Smoothing spatio-temporal data with complex missing data patterns

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