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Multivariate Description

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What Technique? Response variable(s)... Predictors(s) No Predictors(s) Yes... is one distribution summary regression models... are many indirect gradient analysis (PCA, CA, DCA, MDS) cluster analysis direct gradient analysis constrained cluster analysis discriminant analysis (CVA)

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Rotate the Variable Space

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Raw Data

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Linear Regression

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Two Regressions

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Principal Components

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Gulls Variables

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Scree Plot

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Output > gulls.pca2$loadings Loadings: Comp.1 Comp.2 Comp.3 Comp.4 Weight -0.505 -0.343 0.285 0.739 Wing -0.490 0.852 -0.143 0.116 Bill -0.500 -0.381 -0.742 -0.232 H.and.B -0.505 -0.107 0.589 -0.622 > summary(gulls.pca2) Importance of components: Comp.1 Comp.2 Comp.3 Standard deviation 1.8133342 0.52544623 0.47501980 Proportion of Variance 0.8243224 0.06921464 0.05656722 Cumulative Proportion 0.8243224 0.89353703 0.95010425

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Bi-Plot

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Environmental Gradients

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Type of Data Matrix species sites attributes species attributes sites attributes individuals desert macroph inverts uses watervar rain gulls

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Models of Species Response There are (at least) two models:- Linear - species increase or decrease along the environmental gradient Unimodal - species rise to a peak somewhere along the environmental gradient and then fall again

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A Theoretical Model

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Linear

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Unimodal

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Ordination Techniques Linear methodsWeighted averaging (unimodal) Unconstrained (indirect) Principal Components Analysis (PCA) Correspondence Analysis (CA) Constrained (direct) Redundancy Analysis (RDA) Canonical Correspondence Analysis (CCA)

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Inferring Gradients from Attribute Data (e.g. species)

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Indirect Gradient Analysis Environmental gradients are inferred from species data alone Three methods: –Principal Component Analysis - linear model –Correspondence Analysis - unimodal model –Detrended CA - modified unimodal model

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PCA - linear model

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Terschelling Dune Data

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PCA gradient - site plot

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PCA gradient - site/species biplot standard nature biodynamic & hobby

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Making Effective Use of Environmental Variables

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Approaches Use single responses in linear models of environmental variables Use axes of a multivariate dimension reduction technique as responses in linear models of environmental variables Constrain the multivariate dimension reduction into the factor space defined by the environmental variables

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Ordination Constrained by the Environmental Variables

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Constrained?

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Working with the Variability that we Can Explain Start with all the variability in the response variables. Replace the original observations with their fitted values from a model employing the environmental variables as explanatory variables (discarding the residual variability). Carry our gradient analysis on the fitted values.

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Unconstrained/Constrained Unconstrained ordination axes correspond to the directions of the greatest variability within the data set. Constrained ordination axes correspond to the directions of the greatest variability of the data set that can be explained by the environmental variables.

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Dune Data Unconstrained

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Direct Gradient Analysis Environmental gradients are constructed from the relationship between species environmental variables Three methods: –Redundancy Analysis - linear model –Canonical (or Constrained) Correspondence Analysis - unimodal model –Detrended CCA - modified unimodal model

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Direct Gradient Analysis Basic PCA y ik = b 0k + b 1k x i + e ik –x i - the sample scores on the ordination axis –b 1k - the regression coefficients for each species (the species scores on the ordination axis) In RDA there is a further constraint on x i x i = c 1 z i1 + c 2 z i2 Making y ik = b 0k + b 1k c 1 z i1 + b 1k c 2 z i2 + e ik

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Direct Gradient Analysis cca(species_data ~ e1 + e2 +... + en, data=environmental_data) cca(dune ~ Manure + Moisture + A1, data=dune.env)

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Dune Data Constrained

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Lake Nasser - Egypt

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Nasser Data Sites – 23 sampling stations on Lake Nasser 3 Data Frames: –Aquatic macrophytes –Invertebrate classes –Water chemistry

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Lake Nasser Unconstrained

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Lake Nasser Constrained

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Cluster Analysis

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Different types of data example Continuous data:height Categorical data ordered (nominal):growth rate very slow, slow, medium, fast, very fast not ordered:fruit colour yellow, green, purple, red, orange Binary data:fruit / no fruit

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Similarity matrix We define a similarity between units – like the correlation between continuous variables. (also can be a dissimilarity or distance matrix) A similarity can be constructed as an average of the similarities between the units on each variable. (can use weighted average) This provides a way of combining different types of variables.

