# Function for Fitting Single-Species Integrated Spatial Occupancy Models Using Polya-Gamma Latent Variables The function `spIntPGOcc` fits single-species integrated 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. Data integration is done using a joint likelihood framework, assuming distinct detection models for each data source that are each conditional on a single latent occupancy process. ```r spIntPGOcc(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.data, 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 list of symbolic descriptions of the models to be fit for the detection portion of the model using R's model syntax for each data set. Each element in the list is a formula for the detection model of a given data set. 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`, `sites` and `coords`. `y` is a list of matrices or data frames for each data set used in the integrated model. Each element of the list has first dimension equal to the number of sites with that data source and second 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 the number of rows being the number of sites with at least one data source for each column (variable). `det.covs` is a list of variables included in the detection portion of the model for each data source. `det.covs` should have the same number of elements as `y`, where each element is itself a list. Each element of the list for a given data source is a different detection covariate, which can be site-level or observational-level. Site-level covariates are specified as a vector with length equal to the number of observed sites of that data source, while observation-level covariates are specified as a matrix or data frame with the number of rows equal to the number of observed sites of that data source and number of columns equal to the maximum number of replicates at a given site. `coords` is a matrix of the observation site coordinates. Note that `spOccupancy` assumes coordinates are specified in a projected coordinate system. `sites` is a list of site indices with number of elements equal to the number of data sources being modeled. Each element contains a vector of length equal to the number of sites that specific data source contains. Each value in the vector indicates the row in `occ.covs` that corresponds with the specific row of the detection-nondetection data for the data source. This is used to properly link sites across data sets. - `inits`: a list with each tag corresponding to a parameter name. Valid tags are `z`, `beta`, `alpha`, `sigma.sq`, `phi`, `w`, `nu`, `sigma.sq.psi`, `sigma.sq.p`. The value portion of all tags except `alpha` is the parameter's initial value. `sigma.sq.psi` and `sigma.sq.p` are only relevant when including random effects in the occurrence and detection portion of the occupancy model, respectively. The tag `alpha` is a list comprised of the initial values for the detection parameters for each data source. Each element of the list should be a vector of initial values for all detection parameters in the given data source or a single value for each data source to assign all parameters for a given data source the same 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.normal`, `alpha.normal`, `phi.unif`, `sigma.sq.ig`, `sigma.sq.unif`, `nu.unif`, `sigma.sq.psi.ig`, and `sigma.sq.p.ig`. Occurrence (`beta`) and detection (`alpha`) regression coefficients are assumed to follow a normal distribution. For `beta` 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. For the detection coefficients `alpha`, the mean and variance hyperparameters are themselves passed in as lists, with each element of the list corresponding to the specific hyperparameters for the detection parameters in a given data source. If not specified, prior means are set to 0 and prior variances set to 2.72 for normal priors. The spatial variance parameter, `sigma.sq`, is assumed to follow an inverse-Gamma distribution or a uniform distribution (default is inverse-Gamma). `sigma.sq` 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 vector of length two, with the first and second elements corresponding to the **shape** and **scale**, respectively. The hyperparameters of the Uniform are also passed as a vector of length two with the first and second elements corresponding to the lower and upper support, respectively. `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. 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 to run for each chain 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. Default 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.batch * batch.length` samples to discard as burn-in. 