Hierarchical Nonlinear Spatio-temporal Agent-Based Models for Collective Animal Movement

Abstract

Modeling complex collective animal movement presents distinct challenges. In particular, modeling the interactions between animals and the nonlinear behaviors associated with these interactions, while accounting for uncertainty in data, model, and parameters, requires a flexible modeling framework. To address these challenges, we propose a general hierarchical framework for modeling collective movement behavior with multiple stages. Each of these stages can be thought of as processes that are flexible enough to model a variety of complex behaviors. For example, self-propelled particle (SPP) models (e.g., Vicsek et al. in Phys Rev Lett 75:1226–1229, 1995) represent collective behavior and are often applied in the physics and biology literature. To date, the study and application of these models has almost exclusively focused on simulation studies, with less attention given to rigorously quantifying the uncertainty. Here, we demonstrate our general framework with a hierarchical version of the SPP model applied to collective animal movement. This structure allows us to make inference on potential covariates (e.g., habitat) that describe the behavior of agents and rigorously quantify uncertainty. Further, this framework allows for the discrete time prediction of animal locations in the presence of missing observations. Due to the computational challenges associated with the proposed model, we develop an approximate Bayesian computation algorithm for estimation. We illustrate the hierarchical SPP methodology with a simulation study and by modeling the movement of guppies.

Supplementary materials accompanying this paper appear online.

Appendices

Appendix A: Full HiSPP Process Model Equation

The process model (i.e., Eq. 5) from Sect. 4.2 contains various update rules based on the values of the updated locations. In particular, if one or both of the updated x and y locations are outside the boundary of interest, then separate update conditions are invoked. These separate update conditions ensure that the updated locations are within the boundary of interest. We use reflective conditions to push the agents back toward the region of interest. The direction in which the agents reflect back into the region of interest is estimated for the cases where an agent’s updated location is beyond the x or y boundary. If an agent’s updated location is outside both the x and y boundary, then the reflective angle is set to a fixed value (this scenario was not common for our application). Recall that the latent locations, $\mathbf {s}_{i,t}$, are confined to a bounded region such that $\tilde{x}_{i,t} \in [0,x_{m}]$ and $\tilde{y}_{i,t} \in [0,y_{m}]$. The full process model is then given by the following:

