Multivariate Analysis in Ecology
I: Unconstrained Ordination
Jari Oksanen
Oulu
January 2016
http://cc.oulu.fi/ jarioksa/ (Oulu)
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Introduction
January 2016
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What is Ordination?
Multivariate Analysis and Ordination
Basic ordination methods to simplify multivariate data into low dimensional
graphics
Analysis of multivariate dependence and hypotheses
Analyses can be performed in R statistical software using vegan package and
allies
Course homepage http://cc.oulu.fi/~jarioksa/opetus/metodi/
Vegan homepage https://github.com/vegandevs/vegan/
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January 2016
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Introduction
What is Ordination?
Outline
1
Introduction
What is Ordination?
Gradient Analysis
2
Unconstrained Ordination
NMDS
Eigenvector Methods
PCA
CA
Graphics
Environmental Variables
Gradient Model and Ordination
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Introduction
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What is Ordination?
Why Ordination?
Nobody should want to make an ordination, but they are desperate with
multivariate data
Map multidimensional table into low-dimensional display
Cluster Dendrogram
−1.0
14
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1.0
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13
7
6
18
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21
1.5
DCA1
http://cc.oulu.fi/ jarioksa/ (Oulu)
9
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5
24
25
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713
4
−1.0
15 22
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23
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11
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28
27
19
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109
2 12
Height
0.5
0.0
DCA2
1.0
21
25
5
0.7
Ordination
uh.dis
hclust (*, "average")
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January 2016
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Introduction
What is Ordination?
Two Ways of Analysing Data
I
II
Blinacu
5
6
7
III
IV
8
Bryupse
V
5
Chilpol
6
7
8
Dichfal
Response
1.0
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0.0
Fisspus
Fontant
Fontdal
Hygralp
Hygroch
Jungpum
Leptrip
Marsema
1.0
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0.6
0.4
0.2
0.0
1.0
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Palufal
Scapund
Schiaga
Schiriv
1.0
0.8
0.6
0.4
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0.0
5
6
7
8
5
6
7
8
pH
1.5
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Multivariate Analysis in Ecology
Introduction
0.0
0.5
1.0
Gradient Analysis
Dim2
January 2016
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Gradient Analysis
Blinacu
Hygralp
Marsema
Hygrsmi
Bryupse
Scapund
Dichfal
Schiaga
Schiriv
Jungpum
Conduc
Hygrlur
Fisspus
Chilpol
slope Callgig Palufal
Amblflu
Jungexs
Instab
pH
width
−0.5
Fontdal
Gradient Analysis developed Partsize
in 1950s in USA,
with R. H. Whittaker as the
Currvel
Fontant
Depth
main founding father
Platrip
Leptrip
−1.0
Only two or three environmental variables,
or Gradients needed to explain
Hygroch
complicated community patterns
−1.5
Against classification: Species responses smooth along gradients
Against organism analogies: Species responses individualistic
−1.5 theory
−1.0 and
−0.5 praxis:
0.0 Ordination
0.5
1.0 and
1.5 Gradient
2.0
The basis of modern
modelling of
communities
Dim1
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Introduction
Gradient Analysis
The Gradient Model
R. H. Whittaker
(1956)
Smoky
Mountains.
http://cc.oulu.fi/ jarioksa/ (Oulu)
Vegetation
of
The
Ecological
Monographs
26,
Multivariate Analysis in Ecology
Introduction
January 2016
Great
1–80.
7 / 103
Gradient Analysis
Types of Gradients
1
Direct gradients: Influence organims but are not consumed.
Correspond to conditions.
2
Resource gradients: Consumed
Correspond to resources.
Complex gradients. Covarying direct and/or resource gradients: Impossible
to separate effects of single gradients.
Most observed gradients.
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Introduction
Gradient Analysis
Landscapes and Gradients
D
Gradient space
C
A
Landscape
A
C
D
B
B
D
B
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Introduction
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Gradient Analysis
Species responses
0.8
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Response
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Response
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Species have non-linear responses along gradients.
Often assumed to be Gaussian. . .
4.5
5.0
http://cc.oulu.fi/ jarioksa/ (Oulu)
5.5
6.0
6.5
7.0
Multivariate Analysis in Ecology
7.5
8.0
4.5
January 2016
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Introduction
Gradient Analysis
Gaussian Response Function
Height of the response h in the
units of response height µ
++++++++ +++++ ++++++++++++ +
0.8
++
++
 (x − u)2
µ = h × exp−

2t2 

0.6
h
0.2
0.4
t
0.0
Width of the response t in the units
of gradient x
BAUERUBI
Location of the optimum u on the
gradient x
1.0
Gaussian Response Function has three interpretable parameters that define the expected response µ along the gradient x
u
+ + ++
+++++
++ ++ +++++ ++++++
++++++++++++++++++++++++++ + ++++++
1000
1100
1200
1300
Altitude
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Introduction
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Gradient Analysis
Dream of species packing
Species have Gaussian responses and divide the gradient optimally:
Equal heights h.
0.8
0.4
0.0
Response
Equal widths t.
Evenly distributed optima u.
0
1
2
3
4
5
6
Gradient
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Introduction
Gradient Analysis
Evidence for Gaussian Responses
Whittaker reported a large number
of different response types
Only a small proportion were
symmetric, bell shaped responses
Still became the standard of our
times
Comparison of ordination methods
based on simulation, and many of
those use Gaussian responses
We need to use simulation because
then we know the truth that should
be found
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Unconstrained Ordination
Ordination
Ordination maps multivariate data onto low dimensional displays: “Most data
sets have 2.5 dimensions”
Gradients define vegetation: ordination tries to find the underlying gradients
Basic ordination uses only community composition: Indirect Gradient Analysis
Constrained ordination studies only the variation that can be explained by the
available environmental variables: Often called Direct Gradient Analysis
Distinct flavours of tools:
Nonmetric MDS the most robust method
PCA duly despised
Flavours of Correspondence Analysis popular
Canonical method: Constrained Correspondence Analysis
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Unconstrained Ordination
NMDS
Nonmetric Multidimensional Scaling
Rank-order relation with (1) community dissimilarities and (2) ordination
distances: No specified form of regression, but the the best shape is found
from the data.
Non-linear regression can cope with non-linear species responses of various
shapes: Not dependent on Gaussian model.
Iterative solution: No guarantee of convergence.
Must be solved separately for each number of dimensions: A lower
dimensional solutions is not a subset of a higher, but each case is solved
individually.
A test winner, and a natural choice. . .
