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**Multivariate Description**

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**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)

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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 Importance of components:**

> summary(gulls.pca2) Importance of components: Comp Comp Comp Standard deviation Proportion of Variance Cumulative Proportion > gulls.pca2$loadings Loadings: Comp.1 Comp.2 Comp.3 Comp.4 Weight Wing Bill H.and.B

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Bi-Plot

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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: Group means: Wing Weight H.and.B Bill Coefficients of linear discriminants: LD1 Wing Weight H.and.B Bill

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Discriminating

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**Relationship between PCA and LDA**

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CVA

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CVA

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

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**Type of Data Matrix species attributes desert macroph inverts uses**

sites species attributes attributes watervar rain gulls individuals sites

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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 methods Weighted 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 Species (or Attribute) Data**

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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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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 biodynamic & hobby nature

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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 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

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**Direct Gradient Analysis**

cca(species_data ~ e1 + e 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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**Modelling Environmental Variables**

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**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)

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**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)

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**Ways of Evaluating Models**

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**

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 + e 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 + e 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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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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**Distance metrics relevant for continuous variables:**

Euclidean city block or Manhattan 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 0,1 1,1 0,0 1,0 0,1 1,1 0,0 1,0

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**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

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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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**Hierarchical Clustering**

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**Hierarchical Clustering**

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**Summarise by Weighted Averages**

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**Species and Sites as Weighted Averages of each other**

SPP Bel per Jun buf …42.. Jun art Air pra Ele pal Rum ace …23.. Vic lat Bra rut Ran fla Hyp rad Leo aut Pot pal Poa pra …4.. Cal cus Tri pra …2.. Tri rep Ant odo Sal rep Ach mil …2.. Poa tri …45.. Ely rep Sag pro Pla lan …5.. Agr sto Lol per …6.. Alo gen Bro hor …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 Tilia americana Acer saccharum Quercus velutina Juglans nigra

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**Reciprocal Averaging - unimodal**

Site A B C D E F Species Score Species Iteration Prunus serotina Tilia americana Acer saccharum Quercus velutina Juglans nigra Iteration Site Score

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**Reciprocal Averaging - unimodal**

Site A B C D E F Species Score Species Iteration Prunus serotina Tilia americana Acer saccharum Quercus velutina Juglans nigra Iteration Site Score

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**Reciprocal Averaging - unimodal**

Site A B C D E F Species Score Species Iteration Prunus serotina Tilia americana Acer saccharum Quercus velutina Juglans nigra Iteration Site Score

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**Reciprocal Averaging - unimodal**

Site A B C D E F Species Score Species Iteration Prunus serotina Tilia americana Acer saccharum Quercus velutina Juglans nigra Iteration Site Score

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**Reordered Sites and Species**

Site A C E B D F Species Species Score Quercus velutina Prunus serotina Juglans nigra Tilia americana Acer saccharum Site Score

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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 A B C D

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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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**Testing Significance in Ordination**

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Randomisation Tests

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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 < *** Residual Signif. codes: 0 *** ** 0.01 * 0.05

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