The jSDM_poisson_log function performs a Poisson regression with log link function 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.

jSDM_poisson_log(
burnin = 5000,
mcmc = 10000,
thin = 10,
count_data,
site_data,
site_formula,
trait_data = NULL,
trait_formula = NULL,
n_latent = 0,
site_effect = "none",
beta_start = 0,
gamma_start = 0,
lambda_start = 0,
W_start = 0,
alpha_start = 0,
V_alpha = 1,
shape = 0.5,
rate = 5e-04,
mu_beta = 0,
V_beta = 10,
mu_gamma = 0,
V_gamma = 10,
mu_lambda = 0,
V_lambda = 10,
ropt = 0.44,
seed = 1234,
verbose = 1
)

## Arguments

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.

count_data

A matrix $$n_{site} \times n_{species}$$ indicating the abundance of each species on each site.

site_data

A data frame containing the model's explanatory variables by site.

site_formula

A one-sided formula of the form '~x1+...+xp' specifying the explanatory variables for the suitability process of the model,
used to form the design matrix $$X$$ of size $$n_{site} \times np$$.

trait_data

A data frame containing the species traits which can be included as part of the model.
Default to NULL to fit a model without species traits.

trait_formula

A one-sided formula of the form '~ t1 + ... + tk + x1:t1 + ... + xp:tk' specifying the interactions between the environmental variables and the species traits to be considered in the model,
used to form the trait design matrix $$Tr$$ of size $$n_{species} \times nt$$
and to set to $$0$$ the $$\gamma$$ parameters corresponding to interactions not taken into account according to the formula. Default to NULL to fit a model with all possible interactions between species traits found in trait_data and environmental variables defined by site_formula.

n_latent

An integer which specifies the number of latent variables to generate. Defaults to $$0$$.

site_effect

A string indicating whether row effects are included as fixed effects ("fixed"), as random effects ("random"),
or not included ("none") in the model. If fixed effects, then for parameter identifiability the first row effect is set to zero, which analogous to acting as a reference level when dummy variables are used. If random effects, they are drawn from a normal distribution with mean zero and unknown variance, analogous to a random intercept in mixed models. Defaults to "none".

beta_start

Starting values for $$\beta$$ parameters of the suitability process for each species must be either a scalar or a $$np \times n_{species}$$ matrix. If beta_start takes a scalar value, then that value will serve for all of the $$\beta$$ parameters.

gamma_start

Starting values for $$\gamma$$ parameters that represent the influence of species-specific traits on species' responses $$\beta$$, gamma_start must be either a scalar, a vector of length $$nt$$, $$np$$ or $$nt.np$$ or a $$nt \times np$$ matrix. If gamma_start takes a scalar value, then that value will serve for all of the $$\gamma$$ parameters. If gamma_start is a vector of length $$nt$$ or $$nt.np$$ the resulting $$nt \times np$$ matrix is filled by column with specified values, if a $$np$$-length vector is given, the matrix is filled by row.

lambda_start

Starting values for $$\lambda$$ parameters corresponding to the latent variables for each species must be either a scalar or a $$n_{latent} \times n_{species}$$ upper triangular matrix with strictly positive values on the diagonal, ignored if n_latent=0. If lambda_start takes a scalar value, then that value will serve for all of the $$\lambda$$ parameters except those concerned by the constraints explained above.

W_start

Starting values for latent variables must be either a scalar or a $$nsite \times n_latent$$ matrix, ignored if n_latent=0. If W_start takes a scalar value, then that value will serve for all of the $$W_{il}$$ with $$i=1,\ldots,n_{site}$$ and $$l=1,\ldots,n_{latent}$$.

alpha_start

Starting values for random site effect parameters must be either a scalar or a $$n_{site}$$-length vector, ignored if site_effect="none". If alpha_start takes a scalar value, then that value will serve for all of the $$\alpha$$ parameters.

V_alpha

Starting value for variance of random site effect if site_effect="random" or constant variance of the Gaussian prior distribution for the fixed site effect if site_effect="fixed". Must be a strictly positive scalar, ignored if site_effect="none".

shape

Shape parameter of the Inverse-Gamma prior for the random site effect variance V_alpha, ignored if site_effect="none" or site_effect="fixed". Must be a strictly positive scalar. Default to 0.5 for weak informative prior.

rate

Rate parameter of the Inverse-Gamma prior for the random site effect variance V_alpha, ignored if site_effect="none" or site_effect="fixed" Must be a strictly positive scalar. Default to 0.0005 for weak informative prior.

mu_beta

Means of the Normal priors for the $$\beta$$ parameters of the suitability process. mu_beta must be either a scalar or a $$np$$-length vector. If mu_beta takes a scalar value, then that value will serve as the prior mean for all of the $$\beta$$ parameters. The default value is set to 0 for an uninformative prior, ignored if trait_data is specified.

