Function for Fitting Univariate Spatialy-Varying Coefficient GLMMs

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 = 'Poisson',
         n.omp.threads = 1, verbose = TRUE, n.report = 100,
         n.burn = round(.10 * n.batch * batch.length), n.thin = 1,
         n.chains = 1, save.fitted = TRUE, ...)

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, z, coords, and offset. y is a vector, matrix, or data frame of the observed count values. If a vector, the values represent the observed counts at each site. If multiple replicate observations are obtained at the sites (e.g., sub-samples, repeated sampling over multiple seasons), y can be specified as a 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. covs is either a data frame or list containing the variables used in the model. When only fitting a model with site-level data, covs can be specified as a data frame, with each row corresponding to site and each column corresponding to a variable. When multiple abundance values are available at a site, covs is specified as a list, where each list element is a different covariate, which can be site-level or observation-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 replicate observations at a given site. coords is a \(J \times 2\) matrix of the observation coordinates. Note that spAbundance assumes coordinates are specified in a projected coordinate system. For zero-inflated Gaussian models, the tag z is used to specify the binary component of the zero-inflated model and should have the same length as y. offset is an offset to use in the abundance model (e.g., an area offset). This can be either a single value, a vector with an offset for each site (e.g., if survey area differed in size), or a site x replicate matrix if more than one count is available at a given site.
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, kappa. 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, kappa.unif. 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 assuming 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. The negative binomial dispersion parameter kappa is assumed to follow a Uniform distribution. The hyperparameters of the Uniform are passed as a vector of length two with the first and second elements corresponding to the lower and upper support, respectively.
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 list with each tag corresponding to a parameter name, whose value defines the initial variance of the adaptive sampler. Valid tags are 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 the latent abundance process. Currently supports 'NB' (negative binomial), 'Poisson', 'Gaussian', and 'zi-Gaussian'. Default is Poisson.
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.
save.fitted: logical value indicating whether or not fitted values and likelihood values should be saved in the resulting model object. If save.fitted = FALSE, the components y.rep.samples, mu.samples, and like.samples will not be included in the model object, and subsequent functions for calculating WAIC, fitted values, and posterior predictive checks will not work, although they all can be calculated manually if desired. Setting save.fitted = FALSE can be useful when working with very large data sets to minimize the amount of RAM needed when fitting and storing the model object in memory.
...: 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.
kappa.samples: a coda object of posterior samples for the abundance dispersion parameter. Only included when family = 'NB'.
y.rep.samples: a two or three-dimensional object of posterior samples for the abundance replicate (fitted) values with dimensions corresponding to MCMC samples, site, and replicate.
mu.samples: a two or -three-dimensional array of posterior samples for the expected abundance samples with dimensions corresponding to MCMC samples, site, and replicate.
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%