Function for Fitting Multi-Species Multi-Season Occupancy Models

The function tMsPGOcc fits multi-species multi-season occupancy models using Polya-Gamma data augmentation.

Usage

tMsPGOcc(occ.formula, det.formula, data, inits, priors, tuning, 
         n.batch, batch.length, 
         accept.rate = 0.43, n.omp.threads = 1, 
         verbose = TRUE, ar1 = FALSE, 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, and det.covs. y is a four-dimensional array with first dimension equal to the number of species, second dimension equal to the number of sites, third dimension equal to the number of primary time periods, and fourth dimension equal to the maximum number of secondary replicates at a given site. occ.covs is a list of variables included in the occurrence portion of the model. Each list element is a different occurrence covariate, which can be site level or site/primary time period level. Site-level covariates are specified as a vector of length \(J\) while site/primary time period level covariates are specified as a matrix with rows corresponding to sites and columns correspond to primary time periods. Similarly, det.covs is a list of variables included in the detection portion of the model, with each list element corresponding to an individual variable. In addition to site-level and/or site/primary time period-level, detection covariates can also be observational-level. Observation-level covariates are specified as a three-dimensional array with first dimension corresponding to sites, second dimension corresponding to primary time period, and third dimension corresponding to replicate.
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.t, and rho. sigma.sq.t and rho are only relevant when ar1 = TRUE, 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, sigma.sq.t.ig, and rho.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. sigma.sq.t and rho are the AR(1) variance and correlation parameters for the AR(1) zero-mean temporal random effects, respectively. sigma.sq.t is assumed to follow an inverse-Gamma distribution, where the hyperparameters are specified as a list of length two with the first and second elements corresponding to the shape and scale parameters, respectively, which can each be specified as vector equal to the number of species in the model or a single value if the same prior is used for all species. rho is assumed to follow a uniform distribution, where the hyperparameters are specified similarly as a list of length two with the first and second elements corresponding to the lower and upper bounds of the uniform prior, which can each be specified as vector equal to the number of species in the model or a single value if the same prior is used 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 rho. The value portion of each tag defines the initial tuning variance of the Adaptive sampler. See Roberts and Rosenthal (2009) for details.
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.
ar1: logical value indicating whether to include an AR(1) zero-mean temporal random effect in the model for each species. If FALSE, the model is fit without an AR(1) temporal autocovariance structure. If TRUE, a species-specific AR(1) random effect is included in the model to account for temporal autocorrelation across the primary time periods.
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

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 .

Kery, M., & Royle, J. A. (2021). Applied hierarchical modeling in ecology: Analysis of distribution, abundance and species richness in R and BUGS: Volume 2: Dynamic and advanced models. Academic Press. Section 4.6.

Author

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

Value

An object of class tMsPGOcc 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 AR(1) variance (sigma.sq.t) and correlation (rho) parameters. Only included if ar1 = TRUE.
eta.samples: a three-dimensional array of posterior samples for the species-specific AR(1) random effects for each primary time period. Dimensions correspond to MCMC sample, species, and primary time period.
z.samples: a four-dimensional array of posterior samples for the latent occurrence values for each species. Dimensions corresopnd to MCMC sample, species, site, and primary time period.
psi.samples: a four-dimensional array of posterior samples for the latent occupancy probability values for each species. Dimensions correspond to MCMC sample, species, site, and primary time period.
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 four-dimensional array of posterior samples for the likelihood value used for calculating WAIC. Dimensions correspond to MCMC sample, species, site, and time period.
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

# Simulate Data -----------------------------------------------------------
set.seed(500)
J.x <- 8
J.y <- 8
J <- J.x * J.y
# Years sampled
n.time <- sample(3:10, J, replace = TRUE)
# n.time <- rep(10, J)
n.time.max <- max(n.time)
# Replicates
n.rep <- matrix(NA, J, max(n.time))
for (j in 1:J) {
  n.rep[j, 1:n.time[j]] <- sample(2:4, n.time[j], replace = TRUE)
}
N <- 7
# Community-level covariate effects
# Occurrence
beta.mean <- c(-3, -0.2, 0.5)
trend <- FALSE
sp.only <- 0
p.occ <- length(beta.mean)
tau.sq.beta <- c(0.6, 1.5, 1.4)
# Detection
alpha.mean <- c(0, 1.2, -1.5)
tau.sq.alpha <- c(1, 0.5, 2.3)
p.det <- length(alpha.mean)
# Random effects
psi.RE <- list()
p.RE <- list()
# 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]))
}
sp <- FALSE

dat <- simTMsOcc(J.x = J.x, J.y = J.y, n.time = n.time, n.rep = n.rep, N = N,
                 beta = beta, alpha = alpha, sp.only = sp.only, trend = trend,
                 psi.RE = psi.RE, p.RE = p.RE, sp = sp)

y <- dat$y
X <- dat$X
X.p <- dat$X.p
X.re <- dat$X.re
X.p.re <- dat$X.p.re

occ.covs <- list(occ.cov.1 = X[, , 2],
                 occ.cov.2 = X[, , 3])
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)
# 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))
z.init <- apply(y, c(1, 2, 3), function(a) as.numeric(sum(a, na.rm = TRUE) > 0))
inits.list <- list(alpha.comm = 0, beta.comm = 0, beta = 0,
                   alpha = 0, tau.sq.beta = 1, tau.sq.alpha = 1,
                   z = z.init)
# Tuning
tuning.list <- list(phi = 1)