V_beta

Variances of the Normal priors for the $$\beta$$ parameters of the suitability process. V_beta must be either a scalar or a $$np \times np$$ symmetric positive semi-definite square matrix. If V_beta takes a scalar value, then that value will serve as the prior variance for all of the $$\beta$$ parameters, so the variance covariance matrix used in this case is diagonal with the specified value on the diagonal. The default variance is large and set to 10 for an uninformative flat prior.

mu_gamma

Means of the Normal priors for the $$\gamma$$ parameters. mu_gamma must be either a scalar, a vector of length $$nt$$, $$np$$ or $$nt.np$$ or a $$nt \times np$$ matrix. If mu_gamma takes a scalar value, then that value will serve as the prior mean for all of the $$\gamma$$ parameters. If mu_gamma is a vector of length $$nt$$ or $$nt.np$$ the resulting $$nt \times np$$ matrix is filled by column with specified values, if a $$np$$-length vector is given, the matrix is filled by row. The default value is set to 0 for an uninformative prior, ignored if trait_data=NULL.

V_gamma

Variances of the Normal priors for the $$\gamma$$ parameters. V_gamma must be either a scalar, a vector of length $$nt$$, $$np$$ or $$nt.np$$ or a $$nt \times np$$ positive matrix. If V_gamma takes a scalar value, then that value will serve as the prior variance for all of the $$\gamma$$ parameters. If V_gamma is a vector of length $$nt$$ or $$nt.np$$ the resulting $$nt \times np$$ matrix is filled by column with specified values, if a $$np$$-length vector is given, the matrix is filled by row. The default variance is large and set to 10 for an uninformative flat prior, ignored if trait_data=NULL.

mu_lambda

Means of the Normal priors for the $$\lambda$$ parameters corresponding to the latent variables. mu_lambda must be either a scalar or a $$n_{latent}$$-length vector. If mu_lambda takes a scalar value, then that value will serve as the prior mean for all of the $$\lambda$$ parameters. The default value is set to 0 for an uninformative prior.

V_lambda

Variances of the Normal priors for the $$\lambda$$ parameters corresponding to the latent variables. V_lambda must be either a scalar or a $$n_{latent} \times n_{latent}$$ symmetric positive semi-definite square matrix. If V_lambda takes a scalar value, then that value will serve as the prior variance for all of $$\lambda$$ parameters, so the variance covariance matrix used in this case is diagonal with the specified value on the diagonal. The default variance is large and set to 10 for an uninformative flat prior.

ropt

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

seed

The seed for the random number generator. Default 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.

## Value

An object of class "jSDM" acting like a list including :

mcmc.alpha

An mcmc object that contains the posterior samples for site effects $$\alpha_i$$, not returned if site_effect="none".

mcmc.V_alpha

An mcmc object that contains the posterior samples for variance of random site effect, not returned if site_effect="none" or site_effect="fixed".

mcmc.latent

A list by latent variable of mcmc objects that contains the posterior samples for latent variables $$W_l$$ with $$l=1,\ldots,n_{latent}$$, not returned if n_latent=0.

mcmc.sp

A list by species of mcmc objects that contains the posterior samples for species effects $$\beta_j$$ and $$\lambda_j$$ if n_latent>0.

mcmc.gamma

A list by covariates of mcmc objects that contains the posterior samples for $$\gamma_p$$ parameters with $$p=1,\ldots,np$$ if trait_data is specified.

mcmc.Deviance

The posterior sample of the deviance ($$D$$) is also provided, with $$D$$ defined as : $$D=-2\log(\prod_{ij} P(y_{ij}|\beta_j,\lambda_j, \alpha_i, W_i))$$.

log_theta_latent

Predictive posterior mean of the probability to each species to be present on each site, transformed by log link function.

theta_latent

Predictive posterior mean of the probability to each species to be present on each site.

model_spec

Various attributes of the model fitted, including the response and model matrix used, distributional assumptions as link function, family and number of latent variables, hyperparameters used in the Bayesian estimation and mcmc, burnin and thin.

The mcmc. objects can be summarized by functions provided by the coda package.

## Details

We model an ecological process where the presence or absence of species $$j$$ on site $$i$$ is explained by habitat suitability.

