Measuring, mapping, and uncertainty quantification in the space-time cube

Abstract

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, $$\{Y_0(\mathbf{s} _0),Y_1(\mathbf{s} _0),\ldots \}$$ . 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.