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Graphical models have witnessed significant growth and usage in spatial data science for modeling data referenced over a massive number of spatial-temporal coordinates. Much of this literature has focused on a single or relatively few spatially dependent outcomes. Recent attention has focused upon addressing modeling and inference for substantially large number of outcomes. While spatial factor models and multivariate basis expansions occupy a prominent place in this domain, this article elucidates a recent approach, graphical Gaussian Processes, that exploits the notion of conditional independence among a very large number of spatial processes to build scalable graphical models for fully model-based Bayesian analysis of multivariate spatial data. 

Spatial probit generalized linear mixed models (spGLMM) with a linear fixed effect and a spatial random effect, endowed with a Gaussian Process prior, are widely used for analysis of binary spatial data. However, the canonical Bayesian implementation of this hierarchical mixed model can involve protracted Markov Chain Monte Carlo sampling. Alternate approaches have been proposed that circumvent this by directly representing the marginal likelihood from spGLMM in terms of multivariate normal cummulative distribution functions (cdf). We present a direct and fast rendition of this latter approach for predictions from a spatial probit linear mixed model. We show that the covariance matrix of the cdf characterizing the marginal cdf of binary spatial data from spGLMM is amenable to approximation using Nearest Neighbor Gaussian Processes (NNGP). This facilitates a scalable prediction algorithm for spGLMM using NNGP that only involves sparse or small matrix computations and can be deployed in an embarrassingly parallel manner. We demonstrate the accuracy and scalability of the algorithm via numerous simulation experiments and an analysis of species presence-absence data. 

Community-based public health interventions often rely on representative, spatially referenced outcome data to draw conclusions about a finite population. To estimate finite-population parameters, we are posed with two challenges: to correctly account for spatial association among the sampled and nonsampled participants and to correctly model missingness in key covariates, which may be also spatially associated. To accomplish this, we take inspiration from the preferential sampling literature and develop a general Bayesian framework that can specifically account for preferential non-response. This framework is first applied to three missing data scenarios in a simulation study. It is then used to account for missing data patterns seen in reported annual household income in a corner-store intervention project. Through this, we are able to construct finite-population estimates of the percent of income spent on fruits and vegetables. Such a framework provides a flexible way to account for spatial association and complex missing data structures in finite populations.

 

Determining the spatial distributions of species and communities is a key task in ecology and conservation efforts. Joint species distribution models are a fundamental tool in community ecology that use multi‐species detection–nondetection data to estimate species distributions and biodiversity metrics. The analysis of such data is complicated by residual correlations between species, imperfect detection, and spatial autocorrelation. While many methods exist to accommodate each of these complexities, there are few examples in the literature that address and explore all three complexities simultaneously. Here we developed a spatial factor multi‐species occupancy model to explicitly account for species correlations, imperfect detection, and spatial autocorrelation. The proposed model uses a spatial factor dimension reduction approach and Nearest Neighbor Gaussian Processes to ensure computational efficiency for data sets with both a large number of species (e.g., >100) and spatial locations (e.g., 100,000). We compared the proposed model performance to five alternative models, each addressing a subset of the three complexities. We implemented the proposed and alternative models in thespOccupancysoftware, designed to facilitate application via an accessible, well documented, and open‐source R package. Using simulations, we found that ignoring the three complexities when present leads to inferior model predictive performance, and the impacts of failing to account for one or more complexities will depend on the objectives of a given study. Using a case study on 98 bird species across the continental US, the spatial factor multi‐species occupancy model had the highest predictive performance among the alternative models. Our proposed framework, together with its implementation inspOccupancy, serves as a user‐friendly tool to understand spatial variation in species distributions and biodiversity while addressing common complexities in multi‐species detection–nondetection data.

 

Regional aggregates of health outcomes over delineated administrative units (e.g., states, counties, and zip codes), or areal units, are widely used by epidemiologists to map mortality or incidence rates and capture geographic variation. To capture health disparities over regions, we seek “difference boundaries” that separate neighboring regions with significantly different spatial effects. Matters are more challenging with multiple outcomes over each unit, where we capture dependence among diseases as well as across the areal units. Here, we address multivariate difference boundary detection for correlated diseases. We formulate the problem in terms of Bayesian pairwise multiple comparisons and seek the posterior probabilities of neighboring spatial effects being different. To achieve this, we endow the spatial random effects with a discrete probability law using a class of multivariate areally referenced Dirichlet process models that accommodate spatial and interdisease dependence. We evaluate our method through simulation studies and detect difference boundaries for multiple cancers using data from the Surveillance, Epidemiology, and End Results Program of the National Cancer Institute.