What are the important spatial scales in an ecosystem?

Pierre Legendre Département de sciences biologiques

Université de Montréal

Pierre.Legendre@umontreal.ca http://www.bio.umontreal.ca/legendre/

Seminar, School of Environmental Systems Engineering University of Western Australia, Perth, March 20, 2008

MITACS Workshop “Predictive modelling of coastal habitat distributions” Dalhousie University, Halifax, February 23, 2009

Outline of the talk

1. Motivation

2. Variation partitioning

3. Multi-scale analysis

• The PCNM method

• Simulation study (summary)

4. Several applications to real ecological and data

5. Recent developments: MEM, AEM

Setting the stage Ecologists want to understand and model spatial [or temporal] community structures through the analysis of species assemblages. •  Species assemblages are the best response variable available to estimate the impact of [anthropogenic] changes in ecosystems.

• Difficulty: species assemblages form multivariate data tables (sites x species).

Setting the stage Beta diversity is the variation in species composition among sites.

Beta diversity is organized in communities. It displays spatial structures.

Setting the stage Spatial structures in communities indicate that some process has been at work to create them. Two families of mechanisms can generate spatial structures in communities:

•   Induced spatial dependence: forcing (explanatory) variables are responsible for the spatial structures found in the species assemblage. They represent environmental or biotic control of the species assemblages, or historical dynamics. Generally broad-scaled. • Community dynamics: the spatial structures are generated by the species assemblage themselves, creating autocorrelation1 in the response variables (species). Mechanisms: neutral processes such as ecological drift and limited dispersal, interactions among species. Spatial structures are generally fine-scaled.

1 Spatial autocorrelation (SA) is technically defined as the dependence, due to geographic proximity, present in the residuals of a [regression-type] model of a response variable y which takes into account all deterministic effects due to forcing variables. Model: yi = f(Xi) + SAi + εi .

Multivariate variation partitioning Borcard & Legendre 1992 [808 citations]

Borcard & Legendre 1994

and many published application papers

Figure – Venn diagram illustrating a partition of the variation of a response matrix Y (e.g., community composition data) between environmental (matrix X) and spatial (matrix W) explanatory variables. The rectangle represents 100% of the variation of Y.

[a] [b] [c]

[d] = Residuals

Environmental datamatrix X

Spatial datamatrix W

Communitycompositiondata table Y

=

Method described in: Borcard, Legendre & Drapeau 1992, Borcard & Legendre 1994, Legendre & Legendre 1998.

How to combine environmental and spatial variables in modeling community composition data?

• A single response variable: partial multiple regression.

Partial multiple regression is computed as follows: 1. Compute the residuals Xres of the regression of X on W: Xres = X – [W [W'W]–1 W' X] 2. Regress y on Xres

How to combine environmental and spatial variables in modelling community composition data?

• Multivariate data: partial canonical analysis (RDA or CCA).

Response table Explanatory table Explanatory table

XEnvironmental

variables

WSpatial

base functions

YCommunitycomposition

data

Partial canonical analysis is computed as follows: 1. Compute the residuals Xres of the regression of X on W: Xres = X – [W [W'W]–1 W' X] 2. Regress Y on Xres to obtain Yfit . Compute PCA of Yfit .

Geographic base functions

First (simple) representation: Polynomial function of geographic coordinates (polynomial trend-surface analysis).

Example 1: 20 sampling sites in the Thau lagoon, southern France.

z = f(X,Y) = b0 + b1X + b2Y + b3X2 + b4XY + b5Y2 + b6X3 + b7X2Y + b8XY2 + b9Y3^

Small textbook example:

20 sampling sites in the Thau lagoon, southern France.

Response (Y): 2 types of aquatic heterotrophic bacteria (log-transf.)

Environmental (X): NH4, phaeopigments, bacterial production

Spatial (W): selected geographic monomials X2, X3, X2Y, XY2, Y3

In real-life studies, partitioning is carried out on larger data sets.

