Advanced analytical approaches in
ecological data analysis
The world comes in fragments
Early plant succession in the post brown cole mining area Chicken Creak
2005
2010
Succession starts with colonising species from a regional species pool and from the initial
seed bank
Multivariate approaches to biodiversity
Sites
Variables
Interdependence
matrix X
Species trait
matrix T
Species
Species
Species
Traits
Phylogenetic distance
matrix P
Environmental variable
matrix V
Sites
L
Why are species abundant or rare?
What determines community composition?
How does a community function in space and time?
Site GPS location matrix D
Sites
Species
Variables
Traits
Species abundance matrix M
The interplay between traits, phylogeny, and species occurrences
Phylogenetic distance
matrix P
Traits and phylogeny are closely
related.
Trait distance correlates positively
with phylogenetic distance
Niche conservatism
Sites
Species
Species trait
matrix T
Species
Species
Species
Traits
Species abundance matrix M
Competition: niche conservatism causes
phylogenetically close species to cooccur less common.
Habitat filtering causes species with
similar traits (close in phylogeny) to cooccur more frequent.
As species of the same genus have usually, though by no means invariably, some
similarity in habits and constitution, and always in structure, the struggle will generally
be more severe between species of the same genus, when they come into competition
with each other, than between species of distinct genera. (Darwin 1859)
Early succession
Regional pool of
potential colonizers
Regional pool
of species
Environmental
filters
Facilitation
Regional pool
of species
Random colonization
Phylogenetic
Phylogenetic
clumping
segregation
No phylogenetic
No phylogenetic
structure
structure
Local colonizers
Later succession
Positive
interactions
Phylogenetic
clumping
Competition
Phylogenetic
segregation
Local community
structure
Neutral
interactions
No phylogenetic
signal
Species trait
matrix T
Species
Species
Species
Traits
Phylogenetic distance
matrix P
Mantel test
A Mantel test is a correlation between two distance matrices.
We have to transform the trait matrix into a distance matrix and
correlate with the phylogeny matrix.
Mantel test
Agrosti
Achillea Agrosti s_stolo Agrosti
_panno s_capill nifera_ s_vinea
nica
aris
agg.
lis
S
Achillea_pannoni
ca
0.0
Agrostis_capillari
s
179.0
Agrostis_stolonif
era_agg.
179.0
Agrostis_vinealis 179.0
179.0
179.0
179.0
0.0
2.5
2.5
0.0
2.5
2.5
0.0
Agrosti
Achillea Agrosti s_stolo Agrostis
_panno s_capill nifera_ _vineali
nica
aris
agg.
s
S
Achillea_pannon
ica
0.0
Agrostis_capillari
s
179.0
Agrostis_stolonif
era_agg.
179.0
Agrostis_vinealis 179.0
2.5
2.5
179.0 179.0
179.0
0.0
2.5
2.5
2.5
2.5
0.0
2.5
2.5
0.0
R = -0.04; P(r=0) = 0.37
Plant traits and phylogeny in the
year 2011 are not correlated.
Why??
We averaged over all traits
Trait
Trait
10
10
Blue eyes
5
5
10
10
Blue eyes
15
8
Blue eyes
12
15
Blue eyes
Brown eyes
8
12
Brown eyes
The trait ‚brown eyed’ is
phylogenetically conserved.
Closely related species have the same
eye colour
Blue eyes: (10+5+10+15)/3=10
10
10
12
Brown eyes
Blue eyes
Brown eyes
12
Blue eyes
Eye colour is phylogenetically not
conserved.
Blue eyes: (10+5+15++10+ 8+12)/3=20
Compare these phylogenetic distances per trait with those expected from a random
(Brownian motion ) phylogeny.
Brownian motion
Brownian motion is the standard method to
generate random phylogenetic trees
1. Start with a pixel
2. Take a second pixel and move at
random until it sticks to the first
3. Repeat two until the desired number of
end-tips is reached.
Brownian motion
generates random
trees and can simulate
the evolution of traits
along this trees.
