Download Site Description Shannon diversity Simpson diversity Hill numbers

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Vegetation Analysis
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Site Description
Slide 1
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Vegetation Analysis
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Site Description
Slide 2
Shannon diversity
Site Description
H=−
S
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pj logb pj
j=1
Originally information theory with base b = 2: Average length in bits
of code with shortest possible unique coding
1. Diversity indices
2. Species abundance models
• The limit reached when code length is − log2 pi : longer codes for
rare species.
3. Species – area relationship
Biologists use natural logarithms (base b = e), and call it H !
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!2002–2004
Jari Oksanen
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Vegetation Analysis
Dept. Biology, Univ. Oulu, Finland
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February 17, 2004
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Site Description
Slide 3
Information theory makes no sense in ecology: Better to see only as a
variance measure for class data.
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!2002–2004
Jari Oksanen
Dept. Biology, Univ. Oulu, Finland
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Vegetation Analysis
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February 17, 2004
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Site Description
Slide 4
Hill numbers
Simpson diversity
Common measures of diversity are special cases of Rényi entropy;
The probability that two randomly picked individuals belong to the
!S
same species in an infinite community is P = i=1 p2i .
Ha =
Can be changed to a diversity measure (= increases with complexity):
Mark Hill proposed using Na = exp(Ha ) or the “Hill number”:
1. Probability that two individuals belong to different species:
1 − P.
H0
H1
2. Number of species in a community with the same probability P ,
but all species with equal abundances: 1/P .
H2
c
!2002–2004
Jari Oksanen
Dept. Biology, Univ. Oulu, Finland
=
=
=
log(S)
−
S
i=1
− log
pi log p1
S
i=1
p2i
N0
=
S
N1
=
exp(H1 )
=
S
i=1
N2
1/
Number of species
exp Shannon
p2i
Inverse Simpson
Sensitivity to rare species decreases with increasing a: N1 and N2 are
little influenced and nearly linearly related.
Claimed to be ecologically more meaningful than Shannon diversity,
but usually very similar.
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S
"
1
log
pai
1−a
i=1
$
February 17, 2004
All Hill numbers in same units: “virtual species”.
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c
!2002–2004
Jari Oksanen
Dept. Biology, Univ. Oulu, Finland
$
February 17, 2004
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Vegetation Analysis
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Site Description
Slide 5
!
Vegetation Analysis
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Site Description
Slide 6
Evenness
Choice of index
“If everything else remains constant”, diversity increases when
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1. Number of species S increases, or
2. Species abundances pi become more equal.
Evenness: Hidden agenda to separate these two components
4
For a given number of species S, diversity is maximal when all
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probabilities pi = 1/S: in Shannon index Hmax
= log(S)
2
• It is not so important which
index is used, since all sensible indices are very similar.
N2
6
• Diversity indices are only
variances of species abundances.
8
Carabids
Pielou’s evenness is the proportion of observed and maximal diversity
2
4
6
8
10
12
J! =
N1 = exp(H)
c
!2002–2004
Jari Oksanen
!
$
Dept. Biology, Univ. Oulu, Finland
Vegetation Analysis
February 17, 2004
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Site Description
Slide 7
Sample size and diversity
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c
!2002–2004
Jari Oksanen
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$
Dept. Biology, Univ. Oulu, Finland
Vegetation Analysis
February 17, 2004
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Site Description
Slide 8
Logarithmic series
N = 10 , S = 154
100
"
H!
!
Hmax
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800
1000
N
60
Lajiluku
40
20
0
250
Lajiluku
N = 30 , S = 188
0
100
200
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400
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Frekvenssi
N = 110 , S = 255
5
200
400
600
800
1000
0
N
c
!2002–2004
Jari Oksanen
200
0
0
"
150
Frekvenssi
10
0.90
Evenness
100
10 20 30 40 50 60
30
• In larger samples, you may
find more individuals of rare
species, but you find new rare
species
10
0
50
0
S
50
70
Species richness
0
30
• Most species are rare, and
species found only once are the
largest group
1000
25
800
20
600
15
400
0.80
Diversity little influenced by rare species:
a variance measure.
Evenness based on twisted idea.
