Transcript Slajd 1 - Jagiellonian University

Community and gradient analysis:
Matrix approaches in macroecology
The world comes in fragments
Basic metrics of food webs
A pitcher plant
(Nepenthes
albomarginata) food
web
4
S = 19
1
Nepenthes albomarginata
L
= 19x18/2=171
max
2
5
6
12
3
7
8
13
14
15
17
18
19
L = 35
C = 35 / 171 = 0.2
Ch = 100
Li = 40
ChL = 100 / 40 = 2.5
L / S = 35 / 19 = 1.8
9
10
11
16
S = 19 species
Lmax = 19*18/2 = 171 possible links between two species
L = 35 realized links between two species
Connectance: C = 35/171
Ch = 100 total length of all food chains
Li = 40 is the total number of chains
ChL = 100/40 = 2.5 is the average chain length
L/S = 35/19 = 1.8 is the mean number of links per species
S
8
3
9
1
2
6
4
10
5
7
4
1
1
1
1
0
0
0
0
1
0
2
1
1
1
0
0
0
1
0
0
0
7
0
0
1
0
1
1
0
1
0
0
1
0
0
0
1
1
0
0
1
0
0
3
0
1
0
0
0
1
1
0
0
0
5
1
0
1
0
0
1
0
0
0
0
6
1
1
0
0
1
0
0
0
0
0
8
1
0
0
1
0
0
0
0
0
1
S
5
4
4
3
3
3
3
3
N = 28
Fill = 28/80=0.35
Dm=28/10=2.8
Dn=28/8=3.5
S
5
4
4
3
3
3
2
2
1
1
Food web metrics translated into
matrix metrics
Food web
terminology
Links
Connectivity
Linkage density
Web asymmetry
Compartments
Matrix terminology
Metric
Number of incidences N
Matrix fill
Mean marginal total
Matrix shape
Boundary clumping
Morisita, Fractal dimension
Coherence
Diversity
Evenness
Matrix size
Degree distribution
Shared links
Togetherness
Underdispersion
Aggregation
Nestedness
Overdispersion
Turnover
Dependence
Interaction
asymmetry
Niche overlap
Mean togetherness
Nm
d(Ni)/dI
NODFc, NODFr
BR, T, NODF
Bray-Curtis distance, coefficient of
correlation
Metrics of species associations in biogeographic matrices
Species/Site
A
B
C
D
E
F
G
H
I
J
a
1
1
1
1
1
1
1
1
1
1
b
1
0
0
1
1
1
0
1
1
0
c
1
1
1
1
1
0
0
0
0
0
d
1
0
0
1
1
1
0
0
0
0
e
1
1
1
1
0
0
1
0
0
0
f
1
1
1
0
0
0
0
0
0
0
g
1
1
1
0
0
0
0
0
0
0
h
1
1
1
0
0
0
0
0
0
1
The C-score as a metric
of negative associations
CS 
CS 
 1 ... 1 


4  ... ... ...
i, j 
1 ... 1 

Clum ping
Species( Species 1) Sites( Sites  1)
i, j
S ( S  1)
 1 ... 0 


4  ... ... ...
i, j 
0 ... 1 

CS 
Species( Species 1) Sites( Sites  1)
Checkerboards
The Clumping-score as a metric of
positive associations
2 ( N i  N ij )( N j  N ij )
4 ( N i  N ij )( N j  N ij )
i, j
Species ( Species  1) Sites ( Sites  1)
The Togetherness-score as a metric
of niche overlap
 1 ... 0 


