Bayesian prediction of spatial data with non-ignorable missingness
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Bayesian prediction of spatial data with non-ignorable missingness Samira Zahmatkesh1 · Mohsen Mohammadzadeh1 Received: 5 September 2019 / Revised: 13 May 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
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. Keywords Missing values · Spatial data · Joint model · INLA · SPDE
1 Introduction Most of the spatial data sets, the data with spatial references, due to conditions in which measurements are performed contain missing values. According to Rubin (1976) missing data mechanisms are classified into three types: missing completely at random (MCAR), missing at random (MAR) and missing not at random (MNAR). MCAR
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Mohsen Mohammadzadeh [email protected] Department of Statistics, Faculty of Mathematical Sciences, Tarbiat Modares University, Tehran, Iran
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S. Zahmatkesh, M. Mohammadzadeh
occurs when the missingness process does not depend on the observed and unobserved data. MAR is a less strong assumption in which the missingness process depends only on the observed data. In these two cases missing data is said to be ignorable so that the Bayesian and likelihood inferences based on the observed data are valid Little and Rubin (2002). When neither MCAR nor MAR holds the data contain MNAR, i.e., the missingness process is non-ignorable. In this case, the conventional methods that are effective for the ignorable case would produce biased inferences and could not reflect statistical uncertainty. Unfortunately, the missing data mechanism could not be specified from the data set. When there is doubt that the missingness is not at random, under MNAR assumption, the missingness process can be modelled
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