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.