15
Nicolaus C opernicus U niversity – D epartm entofAnim al Ecology Community and gradient analysis: Matrix approaches in macroecology The world comes in fragments
Nicolaus Copernicus University – Department of Animal Ecology
Community and gradient analysis: Matrix approaches in macroecology
The world comes in fragments
Basic metrics of food webs
S = 19 speciesLmax = 19*18/2 = 171 possible links between two speciesL = 35 realized links between two speciesConnectance: C = 35/171Ch = 100 total length of all food chainsLi = 40 is the total number of chainsChL = 100/40 = 2.5 is the average chain lengthL/S = 35/19 = 1.8 is the mean number of links per species
1
2 3
4 5 6 7 8 9 10 11
12 13 14 15 16
17 18 19
S = 19Lmax = 19x18/2=171L = 35C = 35 / 171 = 0.2Ch = 100Li = 40ChL = 100 / 40 = 2.5L / S = 35 / 19 = 1.8
A pitcher plant (Nepenthes albomarginata) food web
Nepenthes albomarginata
Food web terminology
Matrix terminology Metric
Links Number of incidences N Connectivity Matrix fill
Linkage density Mean marginal total
Web asymmetry Matrix shape
Compartments Boundary clumping Morisita, Fractal dimension
Coherence
Diversity Matrix size Nm Evenness Degree distribution d(Ni)/dI
Shared links Togetherness
Underdispersion Aggregation NODFc, NODFr
Nestedness BR, T, NODF
Overdispersion Turnover
Dependence Interaction asymmetry
Niche overlap Mean togetherness Bray-Curtis distance, coefficient of correlation
S 4 2 7 1 3 5 6 8 S
8 1 1 0 0 0 1 1 1 53 1 1 0 0 1 0 1 0 49 1 1 1 0 0 1 0 0 41 1 0 0 1 0 0 0 1 32 0 0 1 1 0 0 1 0 36 0 0 1 0 1 1 0 0 34 0 1 0 0 1 0 0 0 2
10 0 0 1 1 0 0 0 0 25 1 0 0 0 0 0 0 0 17 0 0 0 0 0 0 0 1 1
S 5 4 4 3 3 3 3 3
Food web metrics translated into matrix metrics
N = 28Fill = 28/80=0.35Dm=28/10=2.8Dn=28/8=3.5
Metrics of species associations in biogeographic matrices
Species/Site a b c d e f g hA 1 1 1 1 1 1 1 1B 1 0 1 0 1 1 1 1C 1 0 1 0 1 1 1 1D 1 1 1 1 1 0 0 0E 1 1 1 1 0 0 0 0F 1 1 0 1 0 0 0 0G 1 0 0 0 1 0 0 0H 1 1 0 0 0 0 0 0I 1 1 0 0 0 0 0 0J 1 0 0 0 0 0 0 1
)1(
))((2,
SS
NNNNCS ji
ijjiji
)1()1(
))((4,
SitesSitesSpeciesSpecies
NNNNCS ji
ijjiji
)1()1(1...0.........0...1
4,
SitesSitesSpeciesSpeciesCS
ji
The C-score as a metric of negative associations
The Clumping-score as a metric of positive associations
)1()1(1...1.........1...1
4,
SitesSitesSpeciesSpeciesClumping
ji
Checkerboards
)1()1(0...1.........0...1
4,
SitesSitesSpeciesSpeciesssTogetherne
ji
The Togetherness-score as a metric of niche overlap
S 1 2 3 4 5 6 7 8 Sum Score
1 1 1 1 1 0 0 0 0 4 -1.581
2 1 1 1 0 1 0 0 0 4 -0.913
3 1 1 0 1 0 1 0 0 4 -0.913
4 1 0 1 1 0 0 1 0 4 -0.913
5 1 1 0 0 1 0 0 1 4 -0.002
6 0 0 1 1 0 1 1 0 4 0.002
7 0 0 1 0 1 1 0 1 4 0.913
8 0 1 0 0 0 1 1 1 4 0.913
9 0 0 0 1 1 0 1 1 4 0.913
10 0 0 0 0 1 1 1 1 4 1.581
Sum 5 5 5 5 5 5 5 5
Score -1.414 -0.817 -0.816 -0.816 0.816 0.817 0.817 1.414
AT
AC
AC
The additive nature of the C-score
CMixed = CS – CTurn - CSegr.
