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Ecological Modelling 220 (2009) 2683–2689

Contents lists available at ScienceDirect

Ecological Modelling

journa l homepage: www.e lsev ier .com/ locate /eco lmodel

istance-based eigenvector maps (DBEM) to analyse metapopulation structureith irregular sampling

rancois Munoz a,b,∗

Centre d’Ecologie Fonctionnelle et Evolutive, 1919 route de Mende, F-34293 Montpellier Cedex 5, FranceUniversité Montpellier 2, UMR AMAP (botAnique et bioinforMatique de l’Architecture des Plantes), boulevard de la Lironde, TA A-51/PS2, F-34398 Montpellier Cedex 5, France.

r t i c l e i n f o

rticle history:eceived 1 April 2009eceived in revised form 26 June 2009ccepted 15 July 2009vailable online 19 August 2009

a b s t r a c t

Characterizing spatial patterns due to ecological processes is a major issue for analysing and predictingspecies distributions. Grasping the non-linear nature of population dynamics over networks of discretesuitable sites is here central, as very specific signatures are expected. In the line of promising resultsfrom Fourier analysis of metapopulation maps, we found distance-based eigenvector maps (DBEM) tohelp disentangle the respective signatures of habitat and metapopulation structuring, with the great

eywords:pectral analysisistance-based eigenvector maps (DBEM)moothing

advantage of being applicable to irregular sampling schemes, a common feature of ecological surveys.A smoothing procedure was required to obtain the distinguishable signatures, and this may be a criticalissue for investigating non-contingent and reliable patterns in spatial ecology.

© 2009 Elsevier B.V. All rights reserved.

etapopulationabitat structuringrincipal Component Analysis

. Introduction

The metapopulation theory (Levins, 1969; Hanski and Gilpin,997) has aroused a great interest in ecology by considering thealance of stochastic extinction and colonization events to be a keyeterminant of spatial and temporal population dynamics. The the-ry developed from spatially implicit models relating populationensity to the ratio of colonization and extinction rates (Levins,969), and further studies revealed that population spatial structur-ng is complex and closely depends on non-linear metapopulationynamics (Ovaskainen and Hanski, 2001; Munoz et al., 2007). In thisespect, characterizing spatial occupancy patterns over networks ofuitable habitat sites is an important issue for biological conserva-ion, in the context of increasingly fragmented habitat landscapesFahrig and Merriam, 1994), as well as for predictive modellingf the effect of global changes (Davis et al., 1998). Our hypothe-is is that the population spatial structure due to metapopulationynamics is singular enough to be separable from the habitattructure, and we accordingly aim at characterizing and inferringtochastic population dynamics from reported population distribu-

ions.

Much research has been devoted toward defining useful statis-ics for describing and analysing spatial ecological data (Legendrend Fortin, 1989), especially in the field of landscape ecology (for

∗ Corresponding author. Tel.: +33 04 67 61 75 52; fax: +33 04 67 61 56 68.E-mail address: francois.munoz@cirad.fr.

304-3800/$ – see front matter © 2009 Elsevier B.V. All rights reserved.oi:10.1016/j.ecolmodel.2009.07.008

instance O’Neill et al., 1988, 1989; Cain et al., 1997; Keitt, 2000).Polynoms of geographic coordinates can be used to detect spatialstructures (trend surface analysis, Legendre and Fortin, 1989), butthey still filter out smallest-scale patterns (Borcard and Legendre,2002), and hence they may provide a biased picture of local eco-logical processes. On the other hand, Distance-Based EigenvectorMaps (DBEM) are particular cases of Moran Eigenvector Maps(MEM), derived from spectral graph theory (Chung, 1994; Borcardand Legendre, 2002; Dray et al., 2006), which provide more com-prehensive spatial descriptors covering a wide range of spatialautocorrelation.