Ecological process : $$y_{ij} \sim \mathcal{P}oisson(\theta_{ij})$$ where

 if n_latent=0 and site_effect="none" log$$(\theta_{ij}) = X_i \beta_j$$ if n_latent>0 and site_effect="none" log$$(\theta_{ij}) = X_i \beta_j + W_i \lambda_j$$ if n_latent=0 and site_effect="fixed" log$$(\theta_{ij}) = X_i \beta_j + \alpha_i$$ if n_latent>0 and site_effect="fixed" log$$(\theta_{ij}) = X_i \beta_j + W_i \lambda_j + \alpha_i$$ if n_latent=0 and site_effect="random" log$$(\theta_{ij}) = X_i \beta_j + \alpha_i$$ and $$\alpha_i \sim \mathcal{N}(0,V_\alpha)$$ if n_latent>0 and site_effect="random" log$$(\theta_{ij}) = X_i \beta_j + W_i \lambda_j + \alpha_i$$ and $$\alpha_i \sim \mathcal{N}(0,V_\alpha)$$

In the absence of data on species traits (trait_data=NULL), the effect of species $$j$$: $$\beta_j$$; follows the same a priori Gaussian distribution such that $$\beta_j \sim \mathcal{N}_{np}(\mu_{\beta},V_{\beta})$$, for each species.

If species traits data are provided, the effect of species $$j$$: $$\beta_j$$; follows an a priori Gaussian distribution such that $$\beta_j \sim \mathcal{N}_{np}(\mu_{\beta_j},V_{\beta})$$, where $$\mu_{\beta_jp} = \sum_{k=1}^{nt} t_{jk}.\gamma_{kp}$$, takes different values for each species.

We assume that $$\gamma_{kp} \sim \mathcal{N}(\mu_{\gamma_{kp}},V_{\gamma_{kp}})$$ as prior distribution.

We define the matrix $$\gamma=(\gamma_{kp})_{k=1,...,nt}^{p=1,...,np}$$ such as :

 $$x_0$$ $$x_1$$ ... $$x_p$$ ... $$x_{np}$$ __________ ________ ________ ________ ________ ________ $$t_0$$ | $$\gamma_{0,0}$$ $$\gamma_{0,1}$$ ... $$\gamma_{0,p}$$ ... $$\gamma_{0,np}$$ { effect of | intercept environmental | variables $$t_1$$ | $$\gamma_{1,0}$$ $$\gamma_{1,1}$$ ... $$\gamma_{1,p}$$ ... $$\gamma_{1,np}$$ ... | ... ... ... ... ... ... $$t_k$$ | $$\gamma_{k,0}$$ $$\gamma_{k,1}$$ ... $$\gamma_{k,p}$$ ... $$\gamma_{k,np}$$ ... | ... ... ... ... ... ... $$t_{nt}$$ | $$\gamma_{nt,0}$$ $$\gamma_{nt,1}$$ ... $$\gamma_{nt,p}$$ ... $$\gamma_{nt,np}$$ average trait effect interaction traits environment

## 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.

Ovaskainen, O., Tikhonov, G., Norberg, A., Blanchet, F. G., Duan, L., Dunson, D., Roslin, T. and Abrego, N. (2017) How to make more out of community data? A conceptual framework and its implementation as models and software. Ecology Letters, 20, 561-576.

plot.mcmc, summary.mcmc jSDM_binomial_probit jSDM_binomial_logit

## Author

Jeanne Clément <jeanne.clement16@laposte.net>

## Examples

#==============================================
# jSDM_poisson_log()
# Example with simulated data
#==============================================

#=================
library(jSDM)

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

#= Number of sites
nsite <- 60
#= Number of species
nsp <- 20
#= Set seed for repeatability
seed <- 1234

#= 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)
np <- ncol(X)
set.seed(3*seed)
W <- cbind(rnorm(nsite,0,1),rnorm(nsite,0,1))
n_latent <- ncol(W)
l.zero <- 0
l.diag <- runif(2,0,1)
l.other <- runif(nsp*2-3,-1,1)
lambda.target <- matrix(c(l.diag[1],l.zero,l.other[1],
l.diag[2],l.other[-1]),
byrow=TRUE, nrow=nsp)
beta.target <- matrix(runif(nsp*np,-1,1), byrow=TRUE, nrow=nsp)
V_alpha.target <- 0.5
alpha.target <- rnorm(nsite,0,sqrt(V_alpha.target))
log.theta <- X %*% t(beta.target) + W %*% t(lambda.target) + alpha.target
theta <- exp(log.theta)
Y <- apply(theta, 2, rpois, n=nsite)

