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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
Weight
400 420 440 105 115 125 135
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Wing
Bill
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2022
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H.and.B
Scree Plot
Comp.1 Comp.2 Comp.3 Comp.4
gulls.pca2V
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ces
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Output
> gulls.pca2$loadings
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
> 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
Bi-Plot
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Weight
Wing
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H.and.B
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 1
PCA
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Managmentbiodynamichobbynaturestandard
PCA Plot for Dune Species Data
PCA gradient - site/species biplot
Axis 1
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Ach mil
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J un buf
Leo aut
Lol per
Pla lan
Poa pra
Poa tri
Ran flaRum ace
Sag pro
Tri rep
Bra rut
Biplot for Dune Species Data
standard
nature
biodynamic& hobby
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
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Poapra
Calcus
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Antodo
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How Similar are Objects/Samples/Individuals/Sites?
Similarity approachesor 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 matrixWe 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.
relevant for continuous variables:
Euclidean
city block or Manhattan
Distance metrics
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
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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
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Stoke Ferry
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Driby
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High Muffles
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Cow Green ReservoirBannisdale
Grouping methods
Cluster Analysis
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
Clustering methods
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
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Hierarchical Clustering
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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
Predictors(s)
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
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Linear Regression
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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
Linear
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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
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used to compare graphically two separate ordinations