$$\begin{aligned} f(\mathbf {s}_{i,t-1} ; \Theta )= {\left\{ \begin{array}{ll} ( \tilde{x}_{i,t-1}+u_{i,t}\delta _{x,i,t}, \ \tilde{y}_{i,t-1}+u_{i,t}\delta _{y,i,t} )' \ , \quad &{} \text {if}\ \tilde{x}_{i,t-1}+u_{i,t}\delta _{x,i,t}\in [0,x_{m}], \\ \ &{} \tilde{y}_{i,t-1}+u_{i,t}\delta _{y,i,t} \in [0,y_{m}] \\ (\tilde{x}_{i,t-1}+x_B\delta _{x,i,t}, \ \tilde{y}_{i,t-1}+u_{i,t}\delta _{y,i,t} )' \ , \quad &{} \text {if}\ \ \tilde{x}_{i,t-1}+u_{i,t}\delta _{x,i,t}<0 \\ (\tilde{x}_{i,t-1}+u_{i,t}\delta _{x,i,t}, \ \tilde{y}_{i,t-1}+u_{i,t}y_B )' \ , \quad &{} \text {if}\ \ \tilde{y}_{i,t-1}+u_{i,t}\delta _{y,i,t}<0 \\ \left( \tilde{x}_{i,t-1}+u_{i,t}\left( \frac{\pi }{2}+x_B\right) , \ \tilde{y}_{i,t-1}+u_{i,t}\delta _{y,i,t} \right) ' \ , \quad &{} \text {if}\ \ \tilde{x}_{i,t-1}+u_{i,t}\delta _{x,i,t}>x_{m} \\ \left( \tilde{x}_{i,t-1}+u_{i,t}\delta _{x,i,t}, \ \tilde{y}_{i,t-1}+u_{i,t}\left( \frac{\pi }{2}+y_B\right) \right) ' \ , \quad &{} \text {if}\ \ \tilde{y}_{i,t-1}+u_{i,t}\delta _{y,i,t}>y_{m} \ , \\ \left( \tilde{x}_{i,t-1}+u_{i,t}\frac{5\pi }{4}, \ \tilde{y}_{i,t-1}+u_{i,t} \frac{5\pi }{4}\right) ' \ , \quad &{} \text {if}\ \ \tilde{x}_{i,t-1}+u_{i,t}\delta _{x,i,t}>x_{m}, \\ \ &{} \tilde{y}_{i,t-1}+u_{i,t}\delta _{y,i,t}>y_{m} \ \\ \left( \tilde{x}_{i,t-1}+u_{i,t}\frac{3\pi }{4}, \ \tilde{y}_{i,t-1}+u_{i,t} \frac{3\pi }{4}\right) ' \ , \quad &{} \text {if}\ \ \tilde{x}_{i,t-1}+u_{i,t}\delta _{x,i,t}>x_{m}, \\ \ &{} \tilde{y}_{i,t-1}+u_{i,t}\delta _{y,i,t}<0 \ \\ \left( \tilde{x}_{i,t-1}+u_{i,t}\frac{7\pi }{4}, \ \tilde{y}_{i,t-1}+u_{i,t} \frac{7\pi }{4}\right) ' \ , \quad &{} \text {if}\ \ \tilde{x}_{i,t-1}+u_{i,t}\delta _{x,i,t}<0, \\ \ &{} \tilde{y}_{i,t-1}+u_{i,t}\delta _{y,i,t} >y_{m} \ \\ \left( \tilde{x}_{i,t-1}+u_{i,t}\frac{\pi }{4}, \ \tilde{y}_{i,t-1}+u_{i,t} \frac{\pi }{4}\right) ' \ , \quad &{} \text {if}\ \ \tilde{x}_{i,t-1}+u_{i,t}\delta _{x,i,t}<0, \\ \ &{} \tilde{y}_{i,t-1}+u_{i,t}\delta _{y,i,t} <0. \ \\ \end{array}\right. } \end{aligned}$$

Appendix B: Specification of Priors

Listed below are all of the prior distributions and corresponding hyperparameter values for the HiSPP model from Sect. 4.2.

$\sigma _\epsilon ^2\sim \text {IG}(\alpha _\epsilon ,\beta _\epsilon )$, where $\alpha _\epsilon =.001$ and $\beta _\epsilon =.001$.

$\sigma _\eta ^2\sim \text {IG}(\alpha _\eta ,\beta _\eta )$, where $\alpha _\eta =4.25$ and $\beta _\eta =24.375$.

$r \sim \text {TN}_{[0,r_{\text {max}}]}(\mu _r,\sigma ^2_r)$, where $\mu _r=$0, $\sigma ^2_r=100000$, and $r_{\text {max}}=378.4$.

$\beta _0\sim \text {Gau}(\mu _\beta ,\sigma ^2_\beta )$ and $\beta _1\sim \text {Gau}(\mu _\beta ,\sigma ^2_\beta )$, where $\mu _\beta =0$ and $\sigma ^2_\beta =1000$.

$\lambda _1\sim \text {Gau}(\mu _\lambda ,\sigma ^2_\lambda )$ and $\lambda _2\sim \text {Gau}(\mu _\lambda ,\sigma ^2_\lambda )$, where $\mu _\lambda =0$ and $\sigma ^2_\lambda =1000$.

$x_B\sim \text {U}(a_x,b_x)$, where $a_x=0$ and $b_x=\frac{\pi }{2}$.

$y_B\sim \text {U}(a_y,b_y)$, where $a_y=\frac{\pi }{2}$ and $b_y=\pi$.

$\tau \sim \text {U}(a_\tau ,b_\tau )$, where $a_\tau =0$ and $b_\tau =.75$.

$\rho \sim \text {U}(a_\rho ,b_\rho )$, where $a_\rho =0$ and $b_\rho =6$.

Since many of the hyperparameters above are utilized at the bottom of the hierarchical model, they are given relatively uninformative prior distributions. The hyperparameters for the boundary conditions (i.e., $x_B$ and $y_B$) were selected such that agents would always be reflected back into the region of interest. See supplementary materials (S.5) for a further discussion on the priors used with the HiSPP model.