Poisson log regression model with CAR process

The hSDM.poisson.iCAR function performs a Poisson log regression in a hierarchical Bayesian framework. The suitability process includes a spatial correlation process. The spatial correlation is modelled using an intrinsic CAR model. The hSDM.poisson.iCAR 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.

hSDM.poisson.iCAR(counts, suitability, spatial.entity, data,
n.neighbors, neighbors, suitability.pred=NULL, spatial.entity.pred=NULL,
burnin = 5000, mcmc = 10000, thin = 10, beta.start, Vrho.start, mubeta =
0, Vbeta = 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

counts: A vector indicating the count (or abundance) for each observation.
suitability: A one-sided formula of the form $\sim x_1+...+x_p$ with $p$ terms specifying the explicative variables for the suitability 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.
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.
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.
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 lambda.pred vector. Be careful, setting save.p to 1 might require a large amount of memory.

Value

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 $D=-2\log(\prod_i P(y_i|...))$, 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.
lambda.pred: If save.p is set to 0 (default), lambda.pred is the predictive posterior mean of the abundance associated to the suitability process for each prediction. If save.p is set to 1, lambda.pred is an mcmc object with sampled values of the abundance associated to the suitability process for each prediction.
lambda.latent: Predictive posterior mean of the abundance associated to the suitability process for each observation.

Details

We model an ecological process where the abundance of the species is explained by habitat suitability. The ecological process includes an intrinsic conditional autoregressive (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: $$y_i \sim \mathcal{P}oisson(\lambda_i,t_i)$$ $$log(\lambda_i) = X_i \beta + \rho_{j(i)}$$ $\rho_j$: spatial random effect

$j(i)$: index of the spatial entity for observation $i$.

Spatial autocorrelation:

An intrinsic conditional autoregressive model (iCAR) is assumed: $$\rho_j \sim \mathcal{N}ormal(\mu_j,V_{\rho} / n_j)$$ $\mu_j$: 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 $j$.

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.

Lichstein, J. W.; Simons, T. R.; Shriner, S. A. & Franzreb, K. E. (2002) Spatial autocorrelation and autoregressive models in ecology Ecological Monographs, 72, 445-463.

Diez, J. M. & Pulliam, H. R. (2007) Hierarchical analysis of species distributions and abundance across environmental gradients Ecology, 88, 3144-3152.

Author

Ghislain Vieilledent ghislain.vieilledent@cirad.fr

Examples


if (FALSE) {

#==============================================
# hSDM.poisson.iCAR()
# Example with simulated data
#==============================================

#=================
#== Load libraries                          
library(hSDM)
library(raster)
library(sp)

#===================================
#== Multivariate normal distribution
rmvn <- function(n, mu = 0, V = matrix(1), seed=1234) {
    p <- length(mu)
    if (any(is.na(match(dim(V), p)))) {
        stop("Dimension problem!")
    }
    D <- chol(V)
    set.seed(seed)
    t(matrix(rnorm(n*p),ncol=p)%*%D+rep(mu,rep(n,p)))
}

#==================
#== Data simulation

#= Set seed for repeatability
seed <- 1234

#= Landscape
xLand <- 30
yLand <- 30 
Landscape <- raster(ncol=xLand,nrow=yLand,crs='+proj=utm +zone=1')
Landscape[] <- 0
extent(Landscape) <- c(0,xLand,0,yLand)
coords <- coordinates(Landscape)
ncells <- ncell(Landscape)

#= Neighbors
neighbors.mat <- adjacent(Landscape, cells=c(1:ncells), 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,ncells,ncells)
index.start <- 1
for (i in 1:ncells) {
    index.end <- index.start+n.neighbors[i]-1
    A[i,adj[c(index.start:index.end)]] <- 1
    index.start <- index.end+1
}