Spatial eigenfunctions

Second representation: Principal Coordinates of Neighbor Matrices

(PCNM) leading to multiscale analysis in variation partitioning

Borcard & Legendre 2002, 2004

Dray, Legendre & Peres-Neto 2006

Figure – Eight of the 67 orthogonal PCNM eigenfunctions obtained for 100 equally-spaced points along a transect. Truncation after the first neighbor.

−0.8−0.4

0.00.4

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0.00.40.8

−1.0−0.5

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0.0

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0.000.050.100.15

−0.08−0.04

0.000.040.08

PCOORD 1

PCOORD 2

PCOORD 3

PCOORD 4

PCOORD 33

PCOORD 34

PCOORD 66

PCOORD 67

+

Principal coordinateanalysis

Eigenvectors

–0

Euclidean distances... n–1

1 2 3 4 5 ...1 2 3 4 5 ...

1 2 3 4 5 ...1 2 3 4 5 ...

1 2 3 4 5

1 2 3 4 5 ...

1 2 3 41 2 3

1 21

Truncated matrix of Euclideandistances = neighbor matrix

1...max

Max x 4

1...max1...max

1...max1...max

1...max1...max

1...max1...max

1...max1

XY

Multiple regressionor canonical analysis

Eigenvectors withpositive eigenvalues= PCNM variables

(+)

x(spatial coordinates)

Data

1 2 3 4 5 6 7 8 9 10 11

Observedvariable

y

Figure – Schematic description of PCNM analysis. The descriptors of spatial relationships (PCNM eigenfunctions) are obtained by principal coordinate analysis of a truncated matrix of Euclidean (geographic) distances among the sampling sites.

PCNM eigenfunctions display spatial autocorrelation

A spatial autocorrelation coefficient (Moran’s I) can be computed for each PCNM. These coefficients indicate the sign of the autocorrelation (+ or –) displayed by the PCNMs. Ex. 100-point transect, 67 PCNMs:

°

How to find the truncation distance in 2-dimensional problems?

Compute a minimum spanning tree (or relative neighbourhood graph)

Truncation distance ≥ length of the longest link. Longest link here: D(7, 8) = 3.0414

Technical notes on PCNM eigenfunctions

PCNM variables represent a spectral decomposition of the spatial relationships among the study sites. They can be computed for regular or irregular sets of points in space or time.

PCNM eigenfunctions are orthogonal. If the sampling design is regular, they look like sine waves. This is a property of the eigen-decomposition of the centred form of a distance matrix.

⇒ Consider a matrix of 0/1 connections among points, with • 1 between connected points • 0 between unconnected points and on the diagonal Double-centre the matrix to have the sums of rows and columns equal to 0. The eigenvectors of that matrix, plotted on a map of the geographic coordinates of the points, are sine-shaped.

Simulation study

Type I error study Simulations showed that the procedure is honest. It does not generate more significant results that it should for a given significance level α. Power study Simulations showed that PCNM analysis is capable of detecting spatial structures of many kinds: • random autocorrelated data, •  bumps and sine waves of various sizes, without or with random noise, representing deterministic structures, as long as the structures are larger than the truncation value used to create the PCNM eigenfunctions. Detailed results are found in Borcard & Legendre 2002.

A difficult test case

A difficult test case

.

−6−4−2

02468

1012

g) Data (100%)

0 10 20 30 40 50 60 70 80 90 100

0,00,51,01,5 j) Broad-scale submodel (R2 = 0.058)

PCNM #2

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k) Intermediate-scale submodel (R2 = 0.246) PCNM #6, 8, 14

−3−2−1

0123 l) Fine-scale submodel (R2 = 0.128)

PCNM #28, 33, 35, 41

−4−20246 i) Spatial model with 8 PCNM base functions

(R2 = 0.433)

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8 h) Detrended data

Simulateddata

Step 1:detrending

PCNM analysisof detrended

data

Dep

ende

nt v

aria

ble

A difficult test case

Selection of explanatory variables? Significance of the adjustment of a PCNM model can be tested using the full set of PCNM variables modelling positive spatial correlation, without selection of any kind.

