# Function for Fitting Multi-Species Multi-Season Spatially-Varying Coefficient Occupancy Models The function `svcTMsPGOcc` fits multi-species multi-season 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. ```r svcTMsPGOcc(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, svc.by.sp, 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`, `det.covs`, `coords`, `range.ind`, and `grid.index`. `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. `coords` is a matrix of the observation coordinates used to estimate the SVCs for each site. `coords` has two columns for the easting and northing coordinate, respectively. Typically, each site in the data set will have it's own coordinate, such that `coords` is a $J x 2$ matrix and `grid.index` should not be specified. If you desire to estimate SVCs at some larger spatial level, e.g., if points fall within grid cells and you want to estimate an SVC for each grid cell instead of each point, `coords` can be specified as the coordinate for each grid cell. In such a case, `grid.index` is an indexing vector of length J, where each value of `grid.index` indicates the corresponding row in `coords` that the given site corresponds to. 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`, `nu`, `sigma.sq.t`, and `rho`. `nu` is only specified if `cov.model = "matern"`, `sigma.sq.t` and `rho` are only specified if `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`, `phi.unif`, `nu.unif`, `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 2.73. 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. If desired, the species-specific occupancy coefficients (`beta`) and/or detection coefficients (`alpha`) can also be estimated indepdendently by specifying the tag `independent.betas = TRUE` and/or `independent.alphas = TRUE`, respectively. If specified, this will not estimate species-specific coefficients as random effects from a common-community-level distribution, and rather the values of `beta.comm`/`alpha.comm` and `tau.sq.beta`/`tau.sq.alpha` will be fixed at the specified initial values. This is equivalent to specifying a Gaussian, independent prior for each of the species-specific effects. 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. `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. - `tuning`: a list with each tag corresponding to a parameter name. Valid tags are `phi`, `nu`, and `rho`. 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). - `svc.by.sp`: an optional list with length equal to `length(svc.cols)`. Each element of the list should be a logical vector of length `N` (number of species) where each element is TRUE, which indicates the SVC should be estimated for that species, or 0, which indicates the SVC should be set to 0 and no SVC for that parameter will be estimated. Note the first `n.factors` SVCs for all spatially-varying coefficients must be set to `TRUE`. By default, SVCs are modeled for all species. - `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. - `ar1`: logical value indicating whether to include an AR(1) zero-mean temporal random effect in the model. If `FALSE`, the model is fit without an AR(1) temporal autocovariance structure. If `TRUE`, an 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](https://repositories.lib.utexas.edu/handle/2152/21842). ## References Doser, J. W., Finley, A. O., Saunders, S. P., Kery, M., Weed, A. S., & Zipkin, E. F. (2024A). Modeling complex species-environment relationships through spatially-varying coefficient occupancy models. Journal of Agricultural, Biological and Environmental Statistics. tools:::Rd_expr_doi("10.1007/s13253-023-00595-6") . Doser, J. W., Kery, M., Saunders, S. P., Finley, A. O., Bateman, B. L., Grand, J., Reault, S., Weed, A. S., & Zipkin, E. F. (2024B). Guidelines for the use of spatially varying coefficients in species distribution models. Global Ecology and Biogeography, 33(4), e13814. tools:::Rd_expr_doi("10.1111/geb.13814") . 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") . 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") . ## Author(s) Jeffrey W. Doser doserjef@msu.edu , Andrew O. Finley finleya@msu.edu ## Returns An object of class `svcTMsPGOcc` 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 and the temporal autocorrelation parameters for each species when `ar1 = TRUE`. - **lambda.samples**: a `coda` object of posterior samples for the latent spatial factor loadings for each spatially-varying coefficient. - **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. - **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 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 ```r # 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 <- TRUE svc.cols <- c(1, 2) p.svc <- length(svc.cols) n.factors <- 3 phi <- runif(p.svc * n.factors, 3 / .9, 3 / .3) factor.model <- TRUE cov.model <- 'exponential' 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, factor.model = factor.model, svc.cols = svc.cols, n.factors = n.factors, phi = phi, sp = sp, cov.model = cov.model) y <- dat$y X <- dat$X X.p <- dat$X.p coords <- dat$coords 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, 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 / .9, b = 3 / .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, phi = 3 / .5, 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 # Note that this is just a test case and more iterations/chains may need to # be run to ensure convergence. out <- svcTMsPGOcc(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, NNGP = TRUE, n.neighbors = 5, n.factors = n.factors, svc.cols = svc.cols, cov.model = 'exponential', priors = prior.list, tuning = tuning.list, n.omp.threads = 1, verbose = TRUE, n.report = 1, n.burn = n.burn, n.thin = n.thin, n.chains = 1) summary(out) ```

svcTMsPGOcc function