Function for Fitting Univariate Spatial Abundance GLMs

The function spAbund fits univariate spatial abundance GLMs.

Usage

spAbund(formula, data, inits, priors, tuning,
        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, kappa, sigma.sq.mu, tau.sq. nu is only specified if cov.model = "matern", sigma.sq.mu is only specified if there are random effects in formula, and kappa is only specified when family = 'NB'. tau.sq is only specified when family = 'Gaussian' or family = 'zi-Gaussian'. 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, kappa.unif, sigma.sq.mu.ig, tau.sq.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, is assumed to follow an inverse-Gamma distribution. The spatial decay phi, spatial smoothness nu, and negative binomial dispersion kappa 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.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. tau.sq is the residual variance for Gaussian (or zero-inflated Gaussian) models, and it is assigned an inverse-Gamma prior. The hyperparameters of the inverse-Gamma are passed as a vector of length two, with the first and second element corresponding to the shape and scale parameters, respectively.
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 beta, alpha, beta.star (the abundance random effect values), kappa, phi, and nu. See Roberts and Rosenthal (2009) for details. Note that only phi and nu are the only parameters that require tuning for a Gaussian or zero-inflated Gaussian model.
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'.
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 spAbund that is a list comprised of:

beta.samples: a coda object of posterior samples for the abundance regression coefficients.
kappa.samples: a coda object of posterior samples for the abundance dispersion parameter. Only included when family = 'NB'.
tau.sq.samples: a coda object of posterior samples for the Gaussian residual variance parameter. Only included when family = 'Gaussian' or family = 'zi-Gaussian'.
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 coda object of posterior samples for latent spatial random effects.
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(888)
J.x <- 8
J.y <- 8
J <- J.x * J.y
n.rep <- sample(3, J, replace = TRUE)
beta <- c(0, -1.5, 0.3, -0.8)
p.abund <- length(beta)
mu.RE <- list(levels = c(50, 45),
              sigma.sq.mu = c(1.3, 0.5), 
              beta.indx = c(1, 2))
phi <- 3/.6
sigma.sq <- 2
kappa <- 0.2
sp <- TRUE 
cov.model <- 'exponential'
family <- 'NB'
dat <- simAbund(J.x = J.x, J.y = J.y, n.rep = n.rep, beta = beta, 
                kappa = kappa, mu.RE = mu.RE, sp = sp, phi = phi, 
                sigma.sq = sigma.sq, cov.model = cov.model, family = 'NB')

y <- dat$y
X <- dat$X
X.re <- dat$X.re
coords <- dat$coords

covs <- list(int = X[, , 1], 
             abund.cov.1 = X[, , 2], 
             abund.cov.2 = X[, , 3], 
             abund.cov.3 = X[, , 4],
             abund.factor.1 = X.re[, , 1], 
             abund.factor.2 = X.re[, , 2])

data.list <- list(y = y, covs = covs, coords = coords)

# Priors
prior.list <- list(beta.normal = list(mean = 0, var = 100),
                   phi.unif = c(3 / 1, 3 / .1),
                   sigma.sq.ig = c(2, 1),
                   kappa.unif = c(0.001, 10)) 
# Starting values
inits.list <- list(beta = beta, kappa = kappa, sigma.sq = sigma.sq, phi = phi)

tuning <- list(phi = 0.3, kappa = 0.05, beta = 0.1, beta.star = 0.1, w = 0.1) 
n.batch <- 4
batch.length <- 25
n.burn <- 20 
n.thin <- 1
n.chains <- 1