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Unconstrained Ordination
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NMDS
From Ranks of Dissimilarities to Ordination Distances
Observed dissimilarities:
A
B
C
D
E
B 0.467
C 0.636 0.511
D 0.524 0.356 0.634
E 0.843 0.600 0.753 0.513
F 0.922 0.893 0.852 0.667 0.606
Observed rank: Ordination distance
0.2
E
A
1: 0.133
2: 0.301
0.0
D
5: 0.323
−0.1
B
9: 0.539
4: 0.307
6: 0.414
15: 0.922
11: 0.612
−0.2
7: 0.416
10: 0.605
14: 0.636
F
8: 0.421
3: 0.303
−0.3
NMDS2
0.1
12: 0.615
13: 0.636
Ranks of observed dissimilarities:
A B C D E
B 2
C 9 3
D 5 1 8
E 12 6 11 4
F 15 14 13 10 7
C
−0.4
−0.2
0.0
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NMDS1
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Multivariate Analysis in Ecology
Ordination distances:
A
B
C
D
E
B 0.301
C 0.539 0.303
D 0.323 0.133 0.421
E 0.615 0.414 0.612 0.307
F 0.922 0.636 0.636 0.605 0.416
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Unconstrained Ordination
NMDS
MDS is a map
Map of Europe from road distances
Stockholm
MDS tries to draw a map using
distance data.
●
●
Copenhagen
MDS tries to find an underlying
configuration from
dissimilarities.
Only the configuration counts:
●
Hamburg
●
Hook of Holland
●
●
● Cologne
Calais
Brussels
Cherbourg
●
●
Paris
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●
Munich Vienna
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Lyons
Geneva
No origin, but only the
constellations.
No axes or natural directions,
but only a framework for
points.
Lisbon
Madrid
●
Milan
●
Marseilles
Barcelona
●
●
●
Rome
●
Gibraltar
●
Athens
●
Lambert conical conformal projection
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NMDS
Shepard Diagram
●
3.0
Non−metric fit, R2 = 0.967
Linear fit, R2 = 0.835
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Ordination Distance
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Observed Dissimilarity
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Multivariate Analysis in Ecology
Unconstrained Ordination
NMDS
Iterative Optimization
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Unconstrained Ordination
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NMDS
Recommended procedure
NMDS may be good, but its use needs special care: Not every NMDS
automatically is good
1
Use adequate dissimilarity indices: An adequate index gives a good rank-order
relation between community dissimilarity and gradient distance.
2
No convergence guaranteed: Start with several random starts and inspect
those with lowest stress.
3
Satisfied only if minimum stress configurations are similar.
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Unconstrained Ordination
NMDS
metaMDS I
> vare.mds <- metaMDS(varespec)
Square root transformation
Wisconsin double standardization
Run 0 stress 0.184
Run 1 stress 0.196
Run 2 stress 0.185
... procrustes: rmse 0.0494 max resid 0.158
Run 3 stress 0.209
Run 4 stress 0.215
Run 5 stress 0.235
Run 6 stress 0.196
Run 7 stress 0.234
Run 8 stress 0.196
Run 9 stress 0.222
Run 10 stress 0.185
Run 11 stress 0.195
Run 12 stress 0.229
Run 13 stress 0.184
... New best solution
... procrustes: rmse 3.6e-05 max resid 0.000139
*** Solution reached
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NMDS
metaMDS II
> vare.mds
Call:
metaMDS(comm = varespec)
global Multidimensional Scaling using monoMDS
Data:
wisconsin(sqrt(varespec))
Distance: bray
Dimensions: 2
Stress:
0.184
Stress type 1, weak ties
Two convergent solutions found after 13 tries
Scaling: centring, PC rotation, halfchange scaling
Species: expanded scores based on ‘wisconsin(sqrt(varespec))’
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Multivariate Analysis in Ecology
Unconstrained Ordination
NMDS
Plot metaMDS
0.5
●
+
+
+
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0.0
++
+
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NMDS2
+
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+ +
+
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+
●
+
+
−1.0
+
−0.5
0.0
0.5
1.0
NMDS1
http://cc.oulu.fi/ jarioksa/ (Oulu)
> plot(vare.mds)
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Unconstrained Ordination
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NMDS
Numbers
Badness of fit measure stress is based on the residuals from the non-linear
regression
A proportional measure in the range 0 (perfect) . . . 1 (desperate) related to
goodness of fit measure 1 − R 2
Random configuration typically ≈ 0.4 and 0 degenerate
Often given in percents (but omitting the percent sign: 15 = 0.15, since
cannot be > 1)
Orientation, rotation, scale and origin of the coordinates (scores) are
indeterminate: only the constellation matters
Vegan arbitrarily fixes some of these:
Axes are centred, but the origin has no special meaning
Axes are rotated so that the first is the longest (technically: rotated to
principal components)
Axes are scaled so that one unit corresponds to halving of similarity from the
“replicate similarity”
The sign (direction) of the axes still undefined
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Unconstrained Ordination
NMDS
Half-change Scaling in NMDS
1.2
+
0.8
0.6
+
+ ++
+
+ +
0.0
Linear area of ordination
distance – dissimilarity: 0 . . . 0.8
0.4
Maximum dissimilarity = 1:
nothing in common
+
++++
+
+
+
+ + +++
+
+ +
++ ++ +++ ++ +
Threshold
+
+
+
+ ++
++ + +++++ +
+
++++
+
+
+
+
+
+
+
++ ++++++++++
++++
+
++ +++
Half−change
+
+
+
+
+ + +++++++++ +
+
++ ++++++++
+
+ +
+ +++
+++++
++++++ +
+++
+
+
+++++
++ ++ +
++
+ ++
++ ++
++ ++
+ + + +++
+ ++ + ++
Replicate
dissimilarity
++ +
++
0.2
Replicate similarity: dissimilarity
at ordination distance = 0
Community dissimilarity
1.0
+
+
+
0.0
0.5
1.0
1.5
2.0
Ordination distance
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Unconstrained Ordination
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NMDS
What happened in metaMDS?
1
Square root transformation and Wisconsin double standardization
2
Bray–Curtis dissimilarities
3
monoMDS with several random starts and stopping after finding two identical
minimum stress solutions
4
Solution rotated to PCs
5
Solution scaled to half-change units
6
Species scores as weighted averages
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Unconstrained Ordination
NMDS
0.6
4.5
5.0
5.5
6.0
6.5
7.0
7.5
8.0
6.5
7.0
7.5
8.0
0.2
Response
0.8
1.0
pH
0.0
Should use dissimilarities which
reach their maximum (1) when no
species are shared (like those listed
above)
Indices with no bound maximum are
usually bad (Euclidean distance etc.)
0.4
Dissimilarity
0.2
0.0
Wisconsin double standardization
often helpful
0.6
Bray–Curtis (Steinhaus), Jaccard,
Kulczyński
0.4
Use a dissimilarity that describes
correctly gradient separation
0.8
1.0
Dissimilarity measures
4.5
5.0
5.5
6.0
pH
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Unconstrained Ordination
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NMDS
Procrustes rotation
Procrustes rotation to maximal similarity between two configurations:
Translate the origin.
Rotate the axes.
Deflate or inflate the axis scale.
Single points can move a lot, although the stress is fairly constant: Especially
in large data sets.
