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 insufficient 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. Supplementary materials accompanying this paper appear on-line.

Appendix

1.1 A.1: Simulation Tables

See Tables 3, 4 and 5.

Table 3 Root-mean-squared-prediction-error (rMSPE) for the ST-FPBK predictor, the FPBK predictor, and the SRS estimator for each of the 18 simulation settings

Table 4 Bias (realized current total − predicted current total) for the ST-FPBK predictor, the FPBK predictor, and the SRS estimator for each of the 18 simulation settings

Table 5 Prediction interval coverage for the ST-FPBK predictor, the FPBK predictor, and the SRS for each of the 18 simulation settings

1.2 A.2: Supplementary Analysis

As mentioned in Sect. 3, moose surveys in Alaska are often stratified into “High” and “Low” sites. When using stratum as a covariate in a spatio-temporal (or spatial, if performing a purely spatial analysis) model, we assume that all errors in the model are generated from the same underlying spatio-temporal (or spatial) parameters. However, for many moose surveys, it is more reasonable to allow the sites in the High stratum to have a different set of spatio-temporal (or spatial) parameters than the sites in the Low stratum.

If we allow the strata to have different covariance parameters, then, to construct the ST-FPBK predictor, we simply fit the model once for each stratum. If we assume that there is no cross-covariance (i.e. errors from sites in different strata are not correlated), then the BLUP for $\textbf{b}'_a \textbf{y}_a$ is

$$\begin{aligned} \widehat{\textbf{b}'_a \textbf{y}_a} = \varvec{\lambda }_{o, l}' \textbf{y}_{o, l} + \varvec{\lambda }_{o, h}' \textbf{y}_{o, h}, \end{aligned}$$

(16)

where $\varvec{\lambda }_{o, l}$ and $\varvec{\lambda }_{o, h}'$ are the kriging weights for the Low and High strata, respectively (Eq. 13), and $\textbf{y}_{o, l}$ and $\textbf{y}_{o, h}$ are the vectors of observed responses for the Low and High strata, respectively.

Again assuming that there is no cross-covariance, the prediction variance is simply the sum of the prediction variances of $\varvec{\lambda }_{o, l}' \textbf{y}_{o, l}$ and $\varvec{\lambda }_{o, h}' \textbf{y}_{o, h}$ using Eq. 14.

We can use the purely spatial model and FPBK as well as the spatio-temporal model and ST-FPBK to predict the total moose abundance in 2020, using separate covariance models for the strata in the moose data set in Sect. 3. Table 6 shows the results.

Table 6 Prediction and standard error for total abundance in 2020 using a model that allows errors in separate strata to be modeled with different covariance parameters

The spatio-temporal predictors still have a smaller standard error than their purely spatial model counterparts. Interestingly, the purely spatial FPBK predictor has a slightly lower standard error when fitting strata separately, while the ST-FPBK predictor has a slightly lower standard error when using stratum as a covariate. Whether it makes more sense for stratum to be a covariate or for the strata to be fit separately is application dependent.

For the moose application data, fitting separate covariance models to each stratum is probably the better choice, as the errors for sites in the high stratum have much more overall variability than the errors in the low stratum. However, we chose to have the separate-strata model in the supplementary materials for two reasons. First, the method can be applied to any data set with spatio-temporal covariance and a finite number of sites, and applications in other domains may not have stratification at all. Second, the syntax in the development of the ST-FPBK predictor is much cleaner when stratum is treated as a covariate than when the strata are fit separately. Using the model with stratum as a covariate allows for a better focus on the proposed method itself.