*Ecological Archives* E094-022-A1

J. Andrew Royle, Richard B. Chandler, Kimberly D. Gazenski, Tabitha A. Graves. 2013. Spatial capture–recapture models for jointly estimating population density and landscape connectivity. *Ecology* 94:287–294. http://dx.doi.org/10.1890/12-0413.1

Appendix A. Details for computing the marginal likelihood and obtaining the maximum likelihood estimators (MLEs).

The key operation for computing the likelihood is solving the 2-dimensional integration problem to remove s. There are some general purpose R packages that implement a number of multi-dimensional integration routines including R2Cuba (Hahn et al., 2010). We won't rely on these extraneous R packages but instead will use perhaps less efficient methods in which we replace the integral with a summation over an equal area mesh of points on the state-space *S* and explicitly evaluate the integrand at each point. We invoke the rectangular rule for integration here in which the integrand is evaluated on a regular grid of points of equal area and then averaged. Let *u* = 1, 2, ..., *nG* index a grid of *nG* points, *s*_{u}, where the area of grid cell *u* is constant. In this case, the integrand, i.e., the marginal pmf of *y*_{i}, is approximated by:

(A.1) |

To deal with the fact that *N* is unknown, there are two key issues that need to be addressed. First is that we don't observe the "all-zero" encounter histories (i.e., *y*_{ij} = 0 for all *j*) corresponding to uncaptured individuals, so we have to make sure we compute the probability for that all zero encounter history which we do operationally by tacking a row of zeros onto the encounter history matrix. We include the number of such all-zero encounter histories as an unknown parameter of the model, which we label *n*_{0}. In addition, we have to be sure to include a combinatorial term to account for the fact that of the *n* observed individuals there are () ways to realize a sample of size *n*. The combinatorial term involves the unknown *n*_{0} and thus it must be included in the likelihood.

To compute the integral requires that the bounds of integration are specified, which is equivalent to prescribing the state-space of the underlying point process, i.e., *S*. Given *S*, density is computed as *D*(*S*) = *N*/||*S*||. In our simulation study below we report *N* as the two are equivalent summaries of the data set once the state-space is fixed.

We wrote an R function to evaluate the likelihood which we optimize using the R function optim (or, alternatively, nlm).

Literature Cited

Hahn, T., A. Bouvier, and K. Kiêu, 2010. R2Cuba: Multidimensional Numerical Integration. R package version 1.0-6.