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Unconstrained Ordination
NMDS
Procrustes Rotation
>
>
>
>
>
tmp <- wisconsin(sqrt(varespec))
dis <- vegdist(tmp)
vare.mds0 <- monoMDS(dis, trace = 0)
pro <- procrustes(vare.mds, vare.mds0)
pro
Call:
procrustes(X = vare.mds, Y = vare.mds0)
Procrustes sum of squares:
0.186
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January 2016
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NMDS
Plot Procrustes Rotations
0.4
0.6
Procrustes errors
●
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0.0
Dimension 2
0.2
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−0.2
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●
●
−0.4
●
●
−0.4
−0.2
0.0
0.2
0.4
0.6
Dimension 1
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Multivariate Analysis in Ecology
Unconstrained Ordination
NMDS
Number of dimensions
In NMDS, 2D solution is not a plane in 3D space
Solution must be found separately for each dimensionality
Some people very disturbed: how do they know the correct number
Answer is easy: there is no correct number, although some numbers may be
worse than others
“Most data sets have 2.5 dimensions”
Typically you try with 2 and 3
Do you need more dimensions to explain species patterns and environmental
data?
Is convergence very slow? Try another number of dimensions
Scree plot or stress against the number of dimensions often suggested but
rarely works
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January 2016
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NMDS
Scree Plot
0.15
stress
0.20
0.25
●
0.10
●
0.05
●
●
●
●
1
2
3
4
5
6
No. of Dimensions
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Unconstrained Ordination
Eigenvector Methods
Simplified mapping: Eigen analysis
NMDS uses non-linear mapping for any dissimilarity measure: This is very
difficult
Things are much simpler if we accept only certain dissimilarity indices and
map them linearly onto ordination
Linear mapping is only a rotation, and can be solved using eigenvector
techniques
Sometimes said that certain methods are model-based (CA), but they also
employ a distance
method
metric
mapping
NMDS
MDS
PCA
CA
any
any
Euclidean
Chi-square
nonlinear
linear
linear
weighted linear
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Eigenvector Methods
Why Not PCA?
We admit that PCA is just a rotation, but it is a linear method
PCA works with species space, but we boldly go to gradient space
CA is an optimal scaling method
Sites with similar species composition packed close to each other
Species that occur together simultaneously packed close to each other
CA can handle unimodal species responses, even approximate one
dimensional species packing model
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Unconstrained Ordination
PCA
Species space
60
20
What is the ideal viewing angle to
the species space?
40
Cla.ste
Some species show more of the
configuration than others
80
Graphical presentations of data
matrix: Species are axes and span
the space where sites are points
0
Shows as much as possible of all
species in just two or three
dimensions
0
10
20
30
40
50
60
Cla.ran
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PCA
3
Rotate the axes so that the first axis
(1) is as close to all points as
possible, and (2) explains as much
of the variance as possible
20
Move the origin to the centroid
Cla.arb
2
10
Put sites into species space
0
1
30
40
Rotation in species space
0
10
20
30
40
50
60
Cla.ran
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Unconstrained Ordination
PCA
Goodness of Fit
The total variation (Λ) is the sum of squared distances of points from the
origin
Λ can be expressed as the sum of squares (SS) or variance (SS/n or
SS/(n − 1))
The points are projected on the axis, and the sum of projected squared
distances is the eigenvalue of the axis (λi )
The eigenvalues are ordered and
Ppnon-negative λ1 ≥ λ2 ≥ · · · ≥ λp ≥ 0, and
sum up to total variance Λ = i=1 λi
λi /Λ gives the proportion that an axis explains of the total variance, and λ1
explains the largest proportion
Cumulative sum gives the proportion of variance explained by the first axes:
often emphasized but rather useless statistic
PCA is often used to reduce data into a few linearly independent components
that explain the most of the original variables
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PCA
∑ xijxik
40
40
Euclidean Metric of PCA
+
cosθjk =
Explained
10
sk
+
30
40
50
60
0
http://cc.oulu.fi/ jarioksa/ (Oulu)
djk =
∑ (xij − xik)2
i
θjk
+ + s=
+++
j
+
+
++
+
+
+
Cla.ran
+
+
+
+
+
10
++
20
+
+
0
10
0
+
+
+
+ +
++
+
+
+
+ ++ +
+k
∑ x2ij ∑ x2ik
i
i
30
+
+
+
+
i
20
20
+
i=2 (Cla.arb)
Total
+
0
Cla.arb
30
+
Residual
10
20
∑ x2ij
++ j
i
+
30
40
50
60
i=1 (Cla.ran)
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January 2016
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Unconstrained Ordination
PCA
Running PCA I
> (ord <- rda(dune))
Call: rda(X = dune)
Inertia Rank
Total
84.1
Unconstrained
84.1
19
Inertia is variance
Eigenvalues for unconstrained axes:
PC1
PC2
PC3
PC4
PC5 PC6
PC7
PC8
24.80 18.15 7.63 7.15 5.70 4.33 3.20 2.78
(Showed only 8 of all 19 unconstrained eigenvalues)
> head(summary(ord), 3, 1)
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Unconstrained Ordination
January 2016
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January 2016
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PCA
Running PCA II
Call:
rda(X = dune)
Partitioning of variance:
Inertia Proportion
Total
84.1
1
Unconstrained
84.1
1
Eigenvalues, and their contribution to the variance
Importance of components:
PC1
PC2
PC3
PC4
PC5
PC6
Eigenvalue
24.795 18.147 7.6291 7.153 5.6950 4.3333
Proportion Explained
0.295 0.216 0.0907 0.085 0.0677 0.0515
Cumulative Proportion 0.295 0.510 0.6011 0.686 0.7539 0.8054
PC7
PC8
PC9 PC10
PC11
PC12
Eigenvalue
3.199 2.7819 2.4820 1.854 1.7471 1.3136
Proportion Explained 0.038 0.0331 0.0295 0.022 0.0208 0.0156
Cumulative Proportion 0.843 0.8765 0.9060 0.928 0.9488 0.9644
PC13
PC14
PC15
PC16
PC17
Eigenvalue
0.9905 0.63779 0.55083 0.35058 0.19956
Proportion Explained 0.0118 0.00758 0.00655 0.00417 0.00237
http://cc.oulu.fi/ jarioksa/ (Oulu)
Multivariate Analysis in Ecology
Unconstrained Ordination
PCA
Running PCA III
Cumulative Proportion 0.9762 0.98377 0.99032 0.99448 0.99686
PC18
PC19
Eigenvalue
0.14880 0.11575
Proportion Explained 0.00177 0.00138
Cumulative Proportion 0.99862 1.00000
Scaling 2 for species and site scores
* Species are scaled proportional to eigenvalues
* Sites are unscaled: weighted dispersion equal on all dimensions
* General scaling constant of scores: 6.3229
Species scores
PC1
PC2
PC3
PC4
PC5
PC6
Achimill -0.6038 0.124 0.00846 0.160 0.4087 0.1279
Agrostol 1.3740 -0.964 0.16691 0.266 -0.0877 0.0474
Airaprae 0.0234 0.251 -0.19477 -0.326 0.0557 -0.0796
....
