What are the important spatial scales in an ecosystem?

Size: px

Start display at page:

Download "What are the important spatial scales in an ecosystem?"

Herbert Matthews
5 years ago
Views:

1 What are the important spatial scales in an ecosystem? Pierre Legendre Département de sciences biologiques Université de Montréal 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

2 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

3 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).

4 Setting the stage Beta diversity is the variation in species composition among sites. Beta diversity is organized in communities. It displays spatial structures.

5 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 autocorrelation 1 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: y i = f(x i ) + SA i + ε i.

6 Multivariate variation partitioning Borcard & Legendre 1992 [808 citations] Borcard & Legendre 1994 and many published application papers

7 Environmental data matrix X Spatial data matrix W Community composition data table Y = [a] [b] [c] [d] = Residuals 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. Method described in: Borcard, Legendre & Drapeau 1992, Borcard & Legendre 1994, Legendre & Legendre 1998.

Partial multiple regression is computed as follows: 1.

8 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 X res of the regression of X on W: X res = X [W [W'W] 1 W' X] 2. Regress y on X res

9 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 Y Community composition data X Environmental variables W Spatial base functions Partial canonical analysis is computed as follows: 1. Compute the residuals X res of the regression of X on W: X res = X [W [W'W] 1 W' X] 2. Regress Y on X res to obtain Y fit. Compute PCA of Y fit.

10 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) = b 0 + b 1 X + b 2 Y + b 3 X 2 + b 4 XY + b 5 Y 2 + b 6 X 3 + b 7 X 2 Y + b 8 XY 2 + b 9 Y 3

11 Small textbook example: 20 sampling sites in the Thau lagoon, southern France. Response (Y): 2 types of aquatic heterotrophic bacteria (log-transf.) Environmental (X): NH 4, phaeopigments, bacterial production Spatial (W): selected geographic monomials X 2, X 3, X 2 Y, XY 2, Y 3 In real-life studies, partitioning is carried out on larger data sets.

13 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

14 PCOORD PCOORD PCOORD PCOORD PCOORD PCOORD PCOORD PCOORD Figure Eight of the 67 orthogonal PCNM eigenfunctions obtained for 100 equallyspaced points along a transect. Truncation after the first neighbor.

15 Observed variable y Data x (spatial coordinates) Euclidean distances n Truncated matrix of Euclidean distances = neighbor matrix 1...max 1...max 1...max 1...max Max x max 1...max 1...max 1...max 1...max 1...max 1 Multiple regression or canonical analysis Principal coordinate analysis Y X(+) + 0 Eigenvectors with positive eigenvalues = PCNM variables Eigenvectors 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.

16 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:

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

18 Compute a minimum spanning tree (or relative neighbourhood graph) Truncation distance length of the longest link. Longest link here: D(7, 8) =

19 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 eigendecomposition 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.

20 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.

21 A difficult test case

22 Dependent variable A difficult test case g) Data (100%) Simulated data h) Detrended data Step 1: detrending 6 i) Spatial model with 8 PCNM base functions 4 (R 2 = 0.433) PCNM analysis of detrended data 1,5 j) Broad-scale submodel (R 2 = 0.058) PCNM #2 1,0 0,5 0,0 0,5 1,0 4 3 k) Intermediate-scale submodel (R 2 = 0.246) 2 PCNM #6, 8, l) Fine-scale submodel (R 2 = 0.128) 2 PCNM #28, 33, 35,

23 A difficult test case

24 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 R 2 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 R 2 of the model containing all PCNM eigenfunctions. 2 1 Peres-Neto, P. R., P. Legendre, S. Dray and D. Borcard Variation partitioning of species data matrices: estimation and comparison of fractions. Ecology 87: Blanchet F. G., P. Legendre and D. Borcard Forward selection of explanatory variables. Ecology 89:

25 Example 1 Regular one-dimensional transect in upper Amazonia 1 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: R 2 = 0.102, p = 0.001) 1 Data from Tuomisto & Poulsen 2000, reanalysed in Borcard, Legendre, Avois-Jacquet & Tuomisto 2004.

