Entropy Analysis of Multiple Scale Causality and Qualitative Causal Shifts in Spatial Systems
Spatial systems are typically characterized by multiple controlling factors and processes operating at different spatial and temporal scales (multiple scale causality [MSC]). An entropy decomposition-based approach to MSC is presented here in two contexts. First, given maps or distributions of an observed phenomenon at two or more scales, the contribution at more local or global (relative to the primary scale of observation) controls to the observed entropy can be estimated. Second, a theoretical treatment of the entropy decomposition equations shows that as the range of scale is increased by broadening or narrowing resolutions or by incorporating more controls, the influence of larger or smaller-scale influences not only changes, but may change qualitatively, e.g., in terms of having positive (entropy-increasing) or negative (information-increasing) effects. Such qualitative causal shifts have implications for efforts to use any single causal explanation across the molecular to planetary spatial and instantaneous to geological range of scales relevant to physical geography. The entropy decomposition method is illustrated with an application to soil landscapes in the Ouachita Mountains, Arkansas.