# Function for Fitting Univariate Spatialy-Varying Coefficient GLMMs

`svcAbund.Rd`

The function `svcAbund`

fits univariate spatially-varying coefficient GLMMs.

## Usage

```
svcAbund(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, family = 'Gaussian',
n.omp.threads = 1, verbose = TRUE, n.report = 100,
n.burn = round(.10 * n.batch * batch.length), n.thin = 1,
n.chains = 1, ...)
```

## Arguments

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

- data
a list containing data necessary for model fitting. Valid tags are

`y`

,`covs`

,`coords`

, and`z`

(for`family = 'zi-Gaussian'`

only.`y`

is a vector of the observed count values, where the values represent the observed values at each site.`covs`

is a list, matrix, or data frame of covariates used in the model, where each column (or list element) represents a different covariate.`coords`

is a \(J \times 2\) matrix of the observation coordinates. Note that`svcAbundance`

assumes coordinates are specified in a projected coordinate system.`z`

is used for fitting a zero-inflated Gaussian model. It is a vector where each value indicates the binary component of the model. In the context of abundance models, this can be thought of as the component of the model that indicates whether the species is present at each location, and then the supplied values in`y`

are the observed abundance values at those locations where`z = 1`

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

`beta`

,`sigma.sq`

,`phi`

,`w`

,`nu`

,`tau.sq`

,`sigma.sq.mu`

.`nu`

is only specified if`cov.model = "matern"`

,`sigma.sq.mu`

is only specified if there are random effects in`formula`

, and 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`

,`nu.unif`

,`tau.sq.ig`

,`sigma.sq.mu.ig`

. Abundance (`beta`

) 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 are set to 100. The spatial variance parameter,`sigma.sq`

, and the Gaussian residual variance parameter,`tau.sq`

, are assumed to follow an inverse-Gamma distribution. The spatial decay`phi`

and spatial smoothness`nu`

, parameters are assumed to follow Uniform distributions. The hyperparameters of the inverse-Gamma for`sigma.sq`

is passed as a list of length two with the first and second elements corresponding to the shape and scale parameters of the inverse-Gamma distribution either for each spatially-varying coefficient, or a single value if assumign the same values for all spatially-varying coefficients. The hyperparameters of the inverse-Gamma for`tau.sq`

is 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 list of length two with the first and second elements corresponding to the lower and upper support, respectively, for each SVC or a single value if giving the same prior for each SVC.`sigma.sq.mu`

are the random effect variances for any random effects, 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 effects 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`occ.covs`

(for the intercept, use '(Intercept)').`svc.cols`

default argument of 1 results in a univariate spatial GLMM analogous to`spAbund`

(assuming an intercept is included in the model).- 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 single numeric value representing the initial variance of the adaptive sampler for

`phi`

and`nu`

. See Roberts and Rosenthal (2009) for details.- NNGP
if

`TRUE`

, model is fit with an NNGP. See Datta et al. (2016) and Finley et al. (2019) for more information. Currently only NNGP is supported, functionality for a full GP may be addded in future package development.- 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.- 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.

- family
the distribution to use for abundance. Currently, spatially-varying coefficient models are available for

`family = 'Gaussian'`

and`family = 'zi-Gaussian'`

.- 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.

- ...
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
.

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 `svcAbund`

that is a list comprised of:

- beta.samples
a

`coda`

object of posterior samples for the abundance regression coefficients.- tau.sq.samples
a

`coda`

object of posterior samples for the residual variance parameter.- y.rep.samples
a

`coda`

object of posterior samples for the abundance replicate (fitted) values with dimensions corresponding to MCMC samples and site.- mu.samples
a

`coda`

object of posterior samples for the expected abundance samples with dimensions corresponding to MCMC samples and site.- theta.samples
a

`coda`

object of posterior samples for spatial covariance parameters.- w.samples
a three-dimensional array of posterior samples for the spatially-varying coefficients with dimensions corresponding to MCMC sample, SVC, and site.

- sigma.sq.mu.samples
a

`coda`

object of posterior samples for variances of random effects included in the model. Only included if random effects are specified in`formula`

.- beta.star.samples
a

`coda`

object of posterior samples for the abundance random effects. Only included if random effects 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()`

.

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
# Abundance ---------------------------
beta <- c(5, 0.5, -0.2, 0.75)
p <- length(beta)
mu.RE <- list()
mu.RE <- list(levels = c(35, 40),
sigma.sq.mu = c(0.7, 1.5),
beta.indx = list(1, 1))
# Spatial parameters ------------------
sp <- TRUE
svc.cols <- c(1, 2)
p.svc <- length(svc.cols)
cov.model <- "exponential"
sigma.sq <- runif(p.svc, 0.4, 4)
phi <- runif(p.svc, 3/1, 3/0.6)
tau.sq <- 2
z <- rbinom(J, 1, 0.5)
# Get all the data
dat <- simAbund(J.x = J.x, J.y = J.y, beta = beta, tau.sq = tau.sq,
mu.RE = mu.RE, sp = sp, svc.cols = svc.cols,
family = 'zi-Gaussian', cov.model = cov.model,
sigma.sq = sigma.sq, phi = phi, z = z)
# Get data in format for spAbundance --------------------------------------
y <- dat$y
X <- dat$X
X.re <- dat$X.re
coords <- dat$coords
# Package all data into a list
covs <- cbind(X, X.re)
colnames(covs) <- c('int', 'cov.1', 'cov.2', 'cov.3', 'factor.1', 'factor.2')
# Data list bundle
data.list <- list(y = y, covs = covs, coords = coords, z = z)
# Priors
prior.list <- list(beta.normal = list(mean = 0, var = 1000),
sigma.sq.ig = list(a = 2, b = 1), tau.sq = c(2, 1),
sigma.sq.mu.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 = 3 / 0.5,
tau.sq = 2, sigma.sq.mu = 0.5)
# Tuning
tuning.list <- list(phi = 1)
n.batch <- 10
batch.length <- 25
n.burn <- 100
n.thin <- 1
out <- svcAbund(formula = ~ cov.1 + cov.2 + cov.3 +
(1 | factor.1) + (1 | factor.2),
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,
family = 'zi-Gaussian',
cov.model = "exponential",
tuning = tuning.list,
n.omp.threads = 1,
verbose = TRUE,
NNGP = TRUE,
n.neighbors = 5,
n.report = 25,
n.burn = n.burn,
n.thin = n.thin,
n.chains = 3)
#> ----------------------------------------
#> Preparing to run the model
#> ----------------------------------------
#> No prior specified for tau.sq.
#> Using an inverse-Gamma prior with the shape parameter set to 2 and scale parameter to 0.5.
#> 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 model with 53 sites.
#>
#> Samples per chain: 250 (10 batches of length 25)
#> Burn-in: 100
#> Thinning Rate: 1
#> Number of Chains: 3
#> Total Posterior Samples: 450
#>
#> 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: 10 of 10, 100.00%
#> ----------------------------------------
#> Chain 2
#> ----------------------------------------
#> Sampling ...
#> Batch: 10 of 10, 100.00%
#> ----------------------------------------
#> Chain 3
#> ----------------------------------------
#> Sampling ...
#> Batch: 10 of 10, 100.00%
```