# Function for Fitting Multi-Species Spatial Occupancy Models Using Polya-Gamma Latent Variables

`spMsPGOcc.Rd`

The function `spMsPGOcc`

fits multi-species spatial occupancy models using Polya-Gamma latent variables. Models can be fit using either a full Gaussian process or a Nearest Neighbor Gaussian Process for large data sets.

## Usage

```
spMsPGOcc(occ.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, n.omp.threads = 1,
verbose = TRUE, n.report = 100,
n.burn = round(.10 * n.batch * batch.length), n.thin = 1,
n.chains = 1, k.fold, k.fold.threads = 1, k.fold.seed, ...)
```

## Arguments

- occ.formula
a symbolic description of the model to be fit for the occurrence portion of the model using R's model syntax. Only right-hand side of formula is specified. See example below. Random intercepts 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 are allowed using lme4 syntax (Bates et al. 2015).

- data
a list containing data necessary for model fitting. Valid tags are

`y`

,`occ.covs`

,`det.covs`

,`coords`

.`y`

is a three-dimensional array with first dimension equal to the number of species, second dimension equal to the number of sites, and third dimension equal to the maximum number of replicates at a given site.`occ.covs`

is a matrix or data frame containing the variables used in the occurrence portion of the model, with \(J\) rows for each column (variable).`det.covs`

is a list of variables included in the detection portion of the model. Each list element is a different detection covariate, which can be site-level or observational-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 replicates at a given site.`coords`

is a \(J \times 2\) matrix of the observation coordinates. Note that`spOccupancy`

assumes coordinates are specified in a projected coordinate system.- inits
a list with each tag corresponding to a parameter name. Valid tags are

`alpha.comm`

,`beta.comm`

,`beta`

,`alpha`

,`tau.sq.beta`

,`tau.sq.alpha`

,`sigma.sq.psi`

,`sigma.sq.p`

,`z`

,`sigma.sq`

,`phi`

,`w`

, and`nu`

.`nu`

is only specified if`cov.model = "matern"`

,`sigma.sq.psi`

is only specified if there are random intercepts in`occ.formula`

, and`sigma.sq.p`

is only specified if there are random intercpets in`det.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`

,`alpha.comm.normal`

,`tau.sq.beta.ig`

,`tau.sq.alpha.ig`

,`phi.unif`

,`sigma.sq.ig`

,`sigma.sq.unif`

,`nu.unif`

,`sigma.sq.psi`

,`sigma.sq.p`

. Community-level occurrence (`beta.comm`

) and detection (`alpha.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 set to 2.73. Community-level variance parameters for occupancy (`tau.sq.beta`

) and detection (`tau.sq.alpha`

) 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. The species-specific spatial variance parameter,`sigma.sq`

, is assumed to follow an inverse-Gamma distribution or a uniform distribution (default is inverse-Gamma).`sigma.sq`

of all species can also be fixed at its initial value by setting the prior value to`"fixed"`

. The spatial decay`phi`

and smoothness`nu`

parameters are assumed to follow Uniform distributions. The hyperparameters of the inverse-Gamma are passed as a list of length two, with the list elements being vectors of length N corresponding to the species-specific shape and scale parameters, respectively, or a single value if the same value is assigned for all species. The hyperparameters of the Uniform are also passed as a list with two elements, with both elements being vectors of length N corresponding to the lower and upper support, respectively, or as a single value if the same value is assigned for all species.`sigma.sq.psi`

and`sigma.sq.p`

are the random effect variances for any occurrence 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 or of length one if priors are the same for all random effect variances.- tuning
a list with each tag corresponding to a parameter name. Valid tags are

`phi`

and`nu`

. The value portion of each tag defines the initial variance of the adaptive sampler. We assume the initial variance of the adaptive sampler is the same for each species, although the adaptive sampler will adjust the tuning variances separately for each species. See Roberts and Rosenthal (2009) for details.- 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.- 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 or k-fold cross-validation.- 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 to run for the Adaptive MCMC sampler. See Roberts and Rosenthal (2009) for details.

- accept.rate
target acceptance rate for Adaptive MCMC. Defaul is 0.43. See Roberts and Rosenthal (2009) for details.

- 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.

