phylogenetic diversity and functional diversity
TRANSCRIPT
生物多樣性的量度 (Part 4)Phylogenetic diversity and
functional diversity
趙蓮菊 清華大學統計所http://chao.stat.nthu.edu.tw
2
Phylogenetic diversity系統演化多樣性
(PD) via Hill numbers (consider taxonomic orphylogenetic distance between species)
Functional diversity (FD)功能性或特質多樣性
via Hill numbers (consider species traits) A unified framework: attribute ( diversity (AD) Software: iNEXT.3D (iNterpolation and EXTrapolation)
大綱
Phylogenetic diversity (PD)based on Hill numbers
Chao, A., Chiu C.-H. and Jost, L. (2010). Phylogenetic diversity measures based on Hill numbers. Philosophical Transactions of the Royal Society B., 365, 3599-3609.
Example: quantifying the diversity and differentiation of the vegetation for three dunes habitats Carboni et al. (2010, 2011, 2013)
The data contain a total of 43 vascular plant species collected from vegetation plots in 2002-2009 in three successively less extreme fore dune habitats along the Tyrrhenian coast in west of Italy: Embryo dunes (EM; 17 species)初期沙丘
Mobile dunes (MO; 39 species)流動沙丘
Transition dunes (TR; 42 species)過渡沙丘
EM is closest to the sea, MO is between EM and TR, and TR is farthest from the sea
4
5
Embryo (EM) dune初期沙丘
Mobile (MO) dune流動沙丘
Transition (TR) dune過渡沙丘
How to quantify the diversity and differentiation?
Species EM MO TR Ammophila arenaria 0 0.118 0.006 Anthemis maritima 0.024 0.132 0.046 Asparagus acutifolius 0 0.003 0.003 Bromus diandrus 0 0.005 0.032 Cakile maritima 0.217 0.024 0.004 Calystegia soldanella 0.027 0.026 0.009 Centaurea sphaerocephala 0 0.008 0.015 Chamaesyce peplis 0.097 0.014 0.001 Clematis flammula 0 0.004 0.018 Crucianella maritima 0 0.022 0.080 Cutandia maritima 0.008 0.036 0.095 Cyperus kalli 0.003 0.047 0.037 Daphne gnidium 0 0 0.001 Echinophora spinosa 0.029 0.029 0.004 Elymus farctus 0.161 0.134 0.044 Eryngium maritimum 0.021 0.020 0.001 Euphorbia terracina 0 0.003 0.028 Helicrisum stoechas 0 0.004 0.029 Juniperus oxycedrus 0 0 0.008 Lagurus ovatus 0 0.003 0.022 Lonicera implexa 0 0.001 0.002 Lophocloa pubescens 0 0.005 0.006 Lotus cytisoides 0 0.005 0.062 Medicago littoralis 0 0.021 0.081 Medicago marina 0.003 0.047 0.012 Ononis variegata 0.005 0.070 0.025 Otanthus maritimus 0.043 0.005 0 Pancratium maritimum 0.016 0.042 0.039 Phillirea angustifolia 0 0.002 0.005 Pistacia lentiscus 0 0.001 0.008 Plantago coronopus 0.003 0.013 0.017 Polygonum maritimum 0.038 0.003 0.001 Prasium majus 0 0.001 0.003 Pseudorlaya pumila 0 0.007 0.008 Pycnocomon rutifolium 0 0.003 0.042 Quercus ilex 0 0 0.003 Rubia peregrina 0 0 0.003 Salsola kali 0.193 0.025 0.002 Silene canescens 0 0.058 0.089 Smilax aspera 0 0.003 0.009 Sonchus bulbosus 0 0.006 0.005 Sporobolus virginicus 0.113 0.044 0.021 Vulpia fasciculata 0 0.009 0.073
Species diversityOnly species richness and abundances are involved Embryo dunes
(EM; 17 species) Mobile dunes
(MO; 39 species) Transition dunes
(TR; 42 species)
6
Phylogenetic diversity: adding evolutionary history
Root(325 Myr)
MRCA
7
Tree from PLYLOMATIC
Functional trait Data type Attribute
Life form Nominal 1. Phanerophyte 2. Chamephyte 3. Hemicryptohyte 4. Geophyte 5. Therophyte
Growth form Nominal 1. Short basal 2. Long-semibasal 3. Erect leafy 4. Cushions, tussocks and dwarf shrubs 5. Shrubs, trees and climbers
Leaf texture Nominal 1. Succulents 2. Malacophyllous 3. Semi-sclerophyllous 4. Sclerophyllous
Dispersal mode Nominal 1. Anemochorous 2. Barochorous 3. ZoochorousLeaf persistence Binary 0. Deciduous 1. EvergreenPlant life span Binary 0. Annual 1. Biennal-PerennialPollination system Binary 0. By wind or non-specialized 1. By insects or birds
Vegetative propagation(Clonality) Binary 0. Clonal; 1. Non-clonal
Flowering phenology Ordinal 1. April and before 2. May 3. June 4. July and after
Plant height Quantitative [cm]Leaf size Quantitative [cm2] Leaf thickness Quantitative [mm] Seed mass Quantitative [g]Seed shape Quantitative Variance LDMC Quantitative [mg g-1] SLA Quantitative [mm2 mg-1]
Description of the 16 plant functional traits used in this study (Carboni et al. 2013)
Functional (Ecosystem) Diversity: incorporating species traits
8
Phylogenetic Diversity:
Community 1 Community 2
All else being equal, which community is more diverse?
9
Species in community 2 is more phylogenetically diverse than community 1
Pielou (1975, p. 17) was the first to notice the concept of diversity could be broadened to consider taxonomic/evolutionary difference between species.