The adjusted R2 gives a correct estimate of the variation explained by the PCNMs by correcting for the number of explanatory variables in the model.1

Before constructing submodels, forward selection of PCNMs can be done by combining two criteria during model selection: the alpha significance level and the adjusted R2 of the model containing all PCNM eigenfunctions.2

1 Peres-Neto, P. R., P. Legendre, S. Dray and D. Borcard. 2006. Variation partitioning of species data matrices: estimation and comparison of fractions. Ecology 87: 2614-2625. 2 Blanchet F. G., P. Legendre and D. Borcard. 2008. Forward selection of explanatory variables. Ecology 89: 2623-2632.

Example 1 Regular one-dimensional transect in upper Amazonia1

Data: abundance of the fern Adiantum tomentosum in quadrats. Sampling design: 260 adjacent, square (5 m x 5 m) subplots forming a transect in the region of Nauta, Peru. Questions • At what spatial scales is the abundance of this species structured? • Are these scales related to those of the environmental variables? Pre-treatment • The abundances were square-root transformed • and detrended (significant linear trend: R2 = 0.102, p = 0.001)

1 Data from Tuomisto & Poulsen 2000, reanalysed in Borcard, Legendre, Avois-Jacquet & Tuomisto 2004.

–2

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3(a) R2 = 0.815

1 21 41 61 81 101 121 141 161 181 201 221 241 260

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1 21 41 61 81 101 121 141 161 181 201 221 241 260

(b) Very-broad-scale submodel, 10 PCNMs, R2 = 0.333

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1 21 41 61 81 101 121 141 161 181 201 221 241 260

(c) Medium-scale submodel, 12 PCNMs, R2 = 0.126

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1 21 41 61 81 101 121 141 161 181 201 221 241 260

(d) Fine-scale submodel, 20 PCNMs, R2 = 0.117

Data

PCNM model (50 significant PCNMs out of 176)

Broad-scale submodel, 8 PCNMs, R2 = 0.239

Forward selection

50 PCNM eigenfunctions were selected out of 176 (permutation test, 999 permutations).

The PCNMs were arbitrarily divided into 4 submodels. The submodels are orthogonal to one another.

Significant wavelengths (periodogram analysis):

V-br-scale: 250, 355-440 m Broad-scale: 180 m Medium-scale: 90 m Fine-scale: 50, 65 m

Interpretation: regression on the environmental variables

Use PCNM in variation partitioning: Adiantum tomentosum at Nauta, Peru (R2

a).

Example 2 Regular two-dimensional sampling grid

Chlorophyll a in a brackish lagoon1 Data: Chlorophyll a concentrations at 63 sites on a geographic surface. Sampling design: 63 sites forming a regular grid (1-km mesh) in the Thau marine lagoon (19 km x 5 km). Questions • At what spatial scales is chlorophyll a structured? • Are these scales related to the environmental variables? Pre-treatment • None. Forward selection • 12 PCNM eigenfunctions were selected out of 45. 1 Data first analyzed by Legendre & Troussellier 1988; reanalyzed in Borcard et al. 1992 and

Borcard, Legendre, Avois-Jacquet & Tuomisto 2004.

Example 3 Fish community (multivariate) in the littoral zone of a lake1

Data: Fish community composition (7 sp.) at survey sites, June 2001. Sampling design: The littoral zone of the lake was divided into 90 sites that were surveyed by a diver (visual census). Questions • At what spatial scales is the fish community structured? • Are these scales related to the environmental variables? Pre-treatment • None. Forward selection • 15 PCNM eigenfunctions were selected out of 60.

1 Brind’Amour, A., D. Boisclair, P. Legendre and D. Borcard. 2005 Multiscale spatial distribution of a littoral fish community in relation to environmental variables. Limnology and Oceanography 50: 465-479.

