Joint Temporal Point Pattern Models for Proximate Species Occurrence in a Fixed Area Using Camera Trap Data

Abstract

The distinction between an overlap in species daily activity patterns and proximate co-occurrence of species for a location and time due to behavioral attraction or avoidance is critical when addressing the question of species co-occurrence. We use data from a dense grid of camera traps in a forest in central North Carolina to inform about proximate co-occurrence. Camera trigger times are recorded when animals pass in front of the camera’s field of vision. We view the data as a point pattern over time for each species and model the intensities driving these patterns. These species-specific intensities are modeled jointly in linear time to preserve the notion of co-occurrence. We show that a multivariate log-Gaussian Cox process incorporating both circular and linear time provides a preferred choice for modeling occurrence of forest mammals based on daily activity rhythms. Model inference is obtained under a hierarchical Bayesian framework with an efficient Markov chain Monte Carlo sampling algorithm. After model fitting, we account for imperfect detection of individuals by the camera traps by incorporating species-specific detection probabilities that adjust estimates of occurrence and co-occurrence. We obtain rich inference including assessment of the probability of presence of one species in a particular time interval given presence of another species in the same or adjacent interval, enabling probabilities of proximate co-occurrence. Our results describe the ecology and interactions of four common mammals within this suburban forest including their daily rhythms, responses to temperature and rainfall, and effects of the presence of predator species. Supplementary materials accompanying this paper appear online.

A Appendix

We specified prior distributions for the parameters in the models outlined in Sect. 3.1. Samples from the joint posterior distribution were obtained by a customized Metropolis–Hastings algorithm. For models 1 and 2, each chain was run for 1,000,000 iterations and thinned to every 50th iteration to reduce dependence within the chain. The chain for model 3 was run for 2,000,000 iterations and thinned to every 100th iteration. The first half of the chain was discarded as burn-in and the remaining 10,000 iterations used for posterior inference.

Each species-specific coefficient vector, $\varvec{\beta }^{(r)}$ for $r = 1, \ldots , R$, was modeled as $\varvec{\beta }^{(r)} \sim \text {MVN}(\mathbf {0}, \mathbf {I})$ and was updated in block using a Metropolis step. The diagonal elements of the coregionalization matrix $\mathbf {A}$ were assigned log-normal priors where the mean of each $A_{rr}$ was equal to 1. The off-diagonal element was assigned a mean 0 normal priors. Each element of the lower-triangular matrix of $\mathbf {A}$ was updated univariately using a Metropolis–Hastings algorithm.

The temporal random effects were each modeled using a multivariate normal distribution and updated using an elliptical slice sampler (Murray et al. 2010). The elliptical slice sampler was constructed for sampling $\mathbf {V}_r^*$, the temporal random effects orthogonal to the fixed effects in the model as defined in (4). The decay parameter of the exponential covariance model was fixed to $\phi _r =1/48$ for each r, resulting in a rate of decay of 1/48 per hour or 24/48 per day. The effective range, or, rather, the temporal lag at which the correlation drops below 0.05, was approximately 6 days and was chosen such that it was not on the same scale as the periodic functions or other fixed effects.