Callcusp 0.5385 0.180 0.17509 0.239 0.2553 0.1692
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Unconstrained Ordination
January 2016
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January 2016
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PCA
Running PCA IV
Site scores (weighted sums of species scores)
1
2
3
....
20
PC1
PC2
PC3
PC4
PC5
PC6
-0.857 -0.172 2.608 -1.130 0.4507 -2.4911
-1.645 -1.230 0.887 -0.986 2.0346 1.8106
-0.440 -2.383 0.930 -0.460 -1.0278 -0.0518
2.341
1.299 0.903
http://cc.oulu.fi/ jarioksa/ (Oulu)
0.718 -0.0757 -0.9691
Multivariate Analysis in Ecology
Unconstrained Ordination
PCA
Row and Column scores
The scores are centred (= their mean is zero) and either normalized (= all
have equal spread) or proportional to eigenvalues (= spread is higher when
eigenvalue is high)
Normalized scores give the regression coefficients between the axis and the
variables: often used for species
Scores proportional to the eigenvalue give the true configuration of points in
the space defined by normalized scores: often used for sites (hence in species
space)
Together these scores give a linear least square approximation of the data
Graphical presentation called biplot
However, there are many alternative scaling systems
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Unconstrained Ordination
January 2016
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January 2016
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PCA
Default Plot
2
19
18
17
20
1
11
14
10
0
7 5
PC2
15
6
Anthodor
Planlanc
Bracruta
Salirepe
ScorautuHyporadi
Airaprae
Trifprat Empenigr Callcusp
Achimill
Vicilath
Comapalu
Rumeacet
Ranuflam
Trifrepe Chenalbu
Juncarti
Cirsarve
1 Bromhord
Bellpere
Lolipere
Eleopalu
16
Sagiproc
Juncbufo
Poaprat
−1
Elymrepe
Agrostol
8
2
Alopgeni
Poatriv
12
−2
9
4
13
3
−2
−1
0
1
2
3
PC1
http://cc.oulu.fi/ jarioksa/ (Oulu)
Multivariate Analysis in Ecology
Unconstrained Ordination
PCA
Reading the Plot
Origin: all species (variables) at their average values
The distance from the origin for a row (site) implies how much the point
differs from the average
The distance from the origin for a column (species, variable) implies how
much the point increases to that direction
The change is measured in absolute scale: big changes, long distances from
the origin
Implies a linear model of species response against axes
The angle between two points implies correlations
90◦ means zero correlation, < 90◦ positive correlation, > 90◦ negative
correlation, 0◦ implies r = 1
Arrow biplots often used instead of point biplot
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Unconstrained Ordination
January 2016
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PCA
Arrow Biplot
2
19
18
17
20
1
11
14
10
0
7 5
PC2
15
6
Anthodor
Planlanc
Bracruta
Salirepe
ScorautuHyporadi
Airaprae
Trifprat Empenigr Callcusp
Achimill
Vicilath
Comapalu
Rumeacet
Ranuflam
Trifrepe Chenalbu
Juncarti
Cirsarve
1 Bromhord
Bellpere
Lolipere
Eleopalu
16
Sagiproc
Juncbufo
Poaprat
−1
Elymrepe
Agrostol
8
2
Alopgeni
Poatriv
12
−2
9
4
13
3
−2
−1
0
1
2
3
PC1
http://cc.oulu.fi/ jarioksa/ (Oulu)
Multivariate Analysis in Ecology
Unconstrained Ordination
PCA
2
0
Expected Response
4
6
Linear Model
−2
−1
0
1
2
3
Principal Component 1
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Unconstrained Ordination
January 2016
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PCA
Variances and Correlations
Analysis of raw data explains variances: variables with high variance are most
important
If the variables are standardized to unit variance before analysis
z = (x − x̄)/sx all variables are equally important and the analysis explains
correlations among variables
Standardization can be used when we want all variables to have equal weights
Standardization must be used when variables are measured in different scales,
such as for environmental measurements
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January 2016
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Unconstrained Ordination
PCA
Reducing the Number of Correlated Environmental
Variables I
> (pc <- rda(varechem, scale=TRUE))
Call: rda(X = varechem, scale = TRUE)
Inertia Rank
Total
14
Unconstrained
14
14
Inertia is correlations
Eigenvalues for unconstrained axes:
PC1 PC2 PC3 PC4 PC5 PC6 PC7 PC8 PC9 PC10 PC11 PC12
5.19 3.19 1.69 1.07 0.82 0.71 0.44 0.37 0.17 0.15 0.09 0.07
PC13 PC14
0.04 0.02
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January 2016
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PCA
The Number of Components
PCA is a rotation in species (character) space and retains the original
configuration
The number of PC’s is min(N, S), and all together give the original data
First axes are most important and we may ignore the minor axes
We can either use the axes as variables in other models, or use them to
identify major (almost) independent variables
Often we want to retain a certain proportion of the variance, say 50 %
Sometimes we would like to retain “significant” axes
There really is no way of doing this, but some people suggest comparing
eigenvalues against broken stick distribution
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January 2016
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Unconstrained Ordination
PCA
Broken Stick and Eigenvalues
5
pc
Broken Stick
4
●
Inertia
3
●
2
●
●
●
1
●
●
●
●
●
0
●
PC1 PC2 PC3 PC4 PC5 PC6 PC7 PC8 PC9
http://cc.oulu.fi/ jarioksa/ (Oulu)
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PCA
Two Dimensions, but which?