Community 1 Community 2
10
“I think” Tree of LifeThe first-knownsketch by Charles Darwin of an evolutionary tree describing the relationships among groups of organisms
http://www.amnh.org/exhibitions/darwin/idea/treelg.php
11
p1 p2 p3
p1 p2 p3
Phylogenetic DiversityWe consider not only the relative abundance of species,
phylogenetic relationship (who is related to whom)親緣關係
And, satisfy the essential requirement “replication principle”. 12
Faith’s PD
Branch-length-based measure: Faith’s PD (Faith 1992)
sum of the branch lengths of the phylogeny connecting all species from tips to root (or any reference point on the main trunk)
Faith (1992) PD: total branches length總支脈長度 沒考慮物種豐富度
PD =12 10 9 8 (a) (b) (c) (d)
3 33 32
322
3
2 2
1 1 12
1 1
11
1
1
p2 p3 p4p1 p2 p3 p4p1 p2 p3 p4p1 p2 p3 p4p1 Lineages譜系completely distinct
Theoretical framework
Tip nodes: Current-time species
Expand S species (tip nodes) to Bbranch segments; B: # of branches
Li : length of branch i
Expand relative abundance set (p1, p2, …, pS) to branch abundance(a1, a2, … aS, aS+1,…,aB) ,
ai : the relative abundance descending from branch i =1, 2, …B; with (p1, p2, …, pS) as the first S elements
16
LL
L
p p
L1L2
L7
L3
L4 L5
L8
p1 = a1 p2 = a2 p3 = a3
a7 = p1 + p2 + p3
a8 = p4 + p5
p5=a5p4 = a4
a6 = p2 + p3
L6
root
Extending TD to phylogenetic diversityUltrametric 超度量
Tip nodes: Current-time species
17
Similar extension to non-ultrametric treeBranch length: DNA base-pair change
L1L2
L7
L3
L4
L5
L8
p1 = a1p2 = a2
p3 = a3
root
p5 = a5
p4 = a4L6
L5
a8 = p4 + p5
p5=a5
a8 = p4 + p5
a6 = p2 + p3
a7 = p1 + p2 + p3
Root(根)
Branch(支脈)
Interior nodenode(節點)
Tip(端點)Terminal node
1P 2P 3P 4P0.4 0.3 0.2 0.1
2L1 =
2L 1L3 =
1L5 =
4L4 =
2L6 =
0.5
0.9
)(49.025.011.042.013.014.02aLT ii
abundancetotal=
×+×+×+×+×+×==∑
的平均距離一個 tip到 到RootT =
44到Root 物種距離為此
425.013.05.32.045.0)(T
=×+×+×+×== abundancetotal
的平均距離一個 tip到 到RootT =41tip到Root =
Root
5.0P1 =
4 0.5
2.0P2 =
3.0P3 =
2
1
5.3
5.52tip到Root =33tip到Root =
43.032.05.55.04T =×+×+×=
depth)(treeTaL下,cultrametri為何在Bi
ii =∑∈
Root
1P 2P 3P 4P
321 PPP ++
32 PP +
34321 T)PPP(P ×+++
24321 T)PPP(P ×+++
14321 T)PPP(P ×+++
Faith’s PD (depends on the reference point Tr)
:the set of all branches given any fixed point Tr
Li: the length of branch i Faith PD does not incorporate species
abundances
21
∑∈
=Bi
iLPD
rTBB =
Debates on species vs. phylogeny
There have been intense debates about whether we should preserve many species with recent divergence or fewer species but with longer evolutionary history
“Unfortunately, Noah’s Ark has a limited capacity….and a (limited) budget available for biodiversity preservation…” (Weitzman 1992, 1993, 1998)
What to preserve?
22
The Noah’s Ark: the agony of choice
The woodpecker might have to go!
Courtesy of Ramon Teja, http://www.livepencil.com/ 23
PD and Species RichnessWhen there are no internal nodes and all S branches are equally distinct with branch lengths of unity (i.e., branch lengths are normalized to unity), PD reduces to species richness
Li = 1 for all branch i No internal nodes
(a1, a2, …, aB) = (p1, p2, …, pS)
24
S
∑∈
=Bi
iLPD
Taxonomic Phylogenetic
Species richness Faith PD (Faith 1992)
Entropy Phylogenetic entropy(Allen et al. 2009)
Gini-Simpson Quadratic entropy (Rao 1982)
Hill Numbers Chao, Chiu and Jost (2010)
25
Previous measuresQuadratic entropy (Rao 1982)
mean phylogenetic distance between any two randomlychosen individuals
dij : phylogenetic distance between species i and j, pi and pj denote species relative abundance of species i and j.
Phylogenetic entropy (Allen et. al. 2009)Li : length of branch i, ai : the branch abundance descending from branch i.
26
∑∈
−=Bi
iiip aaLH log
∑=
=S
jijiij ppdQ
1,
Taxonomic and Phylogenetic complexity measures比較
Taxonomic diversity Phylogenetic diversity
q=1
q=2
Generalized or TsallisEntropy
q=0
∑=
−=S
iii ppH
1
1 log
∑=
−=S
iipH
1
22 1
1
11
−
−=
∑=
q
pH
S
i
qi
q
10 −= SH
∑∈
−=Bi
iiip aaLH log
∑=
=S
jijiij ppdQ
1,
∑∈
−=Bi
i TLI0
1−
−=
∑∈
q
aLTI Bi
qii
q
HI到到1L到1T到(1) qqi ===
RP到 到 到I到H(2) qq
( ) 1/ 1(3) 1 ( 1)
( q)q qD q H−
= − −
QppdaaLji
jiijBi
iii ==− ∑∑∈ ,
)1( (proved by Allen et al. 2009)
關係和 ∑∑==
=−=S
jijiij
S
ii ppdQpH
1,1
22 1
)(1,0
01
101110
jiddd ijiiij ≠==
=
∑∑∑ −===<≠
S
iji
jiji
ji pppppQ1
212
Special case
Phylogenetic diversity measures
Except for Faith’s PD, all indices mentioneddo NOT satisfy the “replication principle”.
Chao et al. (2010) were motivated to developa unified class of phylogenetic diversitymeasures based on Hill numbers
Satisfy “replication principle”
29
30
Doubling Property in phylogenetic version
Two completely phylogenetically distinct (no overlapped tree branches) across assemblages, each with diversity measure X
Combine these two, the diversity becomes 2X
30
3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3
Faith’s PD 12 + 12 = 24 Phylogenetic entropy HP ?
