Abstract
In spatial data, especially in geostatistics data where measurements are often provided by satellite scanning, some parts of data may get missed. Due to spatial dependence in the data, these missing values probably are caused by some latent spatial random fields. In this case, ignoring missingness is not logical and may lead to invalid inferences. Thus incorporating the missingness process model into the inferences could improve the results. There are several approaches to take into account the non-ignorable missingness, one of them is the shared parameter model method. In this paper, we extend it for spatial data so that we will have a joint spatial Bayesian shared parameter model. Then the missingness process will be jointly modeled with the measurement process and one or more latent spatial random fields as shared parameters would describe their association. Bayesian inference is implemented by Integrated nested Laplace approximation. A computationally effective approach is applied via a stochastic partial differential equation for approximating latent Gaussian random field. In a simulation study, the proposed spatial joint model is compared with a model that assumes data are missing at random. Based on these two models, the lake surface water temperature data for lake Vänern in Sweden are analyzed. The results of estimation and prediction confirm the efficiency of the spatial joint model.
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Acknowledgements
The authors are thankful to Dr Claire Miller from the School of Mathematics and Statistics, University of Glasgow, for introducing the Arc-Lake project and her guidance about the data set during the research period for this paper. Also, we appreciate Ingelin Steinsland, a professor at the Department of Mathematical Sciences at Norwegian University of Science and Technology and Professor Alfred Stein at the Department of Earth Observation Science at the University of Twente for their helpful guidance. Receiving support from the Center of Excellence in Analysis of Spatio-Temporal Correlated Data at Tarbiat Modares University is also acknowledged.
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Zahmatkesh, S., Mohammadzadeh, M. Bayesian prediction of spatial data with non-ignorable missingness. Stat Papers 62, 2247–2268 (2021). https://doi.org/10.1007/s00362-020-01186-0
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DOI: https://doi.org/10.1007/s00362-020-01186-0