from a Steller sea lion (Eumatopias jubatus) in southeast Alaska.
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Acknowledgements
Funding for this research was provided by NSF (DEB EEID 1414296), NIH (GM116927-01), NOAA (RWO 103), CPW (TO 1304), and NSF (DMS 1614392). Any use of trade, firm, or product names is for descriptive purposes only and does not imply endorsement by the U.S. Government. We thank Brett McClintock, Jay VerHoef, two anonymous reviewers, and an anonymous AE for their helpful suggestions on an earlier draft of this manuscript.
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Appendices
Appendix A: Algorithm for Simulating RSDEs
We here describe an algorithm for simulating from RSDEs constrained to lie within a domain \(\mathcal {D}\).
Appendix B: Simulation Example
In this Appendix, we consider a simple simulation example that illustrates the possible bias incurred by not accounting for constraints in movement. We consider a simple version of the reflected SDE in (8)–(9) in which \(\beta =0\) and \(c(\mathbf {x},\tau )=\sigma \)
and where \(\mathbf {k}_\tau \) is as given in (10). We refer to this constrained process as reflected integrated Brownian motion (RIBM) because the unconstrained version of this process (when \(\mathbf {k}_\tau =0\)) is two-dimensional integrated Brownian motion (IBM). Figure 5a shows a path simulated from RIBM for a given polygonal constraint \(\mathcal {D}\). When the path is far from the boundary \(\partial \mathcal {D}\), the path behaves identically to IBM. When the path is near the border \(\partial \mathcal {D}\), it often ends up identically on the boundary for short periods of time, because the minimal \(\mathbf {k}\) process keeps \(\mathbf {x}_t\) from leaving \(\mathcal {D}\). We also simulated noisy telemetry observations (Fig. 5b) under Gaussian observation error at 300 regularly spaced time points
We considered estimation of model parameters \({\varvec{\theta }}=(\sigma ,\kappa )'\) by specifying diffuse half-normal priors (with variance = 100) for \(\sigma \) and \(\kappa \), and sampling from the posterior distribution \([{\varvec{\theta }}|\mathbf {s}_{1:300}]\) using PMMH algorithms (Andrieu et al. 2010). Code to replicate this simulation study is available upon request. We considered estimation from the true model, RIBM, by constructing a particle filter using the projected simulation approach in Algorithm 1. We estimated model parameters using an unconstrained IBM model for \(\mathbf {x}_\tau \) by constructing a particle filter without any projection or constraint. We note that it is possible to estimate \({\varvec{\theta }}\) under IBM by marginalizing over \(\mathbf {x}_{1:300}\) using the convolution approach of Hooten and Johnson (2017), but we instead used PMMH to allow a more direct comparison between the estimates of \({\varvec{\theta }}\) under constrained (RIBM) and unconstrained (IBM) models. Each PMMH sampler was run for 10,000 iterations, with convergence of Markov chains assessed visually.
Simulation example (IBM). An individual’s path, constrained to occur within the polygon, was simulated using RIBM (a), and iid Gaussian error was added (b) to simulate telemetry observation error. The observation error variance (c) and latent Brownian motion error variance (d) were estimated using PMMH for both RIBM and IBM models with diffuse half-normal priors. Results show some bias in the Brownian motion error variance. Samples from the posterior distribution of animal paths \(\mathbf {x}_{1:T}|\mathbf {s}_{1:T}\) from the RIBM model (e) are constrained to lie within \(\mathcal {D}\), while those from the IBM model (f) are not.
In general, the PMMH algorithm is less computationally efficient for our system than the MCMC algorithm that we develop in Sect. 3.1. The PMMH algorithm essentially attempts a block update of the entire latent path \(\mathbf {x}_{1:300}\) at each MCMC algorithm, while the approach in Sect. 3.1 considers updating each \(\mathbf {x}_t\) one at a time. Code to compare both of these approaches is available upon request.
Figure 5c shows the estimated posterior distributions for the observation error standard deviation \(\kappa \) under IBM and RIBM, and Fig. 5d shows the posterior distributions for the Brownian motion standard deviation \(\sigma \). There is little difference between constrained (RIBM) and unconstrained (IBM) models in the posterior distribution of the observation error \(\kappa \) (Fig. 5c), but the IBM model overestimates the Brownian motion standard deviation \(\sigma \) (Fig. 5d). Figure A.1e shows 10 sample paths from the posterior distribution of \(\mathbf {x}_{1:300}\) under RIBM, and Fig. 1f shows 10 sample paths from the posterior distribution under IBM. From this simulation example, it is clear that parameter estimates obtained without accounting for constraints in movement can show bias under model misspecification, though even under misspecification the 95\(\%\) equal-tailed credible intervals of all model parameters under IBM include the true values simulated under RIBM. This may indicate that estimates obtained by fitting unconstrained movement models may be useful, even when we know the underlying movement process is constrained.
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Hanks, E.M., Johnson, D.S. & Hooten, M.B. Reflected Stochastic Differential Equation Models for Constrained Animal Movement . JABES 22, 353–372 (2017). https://doi.org/10.1007/s13253-017-0291-8
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DOI: https://doi.org/10.1007/s13253-017-0291-8