Function for Fitting Multi-Species Spatially-Varying Coefficient Occupancy Models

The function svcMsPGOcc fits multi-species spatially-varying coefficient 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. Models are implemented using a Nearest Neighbor Gaussian Process.

Usage

svcMsPGOcc(occ.formula, det.formula, data, inits, priors, tuning, 
           svc.cols = 1, cov.model = 'exponential', NNGP = TRUE, 
           n.neighbors = 15, search.type = 'cb', std.by.sp = FALSE, 
           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, ...)

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, and range.ind. 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. range.ind is a matrix with rows corresponding to species and columns corresponding to sites, with each element taking value 1 if that site is within the range of the corresponding species and 0 if it is outside of the range. This matrix is not required, but it can be helpful to restrict the modeled area for each individual species to be within the realistic range of locations for that species when estimating the model parameters. This is applicable when auxiliary data sources are available on the realistic range of the species.
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 2.72. By default, 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 for each spatially-varying coefficient specified in svc.cols. 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 * length(svc.cols) corresponding to the lower and upper support, respectively, or as a single value if the same value is assigned for all factor/SVC combinations. The priors for the factor loadings matrix lambda for each SVC 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 for each SVC. 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.
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 occupancy model analogous to sfMsPGOcc (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. 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.
std.by.sp: a logical value indicating whether the covariates are standardized separately for each species within the corresponding range for each species (TRUE) or not (FALSE). Note that if range.ind is specified in data.list, this will result in the covariates being standardized differently for each species based on the sites where range.ind == 1 for that given species. If range.ind is not specified and std.by.sp = TRUE, this will simply be equivalent to standardizing the covariates across all locations prior to fitting the model. Note that the covariates in occ.formula should still be standardized across all locations. This can be done either outside the function, or can be done by specifying scale() in the model formula around the continuous covariates.
n.factors: the number of factors to use in the spatial factor model approach. Note this corresponds to the number of factors used for each spatially-varying coefficient that is estimated in the model. 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.
...: 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 svcMsPGOcc 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 for each spatially-varying coefficient.
lambda.samples: a coda object of posterior samples for the latent spatial factor loadings for each spatially-varying coefficient.
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 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.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().

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 <- 10
J.y <- 10 
J <- J.x * J.y
n.rep <- sample(5, size = J, replace = TRUE)
N <- 6
# Community-level covariate effects
# Occurrence
beta.mean <- c(0.2, -0.2, 0.3, -0.1, 0.4)
p.occ <- length(beta.mean)
tau.sq.beta <- c(0.6, 1.5, 0.4, 0.5, 0.3)
# Detection
alpha.mean <- c(0, 1.2, -0.5)
tau.sq.alpha <- c(1, 0.5, 1.3)
p.det <- length(alpha.mean)
# Random effects
psi.RE <- list(levels = 15, 
               sigma.sq.psi = 0.7)
p.RE <- list(levels = 20, 
             sigma.sq.p = 0.5)
# 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]))
}
# Number of spatial factors for each SVC
n.factors <- 2
# The intercept and first two covariates have spatially-varying effects
svc.cols <- c(1, 2, 3)
p.svc <- length(svc.cols)
q.p.svc <- n.factors * p.svc
# Spatial decay parameters
phi <- runif(q.p.svc, 3 / 0.9, 3 / 0.1)
# A length N vector indicating the proportion of simulated locations
# that are within the range for a given species.
range.probs <- runif(N, 0.4, 1)
factor.model <- TRUE
cov.model <- 'spherical'
sp <- TRUE

dat <- simMsOcc(J.x = J.x, J.y = J.y, n.rep = n.rep, N = N, beta = beta, alpha = alpha,
                psi.RE = psi.RE, p.RE = p.RE, phi = phi, sp = sp, svc.cols = svc.cols,
                cov.model = cov.model, n.factors = n.factors, 
                factor.model = factor.model, range.probs = range.probs)

y <- dat$y
X <- dat$X
X.re <- dat$X.re
X.p <- dat$X.p
X.p.re <- dat$X.p.re
coords <- dat$coords
range.ind <- dat$range.ind

