A generalized variogram-based framework for multiscale ordination

Multi-scale ordination (MSO) deals with potential scale-dependence in species assemblages, by studying how results from multivariate ordination may be different on 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, applicable to very diverse sampling designs has recently been introduced by H.H. Wagner (2003, 2004). This refers to Principal Component 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 which 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 singleand 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 of available environmental variables. Some aspects of the approach are briefly illustrated with vegetation data from a neotropical rainforest in French Guiana.