4  ... ... ...
i, j 
1 ... 0 

Togetherness 
Species( Species 1) Sites( Sites  1)
The additive nature of the C-score
S
1
2
3
4
5
6
7
8
9
10
1
1
1
1
1
1
0
0
0
0
0
Sum
5
2
1
1
1
0
1
0
0
1
0
0
3
1
1
0
1
0
1
1
0
0
0
4
1
0
1
1
0
1
0
0
1
0
5
0
1
0
0
1
0
1
0
1
1
6
0
0
1
0
0
1
1
1
0
1
7
0
0
0
1
0
1
0
1
1
1
5
5
5
5
5
5
5
-0.817
-0.816
-0.816
0.816
0.817
0.817
1.414
AT
AC
Score -1.414
AC
8
0
0
0
0
1
0
1
1
1
1
Sum
Score
4
-1.581
4
-0.913
4
-0.913
4
-0.913
4
-0.002
4
0.002
4
0.913
4
0.913
4
0.913
4
1.581
Numbers of checkerboards
for entries within the area AT
are a measure of spatial
species turnover.
Numbers of checkerboards
for entries within the area
ATC are a measure of
turnover independent
species segregation.
CMixed = CS – CTurn - CSegr.
The rank correlation of matrix entries is a metric of spatial turnover.
1 1
1 2
1 3
2 1
2 2
…….
7 10
8 10
R2 is a more liberal metric than Cturn.
R2 = 0.347
The correlation of ordination scores is
also a metric of turnover but even less
selective.
Range size coherence
Coherent range
size
Scattered range
size
S
1
2
3
4
5
6
7
8
9
10
1
1
1
1
1
1
0
0
0
0
0
2
1
1
1
0
1
0
0
1
0
0
3
1
1
0
1
0
1
1
0
0
0
4
1
0
1
1
0
1
0
0
1
0
5
0
1
0
0
1
0
1
0
1
1
6
0
0
1
0
0
1
1
1
0
1
7
0
0
0
1
0
1
0
1
1
1
8
0
0
0
0
1
0
1
1
1
1
Sum
5
5
5
5
5
5
5
5
Sum
4
4
4
4
4
4
4
4
4
4
There are 17 embedded absences.
The number of embedded absences is a measure of
species range size coherence.
The metric depends strongly on the ordering of rows and columns
The measurement of nestedness
The distance concept
of nestedness.
A
1
1
1
1
1
0
1
1
1
1
1
1
0
1
0
0
0
0
0
0
C
1
1
1
1
1
0
1
0
0
0
0
1
1
0
1
0
0
1
1
0
1
1
1
0
1
0
1
1
0
0
1
0
1
0
0
1
1
0
0
0
M O P D F H K E B J N L G
1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1
D 1 1 1 1 1 1 1
0 1 1 1 1 1 1 0 1 0 1 1 0
1 1 20;P
1 1 0 0 1 1 0 1 1 0 1
1 1 0 1 1 1 1 1 0 0 0 0 0
d
1 0 1 1 1 1 1 1 0 1 1 0 0
1 0 1 0 0 1 1 0 d 1 13;J
0 1 0 0
1 0 1 1 1 1 0 1 0 1 0 0 0
1 1 X;Y
0 0 1 1 0 1 0 0 0 1 0
1 1 0 0 0 1 0 0 0 1 0 1 0
1 1 0 0 1 0 1 0 0 0 0 0 0
0 1 1 0 1 0 0 0 0 0 0 0 1
0 0 1 1 0 0 0 0 1 0 0 0 0
0 0 0 1 0 0 0 0 1 0 0 0 0
0 0 1 0 0 0 0 0 0 0 0 D0 0
0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0
0 1 0 0 0 0 0 0 0 0 0 0 0
Sum 12 11 10 10 10 10
9
9
9
8
7
6
6
6
5
Sum
16
16
12
11
10
9
9
9
7
6
6
6
5
3
2
1
1
1
1
1
Sort the matrix rows and olumns
according to some gradient.
Define an isocline that divides the
matrix into a perfectly filled and an
empty part.
The normalized squared sum of
relative distances of unexpected
absences and unexpected presences
is now a metric of nestednessis.
4
2

Sp Si 
d ij  
100
1 
T

 

