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Multivariate Description. What Technique?. Raw Data. Linear Regression. Two Regressions. Principal Components. Gulls Variables. Scree Plot. Output. > summary(gulls.pca2) Importance of components:

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> 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


Comp.1 Comp.2 Comp.3 Comp.4Weight -0.505 -0.343 0.285 0.739Wing -0.490 0.852 -0.143 0.116Bill -0.500 -0.381 -0.742 -0.232H.and.B -0.505 -0.107 0.589 -0.622

models of species response
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
indirect gradient analysis
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 gradient site species biplot
PCA gradient - site/species biplot


biodynamic& hobby


  • 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
working with the variability that we can explain
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
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
different types of data
Different types of data


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
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





Distance metrics

relevant for continuous variables:


city block or Manhattan

(also many other variations)

similarity coefficients for binary data









Similarity coefficients for binary data

simple matching

count if both units 0 or both units 1


count only if both units 1

(also many other variants, eg Bray-Curtis)

simple matching can be extended to categorical data

uses of distances
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
non metric multidimensional scaling
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
Procrustes rotation

used to compare graphically two separate ordinations

clustering methods
Clustering methods
  • 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
Agglomerative hierarchical

Single linkage or nearest neighbour

finds the minimum spanning tree:

shortest tree that connects all points

  • chaining can be a problem
agglomerative hierarchical1
Agglomerative hierarchical

Complete linkage or furthest neighbour

  • compact clusters of approximately equal size.
  • (makes compact groups even when none exist)
agglomerative hierarchical2
Agglomerative hierarchical

Average linkage methods

  • between single and complete linkage
building and testing models
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.


Positive Matrix Factorisation