N-mixture model

The hSDM.Nmixture 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) and a \(\mathcal{B}inomial\) observability process (refering to various ecological and methodological issues explaining species detection). The hSDM.Nmixture 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(# Observations
                     counts, observability, site, data.observability,
                     # Habitat
                     suitability, data.suitability,
                     # Predictions
                     suitability.pred = NULL,
                     # Chains
                     burnin = 5000, mcmc = 10000, thin = 10,
                     # Starting values
                     beta.start,
                     gamma.start,
                     # Priors
                     mubeta = 0, Vbeta = 1.0E6,
                     mugamma = 0, Vgamma = 1.0E6,
                     # Various
                     seed = 1234, verbose = 1,
                     save.p = 0, save.N = 0)

Arguments

counts

A vector indicating the count (or abundance) for each observation.

observability

A one-sided formula of the form \(\sim w_1+...+w_q\) with \(q\) terms specifying the explicative variables for the observability process.

site

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.

data.observability

A data frame containing the model's variables for the observability process.

suitability

A one-sided formula of the form \(\sim x_1+...+x_p\) with \(p\) terms specifying the explicative variables for the suitability process.

data.suitability

A data frame containing the model's variables for the suitability process.

suitability.pred

An optional data frame in which to look for variables with which to predict. If NULL, the observations are 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 \(\beta\) parameters of the observability process. This can either be a scalar or a \(q\)-length vector.

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.

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.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.

save.N

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.

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 \(D=-2\log(\prod_{it} P(y_{it},N_i|...))\), is also provided.

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.

N.pred

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.

lambda.latent

Predictive posterior mean of the abundance associated to the suitability process for each observation.

delta.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 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.

Ecological process: $$N_i \sim \mathcal{P}oisson(\lambda_i)$$ $$log(\lambda_i) = X_i \beta$$

Observation process: $$y_{it} \sim \mathcal{B}inomial(N_i, \delta_{it})$$ $$logit(\delta_{it}) = W_{it} \gamma$$

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.

Royle, J. A. (2004) N-mixture models for estimating population size from spatially replicated counts. Biometrics, 60, 108-115.

Author

Ghislain Vieilledent ghislain.vieilledent@cirad.fr

See also

plot.mcmc, summary.mcmc

Examples

if (FALSE) {

#==============================================
# hSDM.Nmixture()
# Example with simulated data
#==============================================

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

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

# Number of observation sites
nsite <- 200

#= Set seed for repeatability
seed <- 4321

#= 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) # Target parameters
log.lambda <- X %*% beta.target
lambda <- exp(log.lambda)
set.seed(seed)
N <- rpois(nsite,lambda)

#= 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) # Target parameters
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)

#================================
#== Parameter inference with hSDM

Start <- Sys.time() # Start the clock
mod.hSDM.Nmixture <- hSDM.Nmixture(# Observations
                           counts=data.obs$Y,
                           observability=~w1+w2,
                           site=data.obs$site,
                           data.observability=data.obs,
                           # Habitat
                           suitability=~x1+x2,
                           data.suitability=data.suit,
                           # Predictions
                           suitability.pred=NULL,
                           # Chains
                           burnin=5000, mcmc=5000, thin=5,
                           # Starting values
                           beta.start=0,
                           gamma.start=0,
                           # Priors
                           mubeta=0, Vbeta=1.0E6,
                           mugamma=0, Vgamma=1.0E6,
                           # Various
                           seed=1234, verbose=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$mcmc)
pdf(file="Posteriors_hSDM.Nmixture.pdf")
plot(mod.hSDM.Nmixture$mcmc)
dev.off()

#= Predictions
summary(mod.hSDM.Nmixture$lambda.latent)
summary(mod.hSDM.Nmixture$delta.latent)
summary(mod.hSDM.Nmixture$lambda.pred)
pdf(file="Pred-Init.pdf")
plot(lambda,mod.hSDM.Nmixture$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$N.pred)
dev.off()

#= Check that Ns are correctly estimated
M <- as.matrix(mod.hSDM.Nmixture$N.pred)
N.est <- apply(M,2,mean)
Y.by.site <- tapply(data.obs$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()

}