0 500 1000 1500 2000 2500 Elevation (m)

Figure 9.3. Estimated response of survival and colonization probability to elevation for the European Crossbill estimated from the Swiss BBS data.

0 500 1000 1500 2000 2500 Elevation (m)

Figure 9.3. Estimated response of survival and colonization probability to elevation for the European Crossbill estimated from the Swiss BBS data.

'rooks' neighborhood'). Sometimes the diagonal cells are included. For an irregular lattice (e.g., counties or other administrative units), it is common to define the neighborhood to be units that share a boundary. Sargeant et al. (2005) apply their model to distribution modeling of kit foxes, in which the lattice consists of townships. Let ni be the cardinality of Ni, and define the spatial auto-covariate

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Figure 9.4. Binary image with auto-logistic parameters a = -0.5 and ft = 1.5 yielding an occupancy process with ^ ~ 0.60. The white cells are unoccupied and the colored cells are occupied.

10 20 30 40

Figure 9.4. Binary image with auto-logistic parameters a = -0.5 and ft = 1.5 yielding an occupancy process with ^ ~ 0.60. The white cells are unoccupied and the colored cells are occupied.

Thus, Xj is the number of occupied neighbors of site i. The auto-logistic model is specified by the conditional distributions [zj|z_j] where z_j denotes the vector of all latent state variables except i. In particular, [zj|z—i] = Bern(ftj) where logit(ftj) = fto + ftiXj.

Other (extrinsic) covariates, e.g., describing landscape or habitat could also be considered (and usually are). Accessible formulations of these models can be found in Hoeting et al. (2000) and Wintle and Bardos (2006).

We illustrate the analysis of these models with simulated data on a rectangular lattice of square sample units (quadrats) and for the special case where the state variable is observed perfectly. Even for this case, the likelihood for this model is intractable and analysis is carried out by approximate methods (e.g., 'pseudo-likelihood') or by MCMC under a fully-Bayesian treatment of the model. Bayesian analysis of the auto-logistic model was considered in the context of distribution modeling by Heikkinen and Hogmander (1994), Wu and Huffer (1997), Huffer and Wu (1998) and, using WinBUGS, Wintle and Bardos (2006).

We suppose the neighbors are the quadrats immediately to the north, south, east and west, so a site has a maximum of 4 neighbors. Let G be the size of the grid (number of grid points), numnn[i] is the number of neighbors each quadrat has, and NN is a matrix having dimension G x max(numnn) where NN[i,j] is the integer identity of neighbor j to quadrat i. This auto-logistic model under perfect observation of the state variable is shown in Panel 9.3. In this case, where z is observed, and the inference problem is only to estimate the parameters ft0 and fti. In the Web Supplement we provide an R script for simulating data and fitting the model in R. It would also be possible to employ this model for the case where not all G quadrats were sampled, in which case a component of the inference problem might be to predict the missing values of Zj.

A realization from this process is shown in Figure 9.4, which has a = —0.5 and ft = 1.5. This generates a mean occupancy of about 0.60. The realization shown here was obtained by Gibbs sampling. Note that, historically, image analysis applications inspired the development of Gibbs sampling algorithms (Geman and Geman, 1984).

An important technical consideration of these models is that the auto-covariate is unobserved. It is unobserved because we may observe the state variable imperfectly, and so we don't know whether any particular neighbor is occupied, even if the model{

logit(psi[i])<- alpha + beta*(x[i,numnn[i]+1]/numnn[i]) z[i]~dbern(psi[i])

Panel 9.3. WinBUGS model specification for an auto-logistic model with perfect observation of the binary state variable z.

neighboring unit was sampled. Also, we may not sample all G spatial units. However, the model is amenable to a Bayesian analysis by MCMC as missing or partially observed variables are treated no differently in the manner by which they are simulated. Almost all contemporary applications of auto-logistic type models are carried out using simulation-based methods.

In the context of binary maps, imperfect detection will cause the appearance of reduced spatial dependence than exists in the process itself. Since, presumably, spatial dependence is primarily the result of ecological processes of relevance (dispersal, migration, diffusive spread), understatement of spatial dependence should be detrimental to inference about such processes.

A hierarchical extension of the model is achieved with a formalization of the observation process. For example, consider the design in which J replicate samples are made of each spatial unit. Then, the observation model for the total number of detections for sample unit i is, for the case where detection probability is constant, yi - Bin(J,pzj).

In general, p may vary among sites based on search intensity and other factors and so additional model structure on p might be desired. Note that this observation model is precisely equivalent to that considered in previous models of occurrence in which imperfect detection was considered, e.g., the models of Chapter 3. Sargeant et al. (2005) used a slightly different observation model. In their study, they sampled for the presence of kit fox tracks, and sampling was conducted until the presence of foxes was confirmed, or until 3 samples were conducted. This is a classic 'removal' design (e.g., see MacKenzie et al., 2006, p. 102).

Under this form of observation error, the model described previously (and implemented in Panel 9.3) is unchanged as a description of the state process. Naturally, since we have seen this observation model so many times before, extending the WinBUGS model specification requires little additional model description. This is shown in Panel 9.4. As before, the R code to simulate data and execute WinBUGS is provided in the Web Supplement. Many modifications to this observation model are possible, as have been elaborated on elsewhere in this book.

While the model, as formulated here, is ideally suited for spatial processes that are defined on a discrete lattice, any region could be rendered into a lattice arbitrarily, with points assigned to grid cells within which they are located. Alternatively, the model can be given a point support formulation by specification of the auto-covariate according to having weights Wj which would typically be a function of distance. In the case considered previously, Wj = 1/n.j if unit j is a neighbor of i and Wj = 0 otherwise. When the spatial process can be viewed as having point support such as locations of trees, or locations of bird nests, then inverse-distance weights have been used (Rathbun and Cressie, 1994) in which:

model{

alpha ~ dnorm(0,.01) beta ~ dnorm(0,.01) p ~ dunif(0,1)

logit(psi[i])<- alpha + beta*(x[i,numnn[i]+1]/numnn[i])

Panel 9.4. WinBUGS model specification for an auto-logistic model with observation of the state variable z subject to imperfect detection.

or, a local density covariate such as

for some threshold distance S. A conceptual issue arises if the locations of all elements of the population are not known. In this case, variation in xj can be largely a result of nuisance sampling processes. In general, one would then need a model describing the probability that an element appears in the sample as a function of its location.

If a geographic area is 'discretized,' then the scale of the lattice relative to the observation network becomes an important issue. The scale of observation has to be roughly consistent with the scale of the lattice under consideration in the sense that we can't have too sparse of an observation network or parameters are poorly identified. When the grid is too fine, then a first order neighborhood won't exhibit much spatial structure. But to increase the neighborhood size induces considerable computational expense. As an example, consider the Swiss bird survey, which is based on sampling several hundred of approximately 41000 1 km quadrats. Using a first order rooks neighborhood would almost certainly produce ft « 0 since there is basically no information about the value of neighboring states from such a sparse observation network. Thus, the coarseness of the lattice has to be consistent in some way with the spatial structure in the process being modeled.

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