Function for fitting multi-season single-species spatial occupancy models using Polya-Gamma latent variables.

## Usage

stPGOcc(occ.formula, det.formula, data, inits, priors,
tuning, cov.model = 'exponential', NNGP = TRUE,
n.neighbors = 15, search.type = 'cb', n.batch,
batch.length, accept.rate = 0.43, n.omp.threads = 1,
verbose = TRUE, ar1 = FALSE, n.report = 100,
n.burn = round(.10 * n.batch * batch.length),
n.thin = 1, n.chains = 1, k.fold, k.fold.threads = 1,
k.fold.seed = 100, k.fold.only = FALSE, ...)

## Arguments

occ.formula

a symbolic description of the model to be fit for the occurrence portion of the model using R's model syntax. Only right-hand side of formula is specified. See example below. Random intercepts are allowed using lme4 syntax (Bates et al. 2015).

det.formula

a symbolic description of the model to be fit for the detection portion of the model using R's model syntax. Only right-hand side of formula is specified. See example below. Random intercepts are allowed using lme4 syntax (Bates et al. 2015).

data

a list containing data necessary for model fitting. Valid tags are y, occ.covs, det.covs, and coords. y is a three-dimensional array with first dimension equal to the number of sites ($$J$$), second dimension equal to the maximum number of primary time periods (i.e., years or seasons), and third dimension equal to the maximum number of replicates at a given site. occ.covs is a list of variables included in the occurrence portion of the model. Each list element is a different occurrence covariate, which can be site level or site/primary timer period level. Site-level covariates are specified as a vector of length $$J$$ while site/primary time period level covariates are specified as a matrix with rows corresponding to sites and columns correspond to primary time periods. Similarly, det.covs is a list of variables included in the detection portion of the model, with each list element corresponding to an individual variable. In addition to site-level and/or site/primary time period-level, detection covariates can also be observational-level. Observation-level covariates are specified as a three-dimensional array with first dimension corresponding to sites, second dimension corresponding to primary time period, and third dimension corresponding to replicate. coords is a $$J \times 2$$ matrix of the observation coordinates. Note that spOccupancy assumes coordinates are specified in a projected coordinate system.

inits

a list with each tag corresponding to a parameter name. Valid tags are z, beta, alpha, sigma.sq, phi, w, nu, sigma.sq.psi, sigma.sq.p, sigma.sq.t, rho. The value portion of each tag is the parameter's initial value. sigma.sq.psi and sigma.sq.p are only relevant when including random effects in the occurrence and detection portion of the occupancy model, respectively. nu is only specified if cov.model = "matern". sigma.sq.t and rho are only relevant when ar1 = TRUE. See priors description for definition of each parameter name. Additionally, the tag fix can be set to TRUE to fix the starting values across all chains. If fix is not specified (the default), starting values are varied randomly across chains.

priors

a list with each tag corresponding to a parameter name. Valid tags are beta.normal, alpha.normal, sigma.sq.psi.ig, sigma.sq.p.ig, phi.unif, sigma.sq.ig, nu.unif, sigma.sq.t.ig, and rho.unif. Occupancy (beta) and detection (alpha) regression coefficients are assumed to follow a normal distribution. The hyperparameters of the normal distribution are passed as a list of length two with the first and second elements corresponding to the mean and variance of the normal distribution, which are each specified as vectors of length equal to the number of coefficients to be estimated or of length one if priors are the same for all coefficients. If not specified, prior means are set to 0 and prior variances set to 2.72. sigma.sq.psi and sigma.sq.p are the random effect variances for any occurrence or detection random effects, respectively, and are assumed to follow an inverse Gamma distribution. The hyperparameters of the inverse-Gamma distribution are passed as a list of length two with first and second elements corresponding to the shape and scale parameters, respectively, which are each specified as vectors of length equal to the number of random intercepts or of length one if priors are the same for all random effect variances. The spatial variance parameter, sigma.sq, is assumed to follow an inverse-Gamma distribution. The spatial decay phi and smoothness nu parameters are assumed to follow Uniform distributions. The hyperparameters of the inverse-Gamma for sigma.sq.ig are passed as a vector of length two, with the first and second elements corresponding to the shape and scale parameters, respectively. The hyperparameters of the uniform are also passed as a vector of length two with the first and second elements corresponding to the lower and upper support, respectively. sigma.sq.t and rho are the AR(1) variance and correlation parameters for the AR(1) zero-mean temporal random effects, respectively. sigma.sq.t is assumed to follow an inverse-Gamma distribution, where the hyperparameters are specified as a vector with elements corresponding to the shape and scale parameters, respectively. rho is assumed to follow a uniform distribution, where the hyperparameters are specified in a vector of length two with elements corresponding to the lower and upper bounds of the uniform prior.

cov.model

a quoted keyword that specifies the covariance function used to model the spatial dependence structure among the observations. Supported covariance model key words are: "exponential", "matern", "spherical", and "gaussian".

tuning

a list with each tag corresponding to a parameter name. Valid tags are phi and nu. The value portion of each tag defines the initial variance of the Adaptive sampler. See Roberts and Rosenthal (2009) for details.

