# Function for Fitting Spatial Factor Multi-Species Occupancy Models

`sfMsPGOcc.Rd`

The function `sfMsPGOcc`

fits multi-species spatial occupancy models with species correlations (i.e., a spatially-explicit joint species distribution model with imperfect detection). We use Polya-Gamma latent variables and a spatial factor modeling approach. Currently, models are implemented using a Nearest Neighbor Gaussian Process. Future development will allow for running the models using full Gaussian Processes.

## Usage

```
sfMsPGOcc(occ.formula, det.formula, data, inits, priors, tuning,
cov.model = 'exponential', NNGP = TRUE,
n.neighbors = 15, search.type = 'cb', n.factors, 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. Random intercepts are allowed using lme4 syntax (Bates et al. 2015). Only right-hand side of formula is specified. See example below.

- 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`

,`phi`

,`lambda`

, and`nu`

.`nu`

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

, and`sigma.sq.psi`

and`sigma.sq.p`

are only specified if random effects are included in`occ.formula`

or`det.formula`

, respectively. 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`

,`sigma.sq.psi`

,`sigma.sq.p`

,`phi.unif`

, and`nu.unif`

. 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 spatial factor model fits`n.factors`

independent spatial processes. The spatial decay`phi`

and smoothness`nu`

parameters for each latent factor 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`n.factors`

corresponding to the lower and upper support, respectively, or as a single value if the same value is assigned for all factors. 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 x n.factors`

matrix 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.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. 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 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.factors
the number of factors to use in the spatial factor model approach. 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 N (the number of species in the 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. 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.

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 `sfMsPGOcc`

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 correlation parameters.- lambda.samples
a

`coda`

object of posterior samples for the latent spatial factor loadings.- 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 latent factor.

- 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`

.- beta.star.samples
a

`coda`

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

.- 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 `fitted()`

.

## 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 <- 8
# 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)
# Random effects
psi.RE <- list()
# Include a non-spatial random effect on occurrence
psi.RE <- list(levels = c(20),
sigma.sq.psi = c(0.5))
p.RE <- list()
# Include a random effect on detection
p.RE <- list(levels = c(40),
sigma.sq.p = c(2))
# 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]))
}
n.factors <- 4
phi <- runif(n.factors, 3/1, 3/.4)
dat <- simMsOcc(J.x = J.x, J.y = J.y, n.rep = n.rep, N = N, beta = beta, alpha = alpha,
phi = phi, sp = TRUE, cov.model = 'exponential',
factor.model = TRUE, n.factors = n.factors, psi.RE = psi.RE,
p.RE = p.RE)
# Number of batches
n.batch <- 10
# Batch length
batch.length <- 25
n.samples <- n.batch * batch.length
y <- dat$y
X <- dat$X
X.p <- dat$X.p
X.p.re <- dat$X.p.re
X.re <- dat$X.re
coords <- as.matrix(dat$coords)
# Package all data into a list
occ.covs <- cbind(X, X.re)
colnames(occ.covs) <- c('int', 'occ.cov', 'occ.re')
det.covs <- list(det.cov.1 = X.p[, , 2],
det.cov.2 = X.p[, , 3],
det.re = X.p.re[, , 1])
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))
# Initial values
lambda.inits <- matrix(0, N, n.factors)
diag(lambda.inits) <- 1
lambda.inits[lower.tri(lambda.inits)] <- rnorm(sum(lower.tri(lambda.inits)))
inits.list <- list(alpha.comm = 0,
beta.comm = 0,
beta = 0,
alpha = 0,
tau.sq.beta = 1,
tau.sq.alpha = 1,
phi = 3 / .5,
lambda = lambda.inits,
z = apply(y, c(1, 2), max, na.rm = TRUE))
# Tuning
tuning.list <- list(phi = 1)
out <- sfMsPGOcc(occ.formula = ~ occ.cov + (1 | occ.re),
det.formula = ~ det.cov.1 + det.cov.2 + (1 | det.re),
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,
n.factors = n.factors,
search.type = 'cb',
n.report = 10,
n.burn = 50,
n.thin = 1,
n.chains = 1)
#> ----------------------------------------
#> Preparing to run the model
#> ----------------------------------------
#> No prior specified for sigma.sq.psi.ig.
#> Setting prior shape to 0.1 and prior scale to 0.1
#> No prior specified for sigma.sq.p.ig.
#> Setting prior shape to 0.1 and prior scale to 0.1
#> sigma.sq.psi is not specified in initial values.