Borcard and Legendre (2002) advocated using a particular kindof DBEM spatial descriptors (Principal Coordinates of NeighbourMatrices, PCNM) as predictors in multiple regression, redundancyanalysis, and other multivariate approaches, as a support for vari-ation partitioning and scale detection in ecological data (as, e.g.,in Griffith and Peres-Neto, 2006; Jombart et al., 2009). Dray etal. (2006) further stressed that the DBEM eigenvalues are relatedto their gradient in spatial autocorrelation, which is analogous tofrequency assortment in spectral analysis (Borcard and Legendre,2002).

We aimed at exposing useful spectral properties of a DBEM-based analysis of ecological data. In this regard, we considered

devising an original smoothing procedure, in analogy with Fourieranalysis (Munoz et al., 2007), for the idea that this would improvethe relevance of the spatial analysis. Following Munoz et al.(2007), we thereby investigated the issue by analysing simulatedmetapopulation maps in varying habitat structures, and com-

2 delling 220 (2009) 2683–2689

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684 F. Munoz / Ecological Mo

ared the relative properties of raw and smooth DBEM-basedpectra.

. Methodological background

.1. DBEM spectral decomposition

From point spatial locations, an adequate matrix transforma-ion provides DBEM orthonormal eigenvectors Wi (see Appendix A)hat have the desirable property of covering a large range of spatialorrelation (Dray et al., 2006), and to such extent they are inde-endent and hierarchically ordered spatial predictors (Borcard andegendre, 2002). As an illustration, within a 100 × 100 lattice repre-enting the study area, we considered two n = 2500-point samplingchemes, one made of regularly spaced points (Fig. 1a), and thether made of randomly and irregularly spread points (Fig. 1b). Cal-

ulating the distance matrix of sampling points is a critical stage,nd we followed the recommendation of Borcard and Legendre2002) to attribute a large constant value 4× dmax for every edgebove the maximal distance dmax to the first neighbour (see Drayt al., 2006, for further discussion about truncation). For the sake

ig. 1. Alternative sampling schemes within a 100 × 100 lattice, featuring (a) 2500egularly spaced points, (b) 2500 irregularly spaced points.

Fig. 2. DBEM Moran’s I values for (a) the regular sampling of Fig. 1a and (b) theirregular sampling of Fig. 1b. Moran’s I is a global autocorrelation statistic, and iscalculated for each DBEM according to the definition of Dray et al. (2006), whichallows values greater than 1.

of comparison, the threshold dmax is common to both samplingschemes, and Fig. 2 represents the gradients in spatial autocorrela-tion of the DBEM, as measured by the Moran’s I statistic of Dray etal. (2006) (formula in Appendix A). When the sampling scheme isregular, the DBEM are sinusoidal waves, and the DBEM-based anal-ysis is then analogous to Fourier analysis (Borcard and Legendre,2002, p. 54). See Appendix A for any more detail on DBEM calcula-tion.

Let us now consider an ecological vector X featuring, at samplingpoints, e.g., binary presence–absence or occupancy informationabout a focal organism, or any other ecological information, includ-ing the factors of some multivariate analysis (as suggested, e.g., byBorcard and Legendre, 2002). In any case, one may wish to relate Xto some other independent, environmental information Y, e.g., fea-turing bioclimate, and/or to the DBEM spatial descriptors Wi, usingunivariate partial regressions or multivariate canonical analyses(Borcard and Legendre, 2002). Calculating the spatial componentof the relationship between X and Y, explained by the DBEM, alongwith the non-spatial counterpart that is independent of the DBEM,may indeed help characterize the nature of the processes involved(Borcard et al., 1992; Griffith and Peres-Neto, 2006).

Instead of a regressive pathway, an algebraic transformationdirectly provides the spectral counterparts SX and SY of X and Yaccording to the DBEM (see Appendix A):

W1 · · · Wne1

⎛w11 · · · w1n

⎞

X = W · SX and Y = W · SY with W = ...

en

⎝ .... . .

...wn1 · · · wnn

⎠SX and SY are raw DBEM-based spectra of X and Y.

