Scaling spatial predictability: An approach to multi‐resolution modeling

We have investigated the dependence of spatial predictability , a statistical measure of the reduction in uncertainty about one spatial variable that can be gained by knowledge of another, on the spatial resolution ( R s ) of the variables. While increasing resolution provides more descriptive information about the patterns in data, it also increases the difficulty of accurately modeling those patterns. By examining the variation of spatial predictability with R s in a number of case studies, we have proposed the existence of an “optimal” R s for specific studies, which balances these two factors. We analyzed land‐use data by resampling map data sets at several different spatial resolutions and measuring predictability at each. Spatial auto‐predictability ( P a ) is the reduction in uncertainty about the state of a cell in a map given knowledge of the state of adjacent cells in that map, and spatial cross‐predictability ( P c ) is the reduction in uncertainty about the state of a cell in a map given knowledge of the state of corresponding cells in other maps. The P a is a measure of the internal pattern in the data, whereas P c is a measure of the ability of some “model” to represent the transition from one map to another. We found a strong linear relationship between the log of P d and the log of R s (measured as the number of cells per square kilometer). While P a generally increases with increasing R s (because more information is being included), P c generally falls or remains stable (because it is easier to model aggregate results than fine‐grained ones). Thus, one can define an “optimal” R s a particular modeling problem that balances the benefit in terms of increasing data predictability (measured by P a ) as one increases resolution, with the cost of decreasing facility of modeling the temporal dynamics (measured by P c ).