We introduce a class of scalable Bayesian hierarchical models for the analysis of massive geostatistical datasets. The underlying idea combines ideas on high-dimensional geostatistics by partitioning the spatial domain and modeling the regions in the partition using a sparsity-inducing directed acyclic graph (DAG). We extend the model over the DAG to a well-defined spatial process, which we call the meshed Gaussian process (MGP). A major contribution is the development of an MGPs on tessellated domains, accompanied by a Gibbs sampler for the efficient recovery of spatial random effects. The source code is available at github.com/mkln/meshgp. 

Joint modeling of spatially-oriented dependent variables are commonplace in the environmentalsciences, where scientists seek to estimate the relationships among a set of environmental outcomesaccounting for dependence among these outcomes and the spatial dependence for each outcome. Suchmodeling is now sought for very large data sets where the variables have been measured at a very largenumber of locations. Bayesian inference, while attractive for accommodating uncertainties through theirhierarchical structures, can become computationally onerous for modeling massive spatial data sets becauseof their reliance on iterative estimation algorithms. This manuscript develops a conjugate Bayesianframework for analyzing multivariate spatial data using analytically tractable posterior distributions thatdo not require iterative algorithms. We discuss differences between modeling the multivariate responseitself as a spatial process and that of modeling a latent process. We illustrate the computational andinferential benefits of these models using simulation studies and real data analyses for a Vege Indicesdataset with observed locations numbering in the millions.

Issue

Geodiversity (i.e., the variation in Earth's abiotic processes and features) has strong effects on biodiversity patterns. However, major gaps remain in our understanding of how relationships between biodiversity and geodiversity vary over space and time. Biodiversity data are globally sparse and concentrated in particular regions. In contrast, many forms of geodiversity can be measured continuously across the globe with satellite remote sensing. Satellite remote sensing directly measures environmental variables with grain sizes as small as tens of metres and can therefore elucidate biodiversity-geodiversity relationships across scales.

Evidence

We show how one important geodiversity variable, elevation, relates to alpha, beta and gamma taxonomic diversity of trees across spatial scales. We use elevation from NASA's Shuttle Radar Topography Mission (SRTM) and c. 16,000 Forest Inventory and Analysis plots to quantify spatial scaling relationships between biodiversity and geodiversity with generalized linear models (for alpha and gamma diversity) and beta regression (for beta diversity) across five spatial grains ranging from 5 to 100 km. We illustrate different relationships depending on the form of diversity; beta and gamma diversity show the strongest relationship with variation in elevation.

Conclusion

With the onset of climate change, it is more important than ever to examine geodiversity for its potential to foster biodiversity. Widely available satellite remotely sensed geodiversity data offer an important and expanding suite of measurements for understanding and predicting changes in different forms of biodiversity across scales. Interdisciplinary research teams spanning biodiversity, geoscience and remote sensing are well poised to advance understanding of biodiversity-geodiversity relationships across scales and guide the conservation of nature.

Aim: We may be able to buffer biodiversity against the effects of ongoing climate change by prioritizing the protection of habitat with diverse physical features (high geodiversity) associated with ecological and evolutionary mechanisms that maintain high biodiversity. Nonetheless, the relationships between biodiversity and habitat vary with spatial and biological context. In this study, we compare how well habitat geodiversity (spatial variation in abiotic processes and features) and climate explain biodiversity patterns of birds and trees. We also evaluate the consistency of biodiversity–geodiversity relationships across ecoregions. Location: Contiguous USA. Time period: 2007–2016. Taxa studied: Birds and trees. Methods: We quantified geodiversity with remotely sensed data and generated biodiversity maps from the Forest Inventory and Analysis and Breeding Bird Survey datasets. We fitted multivariate regressions to alpha, beta and gamma diversity, accounting for spatial autocorrelation among Nature Conservancy ecoregions and relationships among taxonomic, phylogenetic and functional biodiversity. We fitted models including climate alone (temperature and precipitation), geodiversity alone (topography, soil and geology) and climate plus geodiversity. Results: A combination of geodiversity and climate predictor variables fitted most forms of bird and tree biodiversity with < 10% relative error. Models using geodiversity and climate performed better for local (alpha) and regional (gamma) diversity than for turnover-based (beta) diversity. Among geodiversity predictors, variability of elevation fitted biodiversity best; interestingly, topographically diverse places tended to have higher tree diversity but lower bird diversity. Main conclusions: Although climatic predictors tended to have larger individual effects than geodiversity, adding geodiversity improved climate-only models of biodiversity. Geodiversity was correlated with biodiversity more consistently than with climate across ecoregions, but models tended to have a poor fit in ecoregions held out of the training dataset. Patterns of geodiversity could help to prioritize conservation efforts within ecoregions. However, we need to understand the underlying mechanisms more fully before we can build models transferable across ecoregions.