Humdepth
0.5
Baresoil
Mn
PC2
0.0
N
Ca
Mg
K Zn
−0.5
P
Mo
S
pH
−1.0
Fe
Al
−1.0
−0.5
0.0
0.5
PC1
http://cc.oulu.fi/ jarioksa/ (Oulu)
Multivariate Analysis in Ecology
Unconstrained Ordination
PCA
Methods Related to PCA
Metric Scaling a.k.a. Principal Coordinates Analysis
Used dissimilarities instead of raw data
With Euclidean distances equal to PCA, but can use other dissimilarities
Factor Analysis
A statistical method that makes a difference between systematic components
and random error
In PCA we just ignore latter components, but here we really identify the real
components
Much used in human sciences and often referred to in ecology (but usually
misunderstood)
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Unconstrained Ordination
January 2016
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January 2016
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PCA
Confirmatory Factor Analysis
ε
P
0.812
0.340
ε
K
0.399
0.776
ξ
ε
Ca
0.163
ε
Mg
0.310
http://cc.oulu.fi/ jarioksa/ (Oulu)
0.915
0.831
Multivariate Analysis in Ecology
Unconstrained Ordination
CA
Correspondence Analysis
0.20
0.15
0.10
0.05
0.00
Minor variant of PCA: Weighted Principal
Components with Chi-square metric
23
19
22
16
28
13
14
20
25
7
5
6
3
4
2
9
12
10
11
fij − eij
χij = √
eij
27
Chi-square transformation tells how much the
observed proportions fij differ from the expected
proportions eij :
24
Null model: Species composition is identical in all
sampling units
15
Site and species marginal profiles define the
expected abundances
21
Multivariate Analysis in Ecology
Unconstrained Ordination
0.03
18
Cal.vul
Emp.nig
Led.pal
Vac.myr
Vac.vit
Pin.syl
Des.fle
Bet.pub
Vac.uli
Dip.mon
Dic.sp
Dic.fus
Dic.pol
Hyl.spl
Ple.sch
Pol.pil
Pol.jun
Pol.com
Poh.nut
Pti.cil
Bar.lyc
Cla.arb
Cla.ran
Cla.ste
Cla.unc
Cla.coc
Cla.cor
Cla.gra
Cla.fim
Cla.cri
Cla.chl
Cla.bot
Cla.ama
Cla.sp
Cet.eri
Cet.isl
Cet.niv
Nep.arc
Ste.sp
Pel.aph
Ich.eri
Cla.cer
Cla.def
Cla.phy
All sites should have all species in in the same
proportions as in the whole data
http://cc.oulu.fi/ jarioksa/ (Oulu)
0.00
January 2016
55 / 103
CA
Chi-squared metric
Cla.ste
Cla.ran
Cla.arb
eij
Cal.vul
Emp.nig
(fij − eij)
Vac.vit
Cla.unc
Dic.fus
Ple.sch
Vac.myr
Dic.sp
eij
10
2
9
3
http://cc.oulu.fi/ jarioksa/ (Oulu)
4
12
5
6
11
7
13
18
19
21
Multivariate Analysis in Ecology
23
20
14
16
15
27
22
24
25
28
January 2016
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Unconstrained Ordination
CA
2
Relative proportions are axes and
points have weights
3
Chi-square transformation
4
Weighted rotation
5
De-weighting
CA2
Sites in a species space
−0.5
0.0
1
0.5
1.0
CA Rotation
−1.0
−0.5
0.0
0.5
1.0
CA1
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Unconstrained Ordination
January 2016
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January 2016
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CA
Running CA I
> (ord <- cca(dune))
Call: cca(X = dune)
Inertia Rank
Total
2.12
Unconstrained
2.12
19
Inertia is mean squared contingency coefficient
Eigenvalues for unconstrained axes:
CA1
CA2
CA3
CA4
CA5 CA6
CA7
CA8
0.536 0.400 0.260 0.176 0.145 0.108 0.092 0.081
(Showed only 8 of all 19 unconstrained eigenvalues)
> head(summary(ord), 2, 1)
http://cc.oulu.fi/ jarioksa/ (Oulu)
Multivariate Analysis in Ecology
Unconstrained Ordination
CA
Running CA II
Call:
cca(X = dune)
Partitioning of mean squared contingency coefficient:
Inertia Proportion
Total
2.12
1
Unconstrained
2.12
1
Eigenvalues, and their contribution to the mean squared contingency coefficient
Importance of components:
CA1
CA2
CA3
CA4
CA5
CA6
Eigenvalue
0.536 0.400 0.260 0.1760 0.1448 0.108
Proportion Explained 0.253 0.189 0.123 0.0832 0.0684 0.051
Cumulative Proportion 0.253 0.443 0.565 0.6486 0.7170 0.768
CA7
CA8
CA9
CA10 CA11
Eigenvalue
0.0925 0.0809 0.0733 0.0563 0.0483
Proportion Explained 0.0437 0.0382 0.0347 0.0266 0.0228
Cumulative Proportion 0.8117 0.8500 0.8847 0.9113 0.9341
CA12
CA13
CA14
CA15
CA16
Eigenvalue
0.0412 0.0352 0.02053 0.01491 0.00907
Proportion Explained 0.0195 0.0167 0.00971 0.00705 0.00429
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Unconstrained Ordination
January 2016
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January 2016
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CA
Running CA III
Cumulative Proportion 0.9536 0.9702 0.97995 0.98700 0.99129
CA17
CA18
CA19
Eigenvalue
0.00794 0.00700 0.00348
Proportion Explained 0.00375 0.00331 0.00164
Cumulative Proportion 0.99505 0.99836 1.00000
Scaling 2 for species and site scores
* Species are scaled proportional to eigenvalues
* Sites are unscaled: weighted dispersion equal on all dimensions
Species scores
CA1
CA2
CA3
CA4
CA5
CA6
Achimill -0.909 0.0846 -0.586 -0.00892 -0.660 -0.1888
Agrostol 0.934 -0.2065 0.282 0.02429 -0.139 -0.0226
....
Callcusp 1.952 0.5674 -0.859 -0.09897 -0.557 0.2328
Site scores (weighted averages of species scores)
http://cc.oulu.fi/ jarioksa/ (Oulu)
Multivariate Analysis in Ecology
Unconstrained Ordination
CA
Running CA IV
CA1
CA2
CA3
CA4
CA5
CA6
1
-0.812 -1.083 -0.1448 -2.107 -0.393 -1.8346
2
-0.633 -0.696 -0.0971 -1.187 -0.977 0.0658
....
20
1.944 1.069 -0.6660 -0.553 1.596 -1.7029
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Multivariate Analysis in Ecology
Unconstrained Ordination
January 2016
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CA
Goodness of Fit of Scores
Inertia is “mean square contingency coefficient”: Chi-squared of a matrix
standardized to unit sum, or Chi-square of Px x
Eigenvalues are non-negative and ordered like in PCA, but they are bound to
maximum 1
The origin gives the expected abundances for all species and all sites
The deviant species and deviant sites are far away from the origin
CA is weighted analysis, and the weighted sum of squared scores is the
eigenvalue
The species and site scores are (scaled) weighted averages of each other:
proximity matters
Rare species have low weights: they are further away from the origin
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January 2016
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Unconstrained Ordination
CA
Weighted Average?