Brownian motion (random
walk) along a phylogeny
Blue eyes
Blue eyes
Blue eyes
Blue eyes
Blue – brown
boundary
Brown eyes
𝑑𝑋(𝑡) = 𝜎𝑑𝐵(𝑡)
Start
0
1 0.002168
2+B2+0.5*(LOS()-0.5)
Compare the observed pattern of blue and brown with
those generated by many Brownian motion trees.
Moran’s I as a test of phylogenetic signal
𝑛
𝐼=
𝑗
𝑗
𝑗 𝑤𝑖𝑗
𝑗 𝑤𝑖𝑗
𝑖
𝑥𝑖 − 𝑥 𝑥𝑗 − 𝑥
𝑥𝑖 − 𝑥
2
I is similar to a weighed coefficient of correlation.
n
1
2
3
4
5
Mean
X
0.83
3.38
9.18
9.85
4.71
5.59
Diagonal sum
n
59.35
5
Sum
8.99
𝐸𝑥𝑝 𝐼 =
Sum
I
−1
𝑛−1
-11.51
-0.108
M
X-M
-4.76
-2.21
3.59
4.26
-0.88
-4.76
22.66
10.52
-17.09
-20.28
4.19
W
wij
wij
wij
wij
wij
wij
0
0.69
0.46
0.12
0.45
WM
1
1
0
2
7.25852
3
-7.8607
4
-2.4333
5
1.88496
Expected I -0.25
w defines the strength of
distance effects.
where n is the number of data
points.
-2.21
10.52
4.88
-7.93
-9.41
1.94
X-M
3.59
-17.09
-7.93
12.89
15.29
-3.16
4.26
-20.28
-9.41
15.29
18.15
-3.75
-0.88
4.19
1.94
-3.16
-3.75
0.77
wij
0.4
0
0.87
0.03
0.02
wij
0.74
0.15
0
0.56
0.64
wij
0.18
0.53
0.75
0
0.94
wij
0.4
0.47
0.32
0.27
0
2
4.2078
0
-6.9025
-0.2824
0.0389
3
-12.645
-1.1901
0
8.5643
-2.0219
4
-3.65
-4.9897
11.4701
0
-3.5239
5
1.6755
0.9141
-1.011
-1.012
0
Pagel’s lambda
Species
K
P(K)
Lamb P(Lamb
da
da)
Species
Morphological traits
Canopy height (m)
K
P(K)
Lamb P(Lamb
da
da)
Reproductive traits
1.20
0.001
1.00
<0.001
0.07
0.324
0.88
0.001
Leaf mass [mg]
0.11
0.155
1.00
Leaf size [mm2]
0.10
0.138
Life span
0.07
Max releasing height [m]
Average month of
0.11
0.004
0.00
1.000
Duration of flowering
0.04
0.743
0.13
0.135
0.001
Duration of seedling
0.04
0.505
0.00
1.000
0.98
0.003
Early month flowering
0.08
0.028
0.80
<0.001
0.014
0.50
0.020
0.05
0.226
0.00
1.000
2.11
0.001
1.00
<0.001 Latest month flowering
0.14
0.001
0.82
<0.001
Min releasing height [m]
0.78
0.003
1.00
<0.001
0.19
0.001
0.52
0.085
Specific leaf area mm2/mg
0.06
0.048 <0.01
1.000
0.06
0.054
0.63
0.002
0.10
0.012
0.89
<0.001 Mean seed weight
0.13
0.084
1.00
<0.001
Stem erect %
0.11
0.006
0.96
<0.001 Seed bank longevity
0.06
0.074
0.00
1.000
Terminal velocity m/s
0.06
0.120
0.58
<0.001 Type reproduction
0.05
0.179
0.22
0.011
Emergent attached to
substrate
Stem ascending to
prostrate %
seedling
Early month seed
shedding
Latest month seed
shedding
ln (Seeds per shoot)
Species
K
P(K)
Lamb P(Lamb
da
da)
Species
Habitat requirements
K
P(K)
Lamb P(Lamb
da
da)
Molecular traits
Light
0.04
0.506
0.22
0.063
Polyploidy
0.05
0.251
0.40
0.001
Soil fertility
0.06
0.154 <0.01
1.000
Chromosome number
8.62
0.001
1.00
<0.001
pH
0.05
0.251 <0.01
1.000
DNA content
0.39
0.001
0.75
<0.001
Nitrogen
0.05
0.126 <0.01
1.000
Life strategy type
0.06
0.133
0.39
1.000
Grazing tolerance
0.03
0.855 <0.01
1.000
Hemerobic level
0.05
0.142 <0.01
1.000
Morphological and genetic plant traits are phylogenetically more conserved than life
history, reproductive, and ecological traits.