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relevant for continuous variables: Euclidean city block or Manhattan Distance metrics A B A B (also many other variations)

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A Distance Matrix

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Uses of Distances Distance/Dissimilarity can be used to:- Explore dimensionality in data (using PCO) As a basis for clustering/classification

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UK Wet Deposition Network

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Fitting Environmental Variables

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A Map based on Measured Variables

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Fitting Environmental Variables

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Similarity coefficients for binary data simple matching count if both units 0 or both units 1 Jaccard count only if both units 1 (also many other variants) simple matching can be extended to categorical data 0,11,1 0,01,0 0,11,1 0,01,0

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hierarchical divisive put everything together and split monothetic / polythetic agglomerative keep everything separate and join the most similar points (classical cluster analysis) non-hierarchical k-means clustering Clustering methods

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Agglomerative hierarchical Single linkage or nearest neighbour finds the minimum spanning tree: shortest tree that connects all points chaining can be a problem

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Agglomerative hierarchical Complete linkage or furthest neighbour compact clusters of approximately equal size. (makes compact groups even when none exist)

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Agglomerative hierarchical Average linkage methods between single and complete linkage

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From Alexandria to Suez

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Hierarchical Clustering

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Modelling Multi-dimensional Environmental Variables

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Ways of Building Models I.Automated environmental variable selection (stepwise addition or removal of variables from the model – as with multiple regression) mod0 <- cca(nasser.inverts ~ 1, nasser.watervar) mod1 <- cca(nasser.inverts ~., nasser.watervar) op <- options(digits=7) mod <- step(mod0, scope=formula(mod1)) options(op) mod plot(mod)

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Ways of Building Models II.Manual selection of environmental variables using prior knowledge (e.g. example starting with full model and removing terms) mod1 <- cca(nasser.inverts ~., nasser.watervar) mod2 <- cca(nasser.inverts ~. -WMg, nasser.watervar) mod3 <- cca(nasser.inverts ~. -WMg -WEC, nasser.watervar) mod4 <- cca(nasser.inverts ~. -WMg -WEC -WCa, nasser.watervar)

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Ways of Evaluating Models I.Graphically using Procrustes Rotation plot(procrustes(mod2, mod1)) plot(procrustes(mod3, mod2)) plot(procrustes(mod4, mod3)) plot(procrustes(mod4, mod1))

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Procrustes

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Ways of Evaluating Models II.Permutation Tests can be used to assess adequacy of the models using a Pseudo ANOVA or Permutest anova(mod1) anova(mod2) anova(mod3) anova(mod4) permutest.cca(mod1, perm=1000) permutest.cca(mod2, perm=1000) permutest.cca(mod3, perm=1000) permutest.cca(mod4, perm=1000)

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Removing the Effect of Nuisance Variables

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Getting rid of the Variability that is Not of Interest Amongst the explanatory variables there may be variability attributable to: Blocks and other design strata Covariates that we can measure but are not the focus of interest We may want to use only the variability attributable to: Meaningful Environmental Variables

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Partial Analyses Remove the effect of covariates –variables that we can measure but which are of no interest –e.g. block effects, start values, etc. Carry out the gradient analysis on what is left of the variation after removing the effect of the covariates.

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Lichen-rich Forest Understorey

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Forest Data Sites – 28 sites in forests in Finland grazed by reindeer Species Data – 44 heathland plant species (including many lichens and mosses that are very sensitive to their chemical environment) Environmental Data – Soil chemical composition (N P K Ca Mg S Al Fe Mn Zn Mo Baresoil Humdepth pH)

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CCA

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Removing pH Effect cca(species_data ~ e1 + e2 +... + en + Condition(e5), data=environmental_data) cca(varespec ~ Al + P + K + Baresoil + Condition(pH), data=varechem)

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Removing pH Effect

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Interactions in Models cca(species_data ~ e1 + e2 +... + en + Condition(e5), data=environmental_data) cca(varespec ~ Al + P*(K + Baresoil) + Condition(pH), data=varechem)

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CCA

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Removing pH Effect

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Summarise by Weighted Averages

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Species and Sites as Weighted Averages of each other SITES 1 1 1111 1 2111 SPP. 23466185750198304927 Bel per 3.2....2..2..22..... Jun buf.3..........4..…42.. Jun art...3..4..3..4..4.... Air pra........2........3.. Ele pal...8..4..5.....44... Rum ace....6..5....2..…23.. Vic lat..........12.1...... Bra rut..246.22.4242624.342 Ran fla.2.2..2..2.....42... Hyp rad........2..2.....5.. Leo aut 522.3.33223525222623 Pot pal.........2......2... Poa pra 424.34421.44435.…4.. Cal cus...3...........34... Tri pra....5..2........…2.. Tri rep 521.5.22.163322.6232 Ant odo....3..44.4......4.2 Sal rep.............3.5.3.. Ach mil 3...21.22.4.....…2.. Poa tri 79524246..4.5.6…45.. Ely rep 4.4..4.4....6.4..... Sag pro.25...2....22....34. Pla lan....5..52.33.3..…5.. Agr sto.587..4..4..3.454.4. Lol per 5.5.6742..67226.…6.. Alo gen 2524..5.....3.7...8. Bro hor 4.3....2..4.....…2..

