hSDM.Nmixture.iCAR.RdThe hSDM.Nmixture.iCAR function can be used to model
species distribution including different processes in a hierarchical
Bayesian framework: a \(\mathcal{P}oisson\) suitability
process (refering to environmental suitability explaining abundance)
which takes into account the spatial dependence of the observations,
and a \(\mathcal{B}inomial\) observability process
(refering to various ecological and methodological issues explaining
the species detection). The hSDM.Nmixture.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.Nmixture.iCAR(# Observations
counts, observability, site, data.observability,
# Habitat
suitability, data.suitability,
# Spatial structure
spatial.entity,
n.neighbors, neighbors,
# Predictions
suitability.pred = NULL, spatial.entity.pred = NULL,
# Chains
burnin = 5000, mcmc = 10000, thin = 10,
# Starting values
beta.start,
gamma.start,
Vrho.start,
# Priors
mubeta = 0, Vbeta = 1.0E6,
mugamma = 0, Vgamma = 1.0E6,
priorVrho = "1/Gamma",
shape = 0.5, rate = 0.0005,
Vrho.max = 1000,
# Various
seed = 1234, verbose = 1,
save.rho = 0, save.p = 0, save.N = 0)A vector indicating the count (or abundance) for each observation.
A one-sided formula of the form \(\sim w_1+...+w_q\) with \(q\) terms specifying the explicative variables for the observability process.
A vector indicating the site identifier (from one to the total number of sites) for each observation. Several observations can occur at one site. A site can be a raster cell for example.
A data frame containing the model's variables for the observability process.
A one-sided formula of the form \(\sim x_1+...+x_p\) with \(p\) terms specifying the explicative variables for the suitability process.
A data frame containing the model's variables for the suitability process. The number of rows of the data frame should be equal to the total number of spatial entities.
A vector (of length 'nsite') indicating the spatial entity identifier for each site. Values must be between 1 and the total number of spatial entities. Several sites can be found in one spatial entity. A spatial entity can be a raster cell for example.
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.
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).
An optional data frame in which to look for
variables with which to predict. If NULL, the data frame
data.suitability for observations is used.
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.
The number of burnin iterations for the sampler.
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.
The thinning interval used in the simulation. The number of mcmc iterations must be divisible by this value.
Starting values for \(\beta\) parameters of the suitability process. This can either be a scalar or a \(p\)-length vector.
Starting values for \(\beta\) parameters of the observability process. This can either be a scalar or a \(q\)-length vector.
Positive scalar indicating the starting value for the variance of the spatial random effects.
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.
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.
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.
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.
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".
The shape parameter for the Gamma prior on the precision
of the spatial random effects. Default value is shape=0.05 for
uninformative prior.
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.
Upper bound for the uniform prior of the spatial random effect variance. Default set to 1000.
The seed for the random number generator. Default set to 1234.
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.
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.
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.
A switch (0,1) which determines whether or not the
sampled values for the latent count variable N for each observed
cells are saved. Default is 0: the mean (rounded to the closest
integer) is computed and returned in the N.pred vector. Be
careful, setting save.N to 1 might require a large amount of
memory.
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 \(D=-2\log(\prod_{it} P(y_{it},N_i|...))\), is also provided.
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.
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.
If save.N is set to 0 (default), N.pred is
the posterior mean (rounded to the closest integer) of the latent
count variable N for each observed cell. If save.N is set to 1,
N.pred is an mcmc object with sampled values of the
latent count variable N for each observed cell.
Predictive posterior mean of the abundance associated to the suitability process for each observation.
Predictive posterior mean of the probability associated to the observability process for each observation.
The model integrates two processes, an ecological process associated to the abundance 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 abundance of the species at one site depends on the abundance of the species on neighboring sites.
Ecological process: $$N_i \sim \mathcal{P}oisson(\lambda_i)$$ $$log(\lambda_i) = X_i \beta + \rho_i$$ \(\rho_i\): spatial random effect
Spatial autocorrelation:
An intrinsic conditional autoregressive model (iCAR) is assumed: $$\rho_i \sim \mathcal{N}ormal(\mu_i,V_{\rho} / n_i)$$ \(\mu_i\): mean of \(\rho_{i'}\) in the neighborhood of \(i\).
\(V_{\rho}\): variance of the spatial random effects.
\(n_i\): number of neighbors for spatial entity \(i\).