For presence/absence data: weighted
average of a species is in the middle
(“barycentre”) of plots where the species
occurs
3
●
For quantitative data: plots wheres
species is abundant are heavier and the
weighted average is closer to them
●
1
CA2
2
●
●
●
●
●
0
●
● Lolper
●
−1
●
●
−1
Sampling units (SU) are close to species
that occur on them
●
●
●●
CA is a weighted average method: it
tries to put SUs close to the species
that occur in them, and all SUs with
similar species composition close to each
other: Unimodal response
●
●
●
0
1
2
CA1
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Unconstrained Ordination
January 2016
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January 2016
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CA
Default Plot and Effect of Scaling
scaling = 1
5
Empenigr
Airaprae
3
4
Hyporadi
2
19
1
17
Anthodor
0
Callcusp
Comapalu
20
Vicilath Scorautu
Eleopalu
Ranuflam
18 Bracruta
Planlanc
14
15
11
Juncarti
Achimill
16
Sagiproc
6 Trifrepe
10
Trifprat 57
8
Agrostol
Rumeacet 2
12
Poaprat 34 9 13
Bellpere
Lolipere
Bromhord
1
Poatriv
Alopgeni
Juncbufo
Elymrepe
Cirsarve
Chenalbu
−1
CA2
Salirepe
−2
−1
0
1
2
3
4
CA1
http://cc.oulu.fi/ jarioksa/ (Oulu)
scaling =Multivariate
2
Analysis in Ecology
Unconstrained Ordination
CA
Weighted Averages
Species scores are [proportional to] weighted averages of site scores, and
simultaneously
Site scores are [proportional to] weighted averages of species scores
Either one (but not both) of these can be a direct weighted average of other
If sites scores are weighted averages of species scores, site point is in the
middle of points of species that occurs in the site
The location of the point is meaningful whereas in PCA the main things were
distance and direction from the origin (but these, too, matter)
Can approximate unimodal response model and therefore CA is much better
for community ordination than PCA
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Unconstrained Ordination
January 2016
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CA
Linear and Unimodal Models
PCA implies linear relations between axes and species abundances
80
60
0
20
40
cladstel
60
40
20
0
cladstel
80
CA packs species and approximates a unimodal model
−4
−2
0
2
−1
0
PCA1
http://cc.oulu.fi/ jarioksa/ (Oulu)
1
2
3
CA1
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January 2016
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Unconstrained Ordination
CA
Optimal Scaling
0.8
0.6
0.4
0.2
0.0
The species responses should be narrow:
width is measured as SSw
Suboptimal scaling
Response
The locations of species optima (tops)
should be widespread: spread is
measured as SSB
The total variance is their sum
SST = SSB + SSw
1
4
5
0.4
0.0
Response
0.8
Optimal scaling
Scaling is optimal if most of variance is
between species and SSB is high
The criterion of variance is the
eigenvalue maximized in CA:
λ = SSB /SST
0
1
2
3
4
5
Gradient
Multivariate Analysis in Ecology
Unconstrained Ordination
3
Gradient
High SSB means that species have
different optima, and low SSw means
that species have narrow tolerance
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January 2016
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CA
Goodness of Fit Statistics: Repetition
NMDS: stress of nonlinear transformation from observed dissimilarities to
ordination distances
In range 0 . . . 1 (0 . . . 100 %), but in practice 0.4 for random configuration
0.1 is good, and 0.2 is not bad, 0 is suspect
PCA: sum of eigenvalues is variance (or SS)
Upper limit is total variance, large is good
CA: sum of all eigenvalues is (scaled) Chi-square
Single eigenvalue maximum 1
high is good, but λ < 0.2 may not be bad
Eigenvalues λ > 0.7 are suspect: disjunct or very heterogeneous data
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January 2016
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Unconstrained Ordination
CA
Nonlinear and Linear Mapping
●
●
●
●●
0.2
0.4
0.6
0.8
1.0
8
Observed Dissimilarity
10
12
14
16
0.2
0.3
0.4
0.5
0
5
Ordination Distance
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15
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2.5
2.0
1.5
1.0
0.5
0.0
●
Ordination Distance
Non−metric fit, R2 = 0.986
Linear fit, R2 = 0.926
Ordination Distance
CA
0.6
PCA
●
10
3.0
NMDS
18
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0.2
Observed Dissimilarity
0.3
0.4
0.5
0.6
0.7
Observed Dissimilarity
Dune Meadow data
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Unconstrained Ordination
January 2016
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CA
Nonlinear and Linear Mapping: A Difficult Case
CA
0.4
PCA
7
NMDS
0.1
0.2
Ordination Distance
4
2
3
Ordination Distance
3
2
0
0.0
1
1
0
Ordination Distance
5
0.3
6
4
Non−metric fit, R2 = 0.98
Linear fit, R2 = 0.92
0.2
0.4
0.6
0.8
Observed Dissimilarity
1.0
2
4
6
Observed Dissimilarity
8
0.2
0.4
0.6
0.8
Observed Dissimilarity
Bryce Canyon
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Unconstrained Ordination
Graphics
Anatomy of a Plot
0.5
2
CladcervCladstel
Cladphyl
24
Dicrpoly
Cetrisla
Cladchlo
12
Pinusylv
11
0.0
9
4
Flavniva
NMDS2
Polypili
Barbhatc
10
Ptilcili
Pohlnuta
Empenigr
Vaccviti
Cladrang
19
23
Cladunci
Cladgrac
Cladcris
6
13 Claddefo
Cetreric
Cladcorn
20
Cladfimb
Callvulg
Peltapht
15
3
18Cladsp
16
Cladarbu
Cladcocc
7
Polyjuni
14
5Stersp
Diphcomp
21
Cladbotr
Dicrsp
Pleuschr
Betupube
28
Vaccmyrt Rhodtome
22
27
Polycomm
Hylosple
Descflex
Dicrfusc
−0.5
Vacculig
25
Cladamau
Icmaeric
−1.0
Nepharct
−0.5
0.0
0.5
1.0
NMDS1
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Unconstrained Ordination
January 2016
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Graphics
Plotting functions
All vegan ordination functions have a plot function, and ordiplot can be
used for other functions as well
For full control, use first plot(x, type="n") and then add configurable
points or text
Congested plots can displayed with orditorp or edited with orditkplot
Lattice graphics can be made with ordixyplot, ordicloud or ordisplom
Dynamic, spinnable 3D plots can be made with ordirgl function in the
vegan3d package
Items can be added to the plots with ordiarrows, ordihull, ordispider,
ordihull, ordiellipse, ordisegments, or ordigrid
2
24
12
11 (Oulu)
http://cc.oulu.fi/ jarioksa/
10
Multivariate Analysis
in Ecology
21
January 2016
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Unconstrained Ordination
Environmental Variables
Ordination and Environment
We take granted that vegetation is controlled by environment, so
1
Two sites close to each other in ordination have similar vegetation
2
If two sites have similar vegetation, they have similar environment
3
Two sites far away from each other in ordination have dissimilar vegetation,
and perhaps
4
If two sites have different vegetation, they have different environment
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Unconstrained Ordination
January 2016
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Environmental Variables
Fitted Vectors
1.0
0.5
pH
0.0
width
Instab
Currvel
−0.5
Partsize
Depth
−1.0
MDS2
Conduc
slope
−1.5
For every arrow, there is an equally long
arrow into opposite direction:
Decreasing direction of the gradient.
Implies a linear model: Project sample
plots onto the vector for expected value.
1.5
Direction of fitted vector shows the
gradient of the environmental variable,
length shows its importance.