4.16 + 4.16 > 6.24 Rao’s Q ? 2.25 + 2.25 > 2.625
3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3
Phylogenetic entropy HP ?Single: -3(1/4)log(1/4) x 4 (lineages) = 4.16
Pooled: -3(1/8)log(1/8) x 8 (lineages) = 6.24Rao’s Q ?
Single: 12 (pairs) x 3 x (1/4) x (1/4) = 2.25Pooled: 56 (pairs) x 3 x (1/8) x(1/8) = 2.625
∑=
=S
jijiij ppdQ
1,
∑∈
−=Bi
iiip aaLH log
Phylogenetic Diversity Measures:
Two parameters:Order q in Hill number
Time parameter T: Consider the phylogenetic diversity through T years ago t=0
(Present time)
p1+p2+p3+p4
p1+p2+p3
p2+p3
p1 p2 p3 p4
slice 1
slice 2
slice 3
L1
L2 L3
L4
L5
L6
L7
Recall species diversity (Hill Numbers)
All species are taxonomically equally distinct Only species relative abundances are involved
(any abundances or weights must be normalized)
Hill numbers: effective # of species
34
)1/(1
1
qS
i
qi
q pD−
=
= ∑
11 =∑ =
Si ip
Define a phylogenetic attribute
Instead of counting species, we count “unit of branch length” (a phylogenetic attribute)Each branch length is weighted by the “branch abundance” ai
35
Phylogenetic attributesCollection of branches of unit-length (phylogenetic attributes)Total abundance =4 x 0.5+3.5 x 0.2+ 1 x 0.3 +2 x 0.5 = 4
36
Phylogenetic diversityThere are Li “attributes” with relative abundance ai /Here total abundance (Mean branch length)
4 “attributes” (L1) with relative abundance 0.5/4 3.5 “attributes” (L2) with relative abundance 0.2/4 1 “attributes” (L3) with relative abundance 0.3/4 2 “attributes” (L4) with relative abundance 0.5/4
T ∑ == Bi iiaL1
37
T
)1/(1
1)(
qB
i
q
ii
q
TaLTPD
−
=
×= ∑
−== ∑
∈→
Ti
iii
q
q Ta
TaLTPDTPD
Blogexp)(lim)(
1
1
Phylogenetic diversity (PD)Branch length is Li, with “branch abundance” ai Total abundance (Mean branch length
Ultrametric case PD (Effective total length)
Mean-PD (Effective # of lineages)
T
)1/(1
1)(
qB
i
q
ii
q
TaLTPD
−
=
×= ∑
( )( )q
q q PD Tmean PD D TT
= =
∑ == Bi iiaL1
38
TT =
39
)(TDq = )1(/1 q
i
qi
i
T
aTL
−
∈
∑
B=
)1(/11
q
i
qi
iT T
aLT
−
∈
∑
B, q ≥ 0, q ≠ 1;
Mean phylogenetic diversity
−== ∑
∈→Ti
iiiq
qaa
TLTDTD
Blogexp)(lim)(
1
1
Mean-PD
Interpretation of mean-PD
Effective number of equally divergent lineages (species) over T years
Link to traditional diversity (Hill numbers):When all species are completely equally distinct with branch lengths T (including T = 0, ignoring phylogeny)
“Effective number of lineages”
Assemblage: S species{p1, p2, …, pS}Mean-PD =for an order q, time T
Assemblage: lineages with equal relative abundances, completely distinct, all with branch length T
=)(TDq
)(TDq
D1
….
…. D
1D
1D
1
T T T T
Phylogenetic diversity
q = 0, branch diversity reduces to Faith’s PD
PD: the amount of evolutionary history in the assemblage or the effective lineage-years or lineage-length (or other units) contained in the tree in the time period [−T, 0]
TTDTPD q ×= )()(
Species diversity becomes a special caseNo interior nodes, all branch lengths are Li = 1, with relative abundance pi
Total abundance 1== ∑i ip
.1)( )1/(1
1
)1/(1
1 1
qS
i
qi
qS
i
q
Sk k
iq pp
pTPD −
=
−
= =
=
×= ∑∑∑
43
T
Generalize and unify existing measures:
Order q = 0= Total branch lengths in [-T, 0] / T
Order q =1
Order q = 2
)(0 TD
)/exp()(1
THTD p=
TQTD
/11)(
2
−=
122 )(
−
∈
= ∑
Bii
i aTLTD
Allen et al. (2009) proved
QppdaaLji
jiijBi
iii ==− ∑∑∈ ,
)1(
)()1( 22
TDTTaLTaaL
iii
Biiii −=−=− ∑∑
∈
2 1( )1 /
D TQ T
⇒ =−
Special case q = 2
3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3
Faith’s PD 12 + 12 = 24 Phylogenetic entropy HP ?
4.16 + 4.16 > 6.24 Rao’s Q ? 2.25 + 2.25 > 2.625
47
3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3
PD/T 4 + 4 = 8Exp(HP/T) 4 + 4 = 8 1/(1-Q/T) 4 + 4 = 8
PD 12 + 12 = 24HP 4.16 + 4.16 ≠ 6.24 Q 2.25 + 2.25 ≠ 2.625
3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3
PD/T 4 + 4 = 8 Exp(HP/T) 4 + 4 = 8 1/(1-Q/T) 4 + 4 = 8
Figure S1: A hypothetical rooted phylogenetic tree with four species.
t=0(Present time)
p1+p2+p3+p4
p1+p2+p3
p2+p3
p1 p2 p3 p4
L1
L2 L3
L4
L5
L6
L7
Figure S2: Plot of the mean diversity )(TDq
and the phylogenetic diversity qPD(T)
for q = 0, 1 and 2 and T < 6 for the tree in Figure S1 assuming L1 = L6 = 2, L2 = L3 = L5 = 1 and L4 = 4. (a) (p1, p2, p3, p4) = (0.25, 0.25, 0.25, 0.25) (b) (p1, p2, p3, p4) = (0.4, 0.2, 0.3, 0.1) (a) (b)
Phylogenetic tree based on the classical Linnaean taxonomic categories
p2 p3 p4p1
Family
Genus
Speciesp2 p3 p4p1 p2 p3 p4p1 p2 p3 p4p1
Figure S3: A hypothetical rooted phylogenetic tree with four species and four structures. The branch length is shown along each branch. The structure (a) denotes the case of maximally distinct lineages (or species). The plots of their diversities are shown in Figure S5.
