Smoothing spatio-temporal data with complex missing data patterns

We consider spatio-temporal data and functional data with spatial dependence, characterized by complicated missing data patterns. We propose a new method capable to efficiently handle these data structures, including the case where data are missing over large portions of the spatio-temporal domain. The method is based on regression with partial differential equation regularization. The proposed model can accurately deal with data scattered over domains with irregular shapes and can accurately estimate fields exhibiting complicated local features. We demonstrate the consistency and asymptotic normality of the estimators. Moreover, we illustrate the good performances of the method in simulations studies, considering different missing data scenarios, from sparse data to more challenging scenarios where the data are missing over large portions of the spatial and temporal domains and the missing data are clustered in space and/or in time. The proposed method is compared to competing techniques, considering predictive accuracy and uncertainty quantification measures. Finally, we show an application to the analysis of lake surface water temperature data, that further illustrates the ability of the method to handle data featuring complicated patterns of missingness and highlights its potentiality for environmental studies.