Ungulates
10
Body size
Species daily home
range
Large size
Small home range
Large size
Small home range
Large size
Small size
Small home range
Large home range
Without phylogenetic
knowledge we would
link body size to home
range in a functional
manner.
Small size
Large home range
Phylogenetic
pseudoreplication
5
10
10
15
8
12
12
Mammal
predators
Home range and body size are linked by
common phylogeny. They are
phylogenetically preserved.
Phylogenetic regression accounts for the phylogenetic non-independence of variables
Species trait
matrix T
Species
Species
Species
Traits
Phylogenetic distance
matrix P
Eigenvector methods
Every square matrix M has a
vector U so that
M
𝑴𝑼 = 𝜆𝑼
x
U
=l
U
(𝑴 − 𝜆𝑰)𝑼 = 0
U: Eigenvector
l: Eigenvalue
I: Identity matrix
Because both sides of the
equation are equal the right side
contains the same information as
the left side.
The eigenvector U contains the
information in M in a condensed
form.
Species
Leaf
mass
[mg]
Achillea_pannonica 82.3
Agrostis_capillaris
61.0
Agrostis_stolonifera
61.0
_agg.
Agrostis_vinealis
1.4
Ajuga_genevensis
18.0
Leaf
Apera_spica_venti
61.0
Species
mass
Arenaria_serpyllifol [mg]
0.2
ia_agg.
Artemisia_vulgaris_
Achillea_pannonica 1.32
61.0
agg.
Agrostis_capillaris
0.60
Agrostis_stolonifera
0.60
_agg.
Agrostis_vinealis
-1.41
Ajuga_genevensis
-0.85
Apera_spica_venti
Arenaria_serpyllifol
ia_agg.
Artemisia_vulgaris_
agg.
0.60
-1.45
0.60
Specific
ln
Leaf
leaf
Mean Grazing
Life
Soil
Nitro (Seeds
DNA
size
Light
pH
area
seed toleran
fertility
gen
per
content
2 span
2
[mm ]
mm /m
weight
ce
shoot)
g
567.8 5
7
3
6
2
6.3
8.1
19.1
0.1
5.1
1147.9 5
7
0
4
4
5.3
26.8
7.1
0.1
5.0
1147.9
5
8
7
0
5
12.1
22.8
7.0
22.8
5
9
9
3
2
4.3
15.6
6.9
Specific
ln
365.5
5
8
3
7
2
12.1
25.8
5.9
Leaf
leaf
Life
Soil
DNA
1147.9
1 Light
6
6
5 Nitro
0 (Seeds
8.5
0.0
10.8
size
pH
area
fertility
gen
per
content
2 span
2
[mm
/m 1.7
3.6 ] 0.5 8
4
7
0 shoot)
6.0 mm
16.1
g
-0.26 0.58
-0.66
0.76
-0.66
2.31
1147.9
5 -0.58
7
6
0 -0.35
8
12.8 -0.63
0.0
6.0
0.94 0.58 -0.58 -1.80 0.00 0.45 -0.98 1.23
-0.20
0.94
0.58 0.58
-1.39 0.58 1.73
-0.68 0.58 0.58
0.94
-1.73
1.59
-1.43
0.58
1.86
0.94
0.85 -1.51 0.84
0.1
9.0
0.1
2.0
1.8 Grazing
5.1
Mean
0.1 toleran
5.1
seed
weight
ce
0.1
5.1
-0.38
0.1
-0.38
-0.05
5.1
-0.10
1.13
0.83
-0.22
-0.38
2.17
1.61 -0.38 -0.35 -1.27
-0.66 1.13 -0.35 1.13
0.12
1.13
-0.24
-0.45
-0.38
2.65
-1.80
-0.05
0.47
0.04
-1.43
0.57
-0.38
-0.05
-0.28 1.13 -1.14 -0.75
0.17
-1.33
-0.38
-0.05
-1.43
-0.43
-0.38
-0.05
0.38 -1.14
0.58 -0.58 0.47 -1.51 2.03
=(S12ŚREDNIA(S$4:S$11))/ODCH.STAND.POPUL(S$4:S
$11)
1.36
We use Z-transforms to
normalize the trait values
Agros
tis_st Agrosti
Apera
Achillea_p Agrostis_c
Ajuga_gen
Species
olonif s_vine
_spica
annonica apillaris
evensis
era_a alis
_venti
gg.