F. Munoz / Ecological Modelling 220 (2009) 2683–2689 2685

F q (habh stead

2

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dpaidbcDspdn1adbw

ig. 3. Simulated metapopulation maps with varying parameters p (habitat density),abitat cell is white, while unoccupied but suitable habitat cells are grey, at a given

.2. Smoothing procedure

It was proposed to exclude the DBEM associated with negativeigenvalues (Borcard and Legendre, 2002, p. 54) or with spatialtructures of too small amplitude (Dray et al., 2006, p. 486), in ordero circumvent the pitfall of too many predictors in regression anal-ses, and to acknowledge some limitations due to normalisationonstraints (see Appendix A). But spatial signatures of ecologicalrocesses can be multi-scale or cross-scale (Levin, 1992; Perry,995), and, in this respect, the interest of a DBEM-based analysiss to cover a complete range of spatial structures.

On the other hand, a large estimation error is expected whenealing with as many spectral components as the number of sam-ling points, n (see Platt and Denman, 1975, in the context of Fouriernalysis). This may prevent the raw spectrum from being ecolog-cally relevant, because some contingent and noisy signal wouldominate. A statistically sensible solution is to decrease the num-er of estimates by substituting a single synthetic statistic to severalomponents sXj falling in a given “smoothing” window. When usingBEM, consecutive smoothing windows li can be spread along the

patial correlation gradient. A trade-off is to be found between therecision in spectral resolution, by controlling the number of win-ows, and the estimation error in each window, by controlling theumber ni of components sXj per window li (Platt and Denman,

975, p. 195; Appendix in Munoz et al., 2007). The estimation vari-nce of each new “smooth” estimate in li is accordingly expected toecrease by a factor proportional to ni. Finally, the smooth DBEM-ased spectrum includes as many components as the number ofindows, and we considered the following smoothing formula,

itat aggregation) and r (population turnover), within 100 × 100 lattices. An occupiedy-state generation. Always unsuitable habitat is colored in black.

based on quadratic means:

smoothSX (li) =

√∑u ∈ l(sXu)2

ni

This formula may be useful for spatial variance analysis, as thesquare components of smoothSX can be used to partition the vari-ance of X over the range of DBEM structures (see Appendix A). Inour application, we used windows of equal width varying between5 and 125, for n = 2500, i.e., the number of windows varied between20 and 500. Many other smoothing options are thinkable (see, e.g.,Platt and Denman, 1975), according to the same trade-off.

2.3. Application: analysing metapopulation maps

We applied our DBEM-based analysis framework to a set of sim-ulated metapopulation maps, for a range of habitat structure andpopulation turnover. For purpose of comparison, we considered thesimulations to which Munoz et al. (2007) applied Fourier analysis.

2.4. Simulating metapopulation dynamics

Using the C program of Hiebeler (2000), we simulated 100 × 100habitat landcapes of binary suitable/unsuitable cells, with fixed

density p and aggregation q of suitable sites. p and q varied from0.1 to 0.9 by 0.1 increment, under the geometrical constraint thatq ≥ 2 − (1/p) (Hiebeler, 2000; Munoz et al., 2007), and this yielded57 distinct habitat landscapes (see varying habitat contexts inFig. 3).

2 delling 220 (2009) 2683–2689

ttactcptTps10two8qstdbmtuctnm

2

cDspP2egWva(b

3

ptwiedtcat(sAii

Fig. 4. Loadings of the two first Principal Components, PC1 (a) and PC2 (b), for

686 F. Munoz / Ecological Mo

Metapopulation dynamics occur in binary habitat landscapes, sohat any suitable habitat cell is potentially occupied, but only a frac-ion is really occupied at a given date (Fig. 3). Stochastic colonizationnd extinction events further make the local occupancy state tohange in time, while the habitat landscape is set constant. Extinc-ions of populations occur with probability m, while any occupiedell provides neighbouring cells with colonizers with decreasingrobability c(i) = A exp(−i), where i was the Euclidean distance ofhe recipient (i ≤ 5) and A features the global rain of colonizers.he global colonization incidence is C =