NNGP

if TRUE, model is fit with an NNGP. If FALSE, a full Gaussian process is used. See Datta et al. (2016) and Finley et al. (2019) for more information. Currently only NNGP = TRUE is supported for multi-season single-species trend occupancy models.

n.neighbors

number of neighbors used in the NNGP. Only used if NNGP = TRUE. Datta et al. (2016) showed that 15 neighbors is usually sufficient, but that as few as 5 neighbors can be adequate for certain data sets, which can lead to even greater decreases in run time. We recommend starting with 15 neighbors (the default) and if additional gains in computation time are desired, subsequently compare the results with a smaller number of neighbors using WAIC or k-fold cross-validation.

search.type

a quoted keyword that specifies the type of nearest neighbor search algorithm. Supported method key words are: "cb" and "brute". The "cb" should generally be much faster. If locations do not have identical coordinate values on the axis used for the nearest neighbor ordering then "cb" and "brute" should produce identical neighbor sets. However, if there are identical coordinate values on the axis used for nearest neighbor ordering, then "cb" and "brute" might produce different, but equally valid, neighbor sets, e.g., if data are on a grid.

n.batch

the number of MCMC batches in each chain to run for the Adaptive MCMC sampler. See Roberts and Rosenthal (2009) for details.

batch.length

the length of each MCMC batch in each chain to run for the Adaptive MCMC sampler. See Roberts and Rosenthal (2009) for details.

accept.rate

target acceptance rate for Adaptive MCMC. Default is 0.43. See Roberts and Rosenthal (2009) for details.

a positive integer indicating the number of threads to use for SMP parallel processing. The package must be compiled for OpenMP support. For most Intel-based machines, we recommend setting n.omp.threads up to the number of hyperthreaded cores. Note, n.omp.threads > 1 might not work on some systems. Currently only relevant for spatial models.

verbose

if TRUE, messages about data preparation, model specification, and progress of the sampler are printed to the screen. Otherwise, no messages are printed.

ar1

logical value indicating whether to include an AR(1) zero-mean temporal random effect in the model. If FALSE, the model is fit without an AR(1) temporal autocovariance structure. If TRUE, an AR(1) random effect is included in the model to account for temporal autocorrelation across the primary time periods.

n.report

the interval to report MCMC progress.

n.burn

the number of samples out of the total n.samples to discard as burn-in for each chain. By default, the first 10% of samples is discarded.

n.thin

the thinning interval for collection of MCMC samples. The thinning occurs after the n.burn samples are discarded. Default value is set to 1.

n.chains

the number of chains to run in sequence.

k.fold

specifies the number of k folds for cross-validation. If not specified as an argument, then cross-validation is not performed and k.fold.threads and k.fold.seed are ignored. In k-fold cross-validation, the data specified in data is randomly partitioned into k equal sized subsamples. Of the k subsamples, k - 1 subsamples are used to fit the model and the remaining k samples are used for prediction. The cross-validation process is repeated k times (the folds). As a scoring rule, we use the model deviance as described in Hooten and Hobbs (2015). For cross-validation in multi-season models, the data are split along the site dimension, such that each hold-out data set consists of a J / k.fold sites sampled over all primary time periods during which data are available at each given site. Cross-validation is performed after the full model is fit using all the data. Cross-validation results are reported in the k.fold.deviance object in the return list.

number of threads to use for cross-validation. If k.fold.threads > 1 parallel processing is accomplished using the foreach and doParallel packages. Ignored if k.fold is not specified.

k.fold.seed

seed used to split data set into k.fold parts for k-fold cross-validation. Ignored if k.fold is not specified.

k.fold.only

a logical value indicating whether to only perform cross-validation (TRUE) or perform cross-validation after fitting the full model (FALSE). Default value is FALSE.

...

## Note

Some of the underlying code used for generating random numbers from the Polya-Gamma distribution is taken from the pgdraw package written by Daniel F. Schmidt and Enes Makalic. Their code implements Algorithm 6 in PhD thesis of Jesse Bennett Windle (2013) https://repositories.lib.utexas.edu/handle/2152/21842.

## References

Polson, N.G., J.G. Scott, and J. Windle. (2013) Bayesian Inference for Logistic Models Using Polya-Gamma Latent Variables. Journal of the American Statistical Association, 108:1339-1349.

