# ZIB (Zero-Inflated Binomial) model with CAR process taking into account site alteration The `hSDM.ZIB.iCAR.alteration` function can be used to model species distribution including different processes in a hierarchical Bayesian framework: (i) a $Bernoulli$ suitability process (refering to environmental suitability) which takes into account the spatial dependence of the observations, (ii) an alteration process (refering to anthropogenic disturbances), and (iii) a $Binomial$ observability process (refering to various ecological and methodological issues explaining the species detection). The `hSDM.ZIB.iCAR.alteration` function calls a Gibbs sampler written in C code which uses an adaptive Metropolis algorithm to estimate the conditional posterior distribution of hierarchical model's parameters. ```r hSDM.ZIB.iCAR.alteration(presences, trials, suitability, observability, spatial.entity, alteration, data, n.neighbors, neighbors, suitability.pred=NULL, spatial.entity.pred=NULL, burnin = 5000, mcmc = 10000, thin = 10, beta.start, gamma.start, Vrho.start, mubeta = 0, Vbeta = 1e+06, mugamma = 0, Vgamma = 1e+06, priorVrho = "1/Gamma", shape = 0.5, rate = 0.0005, Vrho.max=1000, seed = 1234, verbose = 1, save.rho = 0, save.p = 0) ``` ## Arguments - `presences`: A vector indicating the number of successes (or presences) for each observation. - `trials`: A vector indicating the number of trials for each observation. $t_i$ should be superior to zero and superior or equal to $y_i$, the number of successes for observation $i$. - `suitability`: A one-sided formula of the form $'~x_1+...+x_p'$ with $p$ terms specifying the explicative variables for the suitability process. - `observability`: A one-sided formula of the form $'~w_1+...+w_q'$ with $q$ terms specifying the explicative variables for the observability process. - `spatial.entity`: A vector indicating the spatial entity identifier (from one to the total number of entities) for each observation. Several observations can occur in one spatial entity. A spatial entity can be a raster cell for example. - `alteration`: A vector indicating the proportion of area in the spatial cell which is transformed (by anthropogenic activities for example) for each observation. Must be between 0 and 1. - `data`: A data frame containing the model's variables. - `n.neighbors`: A vector of integers that indicates the number of neighbors (adjacent entities) of each spatial entity. `length(n.neighbors)` indicates the total number of spatial entities. - `neighbors`: A vector of integers indicating the neighbors (adjacent entities) of each spatial entity. Must be of the form c(neighbors of entity 1, neighbors of entity 2, ... , neighbors of the last entity). Length of the `neighbors` vector should be equal to `sum(n.neighbors)`. - `suitability.pred`: An optional data frame in which to look for variables with which to predict. If NULL, the observations are used. - `spatial.entity.pred`: An optional vector indicating the spatial entity identifier (from one to the total number of entities) for predictions. If NULL, the vector `spatial.entity` for observations is used. - `burnin`: The number of burnin iterations for the sampler. - `mcmc`: The number of Gibbs iterations for the sampler. Total number of Gibbs iterations is equal to `burnin+mcmc`. `burnin+mcmc` must be divisible by 10 and superior or equal to 100 so that the progress bar can be displayed. - `thin`: The thinning interval used in the simulation. The number of mcmc iterations must be divisible by this value. - `beta.start`: Starting values for $beta$ parameters of the suitability process. This can either be a scalar or a $p$-length vector. - `gamma.start`: Starting values for $gamma$ parameters of the observability process. This can either be a scalar or a $q$-length