(a) (b) (c) (d)
3 33 32
322
3
2 2
1 1 12
1 1
11
1
1
p2 p3 p4p1 p2 p3 p4p1 p2 p3 p4p1 p2 p3 p4p1
Figure S4: A three-level taxonomic tree with four species and four structures. The mean diversity (or taxonomic diversity) for each of the taxonomic trees in this figure is identical to the mean phylogenetic diversity (or branch diversity) of the corresponding phylogenetic tree in Figure S3.The plots of their diversities are shown in Figure S5.
p2 p3 p4p1
Family
Genus
Speciesp2 p3 p4p1 p2 p3 p4p1 p2 p3 p4p1
T=1T=2
q-profile (1) Phylogenetic diversity profiles as a function of the order q for species abundances (p1, p2, p3,
p4) = (0.25, 0.25, 0.25, 0.25)
a, b, c, dc
d
a, b ab
cd
(2) Phylogenetic diversity profiles as a function of the order q for species abundances (p1, p2, p3,
p4) = (0.1, 0.2, 0.3, 0.4)
c
d
ab
cd
a, ba, b, c, d
Fig. B3. Phylogenetic diversity profile as a function of order of q for a fixed perspective time T = 1, 2, and 3 based on the mean diversity )(TD
q for 50 ≤≤ q for four phylogenetic trees and two
species abundance distributions.
(1) Phylogenetic diversity profiles as a function of time for species abundances (p1, p2, p3, p4) =
(0.25, 0.25, 0.25, 0.25)
ab
c
d
ab
c
d
ab
c
d
(2) Phylogenetic diversity profiles as a function of time for species abundances (p1, p2, p3, p4) =
(0.1, 0.2, 0.3, 0.4)
ab
c
d
ab
c
d
a
bc
d
Fig. B2. Phylogenetic diversity profile as a function of time based on the mean diversity )(TDq
for q = 0, 1 and 2 and 0 < T < 5 for the four phylogenetic trees and two species abundance distributions. The plots show that the effect of tree structures and the diversity ordering follows (a) ≥ (b) ≥ (c) ≥ (d) for the two species abundance distributions.
T-profile
Diversity profiles Species/Taxonomic: Use a profile of Hill
numbers (as a function of order q) to quantify diversity of a community
Phylogenetic: Use a q-profile (from root) and T-profile for three values of q (q = 0, 1, 2); each is a function of time T to quantify phylogenetic diversity
Species EM MO TR Ammophila arenaria 0 0.118 0.006 Anthemis maritima 0.024 0.132 0.046 Asparagus acutifolius 0 0.003 0.003 Bromus diandrus 0 0.005 0.032 Cakile maritima 0.217 0.024 0.004 Calystegia soldanella 0.027 0.026 0.009 Centaurea sphaerocephala 0 0.008 0.015 Chamaesyce peplis 0.097 0.014 0.001 Clematis flammula 0 0.004 0.018 Crucianella maritima 0 0.022 0.080 Cutandia maritima 0.008 0.036 0.095 Cyperus kalli 0.003 0.047 0.037 Daphne gnidium 0 0 0.001 Echinophora spinosa 0.029 0.029 0.004 Elymus farctus 0.161 0.134 0.044 Eryngium maritimum 0.021 0.020 0.001 Euphorbia terracina 0 0.003 0.028 Helicrisum stoechas 0 0.004 0.029 Juniperus oxycedrus 0 0 0.008 Lagurus ovatus 0 0.003 0.022 Lonicera implexa 0 0.001 0.002 Lophocloa pubescens 0 0.005 0.006 Lotus cytisoides 0 0.005 0.062 Medicago littoralis 0 0.021 0.081 Medicago marina 0.003 0.047 0.012 Ononis variegata 0.005 0.070 0.025 Otanthus maritimus 0.043 0.005 0 Pancratium maritimum 0.016 0.042 0.039 Phillirea angustifolia 0 0.002 0.005 Pistacia lentiscus 0 0.001 0.008 Plantago coronopus 0.003 0.013 0.017 Polygonum maritimum 0.038 0.003 0.001 Prasium majus 0 0.001 0.003 Pseudorlaya pumila 0 0.007 0.008 Pycnocomon rutifolium 0 0.003 0.042 Quercus ilex 0 0 0.003 Rubia peregrina 0 0 0.003 Salsola kali 0.193 0.025 0.002 Silene canescens 0 0.058 0.089 Smilax aspera 0 0.003 0.009 Sonchus bulbosus 0 0.006 0.005 Sporobolus virginicus 0.113 0.044 0.021 Vulpia fasciculata 0 0.009 0.073
Species diversityOnly species richness and abundances are involved Embryo dunes
(EM; 17 species) Mobile dunes
(MO; 39 species) Transition dunes
(TR; 42 species)
56
Phylogenetic diversity: add evolutionary history
Root(325 Myr)
57
TD: observed Hill number q-profile
Mean-PD: q-profiles for ref time = root (325 myr)
mean-PD T-profiles for q = 0, 1, 2
60
Embryo (EM) dune
Mobile (MO) dune
Transition (TR) dune
TD, PD:TR > MO > EM(FD later)
Statistical Estimation: essential in diversity analysis
In the theoretical framework, all diversity measures and similarity/differentiation indices are in terms of the true population values of species richness and species relative abundances in an assemblage
In practice, the true values are unknown and estimation or standardization is needed (iNEXT.3D)
61
Functional diversity (FD)based on Hill numbers
Chao, A., Chiu, C.-H., Villéger, S., Sun, I.-F., Thorn, S., Lin, Y.-C., Chiang, J. M. and Sherwin, W. B. (2019). An attribute-diversity approach to functional diversity, functional beta diversity, and related (dis)similarity measures. Ecological Monographs, 89, e01343.