Bates, Douglas, Martin Maechler, Ben Bolker, Steve Walker (2015). Fitting Linear Mixed-Effects Models Using lme4. Journal of Statistical Software, 67(1), 1-48. doi:10.18637/jss.v067.i01 .

Kéry, M., & Royle, J. A. (2021). Applied hierarchical modeling in ecology: Analysis of distribution, abundance and species richness in R and BUGS: Volume 2: Dynamic and advanced models. Academic Press. Section 4.6.

Hooten, M. B., and Hobbs, N. T. (2015). A guide to Bayesian model selection for ecologists. Ecological monographs, 85(1), 3-28.

MacKenzie, D. I., J. D. Nichols, G. B. Lachman, S. Droege, J. Andrew Royle, and C. A. Langtimm. 2002. Estimating Site Occupancy Rates When Detection Probabilities Are Less Than One. Ecology 83: 2248-2255.

## Author

Jeffrey W. Doser doserjef@msu.edu,
Andrew O. Finley finleya@msu.edu

## Value

An object of class tPGOcc that is a list comprised of:

beta.samples

a coda object of posterior samples for the occupancy regression coefficients.

alpha.samples

a coda object of posterior samples for the detection regression coefficients.

z.samples

a three-dimensional array of posterior samples for the latent occupancy values, with dimensions corresponding to posterior sample, site, and primary time period.

psi.samples

a three-dimensional array of posterior samples for the latent occupancy probability values, with dimensions corresponding to posterior sample, site, and primary time period.

theta.samples

a coda object of posterior samples for spatial covariance parameters and temporal covariance parameters if ar1 = TRUE.

w.samples

a coda object of posterior samples for latent spatial random effects.

sigma.sq.psi.samples

a coda object of posterior samples for variances of random intercepts included in the occupancy portion of the model. Only included if random intercepts are specified in occ.formula.

sigma.sq.p.samples

a coda object of posterior samples for variances of random intercpets included in the detection portion of the model. Only included if random intercepts are specified in det.formula.

beta.star.samples

a coda object of posterior samples for the occurrence random effects. Only included if random intercepts are specified in occ.formula.

alpha.star.samples

a coda object of posterior samples for the detection random effects. Only included if random intercepts are specified in det.formula.

eta.samples

a coda object of posterior samples for the AR(1) random effects for each primary time period. Only included if ar1 = TRUE

.

like.samples

a three-dimensional array of posterior samples for the likelihood values associated with each site and primary time period. Used for calculating WAIC.

rhat

a list of Gelman-Rubin diagnostic values for some of the model parameters.

ESS

a list of effective sample sizes for some of the model parameters.

run.time

execution time reported using proc.time().

k.fold.deviance

scoring rule (deviance) from k-fold cross-validation. Only included if k.fold is specified in function call.

The return object will include additional objects used for subsequent prediction and/or model fit evaluation. Note that detection probability estimated values are not included in the model object, but can be extracted using fitted(). Note that if k.fold.only = TRUE, the return list object will only contain run.time and k.fold.deviance.

## Examples

set.seed(500)
# Sites
J.x <- 10
J.y <- 10
J <- J.x * J.y
# Primary time periods
n.time <- sample(10, J, replace = TRUE)
n.time.max <- max(n.time)
# Replicates
n.rep <- matrix(NA, J, max(n.time))
for (j in 1:J) {
n.rep[j, 1:n.time[j]] <- sample(1:4, n.time[j], replace = TRUE)
}
# Occurrence --------------------------
beta <- c(0.4, 0.5, -0.9)
trend <- TRUE
sp.only <- 0
psi.RE <- list()
# Detection ---------------------------
alpha <- c(-1, 0.7, -0.5)
p.RE <- list()
# Spatial -----------------------------
sp <- TRUE
cov.model <- "exponential"
sigma.sq <- 2
phi <- 3 / .4
# Temporal ----------------------------
rho <- 0.5
sigma.sq.t <- 1

# Get all the data
dat <- simTOcc(J.x = J.x, J.y = J.y, n.time = n.time, n.rep = n.rep,
beta = beta, alpha = alpha, sp.only = sp.only, trend = trend,
psi.RE = psi.RE, p.RE = p.RE, sp = TRUE, sigma.sq = sigma.sq,
phi = phi, cov.model = cov.model, ar1 = TRUE,
sigma.sq.t = sigma.sq.t, rho = rho)

