Measuring, mapping, and uncertainty quantification in the space-time cube
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Measuring, mapping, and uncertainty quantification in the space-time cube Noel Cressie1
· Christopher K. Wikle2
Received: 12 February 2020 / Accepted: 4 May 2020 © Universidad Complutense de Madrid 2020
Abstract The space-time cube is not a cube of course, but the idea of one is useful. Its base is a spatial domain, Dt , and the “cube” is traced out by a process of spatial domains, {Dt : t ≥ 0}. Now fill the cube with a spatio-temporal stochastic process {Yt (s) : s ∈ Dt , t ≥ 0}. Assume that {Dt } is fixed and known (but clearly it too could be stochastic). Slicing the cube laterally for a fixed t0 generates a spatial stochastic process {Yt0 (s) : s ∈ Dt0 }. Slicing the cube longitudinally for a fixed s0 generates a temporal process {Yt (s0 ) : t ≥ 0} that, after dicing, yields a time series, {Y0 (s0 ), Y1 (s0 ), . . .}. 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. Keywords Change-of-support · Uncertainty quantification · Space-time interaction · Hierarchical statistical model · Deep neural models · Stochastic PDE Mathematics Subject Classification Primary 62M30; Secondary 62P12
1 Introduction This paper gives a philosophical and etiological discussion of spatio-temporal statistics. We draw on ideas presented in our two recent books on the topic [10,36]. Causation is the “holy grail” of Science, and hence to infer cause-effect relationships (i.e., “why”) it is essential to keep track of “when,” since a cause always precedes an effect. Knowing “where” recognizes the importance of knowing the “lie of the land” and the multi-dimensional world in which we live. Hence, in order to answer the
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Noel Cressie [email protected]
1
University of Wollongong, Wollongong, Australia
2
University of Missouri, Columbia, USA
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N. Cressie, C. K. Wikle
“why” question, science should address the“where” (spatial) and “when” (temporal) questions. Data are fundamental to the advancement of science, and data sets indexed by space (where) and time (when) allow us to climb higher up the knowledge pyramid than if one or both of these indices were missing. Purely spatial data that do not have a temporal dimension can occur for example when data come from a “snapshot” in time (e.g., liver-cancer rates in USA counties in 2019), or they are taken from a process that is not evolving in time (e.g., an iron-ore body in the Pilbara region of Australia). Sometimes the temporal component has simply been discarded and, at worst, the same may have happened to the spatial component as well. Purely temporal data sets are not unusual either. For example, two time series of monthly average carbon dioxide (CO2 ) measurements, one from the Mauna Loa Observatory, Hawaii, and the other from a global average of CO2 , do not have a spatial dimension (for different reasons). Like the temporal component, sometimes t
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