∫c(i) di. The turnover

arameter r = (m/C) (see Levins, 1969) is relevant to characterizeteady-state metapopulation properties (Hanski and Gyllenberg,997), and we focused on making r vary in our simulations from.1 to 0.9, with 0.1 increments. On the other hand, the extinc-ion rate m was always set to 0.09 and every habitat landscapeas randomly sewed with populations randomly occupying 10%

f the suitable area. We simulated metapopulation dynamics over000 generations for each of the (p, q, r) triplet and checked theuasi-stationary population dynamics (e.g., Fig. 3). We further con-idered five independent simulation runs, by repeatedly simulatinghe (p, q, r) triplets, with fixed initial 10% density but varying ran-om distribution of populations. One of the simulation runs is usedelow for illustration and included 225 quasi-stationary, non-nulletapopulation maps, while the metapopulations corresponding

o the remaining triplets (p, q, r) all went to extinction. A metapop-lation could go extinct either deterministically when colonizationould not overcome local extinction, or stochastically during theransitory stage or the quasi-stationary stage. We used the non-ull quasi-stationary maps as illustrations of real-world survivingetapopulations in a variety of habitat contexts.

.5. Principal Component Analysis of DBEM-based spectra

We averaged steady-state metapopulation maps over 1–300onsecutive generations, and calculated both raw and smoothBEM-based spectra. We summarized the variability of so many

pectra (Couteron, 2002) by performing a covariance-based Princi-al Component Analysis and assessed the correlation of resultingrincipal Components with parameters p, q and r (see Munoz et al.,007). This was repeated for varying metapopulation maps (pres-nce/absence at a given time, and time-averaged maps over 2–300enerations), and for five complete replicate runs of simulations.e also compared the results for raw and smooth spectra, and for

arying widths of the smoothing window. Finally, we applied anlternative stepwise regression-based derivation of DBEM spectraBorcard and Legendre, 2002) and compared them to the alge-raically calculated raw spectra.

. Results

When applying the smoothing procedure, the two first Princi-al Components, PC1 and PC2, explained between 92 and 95% ofhe variation in DBEM-based spectra, for time-averaged maps asell as for 0/1 metapopulation maps, and for regular as well as for

rregular sampling. In a clear analogy with Fourier analysis (Munozt al., 2007), the PC1 loadings were all negative and monotonicallyecreased from fine-scale to coarse-scale spatial structures, whilehe PC2 loadings increased from negative to positive values andould feature at sign change a focal scale of habitat aggregationccording to our parameterization (Fig. 4). Furthermore, this pat-ern was quite similar for regular as well as for irregular sampling

Fig. 4, solid vs. empty circles), and hence the nature of spectralignatures was not dependent upon spatial sampling regularity.lthough the DBEM are unique to each sampling scheme, we could

ndeed compare their respective PC loadings using the correspond-ng Moran’s I values as a common reference (Fig. 4).

smooth DBEM-based spectra of 300-generation averaged metapopulation maps,using the regular sampling of Fig. 1a (solid circles) or the irregular sampling of Fig. 1b(empty circles). PC loadings are plotted against middle Moran’s I of correspondingsmoothing windows.

The scores of metapopulation maps on PC1 were clearly corre-lated with p and r parameters but not with q, while scores on PC2were clearly correlated with p and q but not with r, in the line ofthe results of Fourier analysis (Table 1). This structure supportedthe hypothesis that metapopulation dynamics yield self-organizedpatterns (cross-scale signature on PC1), while there was a focal scalein habitat structure (see Munoz et al., 2007). These results held forboth regular and irregular sampling schemes (Table 1a vs. Table 1b)and for presence–absence maps as well as for 2–300-generationtime-averaged maps (Table 1, 300-generation above diagonal vs.presence–absence below diagonal). We however found the cor-relation of r with PC1 to decrease when fewer generations wereconsidered (Table 1a and b). All these results kept consistent overthe five replicate rounds of simulations, and when the width ofthe smoothing window was made to vary between 5 and 125. Theillustrations are given for a windows width of 50.

