Function for Fitting Spatially-Varying Coefficient Multivariate Abundance GLMMs

The function svcMsAbund fits multivariate spatially-varying coefficient GLMs with species correlations (i.e., a spatially-explicit abundace-based joint species distribution model). We use a spatial factor modeling approach. Models are implemented using a Nearest Neighbor Gaussian Process.

Usage

svcMsAbund(formula, data, inits, priors, tuning,
           svc.cols = 1, cov.model = 'exponential', NNGP = TRUE, 
           n.neighbors = 15, search.type = 'cb', n.factors,
           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. y is a matrix with sites corresponding to species and columns corresponding to sites. 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 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 model and should have the same dimensions as y.
inits: a list with each tag corresponding to a parameter name. Valid tags are beta.comm, beta, tau.sq.beta, sigma.sq.mu, phi, lambda, nu, and tau.sq. nu is only specified if cov.model = "matern", tau.sq is only specified for Gaussian and zero-inflated Gaussian models, and sigma.sq.mu is only specified if random effects are included 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.comm.normal, tau.sq.beta.ig, sigma.sq.mu, phi.unif, nu.unif, and tau.sq.ig. Community-level (beta.comm) 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 to 100. Community-level variance parameters (tau.sq.beta) 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, which are each specified as vectors of length equal to the number of coefficients to be estimated or a single value if priors are the same for all parameters. If not specified, prior shape and scale parameters are set to 0.1. If desired, the species-specific regression coefficients (beta) can also be estimated indepdendently by specifying the tag independent.betas = TRUE. If specified, this will not estimate species-specific coefficients as random effects from a common-community-level distribution, and rather the values of beta.comm and tau.sq.beta will be fixed at the specified initial values. This is equivalent to specifying a Gaussian, independent prior for each of the species-specific effects. The spatial factor model fits n.factors independent spatial processes. The spatial decay phi and smoothness nu parameters for each latent factor and spatially-varying coefficient are assumed to follow Uniform distributions. The hyperparameters of the Uniform are passed as a list with two elements, with both elements being vectors of length equal to the number of spatial factors times the number of spatially-varying coefficients corresponding to the lower and upper support, respectively, or as a single value if the same value is assigned for all factors and spatially-varying coefficients. The priors for the factor loadings matrix lambda are fixed following the standard spatial factor model to ensure parameter identifiability (Christensen and Amemlya 2002). The upper triangular elements of the n.sp x n.factors matrix for each spatially-varying coefficient are fixed at 0 and the diagonal elements are fixed at 1. The lower triangular elements are assigned a standard normal prior (i.e., mean 0 and variance 1). sigma.sq.mu are the random effect variances 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 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. tau.sq is the species-specific 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 list of length two, with the first and second element corresponding to the shape and scale parameters, respectively, which are each specified as vectors of length equal to the number of species or a single value if priors are the same for all species.
tuning: a single numeric value representing the initial variance of the adaptive sampler for phi and nu. See Roberts and Rosenthal (2009) for details.
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 spatial factor model analogous to sfMsAbund (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".
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. For spatial factor models, only NNGP = TRUE is currently supported.
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.factors: the number of factors to use in the spatial factor model approach for each spatially-varying coefficient. Typically, the number of factors is set to be small (e.g., 4-5) relative to the total number of species in the community, which will lead to substantial decreases in computation time. However, the value can be anywhere between 1 and the number of species in the modeled community.
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 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. Note this is specified in terms of batches and not overall samples for spatial models.
n.burn: the number of samples out of the total n.samples to discard as burn-in for each chain. 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 chains to run in sequence.
...: currently no additional arguments

References

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.

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 .

Christensen, W. F., and Amemiya, Y. (2002). Latent variable analysis of multivariate spatial data. Journal of the American Statistical Association, 97(457), 302-317.

Author

Jeffrey W. Doser doserjef@msu.edu,
Andrew O. Finley finleya@msu.edu

Value

An object of class svcMsAbund that is a list comprised of:

