The function spPGOcc fits single-species spatial occupancy models using Polya-Gamma latent variables. Models can be fit using either a full Gaussian process or a Nearest Neighbor Gaussian Process for large data sets.

## Usage

spPGOcc(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, 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, ...)

## 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 the detection-nondetection data matrix or data frame with first dimension equal to the number of sites ($$J$$) and second dimension equal to the maximum number of replicates at a given site. occ.covs is a matrix or data frame containing the variables used in the occupancy portion of the model, with $$J$$ rows for each column (variable). det.covs is a list of variables included in the detection portion of the model. Each list element is a different detection covariate, which can be site-level or observational-level. Site-level covariates are specified as a vector of length $$J$$ while observation-level covariates are specified as a matrix or data frame with the number of rows equal to $$J$$ and number of columns equal to the maximum number of replicates at a given site. 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. nu is only specified if cov.model = "matern", sigma.sq.p is only specified if there are random effects in det.formula, and sigma.sq.psi is only specified if there are random effects in occ.formula. The value portion of each tag is the parameter's initial value. 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, phi.unif, sigma.sq.ig, sigma.sq.unif, nu.unif, sigma.sq.psi.ig, and sigma.sq.p.ig. Occurrence (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 2.73. The spatial variance parameter, sigma.sq, is assumed to follow an inverse-Gamma distribution or a uniform distribution (default is inverse-Gamma). sigma.sq can also be fixed at its initial value by setting the prior value to "fixed". The spatial decay phi and smoothness nu parameters are assumed to follow Uniform distributions. The hyperparameters of the inverse-Gamma for sigma.sq are passed as a vector of length two, with the first and second elements corresponding to the shape and scale, 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.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 the 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.

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.

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.

verbose

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

n.report

the interval to report Metropolis sampler acceptance and MCMC progress.

n.burn

the number of samples out of the total n.batch * batch.length samples in each chain to discard as burn-in. 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 MCMC 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). 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.

...

## References

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 .

Datta, A., S. Banerjee, A.O. Finley, and A.E. Gelfand. (2016) Hierarchical Nearest-Neighbor Gaussian process models for large geostatistical datasets. Journal of the American Statistical Association, doi:10.1080/01621459.2015.1044091 .

Finley, A.O., A. Datta, B.D. Cook, D.C. Morton, H.E. Andersen, and S. Banerjee. (2019) Efficient algorithms for Bayesian Nearest Neighbor Gaussian Processes. Journal of Computational and Graphical Statistics, doi:10.1080/10618600.2018.1537924 .

Finley, A. O., Datta, A., and Banerjee, S. (2020). spNNGP R package for nearest neighbor Gaussian process models. arXiv preprint arXiv:2001.09111.

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

Hooten, M. B., and Hefley, T. J. (2019). Bringing Bayesian models to life. CRC Press.

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.

Roberts, G.O. and Rosenthal J.S. (2009) Examples of adaptive MCMC. Journal of Computational and Graphical Statistics, 18(2):349-367.

## Author

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

## Value

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

beta.samples

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

alpha.samples

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

z.samples

a coda object of posterior samples for the latent occurrence values

psi.samples

a coda object of posterior samples for the latent occurrence probability values

theta.samples

a coda object of posterior samples for covariance parameters.

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.

like.samples

a coda object of posterior samples for the likelihood value associated with each site. 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

soring 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 values are not included in the model object, but can be extracted using fitted().

## Examples

set.seed(350)
# Simulate Data -----------------------------------------------------------
J.x <- 8
J.y <- 8
J <- J.x * J.y
n.rep <- sample(2:4, J, replace = TRUE)
beta <- c(0.5, -0.15)
p.occ <- length(beta)
alpha <- c(0.7, 0.4, -0.2)
p.det <- length(alpha)
phi <- 3 / .6
sigma.sq <- 2
dat <- simOcc(J.x = J.x, J.y = J.y, n.rep = n.rep, beta = beta, alpha = alpha,
sigma.sq = sigma.sq, phi = phi, sp = TRUE, cov.model = 'exponential')
y <- dat$y X <- dat$X
X.p <- dat$X.p coords <- as.matrix(dat$coords)

# Package all data into a list
occ.covs <- X[, -1, drop = FALSE]
colnames(occ.covs) <- c('occ.cov')
det.covs <- list(det.cov.1 = X.p[, , 2],
det.cov.2 = X.p[, , 3])
data.list <- list(y = y,
occ.covs = occ.covs,
det.covs = det.covs,
coords = coords)

# Number of batches
n.batch <- 10
# Batch length
batch.length <- 25
n.iter <- n.batch * batch.length
# 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/.1))
# Initial values
inits.list <- list(alpha = 0, beta = 0,
phi = 3 / .5,
sigma.sq = 2,
w = rep(0, nrow(X)),
z = apply(y, 1, max, na.rm = TRUE))
# Tuning
tuning.list <- list(phi = 1)

out <- spPGOcc(occ.formula = ~ occ.cov,
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 = FALSE,
n.neighbors = 5,
search.type = 'cb',
n.report = 10,
n.burn = 50,
n.chains = 1)
#> ----------------------------------------
#> 	Preparing to run the model
#> ----------------------------------------
#> ----------------------------------------
#> 	Model description
#> ----------------------------------------
#> Spatial Occupancy Model with Polya-Gamma latent
#> variable fit with 64 sites.
#>
#> 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.
#>
#> 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:
#> spPGOcc(occ.formula = ~occ.cov, det.formula = ~det.cov.1 + det.cov.2,
#>     data = data.list, inits = inits.list, priors = prior.list,
#>     tuning = tuning.list, cov.model = "exponential", NNGP = FALSE,
#>     n.neighbors = 5, search.type = "cb", n.batch = n.batch, batch.length = batch.length,
#>     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): 0.0036
#>
#> Occurrence (logit scale):
#>               Mean     SD    2.5%    50%  97.5% Rhat ESS
#> (Intercept) 1.0415 0.4029  0.2761 1.0469 1.8607   NA  52
#> occ.cov     0.2147 0.5006 -0.8406 0.1967 1.1852   NA  55
#>
#> Detection (logit scale):
#>                Mean     SD    2.5%     50%  97.5% Rhat ESS
#> (Intercept)  0.3391 0.2017 -0.0722  0.3447 0.6798   NA  76
#> det.cov.1    0.0839 0.1774 -0.2740  0.0953 0.3827   NA 200
#> det.cov.2   -0.1533 0.1877 -0.4954 -0.1592 0.1889   NA 200
#>
#> Spatial Covariance:
#>             Mean     SD   2.5%     50%   97.5% Rhat ESS
#> sigma.sq  1.9194 1.2387 0.5951  1.5286  4.7456   NA  19
#> phi      18.6682 8.5731 3.8986 21.4626 29.2970   NA   7