# Number of batches
n.batch <- 5
# Batch length
batch.length <- 25
n.burn <- 25
n.thin <- 1
n.samples <- n.batch * batch.length

out <- tMsPGOcc(occ.formula = ~ occ.cov.1 + occ.cov.2,
                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,
                n.omp.threads = 1,
                verbose = TRUE,
                n.report = 1,
                n.burn = n.burn,
                n.thin = n.thin,
                n.chains = 1)
#> ----------------------------------------
#> 	Preparing to run the model
#> ----------------------------------------
#> ----------------------------------------
#> 	Model description
#> ----------------------------------------
#> Multi-season Multi-species Occupancy Model with Polya-Gamma latent
#> variables with 64 sites, 7 species, and 10 primary time periods.
#> 
#> Samples per chain: 125 (5 batches of length 25)
#> Burn-in: 25 
#> Thinning Rate: 1 
#> Number of Chains: 1 
#> Total Posterior Samples: 100 
#> 
#> Source compiled with OpenMP support and model fit using 1 thread(s).
#> 
#> Adaptive Metropolis with target acceptance rate: 43.0
#> ----------------------------------------
#> 	Chain 1
#> ----------------------------------------
#> Sampling ... 
#> Batch: 1 of 5, 20.00%
#> -------------------------------------------------
#> Batch: 2 of 5, 40.00%
#> -------------------------------------------------
#> Batch: 3 of 5, 60.00%
#> -------------------------------------------------
#> Batch: 4 of 5, 80.00%
#> -------------------------------------------------
#> Batch: 5 of 5, 100.00%