加入物種在生態系統中物種特質不同的訊息
物種的特質決定各物種在生態系中功能的差異與不同。(若功能太接近,則生態系不穩定; 功能愈多樣, 則生態系較能永續)
功能性(functional)特質(trait)生態系(Ecosystem )多樣性
墾丁和福山樣區比較Functional diversity: based on 3 species traits for leaf和葉子有關的測量值
Thickness (葉厚度) LMA (比葉重) leaf mass per area LDMC (葉乾物質含量) leaf dry matter
content
64
三種沙丘功能性(生態系)指標
沙丘植物功能包含:
三種沙丘 (初期沙丘embryo dunes, 流動沙丘mobile dunes, 中介轉換沙丘transition dunes)
16種植物特質
7定量: 高度、葉子大小、葉子厚度、
種子重量、種子形狀、葉子的生物量
9類別: 授粉方式、繁衍方式、開花方式、
葉子纖維等
65
Example: fish data
• There are nine continuous traits (mean weight, length of first maturity, trophic level, mean temperature preference, maximum weight, growth rate, food consumption and biomass, maximum depth, and generation time);
• Three categorical traits (position in water column, reproductive guild, and body shape).
Traits 特質
11 12 1
21 22 2
1 2
12
K
K
S S SK
sp X X Xsp X X X
spS X X X
S species
K traits
Trait table with S species and K traits.
Gower (1971) distance for species iand j based on trait k Categorical: 相同類別為0, 不同為 1, range(Rk)≡ 1
Quantitative (continuous)
Gower distance for K traits
68
1 [ ]ik jkI X X− =
minmax)(, −==−
ktraitrangeRR
XXk
k
jkik
k
jkikK
k RXX
K−
∑=1
1
SpeciesAmmophila_arenaria
Anthemis_maritima
Asparagus_acutifolius
Bromus_diandrus
Cakile_maritima
Calystegia_soldanella
Centaurea_sphaerocephala
Chamaesyce_peplis
Clematis_flammula
Crucianella_maritima
Cutandia_maritima
Cyperus_kalli
Daphne_gnidium
Echinophora_spinosa
Ammophila_arenaria 0 0.6378163 0.6704218 0.6273421 0.7288881 0.7112156 0.6574736 0.7487045 0.6704297 0.6463197 0.5713132 0.4420351 0.7328366 0.7146479
Anthemis_maritima 0.6378163 0 0.6844813 0.6952136 0.5379003 0.5417153 0.2925521 0.5844223 0.5464927 0.5838375 0.586739 0.6020166 0.5799596 0.4733252
Asparagus_acutifolius 0.6704218 0.6844813 0 0.7401687 0.7237948 0.6194748 0.6732102 0.6755674 0.5950491 0.5790585 0.728719 0.526083 0.5383025 0.7098689
Bromus_diandrus 0.6273421 0.6952136 0.7401687 0 0.5274126 0.7076298 0.7060467 0.5525635 0.6518583 0.7034164 0.3601771 0.6503544 0.6816861 0.7564979
Cakile_maritima 0.7288881 0.5379003 0.7237948 0.5274126 0 0.542463 0.6157531 0.313543 0.6055834 0.6307169 0.4529222 0.6337362 0.6926632 0.6515825
Calystegia_soldanella 0.7112156 0.5417153 0.6194748 0.7076298 0.542463 0 0.607907 0.4505849 0.60629 0.5813926 0.6946999 0.6101482 0.6722993 0.5649852
Centaurea_sphaerocephala 0.6574736 0.2925521 0.6732102 0.7060467 0.6157531 0.607907 0 0.6428435 0.4771694 0.5702954 0.6031748 0.6067833 0.4940238 0.544689
Chamaesyce_peplis 0.7487045 0.5844223 0.6755674 0.5525635 0.313543 0.4505849 0.6428435 0 0.584699 0.6497463 0.5360192 0.6712337 0.6460429 0.6971506
Clematis_flammula 0.6704297 0.5464927 0.5950491 0.6518583 0.6055834 0.60629 0.4771694 0.584699 0 0.6746412 0.5917067 0.6904211 0.401106 0.7150136
Crucianella_maritima 0.6463197 0.5838375 0.5790585 0.7034164 0.6307169 0.5813926 0.5702954 0.6497463 0.6746412 0 0.6563455 0.4467748 0.6339556 0.6264246
Cutandia_maritima 0.5713132 0.586739 0.728719 0.3601771 0.4529222 0.6946999 0.6031748 0.5360192 0.5917067 0.6563455 0 0.5881055 0.7175582 0.7474772
Cyperus_kalli 0.4420351 0.6020166 0.526083 0.6503544 0.6337362 0.6101482 0.6067833 0.6712337 0.6904211 0.4467748 0.5881055 0 0.6772504 0.6003538
Part of Gower pairwise distance matrix (the first 12 species)
69
Form functional groups (clusters) based on given species-pairwise Gower distances
ijd
kld
mnd
Like statistical clustering algorithm, one must first determine a threshold level
Any two species with distance greater than or equal to the specified threshold level are in different clusters and vice versa
Statistical clustering algorithm
The threshold level τ can be chosen to be anypositive value
Any two species with distance beyond this threshold level are regarded as functionally equally-distinct species and in different functional groups and vice versa
τ (tau): level of threshold distinctiveness (threshold level)
Extending Hill numbers to attribute (屬性) diversity:
a unified framework (including TD, PD, FD)
Chao, A., Henderson, P. A., Chiu, C.-H., Moyes, F., Hu, K.-H., Dornelas, M and Magurran, A. E. (2021). Measuring temporal change in alpha diversity: a framework integrating taxonomic, phylogenetic and functional diversity and the iNEXT.3D standardization. To appear in Methods in Ecology and Evolution.