1

23

4

510

9

8 7 6

1 4 4 4 4 4 4 4 1 1 4 4 4 4 4 4 4 1 4 4 4 4 4 4 1 4 4 4 4 4 1 4 4 4 4 1 4 4 4 1 4 4 1 4 1

Site 1 2 3 4 5 6 7 8 9 10 Site12345678910

Truncated Euclidean distance matrix= neighbor matrix

(a)(b)

Principal coordinateanalysis

(c) PCNM 1 (e) PCNM 3

(f) PCNM 4 (g) PCNM 5 (h) PCNM 6

(d) PCNM 2

46.00

46.04

46.08

46.12

46.16

73.44 73.49 73.53 73.58 74.02

Pefl

Relative biomass

Total fish abundanceLegi

Amne

Nocr

Seat

Caco

Fudi

N N N

N N N

N N N

46.00

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73.44 73.49 73.53 73.58 74.02 73.44 73.49 73.53 73.58 74.02

Latitude (N)

Long

itude

(W)

(d) Intermediate200-450 m6 PCNM

SeAt (+), CaCo (+), NoCr (+),FuDi (–), LeGi (–)

(e) Fine scale< 100 m3 PCNM

SeAt (+),IcNe (–), PeFl (–), FuDi (–)

(b) Very broad> 2 km1 PCNM

NoCr (+)

(c) Broad500-1000 m5 PCNM

SeAt (–)

Species codes: CaCo, Catostomus commersoni (white sucker); FuDi, Fundulus diaphanus (bandedkillifish); IcNe, Ictalurus nebulosus (brown bullhead); LeGi, Lepomis gibbosus (pumpkinseed); NoCr,Notemigonus crysoleucas (golden shiner); PeFl, Perca flavescens (yellow perch); SeAt, Semotilusatromaculatus (creek chub).

Community composition Total fish density Relative fish biomassV. broad Broad Meso Broad Meso Broad Meso Fine

R2= 0.513*** 0.263*** 0.198*** 0.171*** 0.047* 0.363* 0.295* 0.052*Riv −0.247** −0.211* −0.409***Lit −0.376***ZS1 −0.214* −0.226*S3S4 −0.220* −0.353***S5S8 −0.167* 0.279**U1U2 0.195* 0.193*U3U4U5Tree -0.217*SubEmer 0.325*** −0.292** 0.195*Cov 0.232** 0.304** −0.371***Fet 0.565*** 0.262* −0.292**Trib 0.315*** −0.228*

June 2001 survey: standardized coefficients (b) of environmental variables which explained significant components of the spatial patterns (PCNM models) of the littoral fish community descriptors at four different scales.

Example 4 Gutianshan forest plot in China

• Evergreen forest in Gutianshan Forest Reserve, Zhejiang Province. • Fully-surveyed 24-ha forest plot in subtropical forest, 29º15'N. • Plot divided into 600 cells of 20 m × 20 m. • 159 tree species. Richness: 19 to 54 species per cell. • Data collection: 2005

Legendre, P., X. Mi, H. Ren, K. Ma, M. Yu, I. F. Sun, and F. He. 2009. Partitioning beta diversity in a subtropical broad-leaved forest of China. Ecology 90: 663-674.

Example 4 Gutianshan forest plot in China

Questions • How much of the variation in species composition among sites (beta diversity) is spatially structured? • Of that, how much is related to the environmental variables?

⇒ Four environmental variables developed in cubic polynomial form: Altitude: altitude, altitude2, altitude3 Convexity: convexity, convexity2, convexity3 Slope: slope, slope2, slope3 Aspect (circular variable): sin(aspect), cos(aspect) (Soil cores collected in 2007. Soil chemistry data not available yet.)

⇒  339 PCNM eigenfunctions. 200 model positive spatial correlation. ⇒  Nearly all of them are significant: spatial variation at all scales.

• 63% of the among-cell variation (Ra2) of the community composition

(159 species) is spatially structured and explained by the 339 PCNMs.

• Nearly half of that 63% is also explained by the four environmental variables. Soil chemistry to be added to the model when available.

• Scales of spatial variation: the dominant structure is broad-scaled.

⇒ Balance between neutral processes and environmental control.