Achillea_pannonica
0.00
3.79
5.37 5.58
5.43
3.77
Agrostis_capillaris
3.79
0.00
4.54 5.61
4.79
4.75
Agrostis_stolonifera_agg.
5.37
4.54
0.00 6.01
5.45
5.44
Agrostis_vinealis
5.58
5.61
6.01 0.00
5.35
6.04
Ajuga_genevensis
5.43
4.79
5.45 5.35
0.00
5.93
Apera_spica_venti
3.77
4.75
5.44 6.04
5.93
0.00
Arenaria_serpyllifolia_agg
5.58
5.06
6.18 4.28
4.69
4.81
.
Artemisia_vulgaris_agg.
5.13
4.76
3.61 5.96
5.98
4.74
Arenari
Artemisia
a_serp
_vulgaris_
yllifolia
agg.
_agg.
5.58
5.06
6.18
4.28
4.69
4.81
5.13
4.76
3.61
5.96
5.98
4.74
0.00
6.52
6.52
0.00
The Euclidean distance matrix
Eigenvalues
-8.80
-6.78
-5.67
-4.74
-4.09
-3.44
-2.84
36.35
Squared
eigenvalues
77.43
45.96
32.16
22.44
16.70
11.81
8.08
1321.43
Explained
variance
0.05
0.03
0.02
0.01
0.01
0.01
0.01
0.86
The dominant
eigenvector
(DEV) contains
86% of variance
in the distance
matrix.
Dominant
2nd
eigenvector eigenvector
0.340
0.139
0.328
0.130
0.356
0.406
0.374
-0.449
0.364
-0.283
0.347
0.139
0.360
-0.513
0.357
0.484
Traits
Eigenvalue
S
Achillea_pannonica
Agrostis_capillaris
Agrostis_stolonifer
Agrostis_vinealis
Ajuga_genevensis
Apera_spica_venti
Arenaria_serpyllifo
DEV phylogeny
1
2011
104786
DEV
0.249299
0.202592
0.202402
0.331781
0.271579
0.202506
0.336897
Phylogeny
Eigenvalue
S
Achillea_pannonica
Agrostis_capillaris
Agrostis_stolonifer
Agrostis_vinealis
Ajuga_genevensis
Apera_spica_venti
Arenaria_serpyllifo
0.6
0.4
0.2
0
0
DEV contains
information on
the average
distance of
species in
niche space
Trait
Spearman's r
Leaf_mass_[mg]
0.12525
Leaf_size_[mm2]
0.055584
Life_span
0.14348
Light
-0.13865
Soil_fertility
0.007277
pH
-0.01992
Nitrogen
-0.06805
ln_(Seeds_per_shoot)
0.10293
0.2
0.4
0.6
0.8
1
Specific_leaf_area_m
m2/mg
0.049542
DEV traits
DNA_content
0.022556
None of the traits is correlated to Mean_seed_weight
0.097499
phylogenetic distance. Grazing_tolerance
-0.03174
R² = 9E-05
0.8
19855
DEV
0.296259
0.318521
0.318521
0.318521
0.323584
0.318354
0.318607
Ecological niches
Species
Achillea_pannonica
Agrostis_capillaris
Agrostis_stolonifera_
agg.
Agrostis_vinealis
Ajuga_genevensis
Apera_spica_venti
Arenaria_serpyllifolia
_agg.
Artemisia_vulgaris_a
gg.