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.
The rank correlation of matrix entries is a metric of spatial turnover.
1 11 21 32 12 2…….7 108 10
R2 = 0.347
R2 is a more liberal metric than Cturn.
The correlation of ordination scores is also a metric of turnover but even less
selective.
S 1 2 3 4 5 6 7 8 Sum
1 1 1 1 1 0 0 0 0 4
2 1 1 1 0 1 0 0 0 4
3 1 1 0 1 0 1 0 0 4
4 1 0 1 1 0 0 1 0 4
5 1 1 0 0 1 0 0 1 4
6 0 0 1 1 0 1 1 0 4
7 0 0 1 0 1 1 0 1 4
8 0 1 0 0 0 1 1 1 4
9 0 0 0 1 1 0 1 1 4
10 0 0 0 0 1 1 1 1 4
Sum 5 5 5 5 5 5 5 5
Range size coherence
There are 17 embedded absences.The number of embedded absences is a measure of
species range size coherence.
Coherent range size
Scattered range size
The metric depends strongly on the ordering of rows and columns
I A C M O P D F H K E B J N L G Sum1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 163 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 167 1 1 1 0 1 1 1 1 1 1 0 1 0 1 1 0 12
15 1 1 0 1 1 1 1 0 0 1 1 0 1 1 0 1 1120 1 1 1 1 1 0 1 1 1 1 1 0 0 0 0 0 104 0 0 0 1 0 1 1 1 1 1 1 0 1 1 0 0 99 1 1 1 1 0 1 0 0 1 1 0 1 0 1 0 0 9
13 1 0 1 1 0 1 1 1 1 0 1 0 1 0 0 0 92 1 0 0 1 1 0 0 1 1 0 1 0 0 0 1 0 7
11 1 0 0 1 1 0 0 0 1 0 0 0 1 0 1 0 617 1 0 1 1 1 0 0 1 0 1 0 0 0 0 0 0 618 1 1 0 0 1 1 0 1 0 0 0 0 0 0 0 1 65 0 1 1 0 0 1 1 0 0 0 0 1 0 0 0 0 58 1 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 3
16 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 26 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1
10 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 112 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 114 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 119 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1
Sum 12 11 10 10 10 10 9 9 9 8 7 6 6 6 5 4
X;Y
13;J
20;Pd
d
D
D
The measurement of nestedness
The distance concept of nestedness.
2Sp Si
ij
i 1 j 1 ij
d100 1T0.04145 SpSi D
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.
-8-6-4-202468
0 50 100
Matrix size
Z-sc
ore
1 1
1 1
0 1
1 1
1 0
1 1
1 1
0 1
1 1
1 0
1 0
1 0
0 1
1 1
1 1
1 0
1 0
0 1
1 1
1 1
1 0
1 0
0 1
1 0
1 1
1 0
1 0
0 1
1 0
1 1
0
0
0
0
1
1
1
0
1
1
0
0
0
0
1
1
1
0
1
1
1 0
1 0
1 1
1 1
0 1
1 0
1 0
1 1
1 1
0 1
1 0
1 0
1 1
1 0
0 1
1 0
1 0
1 1
1 0
0 1
0
0
0
0
1
1
1
1
1
0
0
0
0
0
1
1
1
1
1
0
0 0
0 0
1 1
1 0
1 1
0 0
0 0
1 1
1 0
1 1
0
0
0
0
1
0
0
1
1
1
0
0
0
0
1
0
0
1
1
1
0
0
0
0
1
0
0
1
0
1
0
0
0
0
1
0
0
1
0
1
0
1
1 1 1 0
1 0 1 1
0
1
1 1 1 0
1 0 1 1
0
1
0 1 1 1
1 0 1 1
0
1
0 1 1 1
1 0 1 1
0
1
1 1 0 0
1 0 1 1
0
1
1 1 0 0
1 0 1 1
0
1
1 1 0 0
1 0 1 1
