Model-Based Geostatistics Under Spatially Varying Preferential Sampling

André Victor Ribeiro Amaral*, Elias Teixeira Krainski, Ruiman Zhong, Paula Moraga

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

2 Scopus citations


Geostatistics is concerned with the estimation and prediction of spatially continuous phenomena using data obtained at a discrete set of locations. In geostatistics, preferential sampling occurs when these locations are not independent of the latent spatial field, and common modeling approaches that do not account for such a dependence structure might yield wrong inferences. To overcome this issue, some methods have been proposed to model data collected under preferential sampling. However, while these methods assume a constant degree of preferentiality, real data may present a degree of preferentiality that varies over space. For that reason, we propose a new model that accounts for preferential sampling by including a spatially varying coefficient that describes the dependence strength between the process that models the sampling locations and the latent field. To do so, we approximate the preferentiality component by a set of basis functions with the corresponding coefficients being estimated using the integrated nested Laplace approximation (INLA) method. By doing that, we allow the degree of preferentiality to vary over the domain with low computational burden. We assess our model performance by means of a simulation study and use it to analyze the average PM 2.5 levels in the USA in 2022. We conclude that, given enough observed events, our model, along with the implemented inference routine, retrieves well the latent field itself and the spatially varying preferentiality surface, even under misspecified scenarios. Also, we offer guidelines for the specification and size of the set of basis functions. Supplementary materials accompanying this paper appear online.

Bibliographical note

Publisher Copyright:
© 2023, International Biometric Society.


  • Air pollution
  • Geostatistics
  • INLA
  • Preferential sampling
  • Spatial statistics

ASJC Scopus subject areas

  • Statistics and Probability
  • General Environmental Science
  • Agricultural and Biological Sciences (miscellaneous)
  • General Agricultural and Biological Sciences
  • Statistics, Probability and Uncertainty
  • Applied Mathematics


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