Hutchinson’s duality is the correspondence of points between an input space (e.g. biotope, geographic space) and a feature space (e.g. niche space, environmental space) where variables are axes and measures are coordinates. The outcome of duality is that input points and mapped-back points can display disparate patterns in geographic space (G-space), and it remains unexplored how it methodologically operates in a point-to-point perspective. Herein I describe the Hutchinson’s duality framework on theoretical and empirical multi-scale data.
Results/Conclusions
Duality framework is based on three proprieties: 1) reciprocity between spaces; 2) twofold states of environmental space (E-space); and 3) heterogeneity of environmental variables layers. Variables with small range size (more homogeneous variables) in G-space generate high density of points in E-space in a reciprocal (y ∝ 1/x) relationship. The mechanism is based on independent partition and supports that reciprocity will take place for n-dimensions. E-space has twofold states, and the manifestation of each state depends on the correspondence direction. The inflow correspondence from G-space points into frequency E-space is always one-to-one. When mapping-back correspondence is done, E-space manifests a state of topological space where there is only the point position information, then correspondence is one-to-many; except for data layers with no repeating values (more heterogeneous variables). Duality and correlated features may have consequences on choosing variables for ecological niche modeling as well as for transferability between two geographical extents with different environmental heterogeneities.