0
1
1 1 0 0
1 0 1 1
0
0
0 1 1 1
1 1 1 0
0
0
0 1 1 1
1 1 1 0
0
0
1 1 0 0
1 1 1 0
0
0
1 1 0 0
1 1 1 0
0
0
1 1 0 0
1 1 1 0
0
0
1 1 0 0
1 1 1 0
0
0
1 1 0 0
0 1 1 1
0
0
1 1 0 0
0 1 1 1
0
0
1 1 0 0
0 1 1 1
0
0
1 1 0 0
0 1 1 1
0
0
1 1 0 0
1 1 0 0
0
0
1 1 0 0
1 1 0 0
Nestedness among rowsN
este
dnes
s am
ong
colu
mns
c5c4c3c2c1
r5
r4
r3
r2
r1
0
0
0
0
1
1 1 0 0
1 1 0 0
0 1 1 1
1 1 1 0
1 0 1 1
c5c4c3c2c1
r5
r4
r3
r2
r1
0
0
0
0
1
1 1 0 0
1 1 0 0
0 1 1 1
1 1 1 0
1 0 1 1
c1 c2 c1 c3 c1 c4 c1 c5 c2 c3
c2 c4 c2 c5 c3 c4 c3 c5 c4 c5
Npaired=0 Npaired=67 Npaired=50 Npaired=100 Npaired=67
Npaired=50 Npaired=0 Npaired=100 Npaired=100 Npaired=100
r1
r2
r1
r3
r1
r4
r1
r5
r2
r3
r2
r4
r2
r5
r3
r4
r3
r5
r4
r5
Npaired=67
Npaired=67
Npaired=50
Npaired=50
Npaired=50
Npaired=50
Npaired=0
Npaired=0
Npaired=100
Npaired=100
Ncolumns = 63.4
Nrows = 53.4
NODF = 58.4
1 1
1 1
0 1
1 1
1 0
1 1
1 1
0 1
1 1
1 0
1 0
1 0
0 1
1 1
1 1
1 0
1 0
0 1
1 1
1 1
1 0
1 0
0 1
1 0
1 1
1 0
1 0
0 1
1 0
1 1
0
0
0
0
1
1
1
0
1
1
0
0
0
0
1
1
1
0
1
1
1 0
1 0
1 1
1 1
0 1
1 0
1 0
1 1
1 1
0 1
1 0
1 0
1 1
1 0
0 1
1 0
1 0
1 1
1 0
0 1
0
0
0
0
1
1
1
1
1
0
0
0
0
0
1
1
1
1
1
0
0 0
0 0
1 1
1 0
1 1
0 0
0 0
1 1
1 0
1 1
0
0
0
0
1
0
0
1
1
1
0
0
0
0
1
0
0
1
1
1
0
0
0
0
1
0
0
1
0
1
0
0
0
0
1
0
0
1
0
1
0
1
1 1 1 0
1 0 1 1
0
1
1 1 1 0
1 0 1 1
0
1
0 1 1 1
1 0 1 1
0
1
0 1 1 1
1 0 1 1
0
1
1 1 0 0
1 0 1 1
0
1
1 1 0 0
1 0 1 1
0
1
1 1 0 0
1 0 1 1
0
1
1 1 0 0
1 0 1 1
0
0
0 1 1 1
1 1 1 0
0
0
0 1 1 1
1 1 1 0
0
0
1 1 0 0
1 1 1 0
0
0
1 1 0 0
1 1 1 0
0
0
1 1 0 0
1 1 1 0
0
0
1 1 0 0
1 1 1 0
0
0
1 1 0 0
0 1 1 1
0
0
1 1 0 0
0 1 1 1
0
0
1 1 0 0
0 1 1 1
0
0
1 1 0 0
0 1 1 1
0
0
1 1 0 0
1 1 0 0
0
0
1 1 0 0
1 1 0 0
Nestedness among rowsN
este
dnes
s am
ong
colu
mns
c5c4c3c2c1
r5
r4
r3
r2
r1
0
0
0
0
1
1 1 0 0
1 1 0 0
0 1 1 1
1 1 1 0
1 0 1 1
c5c4c3c2c1
r5
r4
r3
r2
r1
0
0
0
0
1
1 1 0 0
1 1 0 0
0 1 1 1
1 1 1 0
1 0 1 1
c5c4c3c2c1
r5
r4
r3
r2
r1
0
0
0
0
1
1 1 0 0
1 1 0 0
0 1 1 1
1 1 1 0
1 0 1 1
c5c4c3c2c1
r5
r4
r3
r2
r1
0
0
0
0
1
1 1 0 0
1 1 0 0
0 1 1 1
1 1 1 0
1 0 1 1
c1 c2 c1 c3 c1 c4 c1 c5 c2 c3
c2 c4 c2 c5 c3 c4 c3 c5 c4 c5
Npaired=0 Npaired=67 Npaired=50 Npaired=100 Npaired=67
Npaired=50 Npaired=0 Npaired=100 Npaired=100 Npaired=100
r1
r2
r1
r3
r1
r4
r1
r5
r2
r3
r2
r4
r2
r5
r3
r4
r3
r5
r4
r5
Npaired=67
Npaired=67
Npaired=50
Npaired=50
Npaired=50
Npaired=50
Npaired=0
Npaired=0
Npaired=100
Npaired=100
Ncolumns = 63.4
Nrows = 53.4
NODF = 58.4
Nestedness based on Overlap and Decreasing Fill (NODF)
paired
( 1) ( 1)2 2
NNODF
n n m m
NODF is a gap based metric and more conservative than temperature.
