1826 Statistical Reports Ecology, 85(7), 2004, pp. 1826–1832  2004 by the Ecological Society of America DISSECTING THE SPATIAL STRUCTURE OF ECOLOGICAL DATA AT MULTIPLE SCALES DANIEL BORCARD, 1,4 PIERRE LEGENDRE, 1 CAROL AVOIS-JACQUET, 1,2 AND HANNA TUOMISTO 3 1 De ´partment de sciences biologiques, Universite ´ de Montre ´al, C.P. 6128, succursale Centre-ville, Montre ´al, Que ´bec, Canada H3C 3J7 2 Station d’Hydrobiologie Lacustre, INRA, BP 511, F-74203 Thonon-les-Bains Cedex, France 3 Department of Biology, University of Turku, FIN-20014 Turku, Finland Abstract. Spatial structures may not only result from ecological interactions, they may also play an essential functional role in organizing the interactions. Modeling spatial patterns at multiple spatial and temporal scales is thus a crucial step to understand the functioning of ecological communities. PCNM (principal coordinates of neighbor matrices) analysis achieves a spectral decomposition of the spatial relationships among the sampling sites, creating variables that correspond to all the spatial scales that can be perceived in a given data set. The analysis then finds the scales to which a data table of interest responds. The significant PCNM variables can be directly interpreted in terms of spatial scales, or included in a procedure of variation decomposition with respect to spatial and environmental com- ponents. This paper presents four applications of PCNM analysis to ecological data rep- resenting combinations of: transect or surface data, regular or irregular sampling schemes, univariate or multivariate data. The data sets include Amazonian ferns, tropical marine zooplankton, chlorophyll in a marine lagoon, and oribatid mites in a peat bog. In each case, new ecological knowledge was obtained through PCNM analysis. Key words: chlorophyll a; oribatid mites; principal coordinates of neighbor matrices (PCNM); sampling design; scale; spatial analysis; tropical ferns; tropical zooplankton; variation partitioning. INTRODUCTION The importance of spatial ecological structures is now widely recognized in ecological theory (Legendre and Fortin 1989, Legendre 1993, Peterson and Parker 1998). The interactions between living communities and their physical environment, and among the organ- isms themselves, occur at definite spatial and temporal scales, and give rise to spatial patterns that need to be assessed to untangle the processes structuring these communities. This assessment is not trivial when one’s objective is to include in the model all the scales per- ceived in a given data set. Among the methods that have been proposed to include space as an explicit predictor in ecological modeling, Legendre and Trous- sellier (1988) built a matrix of Euclidean distances to be used in a series of Mantel and partial Mantel tests, and Legendre (1990) proposed to use the geographic coordinates directly as explanatory variables in con- strained ordination techniques, augmented by all terms of a cubic trend-surface equation. The latter approach was integrated into a method of variation partitioning, where ecological variation was decomposed into four fractions using partial constrained regression or ordi- nation methods (Borcard et al. 1992, Borcard and Le- gendre 1994, Legendre and Borcard 2004). This tech- nique has proved very successful and is now widely Manuscript received 4 August 2003; revised and accepted 9 January 2004; final version received 24 February 2004. Corre- sponding Editor: G. M. Henebry. 4 E-mail: Daniel.Borcard@umontreal.ca applied in ecology; see references in Legendre and Le- gendre (1998:775). Trend-surface analysis only allows the broad-scale spatial variation (at the scale of the extent of the sam- pling campaign) to be modeled. Therefore, Borcard and Legendre (2002) proposed a new approach devised to identify spatial patterns across the whole range of scales perceptible with a given data set. This method (Fig. A1 in Appendix A) is based on the computation of the principal coordinates of a matrix of geographic neighbors among the sampling sites (PCNM, acronym for principal coordinates of neighbor matrices). The present paper illustrates the application of PCNM anal- ysis to real ecological data observed using various spa- tial designs: linear (transect) and two-dimensional (sur- face), regular or irregular sampling schemes, and pre- sents various ways to obtain new ecological knowledge from the results. THE METHOD OF PCNM ANALYSIS A detailed account of PCNM analysis is found in Borcard and Legendre (2002). The method creates a set of spatial explanatory variables that have structure at all the scales encompassed by the data matrix (akin to the series of sines and cosines used in Fourier anal- ysis), and determines to which of these variables the response data (univariate or multivariate) are statisti- cally responding. The steps to create the PCNM vari- ables are summarized in Table A1 of Appendix A. Note that using this method does not imply that one expects