Lagrangian Spatio-Temporal Nonstationary Covariance Functions

The Lagrangian reference frame has been used to model spatio-temporal dependence of purely spatial second-order stationary random fields that are being transported. This modeling paradigm involves transforming a purely spatial process to spatio-temporal by introducing a transformation in the spatial coordinates. Recently, it has been used to capture dependence in space and time of transported purely spatial random fields with second-order nonstationarity. However, under this modeling framework, the presence of mechanisms enforcing second-order nonstationary behavior introduces considerable challenges in parameter estimation. To address these, we propose a new estimation methodology which includes modeling the second-order nonstationarity parameters by means of thin plate splines and estimating all the parameters via two-step maximum likelihood estimation. In addition, through numerical experiments, we tackle the consequences of model misspecification. That is, we discuss the implications, both in the stationary and nonstationary cases, of fitting Lagrangian spatio-temporal covariance functions to data generated from non-Lagrangian models, and vice versa. Lastly, we apply the Lagrangian models and the new estimation technique to analyze particulate matter concentrations over Saudi Arabia.