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 SitesSpecies 1 2 3 4 5 6 7 8A 1 1 1 1 1 1 1 1 8B 1 1 1 1 1 1 1 1 8C 1 1 1 1 1 1 1 1 8D 1 1 1 1 1 1 1 1 8E 1 1 1 0 1 1 1 1 7F 1 1 1 1 0 0 1 1 6G 1 1 0 1 1 0 1 0 5H 1 0 1 1 1 1 0 0 5I 1 1 1 1 0 0 0 0 4J 1 1 0 0 0 0 1 1 4K 0 1 1 0 1 1 0 0 4L 1 0 1 1 0 1 0 0 4Species 11 10 10 9 8 8 8 7
Discrepancy is a gap counting metric.
How to measure species aggregation?S 50 44 46 43 47 2 7 6 3 4 1 5 15 25 24 23 29 27 26 32 33 34 30 12 28 10 9 8 31 11 13 14 19 16 20 18 35 17 22 21 45 48 49 37 40 36 41 39 42 3850 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 0 043 0 1 0 0 0 1 1 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 041 0 0 0 0 0 1 1 1 1 1 1 1 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 0 040 0 0 0 0 0 1 1 1 1 1 1 1 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 0 049 0 0 0 0 0 0 0 0 1 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 0 033 0 0 1 0 0 1 1 1 1 1 1 1 1 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 0 031 1 0 0 1 0 1 1 1 1 1 1 1 1 1 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 0 039 0 0 0 0 0 1 1 1 1 1 1 1 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 0 034 0 0 0 1 0 1 1 1 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 0 0 0 0 0 0 0 0 0 0 0 0 042 0 0 0 0 0 1 1 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 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 037 0 0 0 1 0 1 1 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 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 035 0 0 0 0 0 1 1 1 1 1 1 1 1 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 0 0 0 0 0 0 0 0 0 036 0 0 0 0 0 1 1 1 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 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 029 0 1 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 1 0 0 0 0 1 1 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 032 0 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 024 0 0 0 0 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 0 0 0 0 1 0 1 1 1 0 1 1 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 027 0 0 0 0 0 1 1 1 1 1 1 1 0 1 1 1 1 1 1 0 0 0 0 1 0 1 1 1 0 1 1 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 026 0 0 0 0 0 1 1 1 1 1 1 1 0 1 1 1 1 1 1 0 0 0 0 1 0 1 1 1 0 1 1 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 023 0 0 0 0 0 1 1 1 1 1 1 1 0 1 1 1 1 1 1 0 0 0 0 1 0 1 1 1 0 1 1 1 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 021 0 0 0 0 0 1 1 1 1 1 1 1 0 1 1 1 1 1 1 0 0 0 0 1 0 1 1 1 0 1 1 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 1 0 0 02 0 1 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 0 1 1 1 0 0 0 0 0 0 0 0 0 047 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 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 05 0 0 1 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 0 1 1 1 0 0 0 0 0 0 0 0 0 03 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 0 1 1 1 0 0 0 0 0 0 0 0 0 011 0 0 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 0 1 1 1 0 0 0 0 0 0 0 0 0 015 0 0 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 0 1 1 1 0 0 0 0 0 0 0 0 0 014 0 0 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 0 1 1 1 0 0 0 0 0 0 0 0 0 013 0 0 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 0 1 1 1 0 0 0 0 0 0 0 0 0 012 0 0 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 0 1 1 1 0 0 0 0 0 0 0 0 0 010 0 0 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 0 1 1 1 0 0 0 0 0 0 0 0 0 08 0 0 