The function svcPGBinom fits single-species spatially-varying coefficient binomial models using Polya-Gamma latent variables. Models are fit using Nearest Neighbor Gaussian Processes.

## Usage

svcPGBinom(formula, data, inits, priors, tuning, svc.cols = 1,
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, k.fold.only = FALSE, ...)

## Arguments

formula

a symbolic description of the model to be fit 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, covs, weights, and coords. y is a numeric vector containing the binomial data with length equal to the total number of sites ($$J$$). covs is a matrix or data frame containing the covariates used in the model, with $$J$$ rows for each column (variable). weights is a numeric vector containing the binomial weights (i.e., the total number of Bernoulli trials) at each site. If weights is not specified, svcPGBinom assumes 1 trial at each site (i.e., presence/absence). 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 beta, sigma.sq, phi, w, nu, and sigma.sq.psi. nu is only specified if cov.model = "matern", and sigma.sq.psi is only specified if there are random effects in 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, phi.unif, sigma.sq.ig, sigma.sq.unif, nu.unif, and sigma.sq.psi.ig. Regression coefficients (beta) 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, for each spatially-varying coefficient is assumed to follow an inverse-Gamma distribution or a uniform distribution (default is inverse-Gamma). 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 list with two elements corresponding to the shape and scale parametters, respetively, with each element comprised of a vector equal to the number of spatially-varying coefficients to be estimated or of length one if priors are the same for all coefficients. The hyperparameters of any uniform priors are also passed as a list of length two with the first and second elements corresponding to the lower and upper support, respectively, which can be passed as a vector equal to the total number of spatially-varying coefficients to be estimated or of length one if priors are the same for all coefficients. sigma.sq.psi are the random effect variances for any 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.

svc.cols

a vector indicating the variables whose effects will be estimated as spatially-varying coefficients. svc.cols can be an integer vector with values indicating the order of covariates specified in the model formula (with 1 being the intercept if specified), or it can be specified as a character vector with names corresponding to variable names in covs (for the intercept, use '(Intercept)').

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, sigma.sq, 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.

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

## 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., and Banerjee, S. (2020). Bayesian spatially varying coefficient models in the spBayes R package. Environmental Modelling and Software, 125, 104608.

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 svcPGBinom that is a list comprised of:

beta.samples

a coda object of posterior samples for the regression coefficients.

y.rep.samples

a coda object of posterior samples for the fitted data values

psi.samples

a coda object of posterior samples for the occurrence probability values

theta.samples

a coda object of posterior samples for spatial covariance parameters.

w.samples

a three-dimensional array of posterior samples for the latent spatial random effects for all spatially-varying coefficients. Dimensions correspond to MCMC sample, coefficient, and sites.

sigma.sq.psi.samples

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

beta.star.samples

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

## Examples

set.seed(1000)
# Sites
J.x <- 10
J.y <- 10
J <- J.x * J.y
# Binomial weights
weights <- sample(10, J, replace = TRUE)
beta <- c(0, 0.5, -0.2, 0.75)
p <- length(beta)
# No unstructured random effects
psi.RE <- list()
# Spatial parameters
sp <- TRUE
# Two spatially-varying covariates.
svc.cols <- c(1, 2)
p.svc <- length(svc.cols)
cov.model <- "exponential"
sigma.sq <- runif(p.svc, 0.4, 1.5)
phi <- runif(p.svc, 3/1, 3/0.2)

# Simulate the data
dat <- simBinom(J.x = J.x, J.y = J.y, weights = weights, beta = beta,
psi.RE = psi.RE, sp = sp, svc.cols = svc.cols,
cov.model = cov.model, sigma.sq = sigma.sq, phi = phi)

# Binomial data
y <- dat$y # Covariates X <- dat$X
# Spatial coordinates
coords <- dat\$coords

# Package all data into a list
# Covariates
covs <- cbind(X)
colnames(covs) <- c('int', 'cov.1', 'cov.2', 'cov.3')

# Data list bundle
data.list <- list(y = y,
covs = covs,
coords = coords,
weights = weights)
# Priors
prior.list <- list(beta.normal = list(mean = 0, var = 2.72),
sigma.sq.ig = list(a = 2, b = 1),
phi.unif = list(a = 3 / 1, b = 3 / 0.1))