IPBES (Intergovernmental Science-Policy Platform on Biodiversity and Ecosystem Services) Conceptual Framework
(Díaz et al. 2015, p. 8)
Biodiversity includes “variation in genetic, phenotypic (表現型), phylogenetic (系統演化), and functional (功能性) attributes, as well as changes in abundance and distribution over time and space, within and among species, biological communities and ecosystems”
Attribute definition
TD: a taxonomic attribute = a species
PD: a phylogenetic attribute = a unit-length branch segment in a phylogeny
FD: a functional attribute = a virtual functional group
75
Attribute diversity (AD)• A hypothetical assemblage which can be decomposed
into M sub-assemblages • The i-th sub-assemblage consists of vi attributes,
each with raw abundance ai, i = 1, 2, …, M. • The attribute diversity (AD) of order q is defined as
the Hill number of order q for the hypothetical assemblage
1
1 1lim exp log
Sq i i
iq i
a aAD AD vV V→ =
= = − ∑
1S
i iiV v a== ∑
Effective total attributes
1/(1 )1/(1 )
1 11
, 0, 1.
qq qqM Mq i i
i iMi ij jj
a aAD v v q q
Vv a
−−
= ==
= = ≥ ≠
∑ ∑∑
Any two species with distance beyond this threshold level are regarded as functionally equally-distinct species and in different functional groups and vice versa
Truncated distance matrix for any τ > 0
Implement τ (tau): level of threshold distinctiveness
( ) [ ( )] [min( , )]ij ijd d∆ τ τ τ≡ =
Assume S species in an assemblage, indexed by i = 1, 2, …, S;
Let ni represent the raw abundance (number of individuals) of species i
The total number of individuals in the assemblage, or assemblage size, is expressed as
Relative abundance of species i:
Theoretical framework
∑ =+ =Si inn 1
+= nnp ii /
∑ = =Si ip1 1
Hill numbers of order q
)1/(1
1
)1/(1
1)/(
qS
i
qi
qS
i
qi
q pnnD−
=
−
=+
=
= ∑∑
−=
−== ∑∑
==++→
S
iii
S
iii
q
qppnnnnDD
111
1 logexp)/log()/(explim
Assume S species, i = 1, 2, …, S; ai(τ): functionally indistinct set at level τ
Species i contribution becomes vi (τ)= ni /ai(τ)
Attribute diversity: incorporating τ
1
( ) ( )( ) 1 1
Sij iji i j
j i j
d da n n
τ ττ
τ τ≠ =
= + − = −
∑ ∑
1 1
( )( ) ( )( )
1 1
i i ii S S
ij ijij j
j j
n n pvd da
n pτ
τ τττ τ= =
= = = − −
∑ ∑
1. If dij= 0, then dij (τ) = 0 for any τ > 0. The two species are always treated as belonging to the same functional group.
2. If dij ≥ τ, then dij (τ) = τ, the two species functionally equally distinct at threshold level τ, and belong to different functional groups with an effective distance τ.
3. If 0 < dij < τ, then dij (τ) = dij , only a proportion of the individuals of species i are functionally equally distinct from species j at τ (and thus are in different functional groups with an effective distance τ), whereas the other individuals of species i are functionally indistinct from species j (i.e., in the same functional group with an effective distance 0).
Three impliications
82
General distance matrix
(2)
ii na =)(τ1)( =τiv
0=ijd
τ≥ijd
=∆
0
0
)(
τττ
τττ
τ
=∆
0000
0000
)(
τ
+==∑ nna ii )(τ
iiii panv == )(/)( ττ
(1)
83
(3) Two groups (1,2) (3,4)
=∆
0000
0000
)(
ττττ
ττττ
τ
2121 )()( nnaa +== ττ
4343 )()( nnaa +== ττ
)(/)( ττ iii anv =
There are S sub-assemblages with the i-th sub-assemblage consisting of functional groups, each with group “relative” abundance , i = 1, 2, …, S.
Functional diversity of order q at level τ
FD as a special case of AD
)(/)( ττ iii anv =
+nai /)(τ
+== ∑∑ === nnavV Si ii
Si i 11 )()()( τττ
)1/(1
1
)()())((
qS
i
q
ii
q
navFD
−
= +
= ∑
τττ∆
1/(1 )1
1 1[1 ( ) / ]
qqS S
i ij ji j
p d pτ τ−−
= =
= − ∑ ∑
The effective number of equally-distinct functional groups (or species) at the threshold distinctiveness of level τ, i.e., distance between any two species-equivalents is at least τ.
The sense of “effective” forfunctional diversity of the actual assemblage is the same as a reference assemblage consisting of x equally-abundant and functionally equally-distinct species with all pairwise distances ≥ τ for any different-species pairs, and distance 0 for any same-species pairs
Interpretation of
xFDq =))(( τ∆
))(( τ∆FDq
q = 0
q =1
q = 2
Special case
∑∑
∑∑=
=== −
===S
iSj jij
iS
i i
iS
ii pdf
pa
nvFD1 111
0
))]((1[)()())((
ττττ∆
( )∑ ∑=
−
= −= Si
Sj ijdfFD 1
1
10 ))]((1[))(( ττ∆
.)]((1[logexp
)(log)()(exp))((lim))((
1 1
11
1
−−=
−==
∑ ∑
∑
= =
= ++→
S
ijij
S
ji
S
i
iii
q
q
pdfp
na
navFDFD
τ
ττττ∆τ∆
ττ∆τττ∆
/))((11
]/)([11))((
1 1
2
QpdpFD S
iSj jiji −
≡−
=∑ ∑= =
.)]([))((1 1∑∑= =
=S
i
S
jjiij ppdQ ττ∆
An attribute‐diversity approach to functional diversity, functional beta diversity, and related (dis)similarity measures
Ecological Monographs, Volume: 89, Issue: 2, First published: 21 November 2018, DOI: (10.1002/ecm.1343)
The q-profile: given τ, plot as a function of q. Three values of τ : dmin: taxonomic diversity dmean: proposed in Chao et al. (2019)dmax: conventional
The τ profile: given q, plot as a function of τ.Three values of q (q = 0, 1, 2)
AUC (Area Under Curve) profile; area under the τ profile in [0, 1] as a function of q ≥ 0
Functional diversity profiles
))(( τ∆FDq
))(( τ∆FDq
Taxonomic (species) diversity, τ ≤ dmin
Each species forms a functional group
Traditional function diversity, τ = dmax
All species in one functional group
Proposed, τ = dmean
FD = effective number of functional groups (species)
An attribute‐diversity approach to functional diversity, functional beta diversity, and related (dis)similarity measures
Ecological Monographs, Volume: 89, Issue: 2, First published: 21 November 2018, DOI: (10.1002/ecm.1343)
Example (Continued) FD for Dunes data
94
Embryo (EM) dune
Mobile (MO) dune
Transition (TR) dune
TD, PD and FD:TR > MO > EM
Interpretation: conform to ecologists’ expectation
EM is closest to the sea, MO is between EM and TR, and TR is farthest from the sea
The vegetation of EM is exposed to wind disturbance, flooding, and salt spray and other harsh environmental factors
The vegetation of the MO is less exposed The vegetation of the TR is the least
exposed
,
.