−1.5
−1.0
−0.5
0.0
0.5
1.0
1.5
2.0
MDS1
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January 2016
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Unconstrained Ordination
Environmental Variables
Interpretation of Arrow
●
0.5
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●
●
NMDS2
●
●
●
●
●
●
●
●
●
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●
−0.5
●
●
●
●
WatrCont
−1.0
●
●
−1.0
−0.5
0.0
0.5
1.0
1.5
NMDS1
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Unconstrained Ordination
January 2016
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Environmental Variables
Alternatives to Vectors
Fitted vectors natural in constrained ordination, since these have linear
constraints.
Distant sites are different, but may be different in various ways:
Environmental variables may have a non-linear relation to ordination.
Linear
0.5
Dim1
http://cc.oulu.fi/ jarioksa/ (Oulu)
1.0
1.5
1.0
0.5
−0.5
Dim2
0.5
−1.5
+
pH
−0.5
+
+ +
+
+
+++ + +
++
++ +
+ +
+
+
+
+ +
+++
+ + ++ pH
+++
+ ++++++ + +++ ++ + +++ +
+
+
+
+
+
+
+
++++++
+ +++
++
+
++
+++
+
+ +++++
+ +
+
+
+
+
++ +++++++ ++++ ++ ++ + +
++ + + + +
+
+
+ + ++
+
+
+
++
1.0
+
+
+
0.0
1.5
1.5
+
+
+
−1.0
Fitted surface
−1.5
+
Observed values
Dim2
+
−1.5
−0.5
Dim2
0.5
1.0
1.5
+
−1.0
0.0
0.5
1.0
Dim1
Multivariate Analysis in Ecology
1.5
+
+
+
+
+
+
+ +
+
+
+++ + +
++
++ +
+ +
+
+
+
+ +
+++
+ + ++ pH
+++
+ ++++++ + +++ ++ + +++ +
+
+
+
+
+
+
+
++++++
+ +++
++
+
++
+++
+
+ +++++
+ +
+
+
+
+
++ +++++++ ++++ ++ ++ + +
++ + + + +
+
+
+ + ++
+
+
+
++
+
+
+
−1.0
0.0
0.5
1.0
1.5
Dim1
January 2016
76 / 103
Unconstrained Ordination
Environmental Variables
Fitting Environmental Vectors I
> (ef <- envfit(vare.mds, varechem, permu = 999))
***VECTORS
NMDS1 NMDS2
r2 Pr(>r)
N
-0.050 -0.999 0.21 0.098
P
0.687 0.727 0.18 0.135
K
0.827 0.562 0.17 0.147
Ca
0.750 0.661 0.28 0.029
Mg
0.697 0.717 0.35 0.015
S
0.276 0.961 0.18 0.143
Al
-0.838 0.546 0.52 0.002
Fe
-0.862 0.507 0.40 0.013
Mn
0.802 -0.597 0.53 0.001
Zn
0.665 0.747 0.18 0.146
Mo
-0.849 0.529 0.05 0.581
Baresoil 0.872 -0.490 0.25 0.035
Humdepth 0.926 -0.377 0.56 0.001
pH
-0.799 0.601 0.26 0.042
--Signif. codes: 0 ‘***’ 0.001 ‘**’
Permutation: free
Number of permutations: 999
http://cc.oulu.fi/ jarioksa/ (Oulu)
.
*
*
**
*
***
*
***
*
0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
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Unconstrained Ordination
January 2016
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January 2016
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Environmental Variables
Plotting Environmental Vectors
0.6
Limit p < 0.1
●
0.4
●
Mg
Ca
●
●
pH
●
●
●
0.0
●
●
●
●
●
●
●
●
●
−0.2
●
●
●
●
Baresoil
●
Humdepth
●
●
Mn
N
−0.4
NMDS2
0.2
Al
Fe
●
−0.4
−0.2
0.0
0.2
0.4
0.6
NMDS1
http://cc.oulu.fi/ jarioksa/ (Oulu)
Multivariate Analysis in Ecology
Unconstrained Ordination
Environmental Variables
Fitting Environmental surfaces
>
>
>
>
>
ef <- envfit(vare.mds ~ Al + Ca, varechem)
plot(vare.mds, display = "sites")
plot(ef)
tmp <- with(varechem, ordisurf(vare.mds, Al, add = TRUE))
tmp <- with(varechem, ordisurf(vare.mds, Ca, add = TRUE, col = "green4"))
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Unconstrained Ordination
January 2016
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January 2016
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Environmental Variables
0.6
Plotting Environmental Surfaces
●
10
0
0.4
●
●
Al
●
Ca
700
0.2
650●
●
●
65
0
550
0.0
●
500
●
450
0
25
●
●
●
●
●
●
350
−0.2
●
●
400
●
●
●
●
200
●
●
150
50
0
−0.4
NMDS2
600
●
−0.4
−0.2
0.0
0.2
0.4
0.6
NMDS1
http://cc.oulu.fi/ jarioksa/ (Oulu)
Multivariate Analysis in Ecology
Unconstrained Ordination
Environmental Variables
Factor Fitting I
> dune.ca <- cca(dune)
> ef <- envfit(dune.ca ~ A1 + Management, data=dune.env, perm=999)
> ef
***VECTORS
CA1
CA2
r2 Pr(>r)
A1 0.9980 0.0606 0.31 0.052 .
--Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Permutation: free
Number of permutations: 999
***FACTORS:
Centroids:
ManagementBF
ManagementHF
ManagementNM
ManagementSF
CA1
-0.73
-0.39
0.65
0.34
CA2
-0.14
-0.30
1.44
-0.68
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Unconstrained Ordination
January 2016
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January 2016
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Environmental Variables
Factor Fitting II
Goodness of fit:
r2 Pr(>r)
Management 0.44 0.003 **
--Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Permutation: free
Number of permutations: 999
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Multivariate Analysis in Ecology
Unconstrained Ordination
Environmental Variables
Plotting Fitted Factors
3
19
2
17
20
1
CA2
ManagementNM
18
14
15
0
11
16
A1
6
10
ManagementBF
5
7
ManagementHF
−1
2
8
12
ManagementSF
4 9
3
13
1
−2
−1
0
1
2
CA1
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Unconstrained Ordination
January 2016
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Environmental Variables
Environmental Interpretation
Environmental variables need not be parallel to ordination axes.
Axes cannot be taken as gradients, but gradients are oblique to axes: You
cannot tear off an axis from an ordination.
Never calculate a correlation between an axis and an environmental variable.
Environmental variables need not be linearly correlated with the ordination,
but locations in ordination can be exceptional.