0.04145 SpSi  i 1 j1  D ij  


Z-score
I
1
3
7
15
20
4
9
13
2
11
17
18
5
8
16
6
10
12
14
19
8
6
4
2
0
-2
-4
-6
-8
0
50
Matrix size
100
Nestedness based on Overlap and Decreasing Fill (NODF)
r1
c2
1
0
c3
1
c4
c5
1
Nestedness among columns
c1
1
r2
1
1
1
0
0
r3
0
1
1
1
0
r4
1
1
0
0
0
r5
1
1
0
0
0
Nestedness among rows
c1 c2
c1 c3
c1 c4
c1 c5
c2 c3
1
0
1
1
1
1
1
1
0
1
1
1
1
1
1
0
1
0
1
1
0
1
0
1
0
1
0
0
1
1
1
1
1
0
1
0
1
0
1
0
1
1
1
0
1
0
1
0
1
0
Npaired=0
Npaired=67
Npaired=50
Npaired=100
Npaired=67
c2 c4
c2 c5
c3 c4
c3 c5
c4 c5
0
1
0
1
1
1
1
1
1
1
1
0
1
0
1
0
1
0
0
0
1
1
1
0
1
1
1
0
1
0
1
0
1
0
0
0
0
0
0
0
1
0
1
0
0
0
0
0
0
0
Npaired=50
r1
1
0
1
1
1
r2
1
1
1
0
0
r1
1
0
1
1
1
r3
0
1
1
1
0
r1
1
0
1
1
1
r4
1
1
0
0
0
r1
1
0
1
1
1
r5
1
1
0
0
0
r2
0
1
r3
Npaired=67
Npaired=67
Npaired=50
1
1
Npaired=50
r2
0
1
1
1
0
r4
1
1
0
0
0
r2
1
1
0
0
0
r5
1
1
0
0
0
r3
1
1
1
0
0
r4
0
1
1
1
0
r3
1
1
1
0
0
r5
1
1
0
0
0
Npaired=0
Npaired=100 Npaired=100 Npaired=100
Npaired=50
Npaired=0
Ncolumns = 63.4
Npaired=0
Nrows = 53.4
Npaired=100
NODF = 58.4
NODF 
N
paired
 n(n  1)   m(m  1) 
 2   

