A generalized, variogram-based framework for multi-scale ordination

Multi-scale ordination (MSO) deals with potential scale dependence in species assemblages by studying how results from multivariate ordination may be different at different spatial scales. MSO methods were initially based on two-term local covariances between species and, therefore, required sampling designs composed of adjacent quadrats. A variogram-based MSO, recently introduced by H. H. Wagner, is applicable to very diverse sampling designs and for use with principal-components analysis, correspondence analysis, and derived, two-table (also called direct) ordination methods, i.e., redundancy analysis and canonical correspondence analysis. In this paper we put forward an enlarged framework for variogram-based MSO that relies on a generalized definition of inter-species covariance and on matrix expression of spatial contiguity between sampling units. This enables us to provide distance-explicit decompositions of variances and covariances (in their generalized meaning) that are consistent with many ordination methods in both their single- and two-table versions. A spatially explicit apportioning of diversity indices is proposed for some particular definitions of variance. Referring to two-table ordination methods allowed the multi-scale study of residual spatial patterns after factoring out available environmental variables. Some aspects of the approach are briefly illustrated with vegetation data from a Neotropical rain forest in French Guiana.