- k.fold
specifies the number of

*k*folds for cross-validation. If not specified as an argument, then cross-validation is not performed and`k.fold.threads`

and`k.fold.seed`

are ignored. In*k*-fold cross-validation, the data specified in`data`

is randomly partitioned into*k*equal sized subsamples. Of the*k*subsamples,*k*- 1 subsamples are used to fit the model and the remaining*k*samples are used for prediction. The cross-validation process is repeated*k*times (the folds). As a scoring rule, we use the model deviance as described in Hooten and Hobbs (2015). Cross-validation is performed after the full model is fit using all the data. Cross-validation results are reported in the`k.fold.deviance`

object in the return list.- k.fold.threads
number of threads to use for cross-validation. If

`k.fold.threads > 1`

parallel processing is accomplished using the foreach and doParallel packages. Ignored if`k.fold`

is not specified.- k.fold.seed
seed used to split data set into

`k.fold`

parts for k-fold cross-validation. Ignored if`k.fold`

is not specified.- ...
currently no additional arguments

## Note

Some of the underlying code used for generating random numbers from the Polya-Gamma distribution is taken from the pgdraw package written by Daniel F. Schmidt and Enes Makalic. Their code implements Algorithm 6 in PhD thesis of Jesse Bennett Windle (2013) https://repositories.lib.utexas.edu/handle/2152/21842.

## 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
.

Finley, A. O., Datta, A., and Banerjee, S. (2020). spNNGP R package
for nearest neighbor Gaussian process models. *arXiv* preprint arXiv:2001.09111.

Polson, N.G., J.G. Scott, and J. Windle. (2013) Bayesian Inference for
Logistic Models Using Polya-Gamma Latent Variables.
*Journal of the American Statistical Association*, 108:1339-1349.

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 .

Hooten, M. B., and Hobbs, N. T. (2015). A guide to Bayesian model
selection for ecologists. *Ecological Monographs*, 85(1), 3-28.

## Author

Jeffrey W. Doser doserjef@msu.edu,

Andrew O. Finley finleya@msu.edu

## Value

An object of class `spMsPGOcc`

that is a list comprised of:

- beta.comm.samples
a

`coda`

object of posterior samples for the community level occurrence regression coefficients.- alpha.comm.samples
a

`coda`

object of posterior samples for the community level detection regression coefficients.- tau.sq.beta.samples
a

`coda`

object of posterior samples for the occurrence community variance parameters.- tau.sq.alpha.samples
a

`coda`

object of posterior samples for the detection community variance parameters.- beta.samples
a

`coda`

object of posterior samples for the species level occurrence regression coefficients.- alpha.samples
a

`coda`

object of posterior samples for the species level detection regression coefficients.- theta.samples
a

`coda`

object of posterior samples for the species level covariance parameters.- z.samples
a three-dimensional array of posterior samples for the latent occurrence values for each species.

- psi.samples
a three-dimensional array of posterior samples for the latent occupancy probability values for each species.

- w.samples
a three-dimensional array of posterior samples for the latent spatial random effects for each species.

- sigma.sq.psi.samples
a

`coda`

object of posterior samples for variances of random intercepts included in the occurrence portion of the model. Only included if random intercepts are specified in`occ.formula`

.- sigma.sq.p.samples
a

`coda`

object of posterior samples for variances of random intercpets included in the detection portion of the model. Only included if random intercepts are specified in`det.formula`

.- alpha.star.samples
a

`coda`

object of posterior samples for the detection random effects. Only included if random intercepts are specified in`det.formula`

.- beta.star.samples
a

`coda`

object of posterior samples for the occurrence random effects. Only included if random intercepts are specified in`occ.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()`

.- k.fold.deviance
vector of scoring rules (deviance) from k-fold cross-validation. A separate value is reported for each species. Only included if

`k.fold`

is specified in function call.

The return object will include additional objects used for subsequent prediction and/or model fit evaluation. Note that detection probability estimated values are not included in the model object, but can be extracted using