vector. - `Vrho.start`: Positive scalar indicating the starting value for the variance of the spatial random effects. - `mubeta`: Means of the priors for the $beta$ parameters of the suitability process. `mubeta` must be either a scalar or a p-length vector. If `mubeta` takes a scalar value, then that value will serve as the prior mean for all of the betas. The default value is set to 0 for an uninformative prior. - `Vbeta`: Variances of the Normal priors for the $beta$ parameters of the suitability process. `Vbeta` must be either a scalar or a p-length vector. If `Vbeta` takes a scalar value, then that value will serve as the prior variance for all of the betas. The default variance is large and set to 1.0E6 for an uninformative flat prior. - `mugamma`: Means of the Normal priors for the $gamma$ parameters of the observability process. `mugamma` must be either a scalar or a p-length vector. If `mugamma` takes a scalar value, then that value will serve as the prior mean for all of the gammas. The default value is set to 0 for an uninformative prior. - `Vgamma`: Variances of the Normal priors for the $gamma$ parameters of the observability process. `Vgamma` must be either a scalar or a p-length vector. If `Vgamma` takes a scalar value, then that value will serve as the prior variance for all of the gammas. The default variance is large and set to 1.0E6 for an uninformative flat prior. - `priorVrho`: Type of prior for the variance of the spatial random effects. Can be set to a fixed positive scalar, or to an inverse-gamma distribution ("1/Gamma") with parameters `shape` and `rate`, or to a uniform distribution ("Uniform") on the interval [0,`Vrho.max`]. Default set to "1/Gamma". - `shape`: The shape parameter for the Gamma prior on the precision of the spatial random effects. Default value is `shape=0.05` for uninformative prior. - `rate`: The rate (1/scale) parameter for the Gamma prior on the precision of the spatial random effects. Default value is `rate=0.0005` for uninformative prior. - `Vrho.max`: Upper bound for the uniform prior of the spatial random effect variance. Default set to 1000. - `seed`: The seed for the random number generator. Default set to 1234. - `verbose`: A switch (0,1) which determines whether or not the progress of the sampler is printed to the screen. Default is 1: a progress bar is printed, indicating the step (in %) reached by the Gibbs sampler. - `save.rho`: A switch (0,1) which determines whether or not the sampled values for rhos are saved. Default is 0: the posterior mean is computed and returned in the `rho.pred` vector. Be careful, setting `save.rho` to 1 might require a large amount of memory. - `save.p`: A switch (0,1) which determines whether or not the sampled values for predictions are saved. Default is 0: the posterior mean is computed and returned in the `prob.p.pred` vector. Be careful, setting `save.p` to 1 might require a large amount of memory. ## Returns - **mcmc**: An mcmc object that contains the posterior sample. This object can be summarized by functions provided by the coda package. The posterior sample of the deviance $D$, with c("$%$", "$\n$", "$ D=-2log(prod_i P(y_i,z_i|...))$"), is also provided. - **rho.pred**: If `save.rho` is set to 0 (default), `rho.pred` is the predictive posterior mean of the spatial random effect associated to each spatial entity. If `save.rho` is set to 1, `rho.pred` is an `mcmc` object with sampled values for each spatial random effect associated to each spatial entity. - **prob.p.pred**: If `save.p` is set to 0 (default), `prob.p.pred` is the predictive posterior mean of the probability associated to the suitability process for each prediction. If `save.p` is set to 1, `prob.p.pred` is an `mcmc` object with sampled values of the probability associated