A GENERALIZED, VARIOGRAM-BASED FRAMEWORK FOR MULTI-SCALE ORDINATION

Abstract

Multi-scale ordination (MSO) deals with potential scale dependence in spe- cies 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 anal- ysis 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 con- sistent 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 re- sidual 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.