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

Type of Data Matrix

Ordination Techniques Linear methods Weighted averaging unconstrained Principal Components Analysis (PCA) Correspondence Analysis (CA) constrained Redundancy Analysis (RDA) Canonical Correspondence Analysis (CCA)

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

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

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

Reciprocal Averaging 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 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 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 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 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

Arches - Artifact or Feature?

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

From Alexandria to Suez

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

Making Effective Use of Environmental Variables

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

CCA - site/species joint plot

CCA - species/environment biplot

Removing the Effect of Nuisance 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.