# Prep data for spOccupancy -----------------------------------------------
# Occurrence covariates
occ.covs <- cbind(X, X.re)
colnames(occ.covs) <- c('int', 'occ.cov.1', 'occ.cov.2', 'occ.cov.3', 
                        'occ.cov.4', 'occ.factor.1')
# Detection covariates
det.covs <- list(det.cov.1 = X.p[, , 2], 
                 det.cov.2 = X.p[, , 3], 
                 det.factor.1 = X.p.re[, , 1]) 
# Data list
data.list <- list(y = y, coords = coords, occ.covs = occ.covs, 
                  det.covs = det.covs, range.ind = range.ind)
# 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)) 
inits.list <- list(alpha.comm = 0, 
                   beta.comm = 0, 
                   beta = 0, 
                   alpha = 0,
                   tau.sq.beta = 1, 
                   tau.sq.alpha = 1, 
                   z = apply(y, c(1, 2), max, na.rm = TRUE)) 
# Tuning
tuning.list <- list(phi = 1)

# Number of batches
n.batch <- 2
# Batch length
batch.length <- 25
n.burn <- 0
n.thin <- 1
n.samples <- n.batch * batch.length

out <- svcMsPGOcc(occ.formula = ~ occ.cov.1 + occ.cov.2 + occ.cov.3 + 
                                  occ.cov.4 + (1 | occ.factor.1),
                  det.formula = ~ det.cov.1 + det.cov.2 + (1 | det.factor.1),
                  data = data.list,
                  inits = inits.list,
                  n.batch = n.batch,
                  n.factors = n.factors,
                  batch.length = batch.length,
                  std.by.sp = TRUE,
                  accept.rate = 0.43,
                  priors = prior.list,
                  svc.cols = svc.cols,
                  cov.model = "spherical",
                  tuning = tuning.list,
                  n.omp.threads = 1,
                  verbose = TRUE,
                  NNGP = TRUE,
                  n.neighbors = 5,
                  search.type = 'cb',
                  n.report = 10,
                  n.burn = n.burn,
                  n.thin = n.thin,
                  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
#> phi is not specified in initial values.
#> Setting initial value to random values from the prior distribution
#> lambda is not specified in initial values.
#> Setting initial values of the lower triangle to random values from a standard normal
#> 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
#> 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 Multispecies Occupancy Model with Polya-Gamma latent
#> variable fit with 100 sites and 6 species.
#> 
#> Samples per chain: 50 (2 batches of length 25)
#> Burn-in: 0 
#> Thinning Rate: 1 
#> Number of Chains: 1 
#> Total Posterior Samples: 50 
#> 
#> Number of spatially-varying coefficients: 3 
#> Using the spherical 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: 2 of 2, 100.00%