26 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, m Broad-scale: 180 m Medium-scale: 90 m Fine-scale: 50, 65 m 3 (a) R 2 Data = PCNM model (50 significant PCNMs out of 176) ,5 (b) Very-broad-scale submodel, 10 PCNMs, R2 = ,5 0-0,5-1 -1, ,5 1 (c) Medium-scale submodel, 12 PCNMs, R2 = ,5 0-0,5-1 -1, ,5 1 0,5 0-0,5-1 -1,5 Broad-scale submodel, 8 PCNMs, R2 = (d) Fine-scale submodel, 20 PCNMs, R2 =

27 Interpretation: regression on the environmental variables

28 Use PCNM in variation partitioning: Adiantum tomentosum at Nauta, Peru (R 2 a ).

29 Example 2 Regular two-dimensional sampling grid Chlorophyll a in a brackish lagoon 1 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 Data first analyzed by Legendre & Troussellier 1988; reanalyzed in Borcard et al and Borcard, Legendre, Avois-Jacquet & Tuomisto 2004.

33 Example 3 Fish community (multivariate) in the littoral zone of a lake 1 Data: Fish community composition (7 sp.) at survey sites, June 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 Brind Amour, A., D. Boisclair, P. Legendre and D. Borcard Multiscale spatial distribution of a littoral fish community in relation to environmental variables. Limnology and Oceanography 50:

35 Truncated Euclidean distance matrix = neighbor matrix (a) Site Site (b) Principal coordinate analysis (c) PCNM 1 (d) PCNM 2 (e) PCNM 3 (f) PCNM 4 (g) PCNM 5 (h) PCNM 6

36 N N N Pefl Nocr Fudi N N N Longitude (W) Legi Seat Total fish abundance N N N Amne Caco Relative biomass Latitude (N)

37 (b) Very broad > 2 km 1 PCNM (c) Broad m 5 PCNM NoCr (+) (d) Intermediate m 6 PCNM (e) Fine scale < 100 m 3 PCNM SeAt ( ) SeAt (+), CaCo (+), NoCr (+), FuDi ( ), LeGi ( ) SeAt (+), IcNe ( ), PeFl ( ), FuDi ( ) Species codes: CaCo, Catostomus commersoni (white sucker); FuDi, Fundulus diaphanus (banded killifish); IcNe, Ictalurus nebulosus (brown bullhead); LeGi, Lepomis gibbosus (pumpkinseed); NoCr, Notemigonus crysoleucas (golden shiner); PeFl, Perca flavescens (yellow perch); SeAt, Semotilus atromaculatus (creek chub).

38 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. Community composition Total fish density Relative fish biomass V. broad Broad Meso Broad Meso Broad Meso Fine R 2 = 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*** Z S * 0.226* S3 S * 0.353*** S5 S * 0.279** U1 U * 0.193* U3 U4 U5 Tree * Sub Emer 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*

39 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 Partitioning beta diversity in a subtropical broad-leaved forest of China. Ecology 90:

41 Questions Example 4 Gutianshan forest plot in China 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, altitude 2, altitude 3 Convexity: convexity, convexity 2, convexity 3 Slope: slope, slope 2, slope 3 Aspect (circular variable): sin(aspect), cos(aspect) (Soil cores collected in 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.

42 Variation explained by Environment = Variation explained by PCNM = Variation in species data Y (beta diversity) = [a] = [b] = [c] = Unexplained (residual) = [d] = % of the among-cell variation (R a2 ) 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.

43 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 W 1, W 2, W 3, containing PCNM submodels.

44 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 Estimating and controlling for spatial structure in the study of ecological communities. Global Ecology and Biogeography 19:

45 Stéphane Dray: MEM analysis 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 0/1 in PCNM analysis. connectivity amongsites matrixhadamard product appliedto linkedges Weights Eigen-decomposition of a spatial weighting matrix (SWM): 1 Dray, S., P. Legendre, and P. R. Peres-Neto Spatial modelling: a comprehensive framework for principal coordinate analysis of neighbour matrices (PCNM). Ecological Modelling 196:

46 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 = d ij if d ij thresh d' ij = 4*thresh if d ij > 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.