beta.comm.samples: a coda object of posterior samples for the community level regression coefficients.
tau.sq.beta.samples: a coda object of posterior samples for the abundance community variance parameters.
beta.samples: a coda object of posterior samples for the species level abundance regression coefficients.
tau.sq.samples: a coda object of posterior samples for the Gaussian residual variance parameter.
theta.samples: a coda object of posterior samples for the spatial correlation parameters.
lambda.samples: a coda object of posterior samples for the latent spatial factor loadings for each spatially-varying coefficient.
y.rep.samples: a three or four-dimensional array of posterior samples for the fitted (replicate) values for each species with dimensions corresponding to MCMC sample, species, site, and replicate.
mu.samples: a three or four-dimensional array of posterior samples for the expected abundance values for each species with dimensions corresponding to MCMC samples, species, site, and replicate.
w.samples: a four-dimensional array of posterior samples for the latent spatial random effects for each spatial factor within each spatially-varying coefficient. Dimensions correspond to MCMC sample, factor, site, and spatially-varying coefficient.
sigma.sq.mu.samples: a coda object of posterior samples for variances of random effects included in the abundance portion of the model. Only included if random effects are specified in abund.formula.
beta.star.samples: a coda object of posterior samples for the abundance random effects. Only included if random effects are specified in abund.formula.
like.samples: a three-dimensional array of posterior samples for the likelihood value associated with each site and species. 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: MCMC sampler 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(332)
J.x <- 10
J.y <- 10
J <- J.x * J.y
n.rep <- rep(1, J)
n.sp <- 6
# Community-level covariate effects
beta.mean <- c(0, 0.25, 0.6)
p.abund <- length(beta.mean)
tau.sq.beta <- c(0.2, 1.2, 0.4)
# Random effects
mu.RE <- list()
# Draw species-level effects from community means.
beta <- matrix(NA, nrow = n.sp, ncol = p.abund)
for (i in 1:p.abund) {
  beta[, i] <- rnorm(n.sp, beta.mean[i], sqrt(tau.sq.beta[i]))
}
sp <- TRUE 
svc.cols <- c(1, 2)
n.factors <- 2
q.p.svc <- length(svc.cols) * n.factors
factor.model <- TRUE
phi <- runif(q.p.svc, 3/1, 3 / .4)
tau.sq <- runif(n.sp, 0.1, 5)
cov.model <- 'exponential'
family <- 'Gaussian'

dat <- simMsAbund(J.x = J.x, J.y = J.y, n.rep = n.rep, n.sp = n.sp, beta = beta, 
                  mu.RE = mu.RE, sp = sp, tau.sq = tau.sq, family = family, 
                  factor.model = factor.model, phi = phi, 
                  cov.model = cov.model, n.factors = n.factors, 
                  svc.cols = svc.cols)

y <- dat$y
X <- dat$X
coords <- dat$coords

covs <- data.frame(abund.cov.1 = X[, 2],
                   abund.cov.2 = X[, 3])
data.list <- list(y = y, covs = covs, coords = coords)
prior.list <- list(beta.comm.normal = list(mean = 0, var = 100),
                   tau.sq.ig = list(a = 2, b = 2), 
       phi.unif = list(a = 3 / 1, b = 3 / .1), 
                   tau.sq.beta.ig = list(a = .1, b = .1)) 
inits.list <- list(beta.comm = 0, 
                   beta = 0, 
                   tau.sq = 1,
                   tau.sq.beta = 1,
                   phi = 3 / 0.5)
tuning.list <- list(phi = 0.5)