[a] =0.029

[b] =0.278

[c] =0.348

Variationexplained byEnvironment= 0.307

Variationexplainedby PCNM

= 0.626

Unexplained (residual) = [d] = 0.344

Variation inspecies data Y(beta diversity)

=

How to use PCNM eigenfunctions? a) We proceeded as follows in the first three examples: • PCNM analysis of the response table Y; • Division of the significant PCNM eigenfunctions into submodels; • Interpretation of the submodels using explanatory variables. The objective was to divide the variation of Y into submodels and relate those to explanatory environmental variables.

b) PCNM eigenfunctions can also be used in the framework of variation partitioning, as in Example #4. The variation of Y is then partitioned with respect to a table of explanatory variables X and (for example) several tables W1, W2, W3, containing PCNM submodels.

How to use PCNM base functions? c) PCNMs can efficiently model spatial structures in data. They can be used to control for spatial autocorrelation in tests of significance of the species-environment relationship (fraction [a]).

Peres-Neto, P. R. and P. Legendre. 2010. Estimating and controlling for spatial structure in the study of ecological communities. Global Ecology and Biogeography 19: 174-184.

The Moran’s eigenvector maps (MEM) method is a generalization of PCNMs to different types of spatial weights. The result is a set of spatial eigenfunctions, as in PCNM analysis.

0 / 1 c o n n e c t i v i t y

m a t r i x a m o n g s i t e s W e i g h t s

a p p l i e d t o l i n k e d g e s ｡ H a d a m a r d

p r o d u c t Stéphane Dray: MEM analysis

1 Dray, S., P. Legendre, and P. R. Peres-Neto. 2006. Spatial modelling: a comprehensive framework for principal coordinate analysis of neighbour matrices (PCNM). Ecological Modelling 196: 483-493.

Eigen-decomposition of a spatial weighting matrix (SWM):

PCNM vs. distance-based Moran’s Eigenvector Maps • D = matrix of geographic distances among the study sites • Set a connexion threshold thresh • Create matrix D' with d'ij = dij if dij ≤ thresh

d'ij = 4*thresh if dij > thresh d'ii = 0 in PCNM, d'ii = 4*thresh in MEM

• Centre D' by rows and columns as in PCoA (Gower 1966) • The eigenvectors of D' are the PCNM eigenfunctions. • In most cases in ecological models, use only the PCNMs modelling positive spatial autocorrelation. They are identical to the MEMs with positive eigenvalues.

4t 4t

4t4tD’PCNM =

D’MEM =4t 4t

4t4t

4t4t

4t

4t

00

00

d12d21

d12d21

Other forms of Moran’s Eigenvector Maps (MEM) can be created (Dray et al. 2006): • Binary MEMs: double-centre matrix A, then compute its eigenvalues and eigenvectors.

• Replace matrix D by some function of the distances.

• Replace matrix D by some other series of weights.

0 / 1 c o n n e c t i v i t y

m a t r i x a m o n g s i t e s W e i g h t s

a p p l i e d t o l i n k e d g e s ｡ H a d a m a r d

p r o d u c t D =

Asymmetric eigenvector maps (AEM) is a spatial eigenfunction method developed to model species spatial distributions generated by an asymmetric, directional physical process.1

Guillaume Blanchet: AEM analysis

1 Blanchet, F.G., P. Legendre and D. Borcard. 2008. Modelling directional spatial processes in ecological data. Ecological Modelling 215: 325-336.

The following slides were provided by F.G. Blanchet from his talk: Modelling directional spatial processes in ecological data. Spatial Ecological Data Analysis with R (SEDAR) Workshop, Université Lyon I, May 26, 2008.