summary(out)
#> 
#> Call:
#> tMsPGOcc(occ.formula = ~occ.cov.1 + occ.cov.2, det.formula = ~det.cov.1 + 
#>     det.cov.2, data = data.list, inits = inits.list, priors = prior.list, 
#>     n.batch = n.batch, batch.length = batch.length, accept.rate = 0.43, 
#>     n.omp.threads = 1, verbose = TRUE, n.report = 1, n.burn = n.burn, 
#>     n.thin = n.thin, n.chains = 1)
#> 
#> Samples per Chain: 125
#> Burn-in: 25
#> Thinning Rate: 1
#> Number of Chains: 1
#> Total Posterior Samples: 100
#> Run Time (min): 0.0062
#> 
#> ----------------------------------------
#> 	Community Level
#> ----------------------------------------
#> Occurrence Means (logit scale): 
#>                Mean     SD    2.5%     50%   97.5% Rhat ESS
#> (Intercept) -3.3951 0.2561 -3.8965 -3.3688 -2.9980   NA  67
#> occ.cov.1    0.2756 0.2465 -0.2455  0.2749  0.7464   NA  51
#> occ.cov.2    0.9911 0.6127 -0.0586  1.0145  2.3914   NA 100
#> 
#> Occurrence Variances (logit scale): 
#>               Mean     SD   2.5%    50%  97.5% Rhat ESS
#> (Intercept) 0.6558 0.6816 0.0831 0.5029 2.2011   NA  17
#> occ.cov.1   0.4368 0.4756 0.0765 0.2968 1.7432   NA  58
#> occ.cov.2   2.7841 1.9843 0.6906 2.2722 7.8407   NA  67
#> 
#> Detection Means (logit scale): 
#>                Mean     SD    2.5%     50%   97.5% Rhat ESS
#> (Intercept)  0.2977 0.2616 -0.2173  0.2969  0.6910   NA  36
#> det.cov.1    1.1457 0.3288  0.3844  1.1673  1.7245   NA 100
#> det.cov.2   -1.5292 0.8089 -2.8766 -1.5298 -0.0014   NA 147
#> 
#> Detection Variances (logit scale): 
#>               Mean     SD   2.5%    50%   97.5% Rhat ESS
#> (Intercept) 0.2899 0.3396 0.0476 0.1997  1.4378   NA  38
#> det.cov.1   0.5948 0.6932 0.0793 0.3576  2.8432   NA  16
#> det.cov.2   6.0538 5.5965 0.9278 4.9073 21.7062   NA  13
#> 
#> ----------------------------------------
#> 	Species Level
#> ----------------------------------------
#> Occurrence (logit scale): 
#>                    Mean     SD    2.5%     50%   97.5% Rhat ESS
#> (Intercept)-sp1 -3.3381 0.2781 -3.8072 -3.3623 -2.8221   NA  13
#> (Intercept)-sp2 -3.8087 0.3364 -4.5895 -3.7667 -3.2830   NA   8
#> (Intercept)-sp3 -3.4782 0.2767 -4.0439 -3.4558 -3.0143   NA  12
#> (Intercept)-sp4 -2.7861 0.2037 -3.1506 -2.7728 -2.3653   NA  20
#> (Intercept)-sp5 -2.9008 0.2052 -3.2888 -2.9090 -2.4694   NA  19
#> (Intercept)-sp6 -3.2567 0.2770 -3.7471 -3.2758 -2.7925   NA  11
#> (Intercept)-sp7 -4.6254 0.6917 -6.0049 -4.6657 -3.4336   NA   4
#> occ.cov.1-sp1    0.1990 0.2319 -0.2268  0.2417  0.6363   NA  19
#> occ.cov.1-sp2   -0.1376 0.2792 -0.7171 -0.1364  0.3086   NA  14
#> occ.cov.1-sp3   -0.3402 0.2547 -0.7901 -0.3369  0.1288   NA  15
#> occ.cov.1-sp4    0.1534 0.1957 -0.1936  0.1549  0.5603   NA  24
#> occ.cov.1-sp5    0.6027 0.2612  0.1821  0.5843  1.0629   NA  17
#> occ.cov.1-sp6    0.5553 0.2275  0.1205  0.5497  1.0428   NA  24
#> occ.cov.1-sp7    0.9313 0.3146  0.4912  0.8504  1.6122   NA  14
#> occ.cov.2-sp1    1.1176 0.2496  0.5983  1.1282  1.5460   NA  20
#> occ.cov.2-sp2    1.7928 0.2196  1.3671  1.7863  2.3106   NA  22
#> occ.cov.2-sp3   -0.5970 0.2756 -1.1362 -0.5582 -0.1028   NA  13
#> occ.cov.2-sp4   -0.3093 0.2761 -0.8506 -0.3490  0.2245   NA  18
#> occ.cov.2-sp5    0.2672 0.2336 -0.1879  0.2839  0.6652   NA  19
#> occ.cov.2-sp6    1.9392 0.2399  1.5127  1.9487  2.4148   NA  27
#> occ.cov.2-sp7    2.9842 0.5896  2.0502  2.8994  4.1770   NA   4
#> 
#> Detection (logit scale): 
#>                    Mean     SD    2.5%     50%   97.5% Rhat ESS
#> (Intercept)-sp1  0.6487 0.3810 -0.0374  0.6079  1.4427   NA  19
#> (Intercept)-sp2 -0.0461 0.2562 -0.5066 -0.0365  0.4261   NA  17
#> (Intercept)-sp3  0.1488 0.4564 -0.8523  0.2275  0.9119   NA  29
#> (Intercept)-sp4  0.2893 0.2634 -0.2456  0.3054  0.7261   NA  42
#> (Intercept)-sp5  0.3673 0.2662 -0.0924  0.3989  0.8349   NA  44
#> (Intercept)-sp6  0.6592 0.2331  0.1994  0.6374  1.0941   NA  49
#> (Intercept)-sp7  0.0337 0.2281 -0.3833  0.0455  0.4328   NA  54
#> det.cov.1-sp1    1.9411 0.5038  1.0673  1.9065  2.9361   NA  21
#> det.cov.1-sp2    0.8524 0.2982  0.4211  0.8139  1.5298   NA  48
#> det.cov.1-sp3    1.0282 0.4706  0.1435  1.0100  2.0621   NA  19
#> det.cov.1-sp4    0.7611 0.3436  0.1435  0.7569  1.5169   NA  28
#> det.cov.1-sp5    1.8650 0.4711  1.2108  1.7717  2.8979   NA  21
#> det.cov.1-sp6    1.0617 0.2706  0.5133  1.0828  1.5632   NA  43
#> det.cov.1-sp7    0.9060 0.2327  0.5224  0.8675  1.3828   NA  54
#> det.cov.2-sp1    0.0701 0.3793 -0.5709  0.0622  0.8415   NA  55
#> det.cov.2-sp2   -0.3658 0.3012 -0.8687 -0.3310  0.1460   NA  53
#> det.cov.2-sp3   -5.9659 1.9654 -9.5462 -6.1462 -2.7888   NA   4
#> det.cov.2-sp4   -3.0455 0.7435 -4.2544 -2.9998 -1.6882   NA  15
#> det.cov.2-sp5   -1.0886 0.4014 -1.9885 -1.0644 -0.4060   NA  34
#> det.cov.2-sp6   -2.0188 0.3918 -2.6893 -1.9970 -1.4653   NA  38
#> det.cov.2-sp7   -1.6556 0.3505 -2.2969 -1.6076 -1.1129   NA  30