Statistical Modeling for Spatio-Temporal Data from Physical Convection-Diffusion Processes
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This paper proposes a statistical modeling approach for spatio-temporal data arising from a generic class of convection-diffusion processes. Such processes are found in various scientific and engineering domains where fundamental physics imposes critical constraints on how data can be modeled and how statistical models should be interpreted. We employ the idea of spectrum decomposition to approximate the physical processes. However, unlike existing models which often assume constant convection-diffusion and zero-mean source-sink, we consider a more realistic scenario with spatially-varying convection-diffusion and nonzero-mean source-sink. As a result, the temporal evolution of spectrum coefficients is closely coupled with each other, which can be seen as the non-linear transfer or redistribution of energy across multiple scales. Because of the spatially-varying convection-diffusion, the space-time covariance is naturally non-stationary in space. A systematic approach is proposed to integrate the theoretical results into the framework of hierarchical dynamical spatio-temporal models. Statistical inference using computationally efficient algorithms is investigated. Some practical considerations related to computational efficiency are discussed in order to make the proposed approach practical. The advantages of the proposed approach are demonstrated by numerical examples, a case study, and comparison studies. Computer code and data are made available.
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