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January 2016
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Unconstrained Ordination
Gradient Model and Ordination
Gradient Model and Ordination
Species packing gradient
Single gradients appear as curves in
linear ordination methods
PCA horseshoe: curve bends inward
and gives wrong ordering of points
on axis 1
PCA
CA
CA arch: axis 1 retains the correct
ordering of sites despite the curve
Environmental interpretation by
vector fitting or surface bound to be
biased
Axes cannot be interpreted as
“gradients”
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Unconstrained Ordination
January 2016
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Gradient Model and Ordination
The birth of the curve
There is a curve in the species space and PCA shows it correctly
CA deals better wit unimodal responses, but the second optimal scaling axis
is folded first axis
Gradient space
Species space
CA1
CA2
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January 2016
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Unconstrained Ordination
Gradient Model and Ordination
Solutions to the Curvature
Detrended Correspondence Analysis (DCA)
CA axis retains the correct ordering: keep that, but instead of orthogonal axes,
use detrended axes
Programme DECORANA additionally rescales axes to sd units approximating t
parameter of the Gaussian model
Distorts space, introduces new artefacts and probably should be avoided
Nonmetric Multidimensional Scaling (NMDS) should be able to cope with
moderately long gradients
Constrained ordination may linearize the responses
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Unconstrained Ordination
January 2016
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January 2016
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Gradient Model and Ordination
Running Detrended Correspondence Analysis
> (ord <- decorana(dune))
Call:
decorana(veg = dune)
Detrended correspondence analysis with 26 segments.
Rescaling of axes with 4 iterations.
DCA1 DCA2
DCA3
DCA4
Eigenvalues
0.512 0.304 0.1213 0.1427
Decorana values 0.536 0.287 0.0814 0.0481
Axis lengths
3.700 3.117 1.3005 1.4789
http://cc.oulu.fi/ jarioksa/ (Oulu)
Multivariate Analysis in Ecology
Unconstrained Ordination
Gradient Model and Ordination
Default plot
3
Empenigr
Airaprae
Hyporadi
2
Salirepe
19
Sagiproc
Scorautu Bracruta
1
17
Vicilath
20
Ranuflam
Cirsarve
Juncarti
Callcusp
15
18
Juncbufo
Agrostol
1416
12
Trifrepe
11
Eleopalu
13 8
Chenalbu
Alopgeni
6
Planlanc
9
4
Comapalu
Rumeacet7
510
3
Trifprat
2 Poatriv
Poaprat
Bellpere
−1
0
DCA2
Anthodor
Lolipere
1
−2
Bromhord
Elymrepe
Achimill
−3
−2
−1
0
1
2
3
DCA1
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Unconstrained Ordination
January 2016
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Gradient Model and Ordination
Community Pattern Simulation
PCA
PCoA
DCA
NMDS
4 x 2.5t
Truth
CA
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January 2016
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Unconstrained Ordination
Gradient Model and Ordination
Short Gradients: Is There a Niche for PCA?
PCA
2 x 1t
Folklore: PCA with short gradients (≤ 2t).
Not based on research, but simulation finds PCA
uniformly worse than CA: With short gradients
about as good as CA, but usually worse.
There should be no species optimum within
gradient: Shortness alone not sufficient.
CA
PCA best used for really linear cases
(environment) or for reduction of variables into
principal components (but see FA).
Noise dominates over signal in homogeneous
data.
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Unconstrained Ordination
January 2016
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Gradient Model and Ordination
Long Gradients: DCA or NMDS
CA
Curvature with long gradients: Need either DCA or
NMDS.
DCA
NMDS is a test winner: More robust than DCA.
DCA more popular.
DCA may produce new artefacts, since it twists the
space.
MDS
8x3t
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January 2016
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Unconstrained Ordination
Gradient Model and Ordination
Extended Dissimilarities and Step-across
0.8
0.7
0.6
0.3
0.4
Flexible shortest path or their
approximations, extended
dissimilarities
0.5
Use step-across points to estimate
their distance
Community Dissimilarity
0.9
1.0
How different are sites that have
nothing in common?
+++ +++ ++++
++
++
++
+ + +++ +
+
++
+
++
++
++
+
++
+
++
++
+ + +++
++
++
++
++
+++
+
+ ++
+
+
+
++
+ ++
+
+
+
++
+
+ ++
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+ ++
++
+
++
+
+ ++
+
+
+
+
+
+
+
+
+
+
+
+
+
+
++ + +++
++
++
+
+
+
+
+
++
+
+
+
+
+
+
+
+
+++
+ +
++ ++
+ ++
++
+
++
+
+
+++ +
++ +
+
++
++
++
+
+
+
+
+
++ +
++
++
++
++
+
++
+
+
+
+
+
+
+
++
++
++ +
+ +
+
+
+ +
++ +
+ ++
+
+
+
+
++
++ +
+
++
++
++
+
+
++
+
+ +
+
++
++
+
+
+
++
+
++
++
+
++++++ ++++ ++ + ++++ ++++ ++++ + +++ +
++ ++ +
+
++
+
+
++
+
++ +
+++++
+
+++ +
+
+
2
4
6
8
http://cc.oulu.fi/ jarioksa/ (Oulu)
2.0
1.5
1.0
Extended Dissimilarity
No shared species since rare species
were not observed: Swan
transformation estimates the
probability of finding an unobserved
species
0.5
Extended dissimilarity: use only
one-site steps, do not update
dissimilarities below a threshold
2.5
Gradient Distance
+
+
+++ +++
+
++
+
++
++
+
+
+
++
++
+
+
+
+
+
++
+
++
++
++
+
+
+
++++
++
+
+++
+
++
+
++
++
++
++
+
+
+
+
+
++
+
+
+
+
+
+
+
+
+
+
++ +++
+
++ +
+ +
++
++ + +
+
++
++ ++
++
++
++
+ ++
++
+ ++
+ +++ ++ +++ +
+
+
+
+
+
+
+
+ +
+
++ +
+
+
++ +
+++
+
++
++
++
++
++ +
+
+
++ ++
+
+ +
++
+
+
++
++
++
++
++
+
++
+
+
++
++++
+
++
+
++
++
+
+
+ +
++
++
+ +
+
++
+
+
++ +
+
++
+++ ++
++
+
+
+
+
+
+
+
+ ++ + +
+
++
++
++
+++ ++ +
+
+
++ +
+
+
+++
++ +++
+ ++
+ ++
++
+
++
++
++
+ +
++
++
++
++
++ +
+
++
+
+
++
+
++
++
+
++
++
+
+
++
++
+
+ ++
++
+
++
+
++
+
+
++
++
++
++
+
+
+
+ +
+
+
+
+
++
+
+
++
+
++
++
+
++
++ +
+
++
+
+
+
+
+
+
+
++
+
+
+ ++
+
++
+
++
+
+
+
+
++
+
++
+
++
++
+
2
+
4
6
Multivariate Analysis in Ecology
Unconstrained Ordination
8
Gradient Distance
January 2016
93 / 103
Gradient Model and Ordination
Strong and Weak Ties
strong
weak
stepacross
swan
Maximum dissimilarities (no shared
species) are tied
Strong tie treatment tries to keep
tied values together and puts
maximum dissimilarites to a circle
Weak tie treatment allows breaking
ties and straightens the axes: now
the default in vegan, whereas earlier
was impossible
8 x 1.5 sd
8 × 1.5 sd units, Gaussian binary response
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January 2016
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