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. - `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.data`: an integer specifying the specific data set to hold out values from. If not specified, data from all data set locations will be incorporated into the k-fold cross-validation. - `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. Hooten, M. B., and Hobbs, N. T. (2015). A guide to Bayesian model selection for ecologists. **Ecological Monographs**, 85(1), 3-28. Hooten, M. B., and Hefley, T. J. (2019). Bringing Bayesian models to life. **CRC Press**. 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. ## Author(s) Jeffrey W. Doser doserjef@msu.edu , Andrew O. Finley finleya@msu.edu ## Returns An object of class `spIntPGOcc` that is a list comprised of: - **beta.samples**: a `coda` object of posterior samples for the occurrence regression coefficients. - **alpha.samples**: a `coda` object of posterior samples for the detection regression coefficients for all data sources. - **z.samples**: a `coda` object of posterior samples for the latent occurrence values - **psi.samples**: a `coda` object of posterior samples for the latent occurrence probability values - **theta.samples**: a `coda` object of posterior samples for covariance parameters. - **w.samples**: a `coda` object of posterior samples for latent spatial random effects. - **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**: execution time reported using `proc.time()`. - **k.fold.deviance**: scoring rule (deviance) from k-fold cross-validation. A separate deviance value is returned for each data source. Only included if `k.fold` is specified in function call. Only a single value is returned if `k.fold.data` is specified. 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 ----------------------------------------------------------- # Number of locations in each direction. This is the total region of interest # where some sites may or may not have a data source. J.x <- 8 J.y <- 8 J.all <- J.x * J.y # Number of data sources. n.data <- 4 # Sites for each data source. J.obs <- sample(ceiling(0.2 * J.all):ceiling(0.5 * J.all), n.data, replace = TRUE) # Replicates for each data source. n.rep <- list() for (i in 1:n.data) { n.rep[[i]] <- sample(1:4, size = J.obs[i], replace = TRUE) } # Occupancy covariates beta <- c(0.5, 0.5) p.occ <- length(beta) # Detection covariates alpha <- list() alpha[[1]] <- runif(2, 0, 1) alpha[[2]] <- runif(3, 0, 1) alpha[[3]] <- runif(2, -1, 1) alpha[[4]] <- runif(4, -1, 1) p.det.long <- sapply(alpha, length) p.det <- sum(p.det.long) sigma.sq <- 2 phi <- 3 / .5 sp <- TRUE # Simulate occupancy data from multiple data sources. dat <- simIntOcc(n.data = n.data, J.x = J.x, J.y = J.y, J.obs = J.obs, n.rep = n.rep, beta = beta, alpha = alpha, sp = sp, sigma.sq = sigma.sq, phi = phi, cov.model = 'exponential') y <- dat$y X <- dat$X.obs X.p <- dat$X.p sites <- dat$sites X.0 <- dat$X.pred psi.0 <- dat$psi.pred coords <- as.matrix(dat$coords.obs) coords.0 <- as.matrix(dat$coords.pred) # Package all data into a list occ.covs <- X[, 2, drop = FALSE] colnames(occ.covs) <- c('occ.cov') det.covs <- list() # Add covariates one by one det.covs[[1]] <- list(det.cov.1.1 = X.p[[1]][, , 2]) det.covs[[2]] <- list(det.cov.2.1 = X.p[[2]][, , 2], det.cov.2.2 = X.p[[2]][, , 3]) det.covs[[3]] <- list(det.cov.3.1 = X.p[[3]][, , 2]) det.covs[[4]] <- list(det.cov.4.1 = X.p[[4]][, , 2], det.cov.4.2 = X.p[[4]][, , 3], det.cov.4.3 = X.p[[4]][, , 4]) data.list <- list(y = y, occ.covs = occ.covs, det.covs = det.covs, sites = sites, coords = coords) J <- length(dat$z.obs) # Initial values inits.list <- list(alpha = list(0, 0, 0, 0), beta = 0, phi = 3 / .5, sigma.sq = 2, w = rep(0, J), z = rep(1, J)) # Priors prior.list <- list(beta.normal = list(mean = 0, var = 2.72), alpha.normal = list(mean = list(0, 0, 0, 0), var = list(2.72, 2.72, 2.72, 2.72)), phi.unif = c(3/1, 3/.1), sigma.sq.ig = c(2, 2)) # Tuning tuning.list <- list(phi = 0.3) # Number of batches n.batch <- 2 # Batch length batch.length <- 25 # Note that this is just a test case and more iterations/chains may need to # be run to ensure convergence. out <- spIntPGOcc(occ.formula = ~ occ.cov, det.formula = list(f.1 = ~ det.cov.1.1, f.2 = ~ det.cov.2.1 + det.cov.2.2, f.3 = ~ det.cov.3.1, f.4 = ~ det.cov.4.1 + det.cov.4.2 + det.cov.4.3), 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 = FALSE, n.report = 10, n.burn = 10, n.thin = 1) summary(out) ```

spIntPGOcc function