Function for Fitting Single-Species Spatially-Explicit Hierarchical Distance Sampling Models

Function for fitting single-sepcies spatially-explicit hierarchical distance sampling models. Spatial models are fit using Nearest Neighbor Gaussian Processes.

Usage

spDS(abund.formula, det.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',
     transect = 'line', det.func = 'halfnormal',
     n.omp.threads = 1, verbose = TRUE,
     n.report = 100, n.burn = round(.10 * n.batch * batch.length), n.thin = 1,
     n.chains = 1, ...)

Arguments

abund.formula: a symbolic description of the model to be fit for the abundance portion of 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).
det.formula: a symbolic description of the model to be fit for the detection portion of 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, dist.breaks, and offset. y is a matrix or data frame of the observed count values, with first dimension equal to the number of sites (\(J\)) and second dimension equal to the number of distance bins. covs is a matrix or data frame containing the variables used in the abundance and/or the detection portion of the model, with \(J\) rows for each column (variable). dist.breaks is a vector of distances that denote the breakpoints of the distance bands. dist.breaks should have length equal to the number of columns in y plus one. offset is an offset that can be used to scale estimates from abundance per transect to density per some desired unit of measure. This can be either a single value or a vector with an offset value for each site (e.g., if transects differ in length). coords is a \(J \times 2\) matrix of the observation coordinates. Note that spAbundance assumes coordinates are specified in a projected coordinate system.
inits: a list with each tag corresponding to a parameter name. Valid tags are N, beta, alpha, kappa, sigma.sq, phi, w, nu, sigma.sq.mu, and sigma.sq.p. The value portion of each tag is the parameter's initial value. sigma.sq.mu and sigma.sq.p are only relevant when including random effects in the abundance and detection portion of the abundance model, respectively. kappa is only relevant when family = 'NB'. nu is only specified if cov.model = "matern". 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, alpha.normal, kappa.unif, phi.unif, sigma.sq.ig, nu.unif, sigma.sq.mu.ig, and sigma.sq.p.ig. Abundance (beta) and detection (alpha) 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 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 and sigma.sq.p are the random effect variances for any abundance or detection 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/slopes or of length one if priors are the same for all random effect variances.
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 adpative sampler. Valid tags include beta, alpha, beta.star (the abundance random effect values), alpha.star (the detection random effect values), kappa, phi, nu, and w. 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 Gaussian Process 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) and 'Poisson'.
transect: the type of transect. Currently supports line transects ('line') or circular transects (i.e., point counts; 'point').
det.func: the detection model used to describe how detection probability varies with distance. In other software, this is often referred to as the key function. Currently supports two functions: half normal ('halfnormal') and negative exponential ('negexp').
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 hypterthreaded cores. Note, n.omp.threads > 1 might not work on some systems. Currently only relevant for spatial models.
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 MCMC progress.
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

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 .

Royle, J. A., Dawson, D. K., & Bates, S. (2004). Modeling abundance effects in distance sampling. Ecology, 85(6), 1591-1597.

Author

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

Value

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

beta.samples: a coda object of posterior samples for the abundance regression coefficients.
alpha.samples: a coda object of posterior samples for the detection regression coefficients.
kappa.samples: a coda object of posterior samples for the abundance dispersion parameter. Only included when family = 'NB'.
N.samples: a coda object of posterior samples for the latent abundance values. Note that these values always represent transect-level abundance, even when an offset is supplied.
mu.samples: a coda object of posterior samples for the latent expected abundance values. When an offset is supplied in the data object, these correspond to expected abundance per unit area (i.e., density).
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 abundance portion of the model. Only included if random effects are specified in abund.formula.
sigma.sq.p.samples: a coda object of posterior samples for variances of random effects included in the detection portion of the model. Only included if random effects are specified in det.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.
alpha.star.samples: a coda object of posterior samples for the detection random effects. Only included if random effects are specified in det.formula.
y.rep.samples: a three-dimensional array of fitted values. Array dimensions correspond to MCMC samples, sites, and distance band.
pi.samples: a three-dimensional array of cell-specific detection probabilities. Array dimensions correspond to MCMC samples, sites, and distance band.
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(123)
J.x <- 10
J.y <- 10
J <- J.x * J.y
# Number of distance bins from which to simulate data.
n.bins <- 5
# Length of each bin. This should be of length n.bins
bin.width <- c(.10, .10, .20, .3, .1)
# Abundance coefficients
beta <- c(1.0, 0.2, 0.3, -0.2)
p.abund <- length(beta)
# Detection coefficients
alpha <- c(-1.0, -0.3)
p.det <- length(alpha)
# Detection decay function
det.func <- 'halfnormal'
mu.RE <- list()
p.RE <- list()
sp <- TRUE
phi <- 3 / .5
sigma.sq <- 0.8
cov.model <- 'exponential'
family <- 'NB'
kappa <- 0.1
offset <- 1.8
transect <- 'point'

