Multivariate Description

What Technique? Response variable(s) ... Predictors(s) No 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)

Rotate the Variable Space

Raw Data

Linear Regression

Two Regressions

Principal Components

Gulls Variables

Scree Plot

Output Importance of components: > 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 > 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

Bi-Plot

Male or Female?

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

Discriminating

Relationship between PCA and LDA

CVA

CVA

Managing Dimensionality (but not acronyms) PCA, CA, RDA, CCA, MDS, NMDS, DCA, DCCA, pRDA, pCCA

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

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

A Theoretical Model

Linear

Unimodal

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

Inferring Gradients from Species (or Attribute) Data

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

PCA - linear model

PCA - linear model

Terschelling Dune Data

PCA gradient - site plot

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

Making Effective Use of Environmental Variables

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

Ordination Constrained by the Environmental Variables

Constrained?

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.

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.

Dune Data Unconstrained

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

Direct Gradient Analysis Basic PCA yik = b0k + b1kxi + eik xi - the sample scores on the ordination axis b1k - the regression coefficients for each species (the species scores on the ordination axis) In RDA there is a further constraint on xi xi = c1zi1 + c2zi2 Making yik = b0k + b1kc1zi1 + b1kc2zi2 + eik

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

Dune Data Constrained

Lake Nasser - Egypt

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

Lake Nasser Unconstrained

Lake Nasser Constrained

Modelling Environmental Variables

Ways of Building Models 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)

Ways of Building Models 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)

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

Procrustes

Ways of Evaluating Models 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)

Removing the Effect of Nuisance Variables

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

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.

Lichen-rich Forest Understorey

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)

CCA

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

Removing pH Effect

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

CCA

Removing pH Effect

Cluster Analysis

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

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.

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

A Distance Matrix

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

UK Wet Deposition Network

Fitting Environmental Variables

A Map based on Measured Variables

Fitting Environmental Variables

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 0,1 1,1 0,0 1,0 0,1 1,1 0,0 1,0

Clustering methods hierarchical non-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

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

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

Agglomerative hierarchical Average linkage methods between single and complete linkage

From Alexandria to Suez

Hierarchical Clustering

Hierarchical Clustering

Hierarchical Clustering

Summarise by Weighted Averages

Species and Sites as Weighted Averages of each other 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..

Species and Sites as Weighted Averages of each other

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

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

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

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

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

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

Gradient Length

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.

A Short Coenocline

A Long Coenocline

Arches - Artifact or Feature?

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

CA - with arch effect (sites)

CA - with arch effect (species)

Long Gradients A B C D

Gradient End Compression

CA - with arch effect (species)

CA - with arch effect (sites)

Detrending by Segments

DCA - modified unimodal

Testing Significance in Ordination

Randomisation Tests

Randomisation Tests

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