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What we Measure vs. What we Want to KnowPowerPoint Presentation

What we Measure vs. What we Want to Know

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### What we Measure vs. What we Want to Know

### Multivariate Description Hyperspace

### How Similar are Environmental VariablesObjects/Samples/Individuals/Sites?

### Similarity approaches Environmental Variablesor what do we mean by similar?

### Grouping methods Environmental Variables

### Cluster Analysis Environmental Variables

### Some Examples Environmental Variables

"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

Gulls Variables Hyperspace

Scree Plot Hyperspace

Output Hyperspace

> 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.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

Bi-Plot Hyperspace

Environmental Gradients Hyperspace

Indirect Gradient Analysis Hyperspace

- 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 Hyperspace

PCA gradient - site plot Hyperspace

Approaches Hyperspace

- 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? Environmental Variables

Working with the Variability that we Environmental VariablesCan 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 Environmental Variables

- 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 Variables

- 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 Environmental Variables

Dune Data Constrained Environmental Variables

Different types of data Environmental Variables

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 Environmental Variables

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 Environmental Variables

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.

A Environmental Variables

B

A

B

Distance metricsrelevant for continuous variables:

Euclidean

city block or Manhattan

(also many other variations)

0,0 Environmental Variables

1,0

0,1

1,1

0,0

1,0

0,1

1,1

Similarity coefficients for binary datasimple 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

A Distance Matrix Environmental Variables

Uses of Distances Environmental Variables

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 Environmental Variables

Clustering methods Environmental Variables

- hierarchical
- divisive
- put everything together and split
- monothetic / polythetic

- agglomerative
- keep everything separate and join the most similar points (classical cluster analysis)

- divisive
- non-hierarchical
- k-means clustering

Agglomerative hierarchical Environmental Variables

Single linkage or nearest neighbour

finds the minimum spanning tree:

shortest tree that connects all points

- chaining can be a problem

Agglomerative hierarchical Environmental Variables

Complete linkage or furthest neighbour

- compact clusters of approximately equal size.
- (makes compact groups even when none exist)

Agglomerative hierarchical Environmental Variables

Average linkage methods

- between single and complete linkage

From Alexandria to Suez Environmental Variables

Hierarchical Clustering Environmental Variables

Hierarchical Clustering Environmental Variables

Hierarchical Clustering Environmental Variables

Building and testing models Environmental Variables

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.

Part of Fig 4. Environmental Variables

What Technique? Environmental Variables

Raw Data Environmental Variables

Linear Regression Environmental Variables

Two Regressions Environmental Variables

Principal Components Environmental Variables

Models of Species Response Environmental Variables

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 Environmental Variables

Unimodal Environmental Variables

Non-metric multidimensional scaling Environmental Variables

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 Environmental Variables

used to compare graphically two separate ordinations

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