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

Abstract

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.

Keywords: forecast; kriging; resource monitoring; spatial; temporal; total.

Fig. 1:

A map of the sites composing the Taylor corridor in eastern-central Alaska. Each site is roughly 4 kilometers in length and roughly 3.5 kilometers in width so that the centroids of two horizontally adjacent sites are about 4 kilometers apart and the centroids of two vertically adjacent sites are about 3.5 kilometers apart. (a). A map of the stratification for the sites in the year 2020. (b). A map of the predictions of sites in 2020 from the spatio-temporal model. A site with a grey dot in the center means that the site was sampled in 2020.

Fig. 2:

Estimated covariance of the errors from the estimated parameters in a spatio-temporal product-sum model. Distance between two sites is calculated from the site centroids; the centroids of two sites directly adjacent to one another are about 3.5 to 4 kilometers apart.

Fig. 3:

Moose abundance predictions for the Taylor Corridor from 2014 through 2021 with the stratified random sampling (StRS) estimator, the spatial FPBK predictor, and the ST-FPBK predictor. Predictions are given with a diamond symbol; the bars surrounding each prediction are standard error bars. Because surveys were not conducted in 2016 and 2021, there is no StRS estimator or spatial FPBK predictor for those years. Also, the standard errors for the ST-FPBK predictor for those years is larger than the standard errors in the other years. The stratification scheme used for 2016 and 2021 in the ST-FPBK analysis was the same scheme used in 2015 and 2020, respectively.

Fig. 4:

The model covariance used in the simulations for the spatio-temporal scenario. Covariance is approximately 0 for errors from data points that are $\sqrt{2}$ distance units apart in space and 1 distance unit apart in time. The spatial dependent error variance $\left({\sigma }_{\delta }^2\right)$, spatial independent error variance $\left({\sigma }_{\gamma }^2\right)$, temporal dependent error variance $\left({\sigma }_{\tau }^2\right)$, and temporal independent error variance $\left({\sigma }_{\eta }^2\right)$ are shown with grey lines.

Fig. 5:

root-mean-squared-prediction-error (rMSPE) for all simulation settings. The ST-FPBK predictor has the smallest rMSPE in all of the ‘all-dev’ and ‘spt-iev’ scenarios while the three methods perform similarly in all of the ‘t-iev’ scenarios.