Indexing and partitioning the spatial linear model for large data sets.

We consider four main goals when fitting spatial linear models: 1) estimating covariance parameters, 2) estimating fixed effects, 3) kriging (making point predictions), and 4) block-kriging (predicting the average value over a region). Each of these goals can present different challenges when analyzing large spatial data sets. Current research uses a variety of methods, including spatial basis functions (reduced rank), covariance tapering, etc, to achieve these goals. However, spatial indexing, which is very similar to composite likelihood, offers some advantages. We develop a simple framework for all four goals listed above by using indexing to create a block covariance structure and nearest-neighbor predictions while maintaining a coherent linear model. We show exact inference for fixed effects under this block covariance construction. Spatial indexing is very fast, and simulations are used to validate methods and compare to another popular method. We study various sample designs for indexing and our simulations showed that indexing leading to spatially compact partitions are best over a range of sample sizes, autocorrelation values, and generating processes. Partitions can be kept small, on the order of 50 samples per partition. We use nearest-neighbors for kriging and block kriging, finding that 50 nearest-neighbors is sufficient. In all cases, confidence intervals for fixed effects, and prediction intervals for (block) kriging, have appropriate coverage. Some advantages of spatial indexing are that it is available for any valid covariance matrix, can take advantage of parallel computing, and easily extends to non-Euclidean topologies, such as stream networks. We use stream networks to show how spatial indexing can achieve all four goals, listed above, for very large data sets, in a matter of minutes, rather than days, for an example data set.

An Application of Spatio-temporal Modeling to Finite Population Abundance Prediction.

Spatio-temporal models can be used to analyze data collected at various spatial locations throughout multiple time points. However, even with a finite number of spatial locations, there may be a lack of resources to collect data from every spatial location at every time point. We develop a spatio-temporal finite-population block kriging (ST-FPBK) method to predict a quantity of interest, such as a mean or total, across a finite number of spatial locations. This ST-FPBK predictor incorporates an appropriate variance reduction for sampling from a finite population. Through an application to moose surveys in the east-central region of Alaska, we show that the predictor has a substantially smaller standard error compared to a predictor from the purely spatial model that is currently used to analyze moose surveys in the region. We also show how the model can be used to forecast a prediction for abundance in a time point for which spatial locations have not yet been surveyed. A separate simulation study shows that the spatio-temporal predictor is unbiased and that prediction intervals from the ST-FPBK predictor attain appropriate coverage. For ecological monitoring surveys completed with some regularity through time, use of ST-FPBK could improve precision. We also give an R package that ecologists and resource managers could use to incorporate data from past surveys in predicting a quantity from a current survey.

sptotal: an R package for predicting totals and weighted sums from spatial data.

In ecological or environmental surveys, it is often desired to predict the mean or total of a variable in some finite region. However, because of time and money constraints, sampling the entire region is often unfeasible. The purpose of the sptotal R package is to provide software that gives a prediction for a quantity of interest, such as a total, and an associated standard error for the prediction. The predictor, referred to as the Finite-Population-Block-Kriging (FPBK) predictor in the literature (J. M. Ver Hoef, 2008), incorporates possible spatial correlation in the data and also incorporates an appropriate variance reduction for sampling from a finite population. In the remainder of the paper, we give an overview of both the background of the method and of the sptotal package.

spmodel: Spatial statistical modeling and prediction in [Formula: see text].

spmodel is an [Formula: see text] package used to fit, summarize, and predict for a variety spatial statistical models applied to point-referenced or areal (lattice) data. Parameters are estimated using various methods, including likelihood-based optimization and weighted least squares based on variograms. Additional modeling features include anisotropy, non-spatial random effects, partition factors, big data approaches, and more. Model-fit statistics are used to summarize, visualize, and compare models. Predictions at unobserved locations are readily obtainable.

A comparison of design-based and model-based approaches for finite population spatial sampling and inference.

The design-based and model-based approaches to frequentist statistical inference rest on fundamentally different foundations. In the design-based approach, inference relies on random sampling. In the model-based approach, inference relies on distributional assumptions. We compare the approaches in a finite population spatial context.We provide relevant background for the design-based and model-based approaches and then study their performance using simulated data and real data. The real data is from the United States Environmental Protection Agency's 2012 National Lakes Assessment. A variety of sample sizes, location layouts, dependence structures, and response types are considered. The population mean is the parameter of interest, and performance is measured using statistics like bias, squared error, and interval coverage.When studying the simulated and real data, we found that regardless of the strength of spatial dependence in the data, the generalized random tessellation stratified (GRTS) algorithm, which explicitly incorporates spatial locations into sampling, tends to outperform the simple random sampling (SRS) algorithm, which does not explicitly incorporate spatial locations into sampling. We also found that model-based inference tends to outperform design-based inference, even for skewed data where the model-based distributional assumptions are violated. The performance gap between design-based inference and model-based inference is small when GRTS samples are used but large when SRS samples are used, suggesting that the sampling choice (whether to use GRTS or SRS) is most important when performing design-based inference.There are many benefits and drawbacks to the design-based and model-based approaches for finite population spatial sampling and inference that practitioners must consider when choosing between them. We provide relevant background contextualizing each approach and study their properties in a variety of scenarios, making recommendations for use based on the practitioner's goals.