# Starting values
inits.list <- list(beta = 0, alpha = 0,
sigma.sq = 1, phi = phi)
# Tuning
tuning.list <- list(phi = 1)

n.batch <- 10
batch.length <- 25
n.burn <- 100
n.thin <- 1

out <- svcPGBinom(formula = ~ cov.1 + cov.2 + cov.3,
svc.cols = c(1, 2),
data = data.list,
n.batch = n.batch,
batch.length = batch.length,
inits = inits.list,
priors = prior.list,
accept.rate = 0.43,
cov.model = "exponential",
tuning = tuning.list,
verbose = TRUE,
NNGP = TRUE,
n.neighbors = 5,
n.report = 2,
n.burn = n.burn,
n.thin = n.thin,
n.chains = 1)
#> ----------------------------------------
#> 	Preparing to run the model
#> ----------------------------------------
#> w is not specified in initial values.
#> Setting initial value to 0
#> ----------------------------------------
#> 	Building the neighbor list
#> ----------------------------------------
#> ----------------------------------------
#> Building the neighbors of neighbors list
#> ----------------------------------------
#> ----------------------------------------
#> 	Model description
#> ----------------------------------------
#> Spatial NNGP Binomial model with Polya-Gamma latent
#> variable fit with 100 sites.
#>
#> Samples per chain: 250 (10 batches of length 25)
#> Burn-in: 100
#> Thinning Rate: 1
#> Number of Chains: 1
#> Total Posterior Samples: 150
#>
#> Number of spatially-varying coefficients: 2
#> Using the exponential spatial correlation model.
#>
#> Using 5 nearest neighbors.
#>
#> Source compiled with OpenMP support and model fit using 1 thread(s).
#>
#> Adaptive Metropolis with target acceptance rate: 43.0
#> ----------------------------------------
#> 	Chain 1
#> ----------------------------------------
#> Sampling ...
#> Batch: 2 of 10, 20.00%
#> 	Coefficient	Parameter	Acceptance	Tuning
#> 	1		phi		60.0		1.03045
#> 	2		phi		64.0		1.03045
#> -------------------------------------------------
#> Batch: 4 of 10, 40.00%
#> 	Coefficient	Parameter	Acceptance	Tuning
#> 	1		phi		32.0		1.03045
#> 	2		phi		44.0		1.05127
#> -------------------------------------------------
#> Batch: 6 of 10, 60.00%
#> 	Coefficient	Parameter	Acceptance	Tuning
#> 	1		phi		36.0		1.03045
#> 	2		phi		52.0		1.07251
#> -------------------------------------------------
#> Batch: 8 of 10, 80.00%
#> 	Coefficient	Parameter	Acceptance	Tuning
#> 	1		phi		44.0		1.05127
#> 	2		phi		60.0		1.09417
#> -------------------------------------------------
#> Batch: 10 of 10, 100.00%

summary(out)
#>
#> Call:
#> svcPGBinom(formula = ~cov.1 + cov.2 + cov.3, data = data.list,
#>     inits = inits.list, priors = prior.list, tuning = tuning.list,
#>     svc.cols = c(1, 2), cov.model = "exponential", NNGP = TRUE,
#>     n.neighbors = 5, n.batch = n.batch, batch.length = batch.length,
#>     accept.rate = 0.43, n.omp.threads = 1, verbose = TRUE, n.report = 2,
#>     n.burn = n.burn, n.thin = n.thin, n.chains = 1)
#>
#> Samples per Chain: 250
#> Burn-in: 100
#> Thinning Rate: 1
#> Number of Chains: 1
#> Total Posterior Samples: 150
#> Run Time (min): 0.0073
#>
#> Occurrence (logit scale):
#>                Mean     SD    2.5%     50%   97.5% Rhat ESS
#> (Intercept) -0.0387 0.1616 -0.2929 -0.0510  0.3403   NA  27
#> cov.1        0.0995 0.2029 -0.2972  0.0908  0.4857   NA  17
#> cov.2       -0.3185 0.1245 -0.5740 -0.3136 -0.0951   NA  40
#> cov.3        0.7424 0.1164  0.5420  0.7371  0.9568   NA  41
#>
#> Spatial Covariance:
#>                         Mean     SD   2.5%     50%   97.5% Rhat ESS
#> sigma.sq-(Intercept)  0.5286 0.2138 0.2338  0.4808  1.0627   NA  13
#> sigma.sq-cov.1        0.5993 0.2753 0.2013  0.5610  1.2730   NA   9
#> phi-(Intercept)      13.4600 5.7156 5.8858 11.0325 26.4691   NA   9
#> phi-cov.1            16.2030 7.4862 4.8745 13.5084 28.7637   NA   5