# Function for Fitting Single-Species Spatially-Varying Coefficient Binomial Models Using Polya-Gamma Latent Variables

`svcPGBinom.Rd`

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.

- 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.- 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.- 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

## 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,
n.omp.threads = 1,
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
```