95
Interpretation 2:
The EM habitat is mainly composed of a few very abundant, specialized phylogenetically related pioneer species with similar functional traits to adapt the extreme environmental filter, leading to lowest diversities in all three dimensions
,
.
96
Interpretation 3:
The species richness and evenness in TR are the highest, the vegetation presents more functionally and evolutionarily diverse species composition, resulting in the highest value in all three dimensions
The MO habitat is between EM and TR, so the diversity in each dimension is between the two extremes
,
.
97
Statistical Estimation: essential in diversity analysis
In the theoretical framework, all diversity measures and similarity/differentiation indices are in terms of the true population values of species richness and species relative abundances in an assemblage
In practice, the true values are unknown and estimation or standardization is needed (iNEXT.3D)
98
iNEXT.3D standardization based on sampling data
Chao, A., Henderson, P. A., Chiu, C.-H., Moyes, F., Hu, K.-H., Dornelas, M and Magurran, A. E. (2021). Measuring temporal change in alpha diversity: a framework integrating taxonomic, phylogenetic and functional diversity and the iNEXT.3D standardization. Methods in Ecology and Evolution. Online
Attribute diversity (AD)• A hypothetical assemblage which can be decomposed
into M sub-assemblages • The i-th sub-assemblage consists of vi attributes,
each with raw abundance ai, i = 1, 2, …, M• The attribute diversity (AD) of order q is defined as
the Hill number of order q for the hypothetical assemblage
1
1 1lim exp log
Sq i i
iq i
a aAD AD vV V→ =
= = − ∑
1S
i iiV v a== ∑
Effective total attributes
1/(1 )1/(1 )
1 11
, 0, 1.
qq qqM Mq i i
i iMi ij jj
a aAD v v q q
Vv a
−−
= ==
= = ≥ ≠
∑ ∑∑
Diversity Number of sub-assemblages (M)
Number of attributes in the i-th sub-assemblage
Abundance of each attribute in the i-th sub-assemblage
Total abundance across all attributes
Effective number of attributes
Unified framework:
Attribute diversity (AD) or Hill-Chao numbers
S or B vi ai 1
Mi ii
V v a=
=∑ 1 (1 )
1
qqMq i
ii
aAD vV
−
=
= ∑
0
1
M
ii
AD v=
=∑
1
1 1lim exp log
Mq i i
iq i
a aAD AD vV V→
=
= = − ∑
2 21
1 ( / )Mi ii
AD v a V=
= ∑
Taxonomic diversity (TD)
S (# of species)
1 (species)
zi (abundance of species i)
1
Sii
V z z+== =∑
(total abundance)
1 (1 )
1
qqSq i
i
zTDz
−
= +
= ∑
Phylogenetic diversity (PD)
B (# of branches/nodes) in a tree with depth T
Li (length of branch/node i)
iz ∗ (abundance of branch/node i)
1
Bi ii
V L z z T∗+=
= =∑ 1 (1 )
1
qqBq i
ii
zPD Lz T
−∗
= +
= ∑
( /q qmean PD PD T= )
Functional diversity* (FD)
S (# of species)
)(τvi (functional groups contributed by species i)
)(τai (abundance of functionally- indistinct set of species i)
( ) ( )1
Si ii
V v a zτ τ +== =∑
(total abundance) ( ) ( )1 (1 )
1
qqSiq
ii
aFD v
zτ
τ
−
= +
= ∑
Unified framework (AD)
Theoretical diversity formulas‡
Interpolation estimator§ (for m < n)
Extrapolation estimator¶ (for a sample of size n+m*)
q = 0
0
1( ) [ ( )]
m
kk
AD m E h m=
=∑
0
1
ˆ( ) ( )m
kk
AD m h m=
=∑
0 * 0( ) obsAD n m AD+ = +
*0 0 ˆ( )[1 (1 ) ]mundetectedAD β− −
q = 1 1 ( )AD m =
1
exp log [ ( )]m
kk
k k E h mmT mT=
− ×
∑
1( )AD m
1
ˆexp log ( )m
kk
k k h mmT mT=
= − ×
∑
1 * 1( ) obsAD n m AD+ = +
*1 1 ˆ( )[1 (1 ) ]mundetectedAD β− −
q =2 2 ( )AD m
2
1
1
[ ( )]m
kk
k E h mmT=
= ×
∑
2( )AD m
2
1
1
ˆ ( )m
kk
k h mmT=
= ×
∑
2 *( )AD n m+ =
*
* *1
1ˆ ˆ ˆ( 1)1 ( 1)ˆ
( 1)
Mi i i
ii
a a an mvn m n n m n n=
−+ −× + + + −
∑
General order q ≥ 0 ( )q AD m
11
1[ ( )]
q qm
kk
k E h mmT
−
=
= × ∑
( )qAD m
1/(1 )
1
ˆ ( )qqm
kk
k h mmT
−
=
= ×
∑
*( )q q
obsAD n m AD+ =
*ˆ( )[1 (1 ) ]q q m
undetectedAD β+ − −
Sample coverage
1( ) 1 (1 )
Sm
i ii
C m p p=
= − −∑ 1
ˆ ( ) 11
iS
i
i
n XmXC m
nnm
=
− = −
−
∑
*ˆ ( )C n m+ * 1
1 1
1 2
( 1)1( 1) 2
mf n fn n f f
+ −
= − − +
The iNEXT.3D standardization
1 1( ) ( ) [ ( ) ] ~ [ ( ) ]
M M
k i i i ii i
h m v m I a m k v I a m k= =
= = =∑ ∑
1[ ( )] 1
k m kMi i
k ii
m a aE h m vk V V
−
=
= −
∑
Software
iNEXT.3D (iNterpolation-EXTrapolation for 3-dimension diversity)
Online version available from Shinyappshttps://chao.shinyapps.io/iNEXT.3D/
104
Summary: 4-step comparison (Chao et al. 2020)
STEP 1. Assessment of sample completeness profile
STEP 2. Asymptotic analysis based on estimating true diversity of entire assemblages(STEP 2a. size-based rarefaction and extrapolation )
STEP 3. Non-asymptotic coverage-based rarefaction and extrapolation analysis