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Species and Sites as Weighted Averages of each other

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Reciprocal Averaging - unimodal Site A B C D E F Species Prunus serotina 6 3 4 6 5 1 Tilia americana 2 0 7 0 6 6 Acer saccharum 0 0 8 0 4 9 Quercus velutina 0 8 0 8 0 0 Juglans nigra 3 2 3 0 6 0

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Reciprocal Averaging - unimodal Site A B C D E F Species Score Species Iteration 1 Prunus serotina 6 3 4 6 5 1 1.00 Tilia americana 2 0 7 0 6 6 0.63 Acer saccharum 0 0 8 0 4 9 0.63 Quercus velutina 0 8 0 8 0 0 0.18 Juglans nigra 3 2 3 0 6 0 0.00 Iteration 1 1.00 0.00 0.86 0.60 0.62 0.99 Site Score

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Reciprocal Averaging - unimodal Site A B C D E F Species Score Species Iteration 1 2 Prunus serotina 6 3 4 6 5 1 1.00 0.68 Tilia americana 2 0 7 0 6 6 0.63 0.84 Acer saccharum 0 0 8 0 4 9 0.63 0.87 Quercus velutina 0 8 0 8 0 0 0.18 0.30 Juglans nigra 3 2 3 0 6 0 0.00 0.67 Iteration 1 1.00 0.00 0.86 0.60 0.62 0.99 Site 2 0.65 0.00 0.88 0.05 0.78 1.00 Score

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Reciprocal Averaging - unimodal Site A B C D E F Species Score Species Iteration 1 2 3 Prunus serotina 6 3 4 6 5 1 1.00 0.68 0.50 Tilia americana 2 0 7 0 6 6 0.63 0.84 0.86 Acer saccharum 0 0 8 0 4 9 0.63 0.87 0.91 Quercus velutina 0 8 0 8 0 0 0.18 0.30 0.02 Juglans nigra 3 2 3 0 6 0 0.00 0.67 0.66 Iteration 1 1.00 0.00 0.86 0.60 0.62 0.99 Site 2 0.65 0.00 0.88 0.05 0.78 1.00 Score 3 0.60 0.01 0.87 0.00 0.78 1.00

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Reciprocal Averaging - unimodal Site A B C D E F Species Score Species Iteration 1 2 3 9 Prunus serotina 6 3 4 6 5 1 1.00 0.68 0.50 0.48 Tilia americana 2 0 7 0 6 6 0.63 0.84 0.86 0.85 Acer saccharum 0 0 8 0 4 9 0.63 0.87 0.91 0.91 Quercus velutina 0 8 0 8 0 0 0.18 0.30 0.02 0.00 Juglans nigra 3 2 3 0 6 0 0.00 0.67 0.66 0.65 Iteration 1 1.00 0.00 0.86 0.60 0.62 0.99 Site 2 0.65 0.00 0.88 0.05 0.78 1.00 Score 3 0.60 0.01 0.87 0.00 0.78 1.00 9 0.59 0.01 0.87 0.00 0.78 1.00

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Reordered Sites and Species Site A C E B D F Species Species Score Quercus velutina 8 8 0 0 0 0 0.004 Prunus serotina 6 3 6 5 4 1 0.477 Juglans nigra 0 2 3 6 3 0 0.647 Tilia americana 0 0 2 6 7 6 0.845 Acer saccharum 0 0 0 4 8 9 0.909 Site Score 0.000 0.008 0.589 0.778 0.872 1.000

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Gradient Length

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Alpha and Beta Diversity alpha diversity is the diversity of a community (either measured in terms of a diversity index or species richness) beta diversity (also known as species turnover or differentiation diversity) is the rate of change in species composition from one community to another along gradients; gamma diversity is the diversity of a region or a landscape.

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A Short Coenocline

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A Long Coenocline

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Arches - Artifact or Feature?

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The Arch Effect What is it? Why does it happen? What should we do about it?

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CA - with arch effect (sites)

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CA - with arch effect (species)

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Long Gradients ABCD

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Gradient End Compression

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CA - with arch effect (species)

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CA - with arch effect (sites)

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Detrending by Segments

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DCA - modified unimodal

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Male or Female?

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Linear Discriminant > gulls.lda <- lda(Sex ~ Wing + Weight + H.and.B + Bill, gulls) lda(Sex ~ Wing + Weight + H.and.B + Bill, data = gulls) Prior probabilities of groups: 0 1 0.5801105 0.4198895 Group means: Wing Weight H.and.B Bill 0 410.0381 871.7619 115.1143 17.62524 1 430.6118 1054.3092 125.9474 19.50789 Coefficients of linear discriminants: LD1 Wing 0.045512619 Weight 0.001887236 H.and.B 0.138127194 Bill 0.444847743

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Discriminating

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Relationship between PCA and LDA

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CVA

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Testing Significance in Ordination

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Randomisation Tests

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Randomisation Example Model: cca(formula = dune ~ Moisture + A1 + Management, data = dune.env) Df Chisq F N.Perm Pr(>F) Model 7 1.1392 2.0007 200 < 0.005 *** Residual 12 0.9761 Signif. codes: 0 *** 0.001 ** 0.01 * 0.05

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