Extreme Value-Based Methods for Modeling Elk Yearly Movements

Abstract

Species range shifts and the spread of diseases are both likely to be driven by extreme movements, but are difficult to statistically model due to their rarity. We propose a statistical approach for characterizing movement kernels that incorporate landscape covariates as well as the potential for heavy-tailed distributions. We used a spliced distribution for distance travelled paired with a resource selection function to model movements biased toward preferred habitats. As an example, we used data from 704 annual elk movements around the Greater Yellowstone Ecosystem from 2001 to 2015. Yearly elk movements were both heavy-tailed and biased away from high elevations during the winter months. We then used a simulation to illustrate how these habitat effects may alter the rate of disease spread using our estimated movement kernel relative to a more traditional approach that does not include landscape covariates. Supplementary materials accompanying this paper appear online.

Appendices

Appendix A: Treatment of Missing Values

We encountered missing values in certain landcover rasters. Aggregating the raster reduced the number of missing values significantly. We observed that missing values were in landscape cells that fell on the boundaries of certain rasters (this is clearly seen in Fig. 2). We also noted that landscape cells with missing values were far away from the region where we observed elk movements. We used a $27 \times 27$ grid to interpolate the value of a missing landscape cell. We weighted the grid giving landscape cells closer to the missing landscape cell higher weights than those further away.

Appendix B: Simulation Study to Test Bias in Estimation

In order to check if there is a systematic bias induced by approximating the normalizing constant using a Monte Carlo approximation, we conducted a simulation study with a single covariate (Elevation) and a gamma distance kernel. We set the true parameter values to $(\beta , k, \theta )=(-1, 0.8615017, 12.63035)$. We simulated 1000 datasets each with 200 yearly movements where the start locations for the 200 movements were chosen randomly from the 704 movements in the dataset. We simulated from our model using importance sampling. We first simulated many end locations $\mathbf {e}_i$ for each start location by simulating bearings and distances from the distance kernel. We then sampled a single end location drawn randomly from the simulated end locations, with probabilities proportional to $\mathrm{e}^{x(\mathbf {e}_i) \beta }$. After simulating 200 yearly movements in this way, we estimated model parameters using the Bayesian method given in Sect. 3. We used 2500 Monte Carlo samples to approximate the normalizing constant. We fit the model to the simulated data running the MCMC sampler for 30,000 iterations and removed the first 5000 runs as burn in. We obtained the posterior mode for each parameter from each dataset and plotted their densities in Fig. 6. The true values are given using a solid vertical line. It is clearly evident from Fig. 6 that there is very little systematic bias induced when using the proposed estimation procedure.

Fig. 6

The distributions of posterior modes for the parameters a$\beta$, b$\log (k)$ and c$\log (\theta )$ for models with a fixed covariate elevation and a gamma distance kernel. The true values of the parameters are $\beta =-1$, $\log (k)=-0.1490782$ and $\log (\theta )=2.536103$. There is no evidence of an estimation bias.

We also simulated 3 datasets using gamma and spliced distance kernels. The true parameters used were $(\beta , k, \theta )=(-1, 0.8615017, 12.63035)$ and $\xi =0.2$ with $u =10$ and $\phi _u=0.3$. The start locations were taken to be those of the actual data. We simulated the end locations similarly to that given above using the relevant distance kernel. We used 5000 Monte Carlo samples in the likelihood estimation. The results are given in Fig. 7. We found that all the true parameters were captured well giving us the assurance that the estimation procedure does well for models with spliced kernels too.

Fig. 7

The posterior distributions of the parameters a$\beta$, b$\log (k)$, c$\log (\theta )$ and d$\xi$ for models with a fixed covariate elevation and a Gamma distance kernel (dotted line), a BTFM distance kernel (dashed line) and a PTFM distance kernel (solid line). The true values of the parameters are $\beta =-1$, $\log (k)=-0.1490782$, $\log (\theta )=2.536103$ and $\xi =0.2$.