On the other hand, when we did not apply the smoothing pro-cedure, the PCA results were strikingly different (PC loadings andscores, Fig. 5). There was no prominent component, as the Prin-cipal Components each explained up to only 8% of the overall

variation in DBEM-based spectra of 300-generation time-averagedmetapopulation maps, and up to 5% of overall variation in case ofpresence–absence maps. Furthermore, there was no clear structurein Principal Component loadings (Fig. 5a and b, two first PC, ran-dom loading distribution). Finally, the Principal Components were

F. Munoz / Ecological Modelling 220 (2009) 2683–2689 2687

Table 1Correlations of Principal Component scores with (p, q, r) parameters.

(a) Smooth spectra – 2500-point regular sampling

PC1 PC2 p q r

PC1 0 −0.323*** 0.117 0.711***

PC2 0 −0.270*** 0.476*** 0.011p −0.678*** −0.278*** 0.204** 0.231***

q −0.011 0.413*** 0.454***

r 0.378*** −0.124

(b) Smooth spectra – 2500-point random irregular sampling

PC1 PC2 p q r

PC1 0 −0.329*** 0.117 0.706***

PC2 0 −0.28*** 0.631*** 0.012p −0.677*** −0.281***

q −0.014 0.548***

r 0.375*** −0.123

(c) Raw spectra – 2500-point regular sampling

PC1 PC2 p q r

PC1 0 0.034 −0.075 0.009PC2 0 −0.169* −0.068 −0.074p 0.060 0.052q 0.081 −0.000r 0.041 0.036

Pearson’s correlations of the two first Principal Components (PC1 and PC2) scores with parameters p, q and r are given for DBEM smooth (a and b) and raw (c) spectra, usingregular (a and c) or irregular (b) sampling, and for 300-generation time-averaged (above diagonal) or presence–absence (below diagonals) maps.

*

sFn(hlFra

4

asisdtmapF

apoTmpmTyfi

Significance label is 5%.** Significance label is 1%.

*** Significance label is 0.1%.

trongly related to particular habitat landscapes (map scores onig. 5c, same kind of pattern for other components), and there waso significant correlation with ecological parameters p, q and rTable 1c). The Principal Components were hence contingent uponabitat map singularities instead of directly depending on the eco-

ogical parameters themselves. We found a pattern very similar toig.5 to occur when using DBEM as spatial predictors in stepwiseegressions to analyse the metapopulation maps (following Borcardnd Legendre, 2002).

. Discussion

Both Distance-Based Eigenvector Maps (DBEM; present paper)nd Fourier analysis (Munoz et al., 2007) allowed separating thepatial signatures of habitat structure and metapopulation dynam-cs. In both cases, there were two major and independent trends inmooth spectra variation, one pertaining to metapopulation innerynamics (correlation with r and not with q) and the other to habi-at aggregation (correlation with q and not with r). This analogy

ight reflect the common mathematical basis of both approaches,s the eigenvector decomposition of an appropriate regular sam-ling scheme provides regular frequencies analogous to a discreteourier frequency decomposition.

The DBEM analysis is still even more advantageous than Fouriernalysis in two important respects. We found that the decouplingroperty was irrespective, in the case of DBEM-based analysis,f the sampling scheme being regular or randomly irregular.his may be well-suited to a majority of field datasets. Further-ore, the decoupling property of DBEM-based analysis held for

resence–absence maps, while it only held for time-averaged

etapopulation maps with Fourier analysis (Munoz et al., 2007).

his is useful for real-world applications as DBEM-based anal-sis of habitat and metapopulation signatures does not requireollowing occupancy maps over several generations. These twomportant advantages open the opportunity to deal with many

naturalist Atlases as data sources. But we should be very care-ful on the ground of statistical robustness, as the correlation ofr with the first Principal Components depicting metapopulationstructuring (Table 1a and b) was decreasing for fewer averagedgenerations.