summary(out)
#> 
#> Call:
#> svcMsPGOcc(occ.formula = ~occ.cov.1 + occ.cov.2 + occ.cov.3 + 
#>     occ.cov.4 + (1 | occ.factor.1), det.formula = ~det.cov.1 + 
#>     det.cov.2 + (1 | det.factor.1), data = data.list, inits = inits.list, 
#>     priors = prior.list, tuning = tuning.list, svc.cols = svc.cols, 
#>     cov.model = "spherical", NNGP = TRUE, n.neighbors = 5, search.type = "cb", 
#>     std.by.sp = TRUE, 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 = n.burn, n.thin = n.thin, 
#>     n.chains = 1)
#> 
#> Samples per Chain: 50
#> Burn-in: 0
#> Thinning Rate: 1
#> Number of Chains: 1
#> Total Posterior Samples: 50
#> Run Time (min): 0.0062
#> 
#> ----------------------------------------
#> 	Community Level
#> ----------------------------------------
#> Occurrence Means (logit scale): 
#>                Mean     SD    2.5%     50%  97.5% Rhat ESS
#> (Intercept)  0.0003 0.8828 -2.2097  0.3088 1.0868   NA   2
#> occ.cov.1    0.1930 0.4557 -0.6595  0.1889 1.2157   NA  32
#> occ.cov.2    0.3875 0.4781 -0.3838  0.4341 1.3478   NA  25
#> occ.cov.3   -0.0827 0.3947 -0.7252 -0.1019 0.6220   NA  28
#> occ.cov.4    0.4362 0.6628 -0.9059  0.4561 1.6089   NA  27
#> 
#> Occurrence Variances (logit scale): 
#>               Mean     SD   2.5%    50%  97.5% Rhat ESS
#> (Intercept) 0.2341 0.1874 0.0572 0.1595 0.7946   NA   8
#> occ.cov.1   1.9632 1.8839 0.0875 1.6484 7.2621   NA   9
#> occ.cov.2   0.8512 0.6872 0.1057 0.5851 2.6347   NA  32
#> occ.cov.3   0.8809 1.0466 0.0465 0.4494 3.0596   NA  12
#> occ.cov.4   2.1776 2.9252 0.1319 1.3970 8.5300   NA  25
#> 
#> Occurrence Random Effect Variances (logit scale): 
#>                Mean     SD   2.5%    50%  97.5% Rhat ESS
#> occ.factor.1 4.5647 2.2304 1.7664 3.9423 8.6058   NA   2
#> 
#> Detection Means (logit scale): 
#>                Mean     SD    2.5%     50%  97.5% Rhat ESS
#> (Intercept) -0.3129 0.4445 -0.9813 -0.3811 0.4507   NA  21
#> det.cov.1    1.3472 0.4493  0.2824  1.3679 2.1247   NA  31
#> det.cov.2   -0.4631 0.4629 -1.4137 -0.4130 0.4867   NA  50
#> 
#> Detection Variances (logit scale): 
#>               Mean     SD   2.5%    50%  97.5% Rhat ESS
#> (Intercept) 1.2072 1.3654 0.0626 0.7661 4.7680   NA  27
#> det.cov.1   2.0009 4.2075 0.1976 1.0119 6.0019   NA  50
#> det.cov.2   2.4840 1.8688 0.3223 1.9282 5.8457   NA  33
#> 
#> Detection Random Effect Variances (logit scale): 
#>                Mean     SD   2.5%   50%  97.5% Rhat ESS
#> det.factor.1 0.9486 1.1996 0.2835 0.423 4.7486   NA   7
#> 
#> ----------------------------------------
#> 	Species Level
#> ----------------------------------------
#> Occurrence (logit scale): 
#>                    Mean     SD    2.5%     50%  97.5% Rhat ESS
#> (Intercept)-sp1 -0.0152 0.8329 -2.1032  0.1534 0.9867   NA   4
#> (Intercept)-sp2 -0.0519 0.8672 -2.0607  0.2322 1.0268   NA   5
#> (Intercept)-sp3  0.2805 0.8770 -1.5017  0.5980 1.3774   NA   3
#> (Intercept)-sp4  0.1295 1.0756 -2.1975  0.1150 1.8442   NA   5
#> (Intercept)-sp5 -0.1775 1.0715 -2.7408  0.1796 1.3202   NA   3
#> (Intercept)-sp6 -0.1938 0.9395 -2.7335  0.1008 0.9183   NA   4
#> occ.cov.1-sp1    0.6958 0.4457 -0.1681  0.7481 1.6253   NA  10
#> occ.cov.1-sp2    2.1056 0.8837  0.2937  2.3944 3.2750   NA   3
#> occ.cov.1-sp3   -0.1563 0.5237 -0.7451 -0.2839 1.2902   NA  10
