jSDM (version 0.1.0)

jSDM_binomial: Binomial logistic regression model

Description

The jSDM_binomial function performs a Binomial logistic regression in a Bayesian framework. The function calls a Gibbs sampler written in C++ code which uses an adaptive Metropolis algorithm to estimate the conditional posterior distribution of model's parameters.

Usage

jSDM_binomial(presences, trials, suitability, data, 
burnin = 5000, mcmc = 10000, thin = 10,
beta_start, mubeta = 0, Vbeta = 1e+06, seed = 1234, ropt = 0.44, verbose = 1)

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. \(t_n\) should be superior or equal to \(y_n\), the number of successes for observation \(n\). If \(t_n=0\), then \(y_n=0\).

suitability

A one-sided formula of the form '~x1+...+xp' with p terms specifying the explicative variables for the suitability process of the model.

data

A data frame containing the model's explicative variables.

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. If beta_start takes a scalar value, then that value will serve for all of the betas.

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.

seed

The seed for the random number generator. Default to 1234.

ropt

Target acceptance rate for the adaptive Metropolis algorithm. Default to 0.44.

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.

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

theta_latent

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

spec

Model's specifications

Details

We model an ecological process where the presence or absence of the species is explained by habitat suitability.

Ecological process: $$y_i \sim \mathcal{B}inomial(\theta_i,t_i)$$ $$logit(\theta_i) = X_i \beta$$

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.

See Also

plot.mcmc, summary.mcmc

Examples

# NOT RUN {
#==============================================
# jSDM_binomial()
# Example with simulated data
#==============================================

#=================
#== Load libraries
library(jSDM)

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

#= Number of sites
nsite <- 200

#= Set seed for repeatability
seed <- 1234

#= Number of visits associated to each site
set.seed(seed)
visits <- rpois(nsite,3)
visits[visits==0] <- 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)
logit.theta <- X %*% beta.target
theta <- inv_logit(logit.theta)
set.seed(seed)
Y <- rbinom(nsite,visits,theta)

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

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

mod_jSDM_binomial <- jSDM_binomial(presences=data.obs$Y,
                                   trials=data.obs$visits,
                                   suitability=~x1+x2,
                                   data=data.obs,
                                   burnin=1000, mcmc=1000, thin=1,
                                   beta_start=0,
                                   mubeta=0, Vbeta=1.0E6,
                                   seed=1234, ropt=0.44, verbose=1)

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

#= Parameter estimates
summary(mod_jSDM_binomial$mcmc)
pdf(file=file.path(tempdir(), "Posteriors_jSDM_binomial.pdf"))
plot(mod_jSDM_binomial$mcmc)
dev.off()

#== glm resolution to compare
mod_glm <- glm(cbind(Y,visits-Y)~x1+x2,family="binomial",data=data.obs)
summary(mod_glm)

#= Predictions
summary(mod_jSDM_binomial$theta_latent)
pdf(file=file.path(tempdir(), "Pred-Init.pdf"))
plot(theta, mod_jSDM_binomial$theta_latent)
abline(a=0 ,b=1, col="red")
dev.off()

# }