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Measuring, mapping, and uncertainty quantification in the space-time cube. (English) Zbl 1451.62106

Summary: The space-time cube is not a cube of course, but the idea of one is useful. Its base is a spatial domain, \(D_t\), and the “cube” is traced out by a process of spatial domains, \(\{D_t : t\ge 0\}\). Now fill the cube with a spatio-temporal stochastic process \(\{Y_t(\mathbf{s}) : \mathbf{s} \in D_t, \,t\ge 0\}\). Assume that \(\{D_t\}\) is fixed and known (but clearly it too could be stochastic). Slicing the cube laterally for a fixed \(t_0\) generates a spatial stochastic process \(\{Y_{t_0}(\mathbf{s}) : \mathbf{s} \in D_{t_0}\}\). Slicing the cube longitudinally for a fixed \(\mathbf{s}_0\) generates a temporal process \(\{Y_t(\mathbf{s}_0) : t\ge 0\}\) that, after dicing, yields a time series, \(\{Y_0(\mathbf{s}_0), Y_1(\mathbf{s}_0), \dots\}\). These are the main highways that traverse the cube but other, less-traveled paths, can be taken. In this paper, we discuss spatio-temporal data and processes whose domain is the space-time cube, and we incorporate them into hierarchical statistical models for spatio-temporal data.

MSC:

62M30 Inference from spatial processes
62A01 Foundations and philosophical topics in statistics
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
62P12 Applications of statistics to environmental and related topics
83A05 Special relativity
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