## Examples

```
set.seed(400)
# Simulate Data -----------------------------------------------------------
J.x <- 7
J.y <- 7
J <- J.x * J.y
n.rep <- sample(2:4, size = J, replace = TRUE)
N <- 5
# Community-level covariate effects
# Occurrence
beta.mean <- c(0.2, -0.15)
p.occ <- length(beta.mean)
tau.sq.beta <- c(0.6, 0.3)
# Detection
alpha.mean <- c(0.5, 0.2, -.2)
tau.sq.alpha <- c(0.2, 0.3, 0.8)
p.det <- length(alpha.mean)
# Draw species-level effects from community means.
beta <- matrix(NA, nrow = N, ncol = p.occ)
alpha <- matrix(NA, nrow = N, ncol = p.det)
for (i in 1:p.occ) {
beta[, i] <- rnorm(N, beta.mean[i], sqrt(tau.sq.beta[i]))
}
for (i in 1:p.det) {
alpha[, i] <- rnorm(N, alpha.mean[i], sqrt(tau.sq.alpha[i]))
}
phi <- runif(N, 3/1, 3/.4)
sigma.sq <- runif(N, 0.3, 3)
sp <- TRUE
dat <- simMsOcc(J.x = J.x, J.y = J.y, n.rep = n.rep, N = N, beta = beta, alpha = alpha,
phi = phi, sigma.sq = sigma.sq, sp = TRUE, cov.model = 'exponential')
# Number of batches
n.batch <- 30
# Batch length
batch.length <- 25
n.samples <- n.batch * batch.length
y <- dat$y
X <- dat$X
X.p <- dat$X.p
coords <- as.matrix(dat$coords)
# Package all data into a list
occ.covs <- X[, 2, drop = FALSE]
colnames(occ.covs) <- c('occ.cov')
det.covs <- list(det.cov.1 = X.p[, , 2],
det.cov.2 = X.p[, , 3])
data.list <- list(y = y,
occ.covs = occ.covs,
det.covs = det.covs,
coords = coords)
# Priors
prior.list <- list(beta.comm.normal = list(mean = 0, var = 2.72),
alpha.comm.normal = list(mean = 0, var = 2.72),
tau.sq.beta.ig = list(a = 0.1, b = 0.1),
tau.sq.alpha.ig = list(a = 0.1, b = 0.1),
phi.unif = list(a = 3/1, b = 3/.1),
sigma.sq.ig = list(a = 2, b = 2))
# Initial values
inits.list <- list(alpha.comm = 0,
beta.comm = 0,
beta = 0,
alpha = 0,
tau.sq.beta = 1,
tau.sq.alpha = 1,
phi = 3 / .5,
sigma.sq = 2,
w = matrix(0, nrow = N, ncol = nrow(X)),
z = apply(y, c(1, 2), max, na.rm = TRUE))
# Tuning
tuning.list <- list(phi = 1)
out <- spMsPGOcc(occ.formula = ~ occ.cov,
det.formula = ~ det.cov.1 + det.cov.2,
data = data.list,
inits = inits.list,
n.batch = n.batch,
batch.length = batch.length,
accept.rate = 0.43,
priors = prior.list,
cov.model = "exponential",
tuning = tuning.list,
n.omp.threads = 1,
verbose = TRUE,
NNGP = TRUE,
n.neighbors = 5,
search.type = 'cb',
n.report = 10,
n.burn = 500,
n.thin = 1,
n.chains = 1)
#> ----------------------------------------
#> Preparing to run the model
#> ----------------------------------------
#> ----------------------------------------
#> Building the neighbor list
#> ----------------------------------------
#> ----------------------------------------
#> Building the neighbors of neighbors list
#> ----------------------------------------
#> ----------------------------------------
#> Model description
#> ----------------------------------------
#> NNGP Multispecies Occupancy Model with Polya-Gamma latent
#> variable fit with 49 sites and 5 species.
#>
#> Samples per chain: 750 (30 batches of length 25)
#> Burn-in: 500
#> Thinning Rate: 1
#> Number of Chains: 1
#> Total Posterior Samples: 250
#>
#> 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 30, 33.33%
#> Species Parameter Acceptance Tuning
#> 1 phi 68.0 1.11628
#> 2 phi 68.0 1.11628
#> 3 phi 68.0 1.11628
#> 4 phi 72.0 1.11628
#> 5 phi 84.0 1.11628
#> -------------------------------------------------
#> Batch: 20 of 30, 66.67%
#> Species Parameter Acceptance Tuning
#> 1 phi 64.0 1.23368
#> 2 phi 76.0 1.20925
#> 3 phi 68.0 1.23368
#> 4 phi 76.0 1.23368
#> 5 phi 48.0 1.20925
#> -------------------------------------------------
#> Batch: 30 of 30, 100.00%
summary(out, level = 'both')
#>
#> Call:
#> spMsPGOcc(occ.formula = ~occ.cov, det.formula = ~det.cov.1 +