STEP 4. Evenness profile
105
Sample Completeness
STEP 1. Assessment of sample completeness profile
Application of the iNEXT.3D standardizationto EM and TR dunes
Asymptotic and empirical PD profiles
Asymptotic and empirical TD profiles(1) Taxonomic Diversity
(2) Phylogenetic Diversity
Asymptotic and empirical FD profiles(3) Functional Diversity
STEP 2. Asymptotic analysis based on estimating true diversity of entire assemblages (STEP 2a. size-based rarefaction and extrapolation )
(Left panels) The sample-size-based rarefaction (solid lines) and extrapolation curves (dashed lines) for TD (row 1), PD (row 2) and FD (row 3) when extrapolation is extended up to double the reference sample size for the 1981 data. Solid dots denote the observed reference data points. All shaded areas denote 95% confidence bands obtained from a bootstrap method with 100 replications. Some bands are invisible due to narrow widths. (Right panels) The asymptotic diversity profiles (solid curves) and empirical diversity profiles (dotted curves) for q between 0 and 2;
Diversity q = 0 q = 1 q = 2
TD 0TDobs = 17, 0TDasy = 21.49
1TDobs = 9.27, 1TDasy = 9.54
2TDobs = 7.22, 2TDasy = 7.34
PD 0PDobs = 8.18,0PDasy = 10.43
1PDobs = 4.13, 1PDasy = 4.18
2PDobs = 2.84, 2PDasy = 2.86
FD 0FDobs = 9.7, 0FDasy = 10.55
1FDobs = 6.22,1FDasy = 6.32
2FDobs = 5.22, 2FDasy = 5.29
Observed and asymptotic diversity estimates in Dunes EM and TR
Diversity q = 0 q = 1 q = 2
TD 0TDobs = 42, 0TDasy = 46
1TDobs = 23.94, 1TDasy = 24.48
2TDobs = 18.61, 2TDasy = 18.92
PD 0PDobs = 16.32,0PDasy = 18.08
1PDobs = 5.9, 1PDasy = 5.95
2PDobs = 3.26, 2PDasy = 3.26
FD 0FDobs = 20.93, 0FDasy = 21.83
1FDobs = 13.5,1FDasy = 13.71
2FDobs = 11.17,2FDasy = 11.32
EM TR
(1) Taxonomic Diversity
(2) Phylogenetic Diversity
(3) Functional Diversity
STEP 3. Non-asymptotic coverage-based rarefaction and extrapolation analysis
The coverage-based rarefaction (solid lines) and extrapolation (dashed lines) curves for TD (row 1), PD (row 2) and FD (row 3) up to the corresponding coverage value of the doubled reference sample size. Solid dots denote observed reference data points. All shaded areas denote 95% confidence bands obtained from a bootstrap method with 100 replications. The extrapolation parts for q = 1 and q = 2 for each dimension are invisible due to little increment on coverage value when reference sample size is doubled. Some confidence bands are invisible due to narrow widths.
Evenness profile under Cmax = 0.996
STEP 4. Evenness profile
We illustrate the proposed iNEXT.3D standardization to the estuarine fishes collected at Bridgwater Bay in UK’s Bristol Channel from 1981 to 2019; see Henderson and Holmes (1991), Magurran and Henderson (2003), and Henderson et al. (2011) for sampling details
A total of 88 species (among 2457 individuals) were caught and their monthly abundances were recorded.
Number of singletons = 8; number of doubletons = 1
Application of the iNEXT.3D standardizationto fish data
PD: phylogenetic
tree of 88 species with normalized
tree depth = 1
FD: Gower distance based on 12 traits
• There are nine continuous traits (mean weight, length of first maturity, trophic level, mean temperature preference, maximum weight, growth rate, food consumption and biomass, maximum depth, and generation time);
• Three categorical traits (position in water column, reproductive guild, and body shape).
Sample Completeness
STEP 1. Assessment of sample completeness profile
Asymptotic and empirical PD profiles
Asymptotic and empirical TD profiles
(1) Taxonomic Diversity
(2) Phylogenetic Diversity
Asymptotic and empirical FD profiles(3) Functional Diversity
(Left panels) The sample-size-based rarefaction (solid lines) and extrapolation curves (dashed lines) for TD (row 1), PD (row 2) and FD (row 3) when extrapolation is extended up to double the reference sample size for the 1981 data. Solid dots denote the observed reference data points. All shaded areas denote 95% confidence bands obtained from a bootstrap method with 100 replications. Some bands are invisible due to narrow widths. (Right panels) The asymptotic diversity profiles (solid curves) and empirical diversity profiles (dotted curves) for q between 0 and 2;
STEP 2. Asymptotic analysis based on estimating true diversity of entire assemblages (STEP 2a. size-based rarefaction and extrapolation )
Diversity q = 0 q = 1 q = 2
TD 0TDobs = 46, 0TDasy = 51.06
1TDobs = 3.15, 1TDasy = 3.16
2TDobs = 2.05, 2TDasy = 2.05
PD 0PDobs = 10.22,