What insights can such consistent results from two types ofspectral analysis bring to ecological research? First, this allows stat-ing that species distribution maps convey enough information toallow separating signatures of habitat structure and metapopula-tions dynamics, and this is clearly important for future inferenceattempts. A most immediate and sensible application could beto perform a multivariate analysis of the spectra of a number ofspecies, and to position them according to their respective signa-tures of habitat requirement and population dynamics. There ishowever no analytical expectation available to predict such signa-tures in the general case, and the decoupling property, though beingrobust when changing initial conditions and considering replicatesimulation rounds, could be still contingent upon the context of thesimulation process. In particular, a perspective would be to exploreother ways to represent habitat structuring, especially in consider-ing a larger range of neutral landscapes (Keitt, 2000). Furthermore,the binary habitat model is itself a particular case, and simulatingmetapopulations over a gradient of habitat suitability and relateddemographic regimes could help investigate potential source-sinkdynamics (Dias, 1996; Thomas and Kunin, 1999).

Second, the striking difference in signatures of raw and smoothspectra is quite important in exposing a necessary trade-offbetween some accurate but potentially contingent characteriza-tion of spatial structures (raw spectra) and some coarser but moresensitive to ecological processes characterization (smooth spectra).

Variation in raw DBEM-based spectra focused on contingent fea-tures of the landscape maps (Fig. 5c), in a way that hindered furtherecological analysis of relevant parameters p, q and r. If consideredwithout care, these results may lead to spurious conclusions. Thesame problem occurred when DBEM were used as spatial predic-

2688 F. Munoz / Ecological Modelling 220 (2009) 2683–2689

Fig. 5. Loadings of the two first Principal Components, PC1 (a) and PC2 (b), for raw DBEM-based spectra of 300-generation time-averaged metapopulation maps, using ther are avq Metapm are 57

tLt

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egular sampling of Fig. 1a. For sake of comparison with Fig. 4a and b, the loadingsuite analogous result when using the irregular sampling of Fig. 1b (not shown). (c)aps for a given habitat landscape of fixed parameters p and q (label inside). There

ors in a stepwise regression-based analysis (following Borcard andegendre, 2002), as no smoothing procedure was applicable to con-rol the contingent variation (pattern similar to Fig. 5).

Although considering DBEM as a basis for spectral analysis isot a usual conception, it clearly opens interesting perspectives inpatial ecology. Much must still be done to more thoroughly char-cterize the mathematical and statistical properties. Unlike Fouriernalysis, which always features sinusoidal spectral components, theBEM spatial structures on which to project ecological data dependpon the sampling scheme, and this may make the interpretationore difficult. But we still can use the gradient in Moran’s I values

ssociated to DBEM as a reference scale for comparing DBEM-basedpectra for different sampling schemes (see Fig. 4). Furthermore,he shape of the gradient (see Fig. 2) is not linear and, accordingo the sampling scheme, the gradient of possible structures may

e more or less biased toward negative, positive, or null autocor-elation values. Hence the range of Moran’s I values relates to theange of spatial structures that we can investigate from the sam-ling scheme, and this can be seen as a sane restriction to theonclusions we can reliably reach. Finally, while the Fourier tech-

eraged in the smoothing windows and plotted against middle Moran’s I. We got aopulation scores are plotted against PC1 and PC2. Ellipses include metapopulation(p, q) combinations.

nique should be restricted to regular sampling schemes, DBEM canbecome a more general approach.

5. Conclusion

We found habitat and metapopulation structures to be distin-guishable after a DBEM-based analysis, but only when using thesmoothing procedure. This generalized the conclusions of Munozet al. (2007), as the philosophy of spectral analysis, promotingsmoothing procedure to remove contingent information, is indeedrelevant for spatial ecology. Furthermore, DBEM-based analysisproved to be an improvement over Fourier analysis by being appli-cable to irregular sampling schemes, a feature of most ecologicaldata, and clearly opens up new perspectives for process inferencein spatial ecology.