2
NODF is a gap based metric and more conservative than temperature.
1
0
1
0
0
0
Npaired=50
r4
1
r5
1
1
1
1
0
0
0
0
0
Npaired=100
The disorder measure of Brualdi and Sanderson
Ho many cells must be filled or emptied to achieve a perfectly ordered matrix.
The Brualdi Sanderson measure is a count of this number
Sites
Species
1
A
1
B
1
C
1
D
1
E
1
F
1
G
1
H
1
I
1
J
1
K
0
L
1
Species 11
Sites
2
1
1
1
1
1
1
1
0
1
1
1
0
10
3
1
1
1
1
1
1
0
1
1
0
1
1
10
4
1
1
1
1
0
1
1
1
1
0
0
1
9
5
1
1
1
1
1
0
1
1
0
0
1
0
8
6
1
1
1
1
1
0
0
1
0
0
1
1
8
7
1
1
1
1
1
1
1
0
0
1
0
0
8
8
1
1
1
1
1
1
0
0
0
1
0
0
7
Discrepancy is a gap counting metric.
8
8
8
8
7
6
5
5
4
4
4
4
How to measure species aggregation?
S
50
43
41
40
49
33
31
39
34
42
37
35
36
29
32
24
27
26
23
21
2
47
5
3
11
15
14
13
12
10
8
9
16
6
4
1
7
17
25
45
38
28
46
44
18
48
19
20
30
22
50
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
44
0
1
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
46
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
43
0
0
0
0
0
0
1
0
1
0
1
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
47
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
2
1
1
1
1
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0
0
0
0
7
0
1
1
1
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0
0
0
0
0
6
0
1
1
1
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0
0
0
0
0
3
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0
0
0
0
4
0
1
1
1
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0
0
0
0
1
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0
0
0
5 15 25
0 0 0
1 0 0
1 0 0
1 0 0
0 0 0
1 1 0
1 1 1
1 1 0
1 1 0
1 0 0
1 0 0
1 1 0
1 1 0
1 1 0
1 1 0
1 0 1
1 0 1
1 0 1
1 0 1
1 0 1
1 1 1
0 0 0
1 1 1
1 1 1
1 1 1
1 1 1
1 1 1
1 1 1
1 1 1
1 1 1
1 1 1
1 1 1
1 1 1
1 1 1
1 1 1
1 1 1
1 1 1
1 1 0
0 1 0
0 0 0
0 0 0
0 1 0
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
1 0 0
0 0 0
24
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
0
0
0
0
0
0
1
0
0
0
0
0
23
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
0
0
0
0
0
1
0
0
0
0
0
0
29
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
27
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0
0
1
0
0
0
26
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0
0
0
1
0
0
32
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
1
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0
0
1
0
0
0
33
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
1
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0
1
0
0
0
0
34
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
0
0
0
0
0
0
1
0
0
0
0
0
30
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
0
1
1
0
0
0
0
0
0
0
12
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
1
1
1
1
1
1
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
1
1
0
0
1
0
1
1
0
1
28
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
1
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
0
1
0
0
1
0
0
0
0
0
10
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
1
1
0
0
1
0
1
1
1
1
9
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
1
1
0
0
1
0
1
1
1
1
8 31 11
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 0 1
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
1 0 0
1 0 0
1 0 1
1 0 1
1 0 1
1 0 1
1 0 1
1 1 1
0 0 0
1 1 1
1 1 1
1 1 1
1 1 1
1 1 1
1 1 1
1 1 1
1 1 1
1 1 1
1 1 1
1 1 1
1 1 1
1 1 1
1 1 1
1 1 1
1 1 1
1 1 1
0 0 0
1 0 1
1 1 1
0 0 0
0 0 0
1 0 1
0 0 0
1 0 1
1 0 1
1 0 1
1 0 1
13
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
0
1
0
1
1
0
1
14
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
1
1
0
0
1
0
1
1
1
1
19
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
0
1
0
0
1
0
1
1
0
1
16
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
1
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
1
1
0
0
1
0
1
1
0
1
dij
20
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
1
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
0
1
1
0
1
0
1
1
0
1
18
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
1
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
0
1
1
0
1
0
1
1
0
1
35
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
1
0
1
1
1
1
17
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
1
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
0
1
0
1
1
0
1
1
0
1
22
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
0
1
0
0
1
0
1
1
0
1
21
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
0
1
0
0
1
0
1
1
0
1
45
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
48
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
49
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
37
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
1
0
1
1
1
1
40
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
1
1
1
1
36
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
1
0
1
1
1
1
41
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
1
1
1
1
39
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
1
0
1
1
1
1
42
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
0
1
1
1
1
38
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
Compartmented matrix
Nearest neighbor metrics
species sites
NND 
 d
i 1
j 1
2
ij
2 fill
Nearest neighbour is a presence –
absence metric
NND has weak power at higher matrix fill
Join count statistics
A sum of cell entries around a focal cell
multiplied by the entry of the focal cell
species1 sites 1 i1i 1 j1 j 1
JoinOcc 
   x
i 2
j  2 i1i 1 j1 j 1
ij
xi1 j1
species sites
 x
i 1
j 1
ij
Join count operates on presence –
absence and abundance matrices
Other metrics proposed:
Morisita
Simpson
Soerensen
Block variance
Ordination score variance
Marginal variances
These metrics have very low power a
moderate to small matrix size and high or
low matrix fill.
Abundance based metrics
The C-score extension
Species
ful
guc 3pog sos 2pog
dabwros gil ter 1pogwil mil swi kor hel lip
wron Sum
Pterostichus nigrita (Paykull)
1
1
2 18 2 5
0 58 53 30 61 39 0 0 0
2
2
Platynus assimilis (Paykull)
48
2
25
9 7 4
0 39 0
0 1
0 0 76 9 117
0
Amara brunea (Gyllenhal)
10
4
0 40 0 5
1 0 0
0 0
0 19 0 3
1
1
Agonum lugens (Duftshmid)
0
0
0
0 0 2
1 3 2
1 2
0 0 0 0
0
1
Loricera pilicornis (Fabricius)
5
0
0
0 1 1
0 5 0
0 1
3 0 0 0
0
0
Pterostichus vernalis (Panzer)
0
0
0
1 0 0
1 0 2
0 21
7 0 0 0
0
1
Amara plebeja (Gyllenhal)
0
0
0
2 0 1
0 5 0
0 0
0 1 0 0
4
0
Badister unipustulatus Bonelli
0
0
0
1 0 0
0 3 0
0 0
0 4 0 0
3
0
Lasoitrechus discus (Fabricius)
0
0
0
0 0 0
0 0 1
0 0
1 0 0 0
0
0
Poecilus cupreus (Linnaeus)
0
0
0
2 0 0
0 0 0
0 0
0 0 1 0
0
0
Sum
4
3
2
7 3 6
3 6 4
2 5
4 3 2 2
5
4
13
11
9
7
6
6
5
4
2
2
a b 

