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

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 >= 0}. Now fill the cube with a spatio-temporal stochastic process {Y-t (s) : s is an element of D-t, t >= 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-t0 (s) : s is an element of D-t0}. Slicing the cube longitudinally for a fixed s(0) generates a temporal process {Y-t (s(0)) : t >= 0} that, after dicing, yields a time series, {Y-0(s(0)), Y-1(s(0)),...}. 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.