#> Setting initial values to random values between 0.5 and 10
#> sigma.sq.p is not specified in initial values.
#> Setting initial values to random values between 0.5 and 10
#> ----------------------------------------
#> Building the neighbor list
#> ----------------------------------------
#> ----------------------------------------
#> Building the neighbors of neighbors list
#> ----------------------------------------
#> ----------------------------------------
#> Model description
#> ----------------------------------------
#> Spatial Factor NNGP Multispecies Occupancy Model with Polya-Gamma latent
#> variable fit with 49 sites and 8 species.
#>
#> Samples per chain: 250 (10 batches of length 25)
#> Burn-in: 50
#> Thinning Rate: 1
#> Number of Chains: 1
#> Total Posterior Samples: 200
#>
#> Using the exponential spatial correlation model.
#>
#> Using 4 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: 10 of 10, 100.00%
summary(out)
#>
#> Call:
#> sfMsPGOcc(occ.formula = ~occ.cov + (1 | occ.re), det.formula = ~det.cov.1 +
#> det.cov.2 + (1 | det.re), data = data.list, inits = inits.list,
#> priors = prior.list, tuning = tuning.list, cov.model = "exponential",
#> NNGP = TRUE, n.neighbors = 5, search.type = "cb", n.factors = n.factors,
#> n.batch = n.batch, batch.length = batch.length, accept.rate = 0.43,
#> n.omp.threads = 1, verbose = TRUE, n.report = 10, n.burn = 50,
#> n.thin = 1, n.chains = 1)
#>
#> Samples per Chain: 250
#> Burn-in: 50
#> Thinning Rate: 1
#> Number of Chains: 1
#> Total Posterior Samples: 200
#> Run Time (min): 0.0165
#>
#> ----------------------------------------
#> Community Level
#> ----------------------------------------
#> Occurrence Means (logit scale):
#> Mean SD 2.5% 50% 97.5% Rhat ESS
#> (Intercept) 0.2642 0.2950 -0.2735 0.2747 0.7882 NA 23
#> occ.cov -0.1290 0.2847 -0.6035 -0.1543 0.4809 NA 54
#>
#> Occurrence Variances (logit scale):
#> Mean SD 2.5% 50% 97.5% Rhat ESS
#> (Intercept) 0.2993 0.2414 0.0584 0.2192 0.8997 NA 65
#> occ.cov 0.5042 0.5414 0.0407 0.3576 1.6568 NA 22
#>
#> Occurrence Random Effect Variances (logit scale):
#> Mean SD 2.5% 50% 97.5% Rhat ESS
#> occ.re 0.3899 0.2824 0.0449 0.3845 1.0109 NA 4
#>
#> Detection Means (logit scale):
#> Mean SD 2.5% 50% 97.5% Rhat ESS
#> (Intercept) -0.0793 0.1846 -0.4726 -0.0749 0.2795 NA 135
#> det.cov.1 0.1411 0.1996 -0.2633 0.1428 0.5309 NA 58
#> det.cov.2 -0.2190 0.3032 -0.7525 -0.2252 0.3853 NA 131
#>
#> Detection Variances (logit scale):
#> Mean SD 2.5% 50% 97.5% Rhat ESS
#> (Intercept) 0.1947 0.1809 0.0338 0.1435 0.6177 NA 89
#> det.cov.1 0.2413 0.1937 0.0387 0.1941 0.8176 NA 101
#> det.cov.2 0.4838 0.4457 0.1298 0.3675 1.7537 NA 115
#>
#> Detection Random Effect Variances (logit scale):
#> Mean SD 2.5% 50% 97.5% Rhat ESS
#> det.re 0.9291 0.3815 0.4909 0.8074 1.8538 NA 4
#>
#> ----------------------------------------
#> Species Level
#> ----------------------------------------