; (a  b, a  c, d  b, d  c)  (a  b, a  c, d  b, d  c)
c
d


The metric CA is a count of the number of abundance checkerboards in the matrix.
CAST 
4CA
m(m  1)n(n  1)
Other 2x2 submatrices catch matrix properties that have not well defined
ecological meaning.
Nestedness in abundance matrices
Species/Sites
ter wron swi dab kor
Pterostichusstrennus(Panzer)
704 36 169 1199 13
Pterostichusmelanarius
8
141
2
9
135
Carabusgranulatus
18
11
12 154 110
Pterostichusoblongopunctatus(Fab)
7
3
22
5
13
Oxypselaphusobscurus(Herbst)
13 166
7
27
48
Pterostichusnigrita(Paykull)
2
5
18
1
2
Pseudoophonusrufipes(DeGeer)
3
13
0
1
2
Pterostichusdiligens(Sturm)
4
12
3
1
1
Patrobusatrorufus(Stroem)
11
2
35
0
6
Synuchusvivalis(Illiger)
51
19
14
1
12
Leistusterminatus(Hellwig)
1
10
3
4
1
Platynusassimilis(Paykull)
9
0
4
76
0
CarabusnemoralisMuller
0
10
16
5
2
Harpalus4-punctatusDejean
69
17
67
9
29
Pterostichusantracinus
46
1
21
0
1
Pterostichusminor(Gyllenhal)
5
1
48
1
7
Amarabrunea(Gyllenhal)
4
3
1
1
0
Badisterbullatus(Schrank)
5
4
1
4
2
wil
60
1
11
5
25
0
1
0
22
2
3
117
12
41
0
2
40
1
sos 2pog 3pog gil
17
26 187 13
6
188
7
180
77
11
25
19
28
30
5
3
0
1
278 27
1
39
30
2
3
1
2
90
1
5
11
4
7
11
0
348
4
24
1
5
3
7
4
3
39
2
7
9
8
0
0
1
9
555
0
0
0
0
1
0
5
2
0
0
10
0
19
0
0
0
7
0
guc
0
4
113
4
85
0
5
5
0
10
0
1
6
6
2
0
1
0
mil
lip
hel
345
4
29
8 1019 11
52
59
0
6
0
14
37
0
96
53
2
61
3
5
0
18
0
1
0
37
2
0
0
0
0
1
0
0
25
48
14
0
6
0
0
77
274
2
0
0
21
0
5
0
0
0
0
0
wros
394
83
0
24
96
58
0
1
9
0
0
0
5
0
11
0
0
0
ful
0
0
11
47
0
0
6
0
81
2
0
0
0
0
11
28
0
0
Species
Weight
15
343
15
218
12
242
11
10
10
815 1175 345
9
681
7
186
17
960
n 1
n
WNODFc, r  100 
i 1
WNODF 
17
454
17
16
16
443 1498 384
kij
j i 1 N j
2(WNODFc  WNODFr )
m(m  1)  n(n  1)
14
902
14
584
13
704
1pog Species
428
15
0
15
1
15
0
15
80
14
0
13
0
13
0
13
0
12
0
12
11
12
0
11
0
11
0
10
0
10
0
10
0
9
2
8
Weight
3624
1802
684
216
986
274
135
67
571
145
51
337
85
879
370
120
84
26
5
522
The metric is a sum of all pairs in the matrix
(first sorted accoding to species richness
then sorted according to weights), where the
weight in the row/column of lower species
richness is smaller than the weight in the
row/column of higher species richness
A complete table of methods for co-occurrence analysis
Segregated
Independent of matrix sorting
C-score
Segregated
Simpson dissimilarity
Soerensen dissimilarity
Other joint occurrence/absence
metrics
Dependent of matrix sorting
Segregated
Other distance based metrics
Segregated
For seriation
Whole matrix
Aggregated
Aggregated
nested
Clumping score
CS/Clumping
NestPairs
Togetherness
Species only
Data type
PA null models
PA
A
A and PA
All
Data type
PA null models
PA
PA
A and PA
A and PA
No fixed - fixed
No fixed - fixed
No fixed - fixed
All
PA
No fixed - fixed
Data type
PA null models
PA
A and PA
A and PA
A and PA
A and PA
PA
PA
All
All
All
All
All
All
All
Aggregated
Data type
PA null models
Morisita
PA
PA
PA
PA
PA
All
All
All
All
All
Aggregated
Simpson similarity
Soerensen dissimilarity
Morisita
Chao
Other joint occurrence/absence
metrics
Whole matrix
Aggregated
Aggregated
nested
NND
Block
Join-coint
Other distance based metrics
NODF
BR
T
Species only
Embedded absences
r2
CTurn
CSegr
All
A null
models
All
All
A null
models
All
All
A null
models
All
All
All
All
A null
models
Pattern detection in large matrices
Pajek: software for social network
analysis
WAND: ecological network analysis
These programs use cluster analysis and
ordination to sort the matrix according to
numbers of occurrences. Didstance metrics
are then used to identify compartments.
KliqueFinder: software for
compartment analysis
They generate hypotheses about matrix
structure.
They do not fully allow for statistical
inference.