# Function for Fitting Multi-Species Spatial Occupancy Models Using Polya-Gamma Latent Variables 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. ```r 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, k.fold.only = FALSE, ...) ``` ## 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 $J x 2$ 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 within chains. 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. - `k.fold.only`: a logical value indicating whether to only perform cross-validation (`TRUE`) or perform cross-validation after fitting the full model (`FALSE`). Default value is `FALSE`. - `...`: 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](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**, tools:::Rd_expr_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**, tools:::Rd_expr_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. tools:::Rd_expr_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(s) Jeffrey W. Doser doserjef@msu.edu , Andrew O. Finley finleya@msu.edu ## Returns 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 `fitted()`. ## Examples ```r 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) # Note that this is just a test case and more iterations/chains may need to # be run to ensure convergence. 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) summary(out, level = 'both') ```

spMsPGOcc function