n.batch <- 5
batch.length <- 25
n.burn <- 0
n.thin <- 1
n.chains <- 1

out <- svcMsAbund(formula = ~ abund.cov.1 + abund.cov.2,
                  data = data.list, 
                  n.batch = n.batch, 
                  inits = inits.list, 
                  priors = prior.list, 
                  tuning = tuning.list,
                  NNGP = TRUE, 
                  svc.cols = c(1, 2),
                  family = 'Gaussian',
                  cov.model = 'exponential', 
                  n.neighbors = 5, 
                  n.factors = n.factors, 
                  batch.length = batch.length, 
                  n.omp.threads = 1,
                  verbose = TRUE, 
                  n.report = 20, 
                  n.burn = n.burn, 
                  n.thin = n.thin, 
                  n.chains = n.chains)
#> ----------------------------------------
#> 	Preparing to run the model
#> ----------------------------------------
#> lambda is not specified in initial values.
#> Setting initial values of the lower triangle to random values from a standard normal
#> 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 Factor NNGP Multi-species Gaussian Model
#> with 100 sites and 6 species.
#> 
#> Samples per chain: 125 (5 batches of length 25)
#> Burn-in: 0 
#> Thinning Rate: 1 
#> Number of Chains: 1 
#> Total Posterior Samples: 125 
#> 
#> Number of spatially-varying coefficients: 2 
#> Using the exponential spatial correlation model.
#> 
#> Using 2 latent spatial factors.
#> 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: 5 of 5, 100.00%
summary(out)
#> 
#> Call:
#> svcMsAbund(formula = ~abund.cov.1 + abund.cov.2, 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.factors = n.factors, n.batch = n.batch, 
#>     batch.length = batch.length, family = "Gaussian", n.omp.threads = 1, 
#>     verbose = TRUE, n.report = 20, n.burn = n.burn, n.thin = n.thin, 
#>     n.chains = n.chains)
#> 
#> Samples per Chain: 125
#> Burn-in: 0
#> Thinning Rate: 1
#> Number of Chains: 1
#> Total Posterior Samples: 125
#> Run Time (min): 0.0052
#> 
#> ----------------------------------------
#> 	Community Level
#> ----------------------------------------
#> Abundance Means: 
#>                Mean     SD    2.5%     50%  97.5% Rhat ESS
#> (Intercept) -0.0542 0.1917 -0.4600 -0.0573 0.3211   NA 125
#> abund.cov.1  1.0396 0.7121 -0.5156  1.0743 2.4272   NA 178
#> abund.cov.2  0.7998 0.3149  0.0721  0.8076 1.4543   NA 125
#> 
#> Abundance Variances: 
#>               Mean     SD   2.5%    50%  97.5% Rhat ESS
#> (Intercept) 0.2153 0.1534 0.0515 0.1680 0.6474   NA 125
#> abund.cov.1 3.4555 2.1769 0.8063 2.9347 8.9049   NA  84
#> abund.cov.2 0.8790 0.7101 0.1894 0.6292 2.6752   NA 125
#> 
#> ----------------------------------------
#> 	Species Level
#> ----------------------------------------
#> Abundance: 
#>                    Mean     SD    2.5%     50%   97.5% Rhat ESS
#> (Intercept)-sp1 -0.2738 0.2220 -0.7279 -0.2698  0.1581   NA  44
#> (Intercept)-sp2 -0.1709 0.1839 -0.5230 -0.1298  0.1139   NA  13
#> (Intercept)-sp3 -0.0559 0.1727 -0.3895 -0.0350  0.2285   NA  13
#> (Intercept)-sp4 -0.2098 0.1726 -0.5356 -0.2027  0.1271   NA 125
#> (Intercept)-sp5  0.5543 0.1344  0.3022  0.5678  0.7906   NA  45
#> (Intercept)-sp6 -0.1313 0.1800 -0.4784 -0.1268  0.2268   NA  33
#> abund.cov.1-sp1  1.5979 0.3086  1.0609  1.6102  2.1489   NA  17
#> abund.cov.1-sp2  3.5170 0.3154  2.8446  3.5041  4.0708   NA   4
#> abund.cov.1-sp3 -0.0710 0.3751 -0.8439 -0.0092  0.5400   NA   5
#> abund.cov.1-sp4  1.0396 0.3031  0.4402  1.0019  1.6204   NA   9
#> abund.cov.1-sp5 -0.8378 0.2672 -1.4312 -0.8019 -0.3783   NA   6
#> abund.cov.1-sp6  0.8199 0.2415  0.4612  0.7867  1.3247   NA  26
#> abund.cov.2-sp1  0.6850 0.1964  0.3121  0.6789  1.0219   NA  64
#> abund.cov.2-sp2  1.7650 0.1424  1.5059  1.7672  2.0264   NA  21
#> abund.cov.2-sp3  0.1106 0.1590 -0.1943  0.1126  0.3979   NA  34
#> abund.cov.2-sp4  0.3916 0.1640  0.0848  0.4093  0.7106   NA  85
#> abund.cov.2-sp5  1.6976 0.1126  1.4576  1.6919  1.8929   NA  60
#> abund.cov.2-sp6  0.1440 0.1328 -0.1169  0.1622  0.4162   NA  83
#> 
#> ----------------------------------------
#> 	Spatial Covariance
#> ----------------------------------------
#>                      Mean     SD   2.5%    50%   97.5% Rhat ESS
#> phi-1-(Intercept)  8.6541 4.0412 3.2935 8.1981 16.4235   NA   5
#> phi-2-(Intercept)  5.2898 1.6956 3.1108 5.2662  9.3440   NA   4
#> phi-1-abund.cov.1  6.1902 1.4954 4.0207 5.8134  8.8614   NA  12
#> phi-2-abund.cov.1 10.5151 5.7228 3.7901 8.3002 23.5328   NA   3
#> 
#> ----------------------------------------
#> 	Residual variance
#> ----------------------------------------
#>       Mean     SD   2.5%    50%  97.5% Rhat ESS
#> sp1 3.5358 0.7356 2.2994 3.4528 5.1541   NA  46
#> sp2 0.8118 0.3225 0.4413 0.7347 1.4865   NA  27
#> sp3 1.1220 0.3979 0.5474 1.0883 1.9401   NA  11
#> sp4 2.4152 0.4358 1.7619 2.3393 3.4294   NA  55
#> sp5 1.1259 0.2518 0.7408 1.1059 1.6417   NA  13
#> sp6 1.9704 0.3284 1.4644 1.9496 2.6776   NA  95