Observation process: $$y_{it} \sim \mathcal{B}inomial(N_i, \delta_{it})$$ $$logit(\delta_{it}) = W_{it} \gamma$$
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.
Royle, J. A. (2004) N-mixture models for estimating population size from spatially replicated counts. Biometrics, 60, 108-115.
if (FALSE) {
#==============================================
# hSDM.Nmixture.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 <- 4321
#= Landscape
xLand <- 20
yLand <- 20
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 <- 150
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)
N <- rpois(nsite,lambda)
#= Relative importance of spatial random effects
RImp <- mean(abs(rho[cells])/abs(X %*% beta.target))
RImp
#= Number of visits associated to each observation point
set.seed(seed)
visits <- rpois(nsite,3)
visits[visits==0] <- 1
# Vector of observation points
sites <- vector()
for (i in 1:nsite) {
sites <- c(sites,rep(i,visits[i]))
}
#= Observation process (detectability)
nobs <- sum(visits)
set.seed(seed)
w1 <- rnorm(nobs,0,1)
set.seed(2*seed)
w2 <- rnorm(nobs,0,1)
W <- cbind(rep(1,nobs),w1,w2)
gamma.target <- c(-1,1,-1)
logit.delta <- W %*% gamma.target
delta <- inv.logit(logit.delta)
set.seed(seed)
Y <- rbinom(nobs,N[sites],delta)
#= Data-sets
data.obs <- data.frame(Y,w1,w2,site=sites)
data.suit <- data.frame(x1,x2,cell=cells)
#================================
#== Parameter inference with hSDM
Start <- Sys.time() # Start the clock
mod.hSDM.Nmixture.iCAR <- hSDM.Nmixture.iCAR(# Observations
counts=data.obs$Y,
observability=~w1+w2,
site=data.obs$site,
data.observability=data.obs,
# Habitat
suitability=~x1+x2, data.suitability=data.suit,
# Spatial structure
spatial.entity=data.suit$cell,
n.neighbors=n.neighbors, neighbors=adj,
# Predictions
suitability.pred=NULL,
spatial.entity.pred=NULL,
# Chains
burnin=5000, mcmc=5000, thin=5,
# Starting values
beta.start=0,
gamma.start=0,
Vrho.start=1,
# Priors
mubeta=0, Vbeta=1.0E6,
mugamma=0, Vgamma=1.0E6,
priorVrho="1/Gamma",
shape=0.5, rate=0.005,
Vrho.max=10,
# Various
seed=1234, verbose=1,
save.rho=1, save.p=0, save.N=1)
Time.hSDM <- difftime(Sys.time(),Start,units="sec") # Time difference
#= Computation time
Time.hSDM
#==========
#== Outputs
#= Parameter estimates
summary(mod.hSDM.Nmixture.iCAR$mcmc)
pdf(file="Posteriors_hSDM.Nmixture.iCAR.pdf")
plot(mod.hSDM.Nmixture.iCAR$mcmc)
dev.off()
#= Predictions
summary(mod.hSDM.Nmixture.iCAR$lambda.latent)
summary(mod.hSDM.Nmixture.iCAR$delta.latent)
summary(mod.hSDM.Nmixture.iCAR$lambda.pred)
pdf(file="Pred-Init.pdf")
plot(lambda,mod.hSDM.Nmixture.iCAR$lambda.pred)
abline(a=0,b=1,col="red")
dev.off()
#= MCMC for latent variable N
pdf(file="MCMC_N.pdf")
plot(mod.hSDM.Nmixture.iCAR$N.pred)
dev.off()
#= Check that Ns are corretly estimated
M <- as.matrix(mod.hSDM.Nmixture.iCAR$N.pred)
N.est <- apply(M,2,mean)
Y.by.site <- tapply(data.obs$Y,data.obs$site,mean) # Mean by site
pdf(file="Check_N.pdf",width=10,height=5)
par(mfrow=c(1,2))
plot(Y.by.site, N.est) ## More individuals are expected (N > Y) due to detection process
abline(a=0,b=1,col="red")
plot(N, N.est) ## N are well estimated
abline(a=0,b=1,col="red")
cor(N, N.est) ## Very close to 1
dev.off()
#= Summary plots for spatial random effects
# rho.pred
rho.pred <- apply(mod.hSDM.Nmixture.iCAR$rho.pred,2,mean)
r.rho.pred <- rasterFromXYZ(cbind(coords,rho.pred))
# plot
pdf(file="Summary_hSDM.Nmixture.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()
}