# Function for Fitting Multi-Season Single-Species Occupancy Models Using Polya-Gamma Latent Variables

`tPGOcc.Rd`

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

## Usage

```
tPGOcc(occ.formula, det.formula, data, inits, priors, tuning,
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`

, and`det.covs`

.`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 time 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.- inits
a list with each tag corresponding to a parameter name. Valid tags are

`z`

,`beta`

,`alpha`

,`sigma.sq.psi`

,`sigma.sq.p`

,`sigma.sq.t`

, and`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.`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`

,`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 unstructured 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.`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.- tuning
a list with each tag corresponding to a parameter name. Valid tags are

`rho`

. The value portion of each tag defines the initial tuning variance of the Adaptive sampler. See Roberts and Rosenthal (2009) for details.- 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.

- n.omp.threads
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. Note this is specified in terms of batches, not MCMC samples.

- 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`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.- k.fold.threads
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`

.- ...
currently no additional arguments

## 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. Note this object will contain predicted occupancy values for sites/primary time periods that were not sampled.

- 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. Note this object will contained predicted occupancy probabilities for sites/primary time periods that were not sampled.

- 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`

.- theta.samples
a

`coda`

object of posterior samples for the AR(1) variance (`sigma.sq.t`

) and correlation (`rho`

) parameters. Only included if`ar1 = TRUE`

.- 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(5:10, 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()
# Temporal parameters -----------------
rho <- 0.7
sigma.sq.t <- 0.6
# 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 = FALSE, 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)
# Priors
prior.list <- list(beta.normal = list(mean = 0, var = 2.72),
alpha.normal = list(mean = 0, var = 2.72),
rho.unif = c(-1, 1),
sigma.sq.t.ig = c(2, 0.5))
# Starting 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)
# Tuning
tuning.list <- list(rho = 0.5)
n.batch <- 50
batch.length <- 25
n.samples <- n.batch * batch.length
n.burn <- 1000
n.thin <- 1
# Run the model
out <- tPGOcc(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,
n.batch = n.batch,
batch.length = batch.length,
verbose = TRUE,
ar1 = TRUE,
n.report = 25,
n.burn = n.burn,
n.thin = n.thin,
n.chains = 1)
#> ----------------------------------------
#> Preparing the data
#> ----------------------------------------
#> rho is not specified in initial values.
#> Setting initial value to random value from the prior distribution
#> sigma.sq.t is not specified in initial values.
#> Setting initial value to random value between 0.5 and 10
#> ----------------------------------------
#> Model description
#> ----------------------------------------
#> Multi-season Occupancy Model with Polya-Gamma latent variable
#> fit with 100 sites and 10 primary time periods.
#>
#> Samples per chain: 1250 (50 batches of length 25)
#> Burn-in: 1000
#> Thinning Rate: 1
#> Number of Chains: 1
#> Total Posterior Samples: 250
#>
#> Using an AR(1) temporal autocorrelation matrix in the occurrence sub-model.
#>
#> Source compiled with OpenMP support and model fit using 1 thread(s).
#>
#> ----------------------------------------
#> Chain 1
#> ----------------------------------------
#> Sampling ...
#> Batch: 25 of 50, 50.00%
#> Parameter Acceptance Tuning
#> rho 68.0 0.63562
#> -------------------------------------------------
#> Batch: 50 of 50, 100.00%
summary(out)
#>
#> Call:
#> tPGOcc(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, n.batch = n.batch, batch.length = batch.length,
#> verbose = TRUE, ar1 = TRUE, n.report = 25, n.burn = n.burn,
#> n.thin = n.thin, n.chains = 1)
#>
#> Samples per Chain: 1250
#> Burn-in: 1000
#> Thinning Rate: 1
#> Number of Chains: 1
#> Total Posterior Samples: 250
#> Run Time (min): 0.0172
#>
#> Occurrence (logit scale):
#> Mean SD 2.5% 50% 97.5% Rhat ESS
#> (Intercept) 0.2761 0.3901 -0.3213 0.2306 1.3294 NA 10
#> trend 0.6303 0.3101 0.0772 0.6068 1.1966 NA 11
#> occ.cov.1 -0.9107 0.1740 -1.2413 -0.9192 -0.5749 NA 37
#>
#> Detection (logit scale):
#> Mean SD 2.5% 50% 97.5% Rhat ESS
#> (Intercept) -0.9884 0.1107 -1.1849 -0.9953 -0.7314 NA 33
#> det.cov.1 0.5571 0.0803 0.4114 0.5563 0.7117 NA 134
#> det.cov.2 -0.5855 0.0998 -0.7972 -0.5730 -0.4248 NA 99
#>
#> Occurrence AR(1) Temporal Covariance:
#> Mean SD 2.5% 50% 97.5% Rhat ESS
#> sigma.sq.t 0.3746 0.2607 0.0994 0.3055 1.0950 NA 47
#> rho 0.4348 0.3761 -0.5284 0.4911 0.8916 NA 18
```