#= Site-occupancy model
# Increase number of iterations (burnin and mcmc) to get convergence
mod <- jSDM_poisson_log(# Chains
burnin=200,
mcmc=200,
thin=1,
# Response variable
count_data=Y,
# Explanatory variables
site_formula=~x1+x2,
site_data=X,
n_latent=n_latent,
site_effect="random",
# Starting values
beta_start=0,
lambda_start=0,
W_start=0,
alpha_start=0,
V_alpha=1,
# Priors
shape=0.5,
rate=0.0005,
mu_beta=0,
V_beta=10,
mu_lambda=0,
V_lambda=10,
# Various
seed=1234,
ropt=0.44,
verbose=1)
#>
#> Running the Gibbs sampler. It may be long, please keep cool :)
#>
#> **********:10.0%, mean accept. rates= beta:0.152 lambda:0.136 W:0.329 alpha:0.217
#> **********:20.0%, mean accept. rates= beta:0.183 lambda:0.176 W:0.328 alpha:0.268
#> **********:30.0%, mean accept. rates= beta:0.270 lambda:0.263 W:0.362 alpha:0.351
#> **********:40.0%, mean accept. rates= beta:0.330 lambda:0.354 W:0.415 alpha:0.396
#> **********:50.0%, mean accept. rates= beta:0.392 lambda:0.401 W:0.427 alpha:0.412
#> **********:60.0%, mean accept. rates= beta:0.438 lambda:0.429 W:0.438 alpha:0.448
#> **********:70.0%, mean accept. rates= beta:0.453 lambda:0.436 W:0.439 alpha:0.443
#> **********:80.0%, mean accept. rates= beta:0.443 lambda:0.423 W:0.420 alpha:0.450
#> **********:90.0%, mean accept. rates= beta:0.444 lambda:0.426 W:0.436 alpha:0.440
#> **********:100.0%, mean accept. rates= beta:0.452 lambda:0.452 W:0.433 alpha:0.450
#' #==========
#' #== Outputs

#= Parameter estimates

## beta_j
# summary(mod$mcmc.sp$sp_1[,1:ncol(X)])
mean_beta <- matrix(0,nsp,np)
pdf(file=file.path(tempdir(), "Posteriors_beta_jSDM_log.pdf"))
par(mfrow=c(ncol(X),2))
for (j in 1:nsp) {
mean_beta[j,] <- apply(mod$mcmc.sp[[j]][,1:ncol(X)], 2, mean) for (p in 1:ncol(X)) { coda::traceplot(mod$mcmc.sp[[j]][,p])
coda::densplot(mod$mcmc.sp[[j]][,p], main = paste(colnames( mod$mcmc.sp[[j]])[p],
", species : ",j))
abline(v=beta.target[j,p],col='red')
}
}
dev.off()
#> agg_png
#>       2

## lambda_j
# summary(mod$mcmc.sp$sp_1[,(ncol(X)+1):(ncol(X)+n_latent)])
# summary(mod$mcmc.sp$sp_2[,(ncol(X)+1):(ncol(X)+n_latent)])
mean_lambda <- matrix(0,nsp,n_latent)
pdf(file=file.path(tempdir(), "Posteriors_lambda_jSDM_log.pdf"))
par(mfrow=c(n_latent*2,2))
for (j in 1:nsp) {
mean_lambda[j,] <- apply(mod$mcmc.sp[[j]] [,(ncol(X)+1):(ncol(X)+n_latent)], 2, mean) for (l in 1:n_latent) { coda::traceplot(mod$mcmc.sp[[j]][,ncol(X)+l])
coda::densplot(mod$mcmc.sp[[j]][,ncol(X)+l], main=paste(colnames(mod$mcmc.sp[[j]])
[ncol(X)+l],", species : ",j))
abline(v=lambda.target[j,l],col='red')
}
}
dev.off()
#> agg_png
#>       2

par(mfrow=c(1,2))
plot(beta.target, mean_beta,
main="species effect beta",
xlab ="obs", ylab ="fitted")
abline(a=0,b=1,col='red')
plot(lambda.target, mean_lambda,
xlab ="obs", ylab ="fitted")
abline(a=0,b=1,col='red')

## W latent variables
par(mfrow=c(1,2))
for (l in 1:n_latent) {
plot(W[,l],
summary(mod$mcmc.latent[[paste0("lv_",l)]])[[1]][,"Mean"], main = paste0("Latent variable W_", l), xlab ="obs", ylab ="fitted") abline(a=0,b=1,col='red') } ## alpha # summary(mod$mcmc.alpha)
par(mfrow=c(1,3))
plot(alpha.target, summary(mod$mcmc.alpha)[[1]][,"Mean"], xlab ="obs", ylab ="fitted", main="site effect alpha") abline(a=0,b=1,col='red') ## Valpha # summary(mod$mcmc.V_alpha)
coda::traceplot(mod$mcmc.V_alpha) coda::densplot(mod$mcmc.V_alpha)
abline(v=V_alpha.target,col='red')

## Deviance
# summary(mod$mcmc.Deviance) plot(mod$mcmc.Deviance)

#= Predictions
# summary(mod$theta_latent) par(mfrow=c(1,2)) plot(log.theta, mod$log_theta_latent,
main="log(theta)",
xlab="obs", ylab="fitted")
abline(a=0 ,b=1, col="red")
plot(theta, mod\$theta_latent,
main="Expected abundance theta",
xlab="obs", ylab="fitted")
abline(a=0 ,b=1, col="red")