Leaf
mass
[mg]
1.32
0.60
Specific
ln
leaf
Mean Grazing
Leaf size Life
Soil
Nitrog (Seeds
DNA
Light
pH
area
seed toleranc
[mm2] span
fertility
en
per
content
2
mm /m
weight
e
shoot)
g
-0.26 0.58 -0.58 -0.66 0.76 -0.35 -0.66
-0.63
2.31
-0.38
-0.05
0.94 0.58 -0.58 -1.80 0.00 0.45 -0.98
1.23
-0.20
-0.38
-0.10
0.60
0.94
-1.41
-0.85
0.60
-1.51 0.84
1.13
0.83
-0.22
-0.38
2.17
-1.39 0.58 1.73
-0.68 0.58 0.58
0.94 -1.59 -1.73
1.61 -0.38 -0.35
-0.66 1.13 -0.35
0.47 0.38 -1.14
-1.27
1.13
0.04
0.12
1.13
-1.43
-0.24
-0.45
0.57
-0.38
2.65
-0.38
-1.80
-0.05
-0.05
-1.45
-1.43 -1.86 0.58
-0.28
1.13 -1.14
-0.75
0.17
-1.33
-0.38
-0.05
0.60
0.94
0.47
-1.51 2.03
1.36
-1.43
-0.43
-0.38
-0.05
Dominant
2nd
eigenvector eigenvector
0.340
0.139
0.328
0.130
0.356
0.406
0.374
-0.449
0.364
-0.283
0.347
0.139
0.360
-0.513
0.357
0.484
0.58 0.58
0.58 -0.58
0.85
Convex hulls, eigenvector ellipses, and
functional attribute diversity
Soil fertility
1.5
1
l1
EV2
EV1
The area of the
eigenvector
ellipse is a
measure of
niche space
l1
0.5
𝐴 = 𝜋𝜆1 𝜆2
0
0.4
0.6
Eigenvalues
1.58
EV1
0.71
-0.71
0.70
EV2
0.71
0.71
Eigenvectors are
always orthogonal.
0.8
1
1.2
Light
1.4
The niche space spanned by
light and soil fertility spans
3.47 units.
Compare these value with those
obtained from a null model
3.47 = 𝜋 × 1.58 × 0.70
1.6
Axes length
are given by
the respective
eigenvalues
The larger the
niche space is
the higher is
the functional
diversity of a
community or
a species.
Convex hulls CH
Soil fertility
1.5
1
The area of a
convex hull is a
measure of total
niche space.
0.5
0
0.4
0.6
0.8
1
1.2
Light
1.4
1.6
Two dimensional convex hulls are easy to calculate.
1
𝐴 = 2 (𝑥1 𝑦2 -𝑥2 𝑦1 + 𝑥2 𝑦3 − 𝑥3 𝑦2 +…+𝑥𝑛−1 𝑦𝑛 − 𝑥𝑛 𝑦𝑛−1 + 𝑥𝑛 𝑦1 − 𝑥1 𝑦𝑛
High dimensional convex hull are difficult to obtain.
Functional attribute diversity FAD
Achillea_
Agrostis_ Agrosti Ajuga_
Agrostis_c
Species
pannoni
stolonife s_vine geneve
apillaris
ca
ra_agg. alis
nsis
Achillea_pannonica
0.00
3.79
5.37
5.58 5.43
Agrostis_capillaris
3.79
0.00
4.54
5.61 4.79
Agrostis_stolonifera_agg.
5.37
4.54
0.00
6.01 5.45
Agrostis_vinealis
5.58
5.61
6.01
0.00 5.35
Ajuga_genevensis
5.43
4.79
5.45
5.35 0.00
Apera_spica_venti
3.77
4.75
5.44
6.04 5.93
Arenaria_serpyllifolia_agg. 5.58
5.06
6.18
4.28 4.69
Apera_ Arenaria_
spica_v serpyllifol
enti
ia_agg.
3.77
5.58
4.75
5.06
5.44
6.18
6.04
4.28
5.93
4.69
0.00
4.81
4.81
0.00
2 𝑑𝑖𝑗
𝐹𝐴𝐷 =
;𝑗 > 1
𝑛(𝑛 − 1)
𝐹𝐴𝐷 =
5.58 + 5.43 + 5.58 + 5.35 + 4.28 + 4.69
= 5.15
6
Raw FAD scores are meaningless.