#> det.cov.2, data = data.list, inits = inits.list, priors = prior.list,
#> tuning = tuning.list, 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 = 10,
#> n.burn = 500, n.thin = 1, n.chains = 1)
#>
#> Samples per Chain: 750
#> Burn-in: 500
#> Thinning Rate: 1
#> Number of Chains: 1
#> Total Posterior Samples: 250
#> Run Time (min): 0.025
#>
#> ----------------------------------------
#> Community Level
#> ----------------------------------------
#> Occurrence Means (logit scale):
#> Mean SD 2.5% 50% 97.5% Rhat ESS
#> (Intercept) -0.6753 0.4680 -1.5356 -0.6980 0.3023 NA 107
#> occ.cov 0.1708 0.3118 -0.4427 0.1637 0.7441 NA 101
#>
#> Occurrence Variances (logit scale):
#> Mean SD 2.5% 50% 97.5% Rhat ESS
#> (Intercept) 0.8273 1.1035 0.0784 0.4570 4.0756 NA 92
#> occ.cov 0.4354 0.4868 0.0550 0.2691 1.8117 NA 129
#>
#> Detection Means (logit scale):
#> Mean SD 2.5% 50% 97.5% Rhat ESS
#> (Intercept) 0.8419 0.3411 0.1076 0.8583 1.4357 NA 89
#> det.cov.1 -0.0983 0.3186 -0.8577 -0.0752 0.4418 NA 163
#> det.cov.2 -0.5556 0.3345 -1.1193 -0.5837 0.2923 NA 133
#>
#> Detection Variances (logit scale):
#> Mean SD 2.5% 50% 97.5% Rhat ESS
#> (Intercept) 0.4318 0.5331 0.0315 0.2654 1.6666 NA 126
#> det.cov.1 0.5055 0.5084 0.0527 0.3449 1.9625 NA 111
#> det.cov.2 0.3928 0.5594 0.0405 0.1938 1.6458 NA 58
#>
#> ----------------------------------------
#> Species Level
#> ----------------------------------------
#> Occurrence (logit scale):
#> Mean SD 2.5% 50% 97.5% Rhat ESS
#> (Intercept)-sp1 -0.3678 0.4314 -1.1104 -0.3898 0.5190 NA 27
#> (Intercept)-sp2 -0.1006 0.4683 -0.9941 -0.1444 0.8299 NA 26
#> (Intercept)-sp3 -1.0037 0.4106 -1.7601 -0.9691 -0.2284 NA 90
#> (Intercept)-sp4 -0.6070 0.4677 -1.6058 -0.5720 0.2984 NA 69
#> (Intercept)-sp5 -1.3604 0.4714 -2.4122 -1.3327 -0.5582 NA 51
#> occ.cov-sp1 -0.4584 0.3722 -1.2386 -0.4522 0.1776 NA 50
#> occ.cov-sp2 0.2642 0.3468 -0.3334 0.2397 1.0028 NA 65
#> occ.cov-sp3 0.2135 0.3624 -0.4122 0.2229 0.8930 NA 91
#> occ.cov-sp4 0.5490 0.3808 -0.1301 0.5108 1.2827 NA 55
#> occ.cov-sp5 0.2689 0.3374 -0.3620 0.2846 0.9329 NA 128
#>
#> Detection (logit scale):
#> Mean SD 2.5% 50% 97.5% Rhat ESS
#> (Intercept)-sp1 1.1325 0.2741 0.6464 1.1135 1.6670 NA 102
#> (Intercept)-sp2 1.2965 0.2912 0.7412 1.2738 1.9205 NA 81
#> (Intercept)-sp3 0.3972 0.4091 -0.4954 0.4290 1.1089 NA 89
#> (Intercept)-sp4 0.8467 0.2962 0.2355 0.8413 1.3927 NA 114
#> (Intercept)-sp5 0.7121 0.4187 -0.2090 0.7484 1.4954 NA 101
#> det.cov.1-sp1 0.1610 0.2615 -0.3235 0.1471 0.7106 NA 135
#> det.cov.1-sp2 -0.5265 0.2314 -0.9965 -0.4956 -0.0950 NA 110
#> det.cov.1-sp3 0.3175 0.3602 -0.3009 0.2878 1.0107 NA 147
#> det.cov.1-sp4 0.2434 0.2513 -0.2312 0.2480 0.7543 NA 161
#> det.cov.1-sp5 -0.6030 0.4387 -1.5919 -0.5609 0.1517 NA 87
#> det.cov.2-sp1 -0.2936 0.2479 -0.7371 -0.3014 0.1707 NA 128
#> det.cov.2-sp2 -0.5095 0.2476 -1.0826 -0.4829 -0.0542 NA 130
#> det.cov.2-sp3 -1.0732 0.4135 -1.9851 -1.0365 -0.3889 NA 93
#> det.cov.2-sp4 -0.8126 0.2734 -1.4042 -0.8005 -0.3131 NA 118
#> det.cov.2-sp5 -0.2467 0.3044 -0.8554 -0.2370 0.3459 NA 51
#>
#> Spatial Covariance:
#> Mean SD 2.5% 50% 97.5% Rhat ESS
#> sigma.sq-sp1 1.8085 1.4403 0.4981 1.3976 5.6380 NA 24
#> sigma.sq-sp2 1.8833 1.3010 0.5853 1.4967 5.4472 NA 13
#> sigma.sq-sp3 1.1590 0.6936 0.4252 0.9649 2.7409 NA 15
#> sigma.sq-sp4 4.5722 3.9608 0.4049 3.3690 14.9193 NA 11
#> sigma.sq-sp5 1.4145 2.3184 0.3086 0.7743 9.2388 NA 12
#> phi-sp1 16.6085 7.3274 3.7099 16.6686 28.3796 NA 11
#> phi-sp2 11.8229 7.9392 3.1365 8.6144 28.8719 NA 13
#> phi-sp3 18.1657 7.2917 6.8418 17.4662 29.8922 NA 17
#> phi-sp4 17.6839 6.4619 6.4113 17.9117 28.0679 NA 37
#> phi-sp5 15.3212 8.5766 3.4542 16.5814 28.5075 NA 8
```