Weighted D-optimal design for spatial regression model in the presence of heteroscedasticity

Abstract

In experimental designs, classical D-optimal design methods rely on homoscedastic and independent error assumptions. In this study, a spatial regression model with heteroscedastic errors was considered and proposed using weighted D-optimal designs as a result. The study proposed a weighted D-optimal design framework that explicitly incorporates spatial epistemic uncertainty derived from an Integrated Nested Laplace Approximation -- Stochastic Partial Differential Equation (INLA--SPDE) model. Spatial dependence through a latent Gaussian field represented by the SPDE approach. Epistemic uncertainty is measured using the posterior variance of the spatial field and is integrated into the D-optimality criterion through location-specific weights. This weighting scheme prioritizes candidate locations with higher predictive uncertainty, thereby targeting regions where additional observations are expected to yield the greatest information gain. The proposed method is applied to spatial data on malaria prevalence among children under 5 years in Nigeria. Results showed substantial spatial heterogeneity in predictive variance, with higher uncertainty concentrated in regions with sparse observations. The weighted D-optimal design allocates sampling points preferentially to these high-uncertainty areas, in contrast to classical D-optimal designs that tend to favour well-sampled regions. Ranking and priority analyses further confirm the internal consistency and transparency of the proposed approach. Overall, the study demonstrates that incorporating spatial epistemic uncertainty into D-optimal design leads to more informative and efficient sampling strategies for spatial regression models under heteroscedasticity and non-normal errors.