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Variation partitioning involving orthogonal spatial eigenfunction submodels.

When partitioning the variation of univariate or multivariate ecological data with respect to several submodels of spatial eigenfunctions (e.g., Moran's eigenvector maps, MEM) acting as explanatory data, a problem occurs: although the submodels are constructed to be orthogonal to one another, the partitioning based on adjusted R2 statistics produces nonzero values in the intersections between spatial submodels. This phenomenon is described and two solutions are proposed. The first solution is to apportion the intersection fractions proportionally to the variation explained by each submodel. The second solution consists in creating a hierarchy among the spatial submodels, in accordance with hierarchy theory. These solutions lead to new partitioning equations that are described in the Appendix. R functions are provided to carry out partitioning with respect to environmental variables and spatial eigenfunction submodels. This development is important for the correct interpretation of spatial modeling results implying explanatory environmental data as well as submodels of spatial eigenfunctions involving two or more spatial scales.