Bayesian design for minimizing prediction uncertainty in bivariate spatial responses with applications to air quality monitoring

Abstract

Model‐based geostatistical design involves the selection of locations to collect data to minimize an expected loss function over a set of all possible locations. The loss function is specified to reflect the aim of data collection, which, for geostatistical studies, could be to minimize the prediction uncertainty at unobserved locations. In this paper, we propose a new approach to design such studies via a loss function derived through considering the entropy about the model predictions and the parameters of the model. The approach includes a multivariate extension to generalized linear spatial models, and thus can be used to design experiments with more than one response. Unfortunately, evaluating our proposed loss function is computationally expensive so we provide an approximation such that our approach can be adopted to design realistically sized geostatistical studies. This is demonstrated through a simulated study and through designing an air quality monitoring program in Queensland, Australia. The results show that our designs remain highly efficient in achieving each experimental objective individually, providing an ideal compromise between the two objectives. Accordingly, we advocate that our approach could be adopted more generally in model‐based geostatistical design.

Prediction and Assessment of Stream Water Quality by Considering Multiple Water Quality Parameters

“…For instance, when dealing with observations that involve both continuous and binary data in a bivariate context, relying on a multivariate normal approximation becomes uncertain for estimating the covariance structure accurately. As such, further work is required to explore alternative methods such as Copula based models for mixed outcomes [34]. In studies where high-dimensional multivariate data are observed, it could be important to use certain Vine-Copulas [35].…”

Bayesian spatial optimal network design for skew Gaussian environmental processes

“…To reduce the uncertainty about responses as well as uncertainty about model parameters or predictions, Caselton and Zidek (1984), Guttorp et al (1993), Zidek et al (2000), and Fuentes et al (2007) developed the maximum entropy theory. In several studies, other aspects of the network design have been considered, including multivariate responses (Li and Zimmerman, 2015), a new entropy-based loss function (Senarathne et al, 2020), and an efficient algorithm for optimization (Leung et al, 2020). It should be noted that some of these studies implement distribution-free methods, e.g.…”