#> Occurrence (logit scale):
#> Mean SD 2.5% 50% 97.5% Rhat ESS
#> (Intercept)-sp1 0.3303 0.4096 -0.5429 0.3194 1.1991 NA 47
#> (Intercept)-sp2 0.4424 0.3899 -0.4142 0.3857 1.2104 NA 63
#> (Intercept)-sp3 0.2711 0.4726 -0.6225 0.2458 1.2290 NA 42
#> (Intercept)-sp4 0.4039 0.4453 -0.3942 0.3542 1.3121 NA 38
#> (Intercept)-sp5 -0.1835 0.4182 -1.0047 -0.1741 0.6183 NA 42
#> (Intercept)-sp6 0.2167 0.4623 -0.7210 0.1754 1.0576 NA 38
#> (Intercept)-sp7 0.4814 0.4661 -0.4284 0.4868 1.5159 NA 30
#> (Intercept)-sp8 0.2586 0.5331 -0.7204 0.2395 1.4776 NA 26
#> occ.cov-sp1 0.3806 0.4412 -0.3967 0.3821 1.3175 NA 46
#> occ.cov-sp2 -0.3329 0.3700 -1.1234 -0.3306 0.3341 NA 56
#> occ.cov-sp3 0.1203 0.4317 -0.6251 0.1039 1.0707 NA 25
#> occ.cov-sp4 -0.1771 0.3722 -0.8058 -0.2062 0.6492 NA 62
#> occ.cov-sp5 -0.8076 0.4566 -1.8465 -0.7262 -0.1490 NA 27
#> occ.cov-sp6 -0.3216 0.3827 -1.1175 -0.3272 0.4619 NA 73
#> occ.cov-sp7 -0.4027 0.4688 -1.2716 -0.3683 0.4972 NA 29
#> occ.cov-sp8 0.3977 0.5469 -0.3596 0.2819 1.7401 NA 36
#>
#> Detection (logit scale):
#> Mean SD 2.5% 50% 97.5% Rhat ESS
#> (Intercept)-sp1 0.0580 0.2514 -0.3748 0.0634 0.5295 NA 97
#> (Intercept)-sp2 -0.3489 0.2888 -0.8884 -0.3275 0.2351 NA 57
#> (Intercept)-sp3 0.1119 0.2839 -0.4776 0.1149 0.6970 NA 74
#> (Intercept)-sp4 0.0660 0.2685 -0.5274 0.0552 0.6317 NA 61
#> (Intercept)-sp5 0.0658 0.3209 -0.4867 0.0621 0.7213 NA 53
#> (Intercept)-sp6 -0.0963 0.3015 -0.7946 -0.0853 0.4244 NA 17
#> (Intercept)-sp7 -0.1632 0.2655 -0.7074 -0.1524 0.2518 NA 65
#> (Intercept)-sp8 -0.4009 0.2675 -0.9441 -0.3816 0.1373 NA 63
#> det.cov.1-sp1 0.3381 0.2553 -0.1713 0.3286 0.8116 NA 96
#> det.cov.1-sp2 0.0223 0.2543 -0.4630 0.0205 0.5105 NA 50
#> det.cov.1-sp3 0.0605 0.2022 -0.3894 0.0516 0.5158 NA 102
#> det.cov.1-sp4 0.7163 0.2958 0.1816 0.7129 1.3733 NA 52
#> det.cov.1-sp5 0.2784 0.2611 -0.2314 0.2793 0.7652 NA 98
#> det.cov.1-sp6 -0.1534 0.2547 -0.6997 -0.1626 0.3139 NA 100
#> det.cov.1-sp7 0.3928 0.2435 -0.0501 0.3741 0.9340 NA 53
#> det.cov.1-sp8 -0.2498 0.2680 -0.7396 -0.2322 0.2482 NA 55
#> det.cov.2-sp1 -0.2438 0.2163 -0.6536 -0.2396 0.1732 NA 249
#> det.cov.2-sp2 -0.8092 0.3178 -1.3988 -0.8174 -0.2568 NA 56
#> det.cov.2-sp3 0.2628 0.2670 -0.2663 0.2776 0.7874 NA 79
#> det.cov.2-sp4 -0.0160 0.2216 -0.4174 -0.0126 0.4248 NA 98
#> det.cov.2-sp5 0.1066 0.3739 -0.7154 0.1110 0.9106 NA 102
#> det.cov.2-sp6 0.3875 0.3132 -0.2339 0.3631 0.9521 NA 115
#> det.cov.2-sp7 -0.9842 0.2994 -1.6330 -0.9623 -0.4248 NA 82
#> det.cov.2-sp8 -0.5067 0.2806 -1.1985 -0.4902 -0.0221 NA 78
#>
#> ----------------------------------------
#> Spatial Covariance
#> ----------------------------------------
#> Mean SD 2.5% 50% 97.5% Rhat ESS
#> phi-1 12.5933 6.1269 4.4529 10.8855 26.3589 NA 19
#> phi-2 14.2373 6.3840 4.2820 13.6586 26.8439 NA 16
#> phi-3 14.8947 8.4997 3.3283 13.8127 28.6755 NA 8
#> phi-4 18.6203 6.8102 4.5221 18.8702 28.5871 NA 9
```