# # Link number

Site number

1 = presence of a link 0 = absence of a link

1 2 3 4 5 6 7 8 9 10 11 12 13 14 … 57

Site 8 0 1 1 1 0 0 0 0 0 1 0 1 0 1 0 0

Spat

ial a

sym

met

ry

Link

Constructing asymmetric eigenvector maps (AEM eigenfunctions)

Constructing asymmetric eigenvector maps (AEM eigenfunctions)

Constructing asymmetric eigenvector maps (AEM eigenfunctions)

Weights can be put on the edges

Constructing asymmetric eigenvector maps (AEM eigenfunctions)

Eigenfunctions (explanatory spatial variables)

SVD

Map the AEM eigenfunctions AEM 1 AEM 2 AEM 3

AEM 22 AEM 23 AEM 24

Negative values

Positive values

[...]

Three applications of AEM analysis in a submitted paper –

Blanchet, F. G., P. Legendre, R. Maranger, D. Monti, and P. Pepin. Modeling the effect of directional spatial ecological processes at different scales.

Example 1 – Roxane Maranger, U. de Montréal Bacterial production in Lake St. Pierre

Sampling was done on the morning of August 18th 2005

Frenette et al. - Limnology & Oceanography (2006)

Current direction

AEM model: R2adj = 51.4% using 4 selected AEMs

94 sites were sampled in Rivière Capesterre

Cur

rent

dire

ctio

n Atya innocous

Example 2 – Dominique Monti, UAG

93 AEM variables were constructed based on this connection diagram; 38 measured positive autocorrelation. 12 were selected.

Atya innocous

AEM model: R2adj = 59.8% using 12 selected AEMs

Example 3: 6 larval stages of Calanus finmarchicus on the Newfoundland and Labrador oceanic shelf

– Pierre Pepin, DFO, St. John’s, NL AEM model: R2

adj = 38.4% using 2 selected AEMs

Computer programs in the R statistical language

On R-Forge: http://r-forge.r-project.org/R/?group_id=195 PCNM library (P. Legendre) AEM library (F. G. Blanchet) spacemakeR: an R package to compute PCNM and MEM (D. Dray) packfor: R package for forward selection of explanatory variables (S. Dray) On the CRAN page: http://cran.r-project.org vegan library: function varpart for multivariate variation partitioning

References, p. 1 Available in PDF at http://www.bio.umontreal.ca/legendre/

Borcard, D., P. Legendre & P. Drapeau. 1992. Partialling out the spatial component of ecological variation. Ecology 73: 1045-1055.

Borcard, D. & P. Legendre. 1994. Environmental control and spatial structure in ecological communities: an example using Oribatid mites (Acari, Oribatei). Environmental and Ecological Statistics 1: 37-61.

Borcard, D. & P. Legendre. 2002. All-scale spatial analysis of ecological data by means of principal coordinates of neighbour matrices. Ecological Modelling 153: 51-68.

Borcard, D., P. Legendre, C. Avois-Jacquet & H. Tuomisto. 2004. Dissecting the spatial structure of ecological data at multiple scales. Ecology 85: 1826-1832.

Dray, S., P. Legendre & P. Peres-Neto. 2006. Spatial modelling: a comprehensive framework for principal coordinate analysis of neighbour matrices (PCNM). Ecological Modelling 196: 483-493.

Peres-Neto, P.R., P. Legendre, S. Dray and D. Borcard. 2006. Variation partitioning of species data matrices: estimation and comparison of fractions. Ecology 87: 2614-2625.

References, p. 2 Available in PDF at http://www.bio.umontreal.ca/legendre/

Blanchet, F. G., P. Legendre, and D. Borcard. 2008. Modelling directional spatial processes in ecological data. Ecological Modelling 215: 325-336.

Blanchet F. G., P. Legendre, and D. Borcard. 2008. Forward selection of explanatory variables. Ecology 89: 2623-2632.

Blanchet, F. G., P. Legendre, R. Maranger, D. Monti, and P. Pepin. Modeling the effect of directional spatial ecological processes at different scales. Oecologia (Submitted).

Peres-Neto, P. R. and P. Legendre. 2010. Estimating and controlling for spatial structure in the study of ecological communities. Global Ecology and Biogeography 19: 174-184.

Guénard, G., P. Legendre, D. Boisclair, and M. Bilodeau. 2010. Assessment of scale-dependent correlations between variables. Ecology (in press).

The End