47 0 d 12 4t 4t D PCNM = d 21 4t 0 0 4t 0 4t d 12 4t 4t D MEM = d 21 4t 4t 4t 4t 4t

48 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. / 1 0 D = d m i g We h t s o n ce c t i v i t y a a t r m H a r d r o d u c pt p l i e d t o i x i n k e ld g e s m o an g s i t e s

Guillaume Blanchet: AEM analysis Asymmetric eigenvector maps (AEM) is a spatial eigenfunction method developed to model species spatial distributions generated by an asymmetric, directional physical

49 Guillaume Blanchet: AEM analysis Asymmetric eigenvector maps (AEM) is a spatial eigenfunction method developed to model species spatial distributions generated by an asymmetric, directional physical process. 1 1 Blanchet, F.G., P. Legendre and D. Borcard Modelling directional spatial processes in ecological data. Ecological Modelling 215: 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.

50 Constructing asymmetric eigenvector maps (AEM eigenfunctions) Link Spatial asymmetry 1 = presence of a link 0 = absence of a link Site # # Link number Site number

51 Constructing asymmetric eigenvector maps (AEM eigenfunctions)

52 Constructing asymmetric eigenvector maps Weights can be put on the edges (AEM eigenfunctions)

53 Constructing asymmetric eigenvector maps (AEM eigenfunctions) SVD Eigenfunctions (explanatory spatial variables)

54 Map the AEM eigenfunctions AEM 1 AEM 2 AEM 3 [...] AEM 22 AEM 23 AEM 24 Negative values Positive values

55 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.

56 Example 1 Roxane Maranger, U. de Montréal Bacterial production in Lake St. Pierre Sampling was done on the morning of August 18 th 2005 Frenette et al. - Limnology & Oceanography (2006) Current direction

57 AEM model: R 2 adj = 51.4% using 4 selected AEMs

58 Example 2 Dominique Monti, UAG Atya innocous 94 sites were sampled in Rivière Capesterre Current direction

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

60 AEM model: R 2 adj = 59.8% using 12 selected AEMs

61 Example 3: 6 larval stages of Calanus finmarchicus on the Newfoundland and Labrador oceanic shelf Pierre Pepin, DFO, St. John s, NL AEM model: R 2 adj = 38.4% using 2 selected AEMs

62 Computer programs in the R statistical language On R-Forge: 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: vegan library: function varpart for multivariate variation partitioning

63 References, p. 1 Available in PDF at Borcard, D., P. Legendre & P. Drapeau Partialling out the spatial component of ecological variation. Ecology 73: Borcard, D. & P. Legendre Environmental control and spatial structure in ecological communities: an example using Oribatid mites (Acari, Oribatei). Environmental and Ecological Statistics 1: Borcard, D. & P. Legendre All-scale spatial analysis of ecological data by means of principal coordinates of neighbour matrices. Ecological Modelling 153: Borcard, D., P. Legendre, C. Avois-Jacquet & H. Tuomisto Dissecting the spatial structure of ecological data at multiple scales. Ecology 85: Dray, S., P. Legendre & P. Peres-Neto Spatial modelling: a comprehensive framework for principal coordinate analysis of neighbour matrices (PCNM). Ecological Modelling 196: Peres-Neto, P.R., P. Legendre, S. Dray and D. Borcard Variation partitioning of species data matrices: estimation and comparison of fractions. Ecology 87:

64 References, p. 2 Available in PDF at Blanchet, F. G., P. Legendre, and D. Borcard Modelling directional spatial processes in ecological data. Ecological Modelling 215: Blanchet F. G., P. Legendre, and D. Borcard Forward selection of explanatory variables. Ecology 89: 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 Estimating and controlling for spatial structure in the study of ecological communities. Global Ecology and Biogeography 19: Guénard, G., P. Legendre, D. Boisclair, and M. Bilodeau Assessment of scale-dependent correlations between variables. Ecology (in press).

65 The End

Temporal eigenfunction methods for multiscale analysis of community composition and other multivariate data

Temporal eigenfunction methods for multiscale analysis of community composition and other multivariate data Pierre Legendre Département de sciences biologiques Université de Montréal Pierre.Legendre@umontreal.ca