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 0 1 1 1 0 0 0 0 0 0 0 0 0 09 0 0 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 0 1 1 1 0 0 0 0 0 0 0 0 0 016 0 0 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 0 1 1 1 0 0 0 0 0 0 0 0 0 06 0 0 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 0 1 1 1 1 0 0 0 0 0 0 0 0 04 0 0 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 0 1 1 1 1 0 0 0 0 0 0 0 0 01 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 0 1 1 1 0 1 0 1 0 0 0 0 0 07 0 0 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 0 1 1 1 0 0 0 0 0 0 0 1 0 017 0 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 0 0 0 0 0 1 0 0 0 025 0 0 0 0 0 1 0 0 1 1 1 0 1 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 0 0 0 0 0 0 0 0 0 045 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 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 038 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 0 1 1 1 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 028 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 0 0 0 0 0 0 0 0 0 046 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 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 044 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 1 0 0 0 0 0 0 0 0 0 0 1 018 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 1 0 1 1 1 1 1 1 1 1 1 1 1 0 0 0 1 1 1 1 1 1 148 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 0 0 0 0 0 0 0 0 0 0 119 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 0 0 0 1 1 1 1 1 1 120 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 0 0 0 1 1 1 1 1 1 130 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 1 1 1 0 1 0 1 0 0 0 0 1 0 0 0 0 0 0 1 1 1 1 1 1 122 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 0 1 1 1 1 1 1 1 1 1 1 1 0 0 0 1 1 1 1 1 1 1
Compartmented matrix
Nearest neighbor metrics
fill
dNND
species
i
sites
jij
21 1
2
dij
species
i
sites
jij
species
i
sites
j
ii
ii
jj
jjjiij
x
xxJoinOcc
1 1
1
2
1
2
11
11
11
1111
Join count statistics
Nearest neighbour is a presence – absence metric
Join count operates on presence – absence and abundance matrices
A sum of cell entries around a focal cell multiplied by the entry of the focal cell
Other metrics proposed:MorisitaSimpson
SoerensenBlock variance
Ordination score varianceMarginal variances
NND has weak power at higher matrix fill
These metrics have very low power a moderate to small matrix size and high or
low matrix fill.
Species ful guc 3pog sos 2pogdabwrosgil ter 1pogwil mil swi kor hel lip wron SumPterostichus nigrita (Paykull) 1 1 2 18 2 5 0 58 53 30 61 39 0 0 0 2 2 13Platynus assimilis (Paykull) 48 2 25 9 7 4 0 39 0 0 1 0 0 76 9 117 0 11Amara brunea (Gyllenhal) 10 4 0 40 0 5 1 0 0 0 0 0 19 0 3 1 1 9Agonum lugens (Duftshmid) 0 0 0 0 0 2 1 3 2 1 2 0 0 0 0 0 1 7Loricera pilicornis (Fabricius) 5 0 0 0 1 1 0 5 0 0 1 3 0 0 0 0 0 6Pterostichus vernalis (Panzer) 0 0 0 1 0 0 1 0 2 0 21 7 0 0 0 0 1 6Amara plebeja (Gyllenhal) 0 0 0 2 0 1 0 5 0 0 0 0 1 0 0 4 0 5Badister unipustulatus Bonelli 0 0 0 1 0 0 0 3 0 0 0 0 4 0 0 3 0 4Lasoitrechus discus (Fabricius) 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 2Poecilus cupreus (Linnaeus) 0 0 0 2 0 0 0 0 0 0 0 0 0 1 0 0 0 2Sum 4 3 2 7 3 6 3 6 4 2 5 4 3 2 2 5 4
Abundance based metrics
The C-score extension
),,,(),,,(; cdbdcabacdbdcabadcba
ST4CACA
m(m 1)n(n 1)
The metric CA is a count of the number of abundance checkerboards in the matrix.
Other 2x2 submatrices catch matrix properties that have not well defined ecological meaning.