#= Spatial effects
Vrho.target <- 5
d <- 1  # Spatial dependence parameter = 1 for intrinsic CAR
Q <- diag(n.neighbors)-d*A + diag(.0001,ncells) # Add small constant to make Q non-singular
covrho <- Vrho.target*solve(Q) # Covariance of rhos
set.seed(seed)
rho <- c(rmvn(1,mu=rep(0,ncells),V=covrho,seed=seed)) # Spatial Random Effects
rho <- rho-mean(rho) # Centering rhos on zero

#= Raster and plot spatial effects
r.rho <- rasterFromXYZ(cbind(coords,rho))
plot(r.rho)

#= Sample the observation sites in the landscape
nsite <- 250
set.seed(seed)
x.coord <- runif(nsite,0,xLand)
set.seed(2*seed)
y.coord <- runif(nsite,0,yLand)
sites.sp <- SpatialPoints(coords=cbind(x.coord,y.coord))
cells <- extract(Landscape,sites.sp,cell=TRUE)[,1]

#= Ecological process (suitability)
set.seed(seed)
x1 <- rnorm(nsite,0,1)
set.seed(2*seed)
x2 <- rnorm(nsite,0,1)
X <- cbind(rep(1,nsite),x1,x2)
beta.target <- c(-1,1,-1)
log.lambda <- X %*% beta.target + rho[cells]
lambda <- exp(log.lambda)
set.seed(seed)
Y <- rpois(nsite,lambda)

#= Relative importance of spatial random effects
RImp <- mean(abs(rho[cells])/abs(X %*% beta.target))
RImp

#= Data-sets
data.obs <- data.frame(Y,x1,x2,cell=cells)

#==================================
#== Site-occupancy model

Start <- Sys.time() # Start the clock
mod.hSDM.poisson.iCAR <- hSDM.poisson.iCAR(counts=data.obs$Y,
                                           suitability=~x1+x2,
                                           spatial.entity=data.obs$cell,
                                           data=data.obs,
                                           n.neighbors=n.neighbors,
                                           neighbors=adj,
                                           suitability.pred=NULL,
                                           spatial.entity.pred=NULL,
                                           burnin=5000, mcmc=5000, thin=5,
                                           beta.start=0,
                                           Vrho.start=1,
                                           mubeta=0, Vbeta=1.0E6,
                                           priorVrho="1/Gamma",
                                           shape=0.5, rate=0.0005,
                                           seed=1234, verbose=1,
                                           save.rho=1, save.p=0)
Time.hSDM <- difftime(Sys.time(),Start,units="sec") # Time difference

#= Computation time
Time.hSDM

#==========
#== Outputs

#= Parameter estimates 
summary(mod.hSDM.poisson.iCAR$mcmc)
pdf("Posteriors_hSDM.poisson.iCAR.pdf")
plot(mod.hSDM.poisson.iCAR$mcmc)
dev.off()

#= Predictions
summary(mod.hSDM.poisson.iCAR$lambda.latent)
summary(mod.hSDM.poisson.iCAR$lambda.pred)
pdf(file="Pred-Init.pdf")
plot(lambda,mod.hSDM.poisson.iCAR$lambda.pred)
abline(a=0,b=1,col="red")
dev.off()

#= Summary plots for spatial random effects

# rho.pred
rho.pred <- apply(mod.hSDM.poisson.iCAR$rho.pred,2,mean)
r.rho.pred <- rasterFromXYZ(cbind(coords,rho.pred))

# plot
pdf(file="Summary_hSDM.poisson.iCAR.pdf")
par(mfrow=c(2,2))
# rho target
plot(r.rho, main="rho target")
plot(sites.sp,add=TRUE)
# rho estimated
plot(r.rho.pred, main="rho estimated")
# correlation and "shrinkage"
Levels.cells <- sort(unique(cells))
plot(rho[-Levels.cells],rho.pred[-Levels.cells],
     xlim=range(rho),
     ylim=range(rho),
     xlab="rho target",
     ylab="rho estimated")
points(rho[Levels.cells],rho.pred[Levels.cells],pch=16,col="blue")
legend(x=-3,y=4,legend="Visited cells",col="blue",pch=16,bty="n")
abline(a=0,b=1,col="red")
dev.off()

}