Acknowledgments

I warmly thank Pierre-Olivier Cheptou, Pierre Couteron,Stéphane Dray and Finn Kjellberg for their valuable comments anddiscussion on the issue.

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F. Munoz / Ecological Mo

ppendix A.

.1. How to get a raw DBEM spectrum

Following Borcard and Legendre (2002) and Dray et al. (2006),ere are the main steps of the DBEM-based analysis we used:

First build a Euclidean distance matrix D = (dij) of sampling points,usually according to geographical distances.Truncate the distance matrix over a threshold dmax, so as to keepall nodes interconnected within distance dmax, while larger dis-tances are set to an arbitrary large value, say, 4dmax. The resultingtruncated matrix is D∗ = (d∗

ij).

Transform D* into a matrix of elements −(1/2)d∗ij2

, and centre both

lines and columns, to get the weighting matrix �.Finally get the eigenvalues �j and the corresponding 1-normedDBEM Wj of �.

In the special case of the PCNM version (Borcard and Legendre,002), Wj should scale to length

√�j and hence, for negative eigen-

alues �j, Wj would be imaginary. That is why Borcard and Legendre2002) primarily advocated excluding eigenvectors associated toegative eigenvalues. But in the present application, eigenvectorsre scaled to 1, so that the mathematical constraint on PCNM doesot hold and all DBEMs can be kept.

The DBEM Wj are algebraically independent variables with coor-inates wij at sampling points Ei, such as the following matrix,,

=

W1 · · · WnE1...

En

⎛⎝ w11 · · · w1n

.... . .

...wn1 · · · wnn

⎞⎠

s the transition matrix from the location data to the DBEM coun-erpart. Dray et al. (2006) noticed that the DBEM lay on a range of

oran’s I spatial correlation, I(Wj), which reads, in our case:

(Wj) = nW′

j�Wj

1′�1

his provides a useful reference scale of spatial structures.Vectors X and Y classically represent population/community

nd environmental data, respectively. We relate X and Y to theirpectral counterparts SX and SY, according to the following linearransformation:

=

⎛⎝ x1

...xn

⎞⎠ → SX =

⎛⎝ sX1

...sXn

⎞⎠

=n∑

i=1

xi · Ei =n∑

j=1

sXj · Wj =n∑

j=1

sXj ·n∑

i=1

wijEi

i =n∑

j=1

wij · sXj = (W · SX)i, hence X = W · SX

ikewise, SX = W−1·X (bijection).W is the transition matrix between two orthonormal bases

E1. . .En) and (W1. . .Wn) and hence W−1 = W′. Coordinates sXi arehe projection of X on each DBEM Wi, and the magnitude of absoluteXi represents the strength of the signature of X on the correspond-ng Wi. We denote SX the raw DBEM spectrum of X. The componentf SX associated to Wn = 1/

√n 1 is the spatial mean of variable

220 (2009) 2683–2689 2689

X, which is null whenever the mean of X is set to 0 (centreddata).

A.2. Variance and covariance decomposition

In the case of two centred, 0-mean, X and Y variables and theirrespective raw spectra SX and SY, the spatial variance and covari-ance are readily related to the DBEM-based spectra:

Variance : Var(X) = X′ · X = S′X · W′W · SX = S′

XSX

Covariance : Cov(X, Y) = X′ · Y = Y′ · X = S′X · W′W · SY = S′

XSY

The square spectrum components s2Xi

or s2Yi

are elements of apartition of the variance of X or Y over the range of DBEM spatialstructures, while the product components sXisYi likewise providea partition of the covariance of X and Y, which is analogous to acospectrum. Cospectrum analysis allows detecting scales at whichtwo variables may significantly covary.

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