# Package all data into a list
# Occurrence
occ.covs <- list(int = dat$X[, , 1], trend = dat$X[, , 2],
occ.cov.1 = dat$X[, , 3]) # Detection det.covs <- list(det.cov.1 = dat$X.p[, , , 2],
det.cov.2 = dat$X.p[, , , 3]) # Data list bundle data.list <- list(y = dat$y,
occ.covs = occ.covs,
det.covs = det.covs,
coords = dat$coords) # Priors prior.list <- list(beta.normal = list(mean = 0, var = 2.72), alpha.normal = list(mean = 0, var = 2.72), sigma.sq.ig = c(2, 2), phi.unif = c(3 / 1, 3 / 0.1), rho.unif = c(-1, 1), sigma.sq.t.ig = c(2, 1)) # Initial values z.init <- apply(dat$y, c(1, 2), function(a) as.numeric(sum(a, na.rm = TRUE) > 0))
inits.list <- list(beta = 0, alpha = 0, z = z.init, phi = 3 / .5, sigma.sq = 2,
w = rep(0, J), rho = 0, sigma.sq.t = 0.5)
# Tuning
tuning.list <- list(phi = 1, rho = 1)
# Number of batches
n.batch <- 10
# Batch length
batch.length <- 25
n.iter <- n.batch * batch.length

# Run the model
out <- stPGOcc(occ.formula = ~ trend + occ.cov.1,
det.formula = ~ det.cov.1 + det.cov.2,
data = data.list,
inits = inits.list,
n.batch = n.batch,
batch.length = batch.length,
priors = prior.list,
cov.model = "exponential",
tuning = tuning.list,
NNGP = TRUE,
ar1 = TRUE,
n.neighbors = 5,
search.type = 'cb',
n.report = 10,
n.burn = 50,
n.chains = 1)
#> ----------------------------------------
#> 	Preparing the data
#> ----------------------------------------
#> ----------------------------------------
#> 	Building the neighbor list
#> ----------------------------------------
#> ----------------------------------------
#> Building the neighbors of neighbors list
#> ----------------------------------------
#> ----------------------------------------
#> 	Model description
#> ----------------------------------------
#> Spatial NNGP Multi-season Occupancy Model with Polya-Gamma latent
#> variable fit with 100 sites and 10 primary time periods.
#>
#> Samples per chain: 250 (10 batches of length 25)
#> Burn-in: 50
#> Thinning Rate: 1
#> Number of Chains: 1
#> Total Posterior Samples: 200
#>
#> Using the exponential spatial correlation model.
#>
#> Using 5 nearest neighbors.
#>
#> Using an AR(1) temporal autocorrelation matrix.
#>
#> Source compiled with OpenMP support and model fit using 1 thread(s).
#>
#> Adaptive Metropolis with target acceptance rate: 43.0
#> ----------------------------------------
#> 	Chain 1
#> ----------------------------------------
#> Sampling ...
#> Batch: 10 of 10, 100.00%

summary(out)
#>
#> Call:
#> stPGOcc(occ.formula = ~trend + occ.cov.1, det.formula = ~det.cov.1 +
#>     det.cov.2, data = data.list, inits = inits.list, priors = prior.list,
#>     tuning = tuning.list, cov.model = "exponential", NNGP = TRUE,
#>     n.neighbors = 5, search.type = "cb", n.batch = n.batch, batch.length = batch.length,
#>     ar1 = TRUE, n.report = 10, n.burn = 50, n.chains = 1)
#>
#> Samples per Chain: 250
#> Burn-in: 50
#> Thinning Rate: 1
#> Number of Chains: 1
#> Total Posterior Samples: 200
#> Run Time (min): 0.0082
#>
#> Occurrence (logit scale):
#>                Mean     SD    2.5%     50%   97.5% Rhat ESS
#> (Intercept)  1.2084 0.6023  0.2504  1.0865  2.1566   NA   3
#> trend       -0.4900 0.3418 -1.1381 -0.5272  0.2003   NA  27
#> occ.cov.1   -1.0684 0.2009 -1.4522 -1.0775 -0.6865   NA  16
#>
#> Detection (logit scale):
#>                Mean     SD    2.5%     50%   97.5% Rhat ESS
#> (Intercept) -0.9912 0.0869 -1.1620 -0.9915 -0.8295   NA  53
#> det.cov.1    0.6118 0.0816  0.4575  0.6140  0.7586   NA  96
#> det.cov.2   -0.5477 0.0858 -0.7200 -0.5454 -0.4020   NA  90
#>
#> Spatio-temporal Covariance:
#>              Mean     SD    2.5%    50%   97.5% Rhat ESS
#> sigma.sq   2.9723 1.2126  1.3673 2.7057  5.8712   NA  11
#> phi        7.9051 3.9239  3.2837 5.8518 15.3431   NA   3
#> sigma.sq.t 0.8014 0.5526  0.1785 0.6592  2.2398   NA  65
#> rho        0.0234 0.2477 -0.3954 0.0496  0.4580   NA  35