out <- spAbund(formula = ~ abund.cov.1 + abund.cov.2 + abund.cov.3 +
                           (1 | abund.factor.1) + (abund.cov.1 | abund.factor.2),
               data = data.list, 
               n.batch = n.batch, 
               batch.length = batch.length, 
               inits = inits.list, 
               tuning = tuning,
               priors = prior.list, 
               NNGP = TRUE, 
               cov.model = 'exponential',
               search.type = 'cb',
               n.neighbors = 5,
               accept.rate = 0.43, 
               n.omp.threads = 1, 
               verbose = TRUE, 
               n.report = 1,
               n.burn = n.burn,
               n.thin = n.thin,
               n.chains = n.chains) 
#> ----------------------------------------
#> 	Preparing to run the model
#> ----------------------------------------
#> No prior specified for sigma.sq.mu.ig.
#> Setting prior shape to 0.1 and prior scale to 0.1
#> sigma.sq.mu is not specified in initial values.
#> Setting initial values to random values between 0.05 and 1
#> 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 Poisson Abundance model fit with 64 sites.
#> 
#> Samples per Chain: 100 (4 batches of length 25)
#> Burn-in: 20 
#> Thinning Rate: 1 
#> Number of Chains: 1 
#> Total Posterior Samples: 80 
#> 
#> 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: 1 of 4, 25.00%
#> 	Parameter	Acceptance	Tuning
#> 	beta[1]		48.0		0.10202
#> 	beta[2]		36.0		0.10000
#> 	beta[3]		40.0		0.10000
#> 	beta[4]		40.0		0.09802
#> 	phi		88.0		0.30606
#> -------------------------------------------------
#> Batch: 2 of 4, 50.00%
#> 	Parameter	Acceptance	Tuning
#> 	beta[1]		32.0		0.10101
#> 	beta[2]		24.0		0.09900
#> 	beta[3]		48.0		0.10101
#> 	beta[4]		68.0		0.09900
#> 	phi		92.0		0.30914
#> -------------------------------------------------
#> Batch: 3 of 4, 75.00%
#> 	Parameter	Acceptance	Tuning
#> 	beta[1]		28.0		0.10000
#> 	beta[2]		12.0		0.09802
#> 	beta[3]		40.0		0.10000
#> 	beta[4]		44.0		0.10000
#> 	phi		88.0		0.31224
#> -------------------------------------------------
#> Batch: 4 of 4, 100.00%
summary(out)
#> 
#> Call:
#> spAbund(formula = ~abund.cov.1 + abund.cov.2 + abund.cov.3 + 
#>     (1 | abund.factor.1) + (abund.cov.1 | abund.factor.2), data = data.list, 
#>     inits = inits.list, priors = prior.list, tuning = tuning, 
#>     cov.model = "exponential", NNGP = TRUE, n.neighbors = 5, 
#>     search.type = "cb", n.batch = n.batch, batch.length = batch.length, 
#>     accept.rate = 0.43, n.omp.threads = 1, verbose = TRUE, n.report = 1, 
#>     n.burn = n.burn, n.thin = n.thin, n.chains = n.chains)
#> 
#> Samples per Chain: 100
#> Burn-in: 20
#> Thinning Rate: 1
#> Number of Chains: 1
#> Total Posterior Samples: 80
#> Run Time (min): 0.0013
#> 
#> Abundance (log scale): 
#>                Mean     SD    2.5%     50%   97.5% Rhat ESS
#> (Intercept) -0.2628 0.2131 -0.5803 -0.2921  0.2442   NA   3
#> abund.cov.1 -2.4345 0.0970 -2.5822 -2.4204 -2.3191   NA   2
#> abund.cov.2  0.7826 0.0734  0.6765  0.7587  1.0044   NA   5
#> abund.cov.3 -0.7686 0.0778 -0.9057 -0.7703 -0.6091   NA   8
#> 
#> Abundance Random Effect Variances (log scale): 
#>                              Mean     SD   2.5%    50%  97.5% Rhat ESS
#> (Intercept)-abund.factor.1 1.0911 0.3591 0.6091 0.9995 1.9510   NA   4
#> abund.cov.1-abund.factor.2 1.9088 0.4733 1.2096 1.8442 2.9181   NA  80
#> 
#> Spatial Covariance: 
#>             Mean     SD   2.5%     50%   97.5% Rhat ESS
#> sigma.sq  0.5528 0.2002 0.1989  0.5352  0.9803   NA   2
#> phi      11.6883 2.5536 8.2529 11.2674 16.8277   NA  10