dat <- simDS(J.x = J.x, J.y = J.y, n.bins = n.bins, bin.width = bin.width,
             beta = beta, alpha = alpha, det.func = det.func, kappa = kappa,
             mu.RE = mu.RE, p.RE = p.RE, sp = sp,
             offset = offset, transect = transect, phi = phi, sigma.sq = sigma.sq,
             cov.model = cov.model)
#> overdispersion parameter (kappa) is ignored when family == 'Poisson'

y <- dat$y
X <- dat$X
X.re <- dat$X.re
X.p <- dat$X.p
X.p.re <- dat$X.p.re
dist.breaks <- dat$dist.breaks
coords <- dat$coords

covs <- cbind(X, X.p)
colnames(covs) <- c('int.abund', 'abund.cov.1', 'abund.cov.2', 'abund.cov.3',
                    'int.det', 'det.cov.1')

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

# Priors
prior.list <- list(beta.normal = list(mean = 0, var = 10),
                   alpha.normal = list(mean = 0,
                                       var = 10),
                   kappa.unif = c(0, 100),
                   phi.unif = c(3 / 1, 3 / .1),
                   sigma.sq.ig = c(2, 1))
# Starting values
inits.list <- list(alpha = 0,
                   beta = 0,
                   kappa = 1,
                   phi = 3 / .5,
                   sigma.sq = 1)
# Tuning values
tuning <- list(beta = 0.1, alpha = 0.1, beta.star = 0.3, alpha.star = 0.1,
               kappa = 0.2, phi = 1, w = 1)

out <- spDS(abund.formula = ~ abund.cov.1 + abund.cov.2 + abund.cov.3,
            det.formula = ~ det.cov.1,
            data = data.list,
            n.batch = 10,
            batch.length = 25,
            inits = inits.list,
            family = 'NB',
            det.func = 'halfnormal',
            transect = 'point',
            cov.model = 'exponential',
            NNGP = TRUE,
      n.neighbors = 5,
            tuning = tuning,
            priors = prior.list,
            accept.rate = 0.43,
            n.omp.threads = 1,
            verbose = TRUE,
            n.report = 100,
            n.burn = 100,
            n.thin = 1,
            n.chains = 1)
#> ----------------------------------------
#> 	Preparing to run the model
#> ----------------------------------------
#> **NOTE**: spatial negative binomial models can be difficult to
#> estimate as they contain two forms of overdispersion. If experiencing
#> very poor mixing/convergence of MCMC chains (particularly kappa and phi),
#> consider using a spatial Poisson model or more informative
#> priors on kappa or phi.
#> N is not specified in initial values.
#> Setting initial values based on observed data
#> 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 Negative Binomial HDS model 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 
#> 
#> 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%
summary(out)
#> 
#> Call:
#> spDS(abund.formula = ~abund.cov.1 + abund.cov.2 + abund.cov.3, 
#>     det.formula = ~det.cov.1, data = data.list, inits = inits.list, 
#>     priors = prior.list, tuning = tuning, cov.model = "exponential", 
#>     NNGP = TRUE, n.neighbors = 5, n.batch = 10, batch.length = 25, 
#>     accept.rate = 0.43, family = "NB", transect = "point", det.func = "halfnormal", 
#>     n.omp.threads = 1, verbose = TRUE, n.report = 100, n.burn = 100, 
#>     n.thin = 1, n.chains = 1)
#> 
#> Samples per Chain: 250
#> Burn-in: 100
#> Thinning Rate: 1
#> Number of Chains: 1
#> Total Posterior Samples: 150
#> Run Time (min): 0.0043
#> 
#> Abundance (log scale): 
#>                Mean     SD    2.5%     50%   97.5% Rhat ESS
#> (Intercept)  1.2593 0.0977  1.0992  1.2442  1.4438   NA   5
#> abund.cov.1 -0.1356 0.0850 -0.3000 -0.1491  0.0248   NA   7
#> abund.cov.2  0.3655 0.1048  0.1994  0.3449  0.5526   NA   5
#> abund.cov.3 -0.2982 0.0919 -0.4746 -0.2927 -0.1503   NA   8
#> 
#> Detection (log scale): 
#>                Mean     SD    2.5%     50%   97.5% Rhat ESS
#> (Intercept) -0.9430 0.0587 -1.0486 -0.9376 -0.8384   NA   5
#> det.cov.1   -0.3936 0.0595 -0.5057 -0.3791 -0.2949   NA   8
#> 
#> Spatial Covariance: 
#>            Mean     SD   2.5%    50%   97.5% Rhat ESS
#> sigma.sq 0.5518 0.1507 0.3180 0.5403  0.8949   NA  14
#> phi      7.8430 3.8517 3.3891 7.1978 18.3146   NA   6
#> 
#> NB overdispersion: 
#>          Mean      SD   2.5%     50%   97.5% Rhat ESS
#> kappa 63.1189 21.8282 23.721 69.2224 92.0465   NA   2