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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

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Ordination Techniques Linear methodsWeighted averaging unconstrainedPrincipal Components Analysis (PCA) Correspondence Analysis (CA) constrainedRedundancy Analysis (RDA) Canonical Correspondence Analysis (CCA)

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

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Reciprocal Averaging Site A B C D E F Species Score Species Iteration 1 Prunus serotina Tilia americana Acer saccharum Quercus velutina Juglans nigra Iteration Site Score

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Reciprocal Averaging Site A B C D E F Species Score Species Iteration 1 2 Prunus serotina Tilia americana Acer saccharum Quercus velutina Juglans nigra Iteration Site Score

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Reciprocal Averaging 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 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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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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From Alexandria to Suez

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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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Making Effective Use of Environmental Variables

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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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CCA - site/species joint plot

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CCA - species/environment biplot

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Removing the Effect of Nuisance 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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