Nestedness in abundance matricesSpecies/Sites ter wron swi dab kor wil sos 2pog 3pog gil guc mil lip hel wros ful 1pog Species WeightPterostichusstrennus(Panzer) 704 36 169 1199 13 60 17 26 187 13 0 345 4 29 394 0 428 15 3624Pterostichusmelanarius 8 141 2 9 135 1 6 188 7 180 4 8 1019 11 83 0 0 15 1802Carabusgranulatus 18 11 12 154 110 11 77 11 25 19 113 52 59 0 0 11 1 15 684Pterostichusoblongopunctatus(Fab) 7 3 22 5 13 5 28 30 5 3 4 6 0 14 24 47 0 15 216Oxypselaphusobscurus(Herbst) 13 166 7 27 48 25 0 1 278 27 85 37 0 96 96 0 80 14 986Pterostichusnigrita(Paykull) 2 5 18 1 2 0 1 39 30 2 0 53 2 61 58 0 0 13 274Pseudoophonusrufipes(DeGeer) 3 13 0 1 2 1 3 1 2 90 5 3 5 0 0 6 0 13 135Pterostichusdiligens(Sturm) 4 12 3 1 1 0 1 5 11 4 5 18 0 1 1 0 0 13 67Patrobusatrorufus(Stroem) 11 2 35 0 6 22 7 11 0 348 0 0 37 2 9 81 0 12 571Synuchusvivalis(Illiger) 51 19 14 1 12 2 4 24 1 5 10 0 0 0 0 2 0 12 145Leistusterminatus(Hellwig) 1 10 3 4 1 3 3 7 4 3 0 0 1 0 0 0 11 12 51Platynusassimilis(Paykull) 9 0 4 76 0 117 39 2 7 9 1 0 25 48 0 0 0 11 337CarabusnemoralisMuller 0 10 16 5 2 12 8 0 0 1 6 14 0 6 5 0 0 11 85Harpalus4-punctatusDejean 69 17 67 9 29 41 9 555 0 0 6 0 0 77 0 0 0 10 879Pterostichusantracinus 46 1 21 0 1 0 0 0 1 0 2 274 2 0 11 11 0 10 370Pterostichusminor(Gyllenhal) 5 1 48 1 7 2 5 2 0 0 0 0 21 0 0 28 0 10 120Amarabrunea(Gyllenhal) 4 3 1 1 0 40 10 0 19 0 1 5 0 0 0 0 0 9 84Badisterbullatus(Schrank) 5 4 1 4 2 1 0 0 7 0 0 0 0 0 0 0 2 8 26
Species 17 17 17 16 16 15 15 14 14 13 12 11 10 10 9 7 5Weight 960 454 443 1498 384 343 218 902 584 704 242 815 1175 345 681 186 522
1
1 1
100,n
i
n
ij j
ij
Nk
rWNODFc
)1()1()(2
nnmm
WNODFrWNODFcWNODF
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
Whole matrix
Segregated Aggregated Aggregated nested Data type PA null models A null
modelsIndependent of matrix sorting C-score Clumping score CS/Clumping PA All
NestPairs A All Togetherness A and PA All All Species only
Segregated Aggregated Data type PA null models A null models
Simpson dissimilarity Simpson similarity PA No fixed - fixed Soerensen dissimilarity Soerensen dissimilarity PA No fixed - fixed Morisita A and PA No fixed - fixed All Chao A and PA All All
Other joint occurrence/absence metrics
Other joint occurrence/absence metrics PA No fixed - fixed
Dependent of matrix sorting Whole matrix
Segregated Aggregated Aggregated nested Data type PA null models A null
models NND PA All Block A and PA All All Join-coint A and PA All All Other distance based metrics Other distance based metrics A and PA All All NODF A and PA All All BR PA All T PA All Species only
Segregated Aggregated Data type PA null models A null models
Embedded absences PA All For seriation r2 PA All
CTurn PA All CSegr PA All Morisita PA All
A complete table of methods for co-occurrence analysis
Pattern detection in large matrices
These programs use cluster analysis and ordination to sort the matrix according to
numbers of occurrences. Didstance metrics are then used to identify compartments.
They generate hypotheses about matrix structure.
They do not fully allow for statistical inference.
WAND: ecological network analysisPajek: software for social network analysis
KliqueFinder: software for compartment analysis