200
N
J
• Evenness decreases
0
Lajiluku
3.0
2.5
• Number of species S increases
• Diversity (N1 or N2 ) stabilizes
• R.A. Fisher in 1940’s
2.0
With increasing sample size
H
3.5
80
Diversity
Dept. Biology, Univ. Oulu, Finland
$
February 17, 2004
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!2002–2004
Jari Oksanen
500
1000
Frekvenssi
Dept. Biology, Univ. Oulu, Finland
1500
2000
$
February 17, 2004
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5
500
• Linear: Pre-emption model
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Oktaavi
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February 17, 2004
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Site Description
Slide 11
Fitting RAD models
0
50
100
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!2002–2004
Jari Oksanen
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Vegetation Analysis
150
200
250
Rank
• Sigmoid: Log-normal or brokenstick
0
2
Dept. Biology, Univ. Oulu, Finland
Vegetation Analysis
Runsaus
30
25
20
The shape of abundance distribution
clearly visible:
a
R0
• Canonical standard model of
our times
c
!2002–2004
Jari Oksanen
a = 3.98
10
• Modal class in higher octaves,
and not so many rare species
"
• Vertical axis: Logarithmic abundance
S0 = 31.2
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Lajien lukumäärä
• Plotted number of species
against ‘octaves’:
doubling
classes of abundance
S0
Ranked abundance diagrams
• Horizontal axis: ranked species
R0 = 4.46
35
• Preston did not accept Fisher’s
log-series, but assumed that
rare species end with sampling
Slide 10
100
Log-Normal model
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Site Description
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Vegetation Analysis
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Slide 9
5
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Site Description
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!
Vegetation Analysis
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Dept. Biology, Univ. Oulu, Finland
February 17, 2004
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Site Description
Slide 12
Broken Stick
• Pre-emption model
dis-
– Sigmoid: excess of both
abundant and rare species
to pre-emption model.
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c
!2002–2004
Jari Oksanen
• No real hierarchy, but chips
arranged in rank order:
+
+
+ + +
+ + +
+ + +
• Result looks sigmoid, and
can be fitted with logNormal model.
+ + + +
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Dept. Biology, Univ. Oulu, Finland
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15
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Rank
20
10
5
+
Abundance
20
10
+
5
Abundance
+
2
– Species abundances
tributed Normally
+
2
• Log-normal model
+
1
– A line in the ranked abundance diagram.
Carabid site 6
• Species ‘break’ a community (‘stick’) simultaneously in S pieces.
Carabid, site 6
1
– Species abundances decay
by constant proportion.
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Rank
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February 17, 2004
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!2002–2004
Jari Oksanen
Dept. Biology, Univ. Oulu, Finland
$
February 17, 2004
!
Vegetation Analysis
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Site Description
Slide 13
Hubbell’s abundance model
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Vegetation Analysis
• θ = 2JM ν, where JM is metacommunity size and ν evolution speed
Carabid site 6
20
10
5
Abundance
• Rare species have a huge impact in species richness.
2
• Rarefaction: Removing the effects of varying sample size.
5
10
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!
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• Plants often difficult to count.
Rank
$
Dept. Biology, Univ. Oulu, Finland
Vegetation Analysis
• Sample size must be known in individuals: Equal area does not
imply equal number of individuals.
1
Species generator θ/(θ +j −1) gives
the probability that jth individual
is a new species for the community.
c
!2002–2004
Jari Oksanen
• Species richness increases with sample size: can be compared
only with the same size.
!=8
• Simulations can be used for estimating θ.
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Slide 14
Species richness: The trouble begins
Ultimate diversity parameter θ
• θ and J define the abundance
distribution
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Site Description
February 17, 2004
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Site Description
Slide 15
Rarefaction
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c
!2002–2004
Jari Oksanen
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Dept. Biology, Univ. Oulu, Finland
Vegetation Analysis
S
15
20
Species never end, but the rate
of increase slows down.
10
Fisher log-series predicts:
#
$
N
S = α ln 1 +
α
5
0.3 0.4 0.5 0.6 0.7 0.8 0.9
1
Species richness
Rarefied to N=4
E(S|N=2) ! 1
20
10
5
+
2
E(S|N=4)
Carabids
! = 3.82
+ + ++ + +
+
+++++++ ++ ++ + +
+ + +
++
++ +
+ ++ ++ ++ ++
+
++ +
+
+
+++++ ++ + +
+ + +
+
+
+ ++
0
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c
!2002–2004
Jari Oksanen
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100 200
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Slide 16
Species richness and sample size
Carabids
10
February 17, 2004
Site Description
Rarefy to a lower, equal number of individuals
Only a variant of Simpson’s index
+
$
500
0.3
0.4
N
0.5
0.6
0.7
Simpson index
Dept. Biology, Univ. Oulu, Finland
0.8
0.9
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Number of individuals
$
February 17, 2004
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c
!2002–2004
Jari Oksanen
Dept. Biology, Univ. Oulu, Finland
$
February 17, 2004
!
Vegetation Analysis
#
Site Description
Slide 17
Species – Area models
• Island biogeography: S = cAz .
Carabids
• Assuming that doubling area A
brings along a constant number of
new species fits often better.
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c
!2002–2004
Jari Oksanen
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S
10
Arrhenius 0.19
Doubling 1.40
5
• Regarded as universally good: Often the only model studied, so no
alternatives inspected.
20
• Parameter c is uninteresting, but z
should describe island isolation.
0
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Dept. Biology, Univ. Oulu, Finland
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Number of individuals
$
February 17, 2004