hSDM.ZIB: ZIB (Zero-Inflated Binomial) model

Description

The hSDM.ZIB function can be used to model species distribution including different processes in a hierarchical Bayesian framework: a $\mathcal{B}ernoulli$ suitability process (refering to environmental suitability) and a $\mathcal{B}inomial$ observability process (refering to various ecological and methodological issues explaining the species detection). The hSDM.ZIB function calls a Gibbs sampler written in C code which uses a Metropolis algorithm to estimate the conditional posterior distribution of hierarchical model's parameters.

Usage

hSDM.ZIB(presences, trials, suitability,
observability, data, suitability.pred=NULL, burnin = 5000, mcmc = 10000,
thin = 10, beta.start, gamma.start, mubeta = 0, Vbeta = 1e+06, mugamma =
0, Vgamma = 1e+06, seed = 1234, verbose = 1, save.p = 0)

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 $D=-2\log(\prod_i P(y_i,z_i|...))$, is also provided.
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.

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 $\sim x_1+...+x_p$ with $p$ terms specifying the explicative variables for the suitability process.
observability: A one-sided formula of the form $\sim w_1+...+w_q$ with $q$ terms specifying the explicative variables for the observability process.
data: A data frame containing the model's variables.
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 prob.p.pred vector. Be careful, setting save.p to 1 might require a large amount of memory.

Author

Ghislain Vieilledent ghislain.vieilledent@cirad.fr

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.

Ecological process: $$z_i \sim \mathcal{B}ernoulli(\theta_i)$$ $$logit(\theta_i) = X_i \beta$$

Observation process: $$y_i \sim \mathcal{B}inomial(z_i * \delta_i, t_i)$$ $$logit(\delta_i) = W_i \gamma$$

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.

Examples

Run this code


if (FALSE) { 

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

#============
#== Preambule
library(hSDM)

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

# Set seed for repeatability
seed <- 1234

# Number of observations
nobs <- 1000

# 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)

# Covariates for "suitability" process
set.seed(seed)
X1 <- rnorm(n=nobs,0,1)
set.seed(2*seed)
X2 <- rnorm(n=nobs,0,1)
X <- cbind(rep(1,nobs),X1,X2)

# 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 <- X %*% beta.target
theta.1 <- inv.logit(logit.theta.1)
set.seed(seed)
y.1 <- rbinom(nobs,1,theta.1)

# 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*y.1

#== Data-set
Data <- data.frame(Y,trials,X1,X2)
## Uncomment if you want covariates on the observability process 
## Data <- data.frame(Y,trials,X1,X2,W1,W2)

#==================================
#== ZIB model

mod.hSDM.ZIB <- hSDM.ZIB(presences=Data$Y,
                         trials=Data$trials,
                         suitability=~X1+X2,
                         observability=~1, #=~1+W1+W2 if covariates
                         data=Data,
                         suitability.pred=NULL,
                         burnin=1000, mcmc=1000, thin=5,
                         beta.start=0,
                         gamma.start=0,
                         mubeta=0, Vbeta=1.0E6,
                         mugamma=0, Vgamma=1.0E6,
                         seed=1234, verbose=1,
                         save.p=0)

#==========
#== Outputs
pdf(file="Posteriors_hSDM.ZIB.pdf")
plot(mod.hSDM.ZIB$mcmc)
dev.off()
summary(mod.hSDM.ZIB$prob.p.latent)
summary(mod.hSDM.ZIB$prob.q.latent)
summary(mod.hSDM.ZIB$prob.p.pred)

}

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