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

`spDS.Rd`

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 single numeric value representing the initial variance of the adaptive sampler for

`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
```