What we Measure vs. What we Want to Know "Not everything that counts can be counted, and not everything that can be counted counts." - Albert Einstein

Scales, Transformations, Vectors and Multi-Dimensional Hyperspace All measurement is a proxy for what is really of interest - The Relationship between them The scale of measurement and the scale of analysis and reporting are not always the same - Transformations We often make measurements that are highly correlated - Multi-component Vectors

Multivariate Description

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

Environmental Gradients

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

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

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

Dimension Reduction (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.

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

Dune Data Unconstrained

Dune Data Constrained

How Similar are Objects/Samples/Individuals/Sites?

Similarity approaches or what do we mean by similar?

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

Different scales of measurement example Large Range : soil ion concentrations Restricted Range : air pressure Constrained : proportions Large numbers : altitude Small numbers : attribute counts Do we standardise measurement scales to make them equivalent? If so what do we lose?

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)

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, eg Bray-Curtis) simple matching can be extended to categorical data 0,1 1,1 0,0 1,0 0,1 1,1 0,0 1,0

A Distance Matrix

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

UK Wet Deposition Network

Shown with Environmental Variables

A Map based on Measured Variables

Fitting Environmental Variables

Grouping methods

Discriminating If you have continuous measurements and you know which 2 groups you are looking for (e.g. male and female in the gulls data), linear discriminant analysis will find a function of the measurements which will help to allocate new subjects to the groups

Canonical Variate Analysis For more than 2 groups canonical variate analysis maximises the between group to within group variances – this is related to a multivariate analysis of variance (MANOVA)

Cluster Analysis

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

Building and testing models Basically you just approach this in the same way as for multiple regression – so there are the same issues of variable selection, interactions between variables, etc. However the basis of any statistical tests using distributional assumptions are more problematic, so there is much greater use of randomisation tests and permutation procedures to evaluate the statistical significance of results.

Some Examples

Part of Fig 4.

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)

Raw Data

Linear Regression

Two Regressions

Principal Components

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

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)

Non-metric multidimensional scaling NMDS maps the observed dissimilarities onto an ordination space by trying to preserve their rank order in a low number of dimensions (often 2) – but the solution is linked to the number of dimensions chosen it is like a non-linear version of PCO define a stress function and look for the mapping with minimum stress (e.g. sum of squared residuals in a monotonic regression of NMDS space distances between original and mapped dissimilarities) need to use an iterative process, so try with many different starting points and convergence is not guaranteed

Procrustes rotation used to compare graphically two separate ordinations