1. Multivariate Description

2. 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)

3. Rotate the Variable Space

4. Raw Data

5. Linear Regression

6. Two Regressions

7. Principal Components

8. Gulls Variables

9. Scree Plot

10. Output Importance of components:
> summary(gulls.pca2)
Importance of components:
Comp Comp Comp Standard deviation Proportion of Variance Cumulative Proportion
> gulls.pca2$loadings
Loadings:
Comp.1 Comp.2 Comp.3 Comp.4 Weight Wing Bill H.and.B

11. Bi-Plot

12. Male or Female?

13. Linear Discriminant
> gulls.lda <- lda(Sex ~ Wing + Weight + H.and.B + Bill, gulls)
lda(Sex ~ Wing + Weight + H.and.B + Bill, data = gulls)
Prior probabilities of groups:
Group means:
Wing Weight H.and.B Bill
Coefficients of linear discriminants:
LD1 Wing Weight H.and.B Bill

14. Discriminating

15. Relationship between PCA and LDA

16. CVA

17. CVA

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

19. Type of Data Matrix species attributes desert macroph inverts uses
sites
species
attributes
attributes
watervar
rain
gulls
individuals
sites

20. 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

21. A Theoretical Model

22. Linear

23. Unimodal

24. 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)

25. Inferring Gradients from Species (or Attribute) Data

26. 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

27. PCA - linear model

28. PCA - linear model

29. Terschelling Dune Data

30. PCA gradient - site plot

31. PCA gradient - site/species biplot
standard
biodynamic & hobby
nature

32. Making Effective Use of Environmental Variables

33. 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

34. Ordination Constrained by the Environmental Variables

35. Constrained?

36. 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.

37. 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.

38. Dune Data Unconstrained

39. 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

40. Direct Gradient Analysis
Basic PCA yik = b0k + b1kxi + eik
xi - the sample scores on the ordination axis
b1k - the regression coefficients for each species (the species scores on the ordination axis)
In RDA there is a further constraint on xi xi = c1zi1 + c2zi2
Making yik = b0k + b1kc1zi1 + b1kc2zi2 + eik

41. Direct Gradient Analysis
cca(species_data ~ e1 + e en, data=environmental_data)
cca(dune ~ Manure + Moisture + A1, data=dune.env)

42. Dune Data Constrained

43. Lake Nasser - Egypt

44. Nasser Data Sites – 23 sampling stations on Lake Nasser 3 Data Frames:
Aquatic macrophytes
Invertebrate classes
Water chemistry

45. Lake Nasser Unconstrained

46. Lake Nasser Constrained

47. Modelling Environmental Variables

48. Ways of Building Models
Automated environmental variable selection (stepwise addition or removal of variables from the model – as with multiple regression)
mod0 <- cca(nasser.inverts ~ 1, nasser.watervar)
mod1 <- cca(nasser.inverts ~ ., nasser.watervar)
op <- options(digits=7)
mod <- step(mod0, scope=formula(mod1))
options(op)
mod
plot(mod)

49. Ways of Building Models
Manual selection of environmental variables using prior knowledge (e.g. example starting with full model and removing terms)
mod1 <- cca(nasser.inverts ~ ., nasser.watervar)
mod2 <- cca(nasser.inverts ~ . -WMg, nasser.watervar)
mod3 <- cca(nasser.inverts ~ . -WMg -WEC, nasser.watervar)
mod4 <- cca(nasser.inverts ~ . -WMg -WEC -WCa, nasser.watervar)

50. Ways of Evaluating Models
Graphically using Procrustes Rotation
plot(procrustes(mod2, mod1))
plot(procrustes(mod3, mod2))
plot(procrustes(mod4, mod3))
plot(procrustes(mod4, mod1))

51. Procrustes

52. Ways of Evaluating Models
Permutation Tests can be used to assess adequacy of the models using a Pseudo ANOVA or Permutest
anova(mod1)
anova(mod2)
anova(mod3)
anova(mod4)
permutest.cca(mod1, perm=1000)
permutest.cca(mod2, perm=1000)
permutest.cca(mod3, perm=1000)
permutest.cca(mod4, perm=1000)

53. Removing the Effect of Nuisance Variables

54. Getting rid of the Variability that is Not of Interest
Amongst the explanatory variables there may be variability attributable to:
Blocks and other design strata
Covariates that we can measure but are not the focus of interest
We may want to use only the variability attributable to:
Meaningful Environmental Variables

55. 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.