#> occ.cov.1-sp4   -0.5689 0.7422 -2.0949 -0.4914 0.4260   NA   9
#> occ.cov.1-sp5   -0.2410 0.6413 -1.4663 -0.0956 0.7205   NA   8
#> occ.cov.1-sp6   -0.9817 0.6735 -2.3541 -0.9374 0.0746   NA   6
#> occ.cov.2-sp1    0.1775 0.4411 -0.8308  0.2006 0.8097   NA   8
#> occ.cov.2-sp2    1.2458 0.5921 -0.0281  1.2656 2.2964   NA  13
#> occ.cov.2-sp3   -0.3038 0.4503 -0.9594 -0.2925 0.4799   NA   7
#> occ.cov.2-sp4    0.3335 0.7223 -0.9561  0.3955 1.6625   NA  10
#> occ.cov.2-sp5    0.7130 0.5432 -0.4812  0.9303 1.4144   NA   6
#> occ.cov.2-sp6    0.8212 0.5876 -0.0769  0.8392 1.9877   NA  55
#> occ.cov.3-sp1   -0.5717 0.3999 -1.2736 -0.6329 0.2900   NA  12
#> occ.cov.3-sp2   -0.1207 0.3711 -0.7045 -0.0796 0.6124   NA  16
#> occ.cov.3-sp3   -0.1835 0.4174 -0.8854 -0.1517 0.4798   NA  33
#> occ.cov.3-sp4   -0.0496 0.5699 -1.0193 -0.0783 1.0665   NA   8
#> occ.cov.3-sp5    0.8718 0.7120 -0.3165  0.7681 2.3946   NA   6
#> occ.cov.3-sp6   -0.6365 0.5429 -1.9309 -0.6078 0.1333   NA   9
#> occ.cov.4-sp1    1.9809 0.6760  0.6441  2.0003 3.2657   NA   9
#> occ.cov.4-sp2    0.5549 0.4780 -0.4102  0.5907 1.3740   NA  11
#> occ.cov.4-sp3   -0.8335 0.5077 -1.8004 -0.8183 0.3820   NA  16
#> occ.cov.4-sp4    1.5694 0.9573 -0.1290  1.7278 2.9211   NA   6
#> occ.cov.4-sp5    0.3708 0.5596 -0.7020  0.2925 1.2425   NA  11
#> occ.cov.4-sp6    0.4799 0.6920 -0.8997  0.4878 1.7226   NA  17
#> 
#> Detection (logit scale): 
#>                    Mean     SD    2.5%     50%   97.5% Rhat ESS
#> (Intercept)-sp1  0.3148 0.4147 -0.6349  0.4058  0.8854   NA   9
#> (Intercept)-sp2 -0.0422 0.3982 -0.7970 -0.0186  0.6023   NA  15
#> (Intercept)-sp3  0.2991 0.3724 -0.6028  0.4018  0.8217   NA   6
#> (Intercept)-sp4 -1.3487 0.3489 -1.8054 -1.3477 -0.7432   NA  18
#> (Intercept)-sp5 -0.3042 0.3056 -0.8432 -0.2893  0.1826   NA   4
#> (Intercept)-sp6 -0.9327 0.3486 -1.7058 -0.8760 -0.2856   NA  24
#> det.cov.1-sp1    0.4613 0.2064  0.0372  0.4789  0.7814   NA  32
#> det.cov.1-sp2    3.0573 1.1096  1.4463  2.7479  5.5803   NA   5
#> det.cov.1-sp3    0.7365 0.1800  0.4192  0.7399  1.0228   NA  20
#> det.cov.1-sp4    1.9913 0.5020  0.8353  2.0092  2.8268   NA  14
#> det.cov.1-sp5    0.8767 0.2265  0.4867  0.8919  1.3168   NA  48
#> det.cov.1-sp6    1.4186 0.4384  0.6928  1.4048  2.2438   NA  22
#> det.cov.2-sp1   -0.6952 0.1866 -1.0001 -0.6928 -0.4181   NA  31
#> det.cov.2-sp2   -2.3564 0.7360 -3.6645 -2.3146 -0.8637   NA  10
#> det.cov.2-sp3   -1.0150 0.2096 -1.4032 -1.0397 -0.6485   NA  22
#> det.cov.2-sp4   -0.9315 0.3602 -1.4647 -0.9405 -0.2239   NA  21
#> det.cov.2-sp5    0.4577 0.2409 -0.0549  0.4862  0.8713   NA  50
#> det.cov.2-sp6    1.5097 0.4703  0.8174  1.4415  2.4276   NA  12
#> 
#> ----------------------------------------
#> 	Spatial Covariance
#> ----------------------------------------
#>                      Mean     SD   2.5%     50%   97.5% Rhat ESS
#> phi-1-(Intercept) 16.5206 7.8594 7.2168 14.4146 28.5076   NA   5
#> phi-2-(Intercept) 17.5785 8.3706 7.4059 13.9784 28.9864   NA   3
#> phi-1-occ.cov.1   12.9727 4.9110 6.0180 11.5222 23.0404   NA   8
#> phi-2-occ.cov.1   15.4947 5.3038 6.7308 15.6198 25.8761   NA  10
#> phi-1-occ.cov.2   14.2845 6.1628 6.3465 13.6324 24.6283   NA   6
#> phi-2-occ.cov.2    8.1886 4.2072 5.2611  6.0088 18.6819   NA  11