You have to compare these scores with an appropriate null model
of species occurrences.
Sites
1
0
0
1
1
0
1
Total raw functional attribute
diversity (grey bars) increased
while the respective SES scores of
FAD (red bars) and convex hulls
(blue bars) decrease during
succession when compared to a
neutral null model .
High plant cover decreased an species
richness increases standardized effect
sizes of FAD (neutral null model) in all
study years.
2011
Soil characteristics did not significantly
influence SES FAD (functional
diversity).
2006
Regression coefficients
The evolutionary dimension of species occurrences
Eigenvector mapping, eigenvector regression, logistic eigenvector regressiuon
EV1EV2
Sites
Species
Species
Species
Phylogenetic
distance
matrix P
Significant
eigenvectors
Species
abundance matrix
M
𝑨 = 𝑼𝜷
The explained variance r2 of this regression is a measure of the influence of evolutionary
history on species abundances.
Net relatedness index
S
Achillea_pannonica
Agrostis_capillaris
Agrostis_stolonifera_agg
.
Agrostis_vinealis
Ajuga_genevensis
Apera_spica_venti
Agrostis_st
Achillea_p Agrostis_c
Agrostis_vi Ajuga_gen Apera_spic
olonifera_
annonica apillaris
nealis
evensis
a_venti
agg.
0.0
179.0
179.0
179.0
117.0
179.0
179.0
0.0
2.5
2.5
179.0
6.7
M7-2
0.5
0.5
179.0
2.5
0.0
2.5
179.0
6.7
0
179.0
117.0
179.0
2.5
179.0
6.7
2.5
179.0
6.7
0.0
179.0
6.7
179.0
0.0
179.0
6.7
179.0
0.0
0
Phylogenetic distance matrix
179 + 117 + 179
𝑅𝑎𝑤 𝑁𝑅𝐼 =
= 158
3
The raw net relatedness index is the
phylogenetic distance of all species present at
a focal site.
𝑟𝑎𝑤 𝑁𝑅𝐼 − 𝑟𝑎𝑛𝑑𝑜𝑚𝑖𝑧𝑒𝑑 𝑟𝑎𝑤 𝑁𝑅𝐼
𝑁𝑅𝐼 = −
𝜎𝑟𝑎𝑛𝑑𝑜𝑚𝑖𝑧𝑒𝑑 𝑟𝑎𝑤 𝑁𝑅𝐼
NRI increases with increasing
phylogenetic clustering
0.5
0
Abundance vector
M7-2
Calculate the raw
NRI from 1000
randomized
abundance
matrices
0
0
0.5
0.5
0.5
0
Randomized
bundance vector
NRI2011
4
NTI2011
4
3.5
3.5
3
3
2.5
2.5
2
2
1.5
1.5
1
1
0.5
0.5
0
0
-0.5
-0.5
-1
-1
-1.5
-1.5
-2
-2
-2.5
-2.5
-3
-3
-3.5
-3.5
-4
-4
-4.5
-4.5
-5
-5
-5.5
-5.5
-6
-6
-6.5
-6.5
-7
-7
-7.5
-7.5
-8
-8
-8.5
-8.5
-9
-9
-9.5
-9.5
Relating phylogenetic patterns to environmental variables
NRI
Variables
Constant
Study year
Species
richness
Abundance
Soil carbon
Sand
pH
B
Error B
t
p
1305.600 121.690 10.729 <0.0001
-0.651 0.061 -10.746 <0.0001
Phylogenetic eigenvector regression
r2
B
Error B
t
p
-15.484 3.2963 -4.697 <0.0001
0.008 0.0016 4.748 <0.0001
0.193
0.016
11.947 <0.0001
0.001
0.0004
2.538
0.011
-0.023
0.345
0.008
-0.008
0.002
0.096
0.011
0.127
-13.173 <0.0001
3.588 <0.0001
0.699
0.484
-0.066 0.947
0.001
-0.008
-0.001
-0.002
0.0000
0.0026
0.0003
0.0034
22.931 <0.0001
-3.118 0.002
-3.746 <0.0001
-0.610 0.542