56. Lichen-rich Forest Understorey

57. Forest Data Sites – 28 sites in forests in Finland grazed by reindeer
Species Data – 44 heathland plant species (including many lichens and mosses that are very sensitive to their chemical environment)
Environmental Data – Soil chemical composition (N P K Ca Mg S Al Fe Mn Zn Mo Baresoil Humdepth pH)

58. CCA

59. Removing pH Effect
cca(species_data ~ e1 + e en + Condition(e5), data=environmental_data)
cca(varespec ~ Al + P + K + Baresoil + Condition(pH), data=varechem)

60. Removing pH Effect

61. Interactions in Models
cca(species_data ~ e1 + e en + Condition(e5), data=environmental_data)
cca(varespec ~ Al + P*(K + Baresoil) + Condition(pH), data=varechem)

62. CCA

63. Removing pH Effect

64. Cluster Analysis

65. 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

66. 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.

67. Distance metrics relevant for continuous variables:
Euclidean
city block or Manhattan A B A B
(also many other variations)

68. A Distance Matrix

69. Uses of Distances Distance/Dissimilarity can be used to:-
Explore dimensionality in data (using PCO)
As a basis for clustering/classification

70. UK Wet Deposition Network

71. Fitting Environmental Variables

72. A Map based on Measured Variables

73. Fitting Environmental Variables

74. 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)
simple matching can be extended to categorical data
0,1
1,1
0,0
1,0
0,1
1,1
0,0
1,0

75. 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

76. Agglomerative hierarchical
Single linkage or nearest neighbour
finds the minimum spanning tree:
shortest tree that connects all points
chaining can be a problem

77. Agglomerative hierarchical
Complete linkage or furthest neighbour
compact clusters of approximately equal size.
(makes compact groups even when none exist)

78. Agglomerative hierarchical
Average linkage methods
between single and complete linkage

79. From Alexandria to Suez

80. Hierarchical Clustering

81. Hierarchical Clustering

82. Hierarchical Clustering

83. Summarise by Weighted Averages

84. Species and Sites as Weighted Averages of each other
SPP Bel per Jun buf …42.. Jun art Air pra Ele pal Rum ace …23.. Vic lat Bra rut Ran fla Hyp rad Leo aut Pot pal Poa pra …4.. Cal cus Tri pra …2.. Tri rep Ant odo Sal rep Ach mil …2.. Poa tri …45.. Ely rep Sag pro Pla lan …5.. Agr sto Lol per …6.. Alo gen Bro hor …2..

85. Species and Sites as Weighted Averages of each other

86. Reciprocal Averaging - unimodal
Site A B C D E F Species Prunus serotina Tilia americana Acer saccharum Quercus velutina Juglans nigra

87. Reciprocal Averaging - unimodal
Site A B C D E F Species Score Species Iteration Prunus serotina Tilia americana Acer saccharum Quercus velutina Juglans nigra
Iteration Site Score

88. Reciprocal Averaging - unimodal
Site A B C D E F Species Score Species Iteration Prunus serotina Tilia americana Acer saccharum Quercus velutina Juglans nigra
Iteration Site Score

89. Reciprocal Averaging - unimodal
Site A B C D E F Species Score Species Iteration Prunus serotina Tilia americana Acer saccharum Quercus velutina Juglans nigra
Iteration Site Score

90. Reciprocal Averaging - unimodal
Site A B C D E F Species Score Species Iteration Prunus serotina Tilia americana Acer saccharum Quercus velutina Juglans nigra
Iteration Site Score

91. Reordered Sites and Species
Site A C E B D F Species Species Score Quercus velutina Prunus serotina Juglans nigra Tilia americana Acer saccharum Site Score

92. Gradient Length

93. 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.

94. A Short Coenocline

95. A Long Coenocline

96. Arches - Artifact or Feature?

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

98. CA - with arch effect (sites)

99. CA - with arch effect (species)

100. Long Gradients A B C D

101. Gradient End Compression

102. CA - with arch effect (species)

103. CA - with arch effect (sites)

104. Detrending by Segments

105. DCA - modified unimodal

106. Testing Significance in Ordination

107. Randomisation Tests

108. Randomisation Tests

109. Randomisation Example
Model: cca(formula = dune ~ Moisture + A1 + Management, data = dune.env)
Df Chisq F N.Perm Pr(>F)
Model < ***
Residual
Signif. codes: 0 *** ** 0.01 * 0.05