to the suitability process for each prediction. - **prob.p.latent**: Predictive posterior mean of the probability associated to the suitability process for each observation. - **prob.q.latent**: Predictive posterior mean of the probability associated to the observability process for each observation. ## Details The model integrates two processes, an ecological process associated to the presence or absence of the species due to habitat suitability and an observation process that takes into account the fact that the probability of detection of the species is inferior to one. The ecological process includes an intrinsic conditional autoregressive model (iCAR) model for spatial autocorrelation between observations, assuming that the probability of presence of the species at one site depends on the probability of presence of the species on neighboring sites. Ecological process: $$ z_i \sim \mathcal{B}ernoulli(\theta_i)z_i ~ Bernoulli(theta_i) $$ $$ logit(\theta_i) = X_i \beta + \rho_{j(i)}logit(theta_i) = X_i *beta + rho_j(i) $$ $rho_j$: spatial random effect $j(i)$: index of the spatial entity for observation $i$. Spatial autocorrelation: An intrinsic conditional autoregressive model (iCAR) is assumed: $$ \rho_j \sim \mathcal{N}ormal(\mu_j,V_{\rho} / n_j)rho_j ~Normal(mu_j,V_rho / n_j) $$ $mu_j$: mean of $rho_j'$ in the neighborhood of $j$. $V_rho$: variance of the spatial random effects. $n_j$: number of neighbors for spatial entity $j$. Observation process: $$ y_i \sim \mathcal{B}inomial(z_i * \delta_i, t_i)y_i ~ Binomial(z_i* delta_i, t_i) $$ $$ logit(\delta_i) = W_i \gammalogit(delta_i) = W_i * gamma $$ ## References Gelfand, A. E.; Schmidt, A. M.; Wu, S.; Silander, J. A.; Latimer, A. and Rebelo, A. G. (2005) Modelling species diversity through species level hierarchical modelling. **Applied Statistics**, 54, 1-20. Latimer, A. M.; Wu, S. S.; Gelfand, A. E. and Silander, J. A. (2006) Building statistical models to analyze species distributions. **Ecological Applications**, 16, 33-50. MacKenzie, D. I.; Nichols, J. D.; Lachman, G. B.; Droege, S.; Andrew Royle, J. and Langtimm, C. A. (2002) Estimating site occupancy rates when detection probabilities are less than one. **Ecology**, 83, 2248-2255. ## Author(s) Ghislain Vieilledent ghislain.vieilledent@cirad.fr ## See Also `plot.mcmc`, `summary.mcmc` ## Examples ```r ## Not run: #============================================== # hSDM.ZIB.iCAR.alteration() # Example with simulated data #============================================== #============ #== Preambule library(hSDM) library(raster) library(sp) library(mvtnorm) #================== #== Data simulation # Set seed for repeatability seed <- 1234 # Target parameters beta.target <- matrix(c(0.2,0.5,0.5),ncol=1) gamma.target <- matrix(c(1),ncol=1) ## Uncomment if you want covariates on the observability process ## gamma.target <- matrix(c(0.2,0.5,0.5),ncol=1) Vrho.target <- 1 # Spatial Variance # Landscape Landscape <- raster(ncol=20,nrow=20,crs='+proj=utm +zone=1') ncell <- ncell(Landscape) # Neighbors neighbors.mat <- adjacent(Landscape, cells=c(1:ncell), directions=8, pairs=TRUE, sorted=TRUE) n.neighbors <- as.data.frame(table(as.factor(neighbors.mat[,1])))[,2] adj <- neighbors.mat[,2] # Generate symmetric adjacency matrix, A A <- matrix(0,ncell,ncell) index.start <- 1 for (i in 1:ncell) { index.end <- index.start+n.neighbors[i]-1 A[i,adj[c(index.start:index.end)]] <- 1 index.start <- index.end+1 } # Spatial effects d <- 1 # Spatial dependence parameter = 1 for intrinsic CAR Q <- diag(n.neighbors)-d*A + diag(.0001,ncell) # Add small constant to make Q non-singular covrho <- Vrho.target*solve(Q) # Covariance of rhos set.seed(seed) rho <- c(rmvnorm(1,sigma=covrho)) # Spatial Random Effects rho <- rho-mean(rho) # Centering rhos on zero # Visited cells n.visited <- 150 # Compare with 400, 350 and 100 for example set.seed(seed) visited.cells <- sort(sample(1:ncell,n.visited,replace=FALSE)) # Draw visited cells at random notvisited.cells <- c(1:ncell)[-visited.cells] # Number of observations nobs <- 300 # Cell vector set.seed(seed) cells <- c(visited.cells,sample(visited.cells,nobs-n.visited,replace=TRUE)) coords <- xyFromCell(Landscape,cells) # Get coordinates # Covariates for "suitability" process set.seed(seed) X1.cell <- rnorm(n=ncell,0,1) set.seed(2*seed) X2.cell <- rnorm(n=ncell,0,1) X1 <- X1.cell[cells] X2 <- X2.cell[cells] X <- cbind(rep(1,nobs),X1,X2) # Alteration U <- runif(n=nobs,min=0,max=1) # Covariates for "observability" process W <- cbind(rep(1,nobs)) ## Uncomment if you want covariates on the observability process ## set.seed(3*seed) ## W1 <- rnorm(n=nobs,0,1) ## set.seed(4*seed) ## W2 <- rnorm(n=nobs,0,1) ## W <- cbind(rep(1,nobs),W1,W2) #== Simulating latent variables # Suitability logit.theta.1 <- vector() for (n in 1:nobs) { logit.theta.1[n] <- X[n,]%*%beta.target+rho[cells[n]] } theta.1 <- inv.logit(logit.theta.1) set.seed(seed) y.1 <- rbinom(nobs,1,theta.1) # Alteration u <- rbinom(nobs,1,U) # Observability set.seed(seed) trials <- rpois(nobs,5) # Number of trial associated to each observation trials[trials==0] <- 1 logit.theta.2 <- W%*%gamma.target theta.2 <- inv.logit(logit.theta.2) set.seed(seed) y.2 <- rbinom(nobs,trials,theta.2) #== Simulating response variable Y <- y.2*(1-u)*y.1 #== Data-set Data <- data.frame(Y,trials,cells,X1,X2,U) ## Uncomment if you want covariates on the observability process ## Data <- data.frame(Y,trials,cells,X1,X2,W1,W2,U) Data <- SpatialPointsDataFrame(coords=coords,data=Data) plot(Data) #== Data-set for predictions (suitability on each spatial cell) Data.pred <- data.frame(X1=X1.cell,X2=X2.cell,cells=c(1:ncell)) #================================== #== Site-occupancy model mod.hSDM.ZIB.iCAR.alteration <- hSDM.ZIB.iCAR.alteration(presences=Data$Y, trials=Data$trials, suitability=~X1+X2, observability=~1, spatial.entity=Data$cells, alteration=Data$U, data=Data, n.neighbors=n.neighbors, neighbors=adj, ## suitability.pred=NULL, ## spatial.entity.pred=NULL, suitability.pred=Data.pred, spatial.entity.pred=Data.pred$cells, burnin=5000, mcmc=5000, thin=5, beta.start=0, gamma.start=0, Vrho.start=10, priorVrho="1/Gamma", #priorVrho="Uniform", #priorVrho=10, mubeta=0, Vbeta=1.0E6, mugamma=0, Vgamma=1.0E6, shape=0.5, rate=0.0005, #Vrho.max=1000, seed=1234, verbose=1, save.rho=1, save.p=0) #========== #== Outputs #= Parameter estimates summary(mod.hSDM.ZIB.iCAR.alteration$mcmc) #= MCMC and posteriors pdf(file="Posteriors_hSDM.ZIB.iCAR.alteration.pdf") plot(mod.hSDM.ZIB.iCAR.alteration$mcmc) dev.off() pdf(file="Posteriors.rho_hSDM.ZIB.iCAR.alteration.pdf") plot(mod.hSDM.ZIB.iCAR.alteration$rho.pred) dev.off() #= Summary plots # rho r.rho <- r.rho.pred <- r.visited <- Landscape r.rho[] <- rho rho.pred <- apply(mod.hSDM.ZIB.iCAR.alteration$rho.pred,2,mean) r.rho.pred[] <- rho.pred r.visited[] <- 0 r.visited[visited.cells] <- 1 # prob.p r.prob.p <- Landscape r.prob.p[] <- mod.hSDM.ZIB.iCAR.alteration$prob.p.pred pdf(file="Summary_hSDM.ZIB.iCAR.alteration.pdf") par(mfrow=c(3,2)) plot(r.rho, main="rho target") plot(r.visited,main="Visited cells and presences") plot(Data[Y>0,],add=TRUE,pch=16,cex=0.5) plot(r.rho.pred, main="rho estimated") plot(rho[visited.cells],rho.pred[visited.cells], xlab="rho target", ylab="rho estimated") points(rho[notvisited.cells],rho.pred[notvisited.cells],pch=16,col="blue") legend(x=-4,y=3.5,legend="Unvisited cells",col="blue",pch=16,bty="n") abline(a=0,b=1,col="red") plot(r.prob.p,main="Proba of presence") plot(Data[Y>0,],add=TRUE,pch=16,cex=0.5) dev.off() ## End(Not run) ```

hSDM.ZIB.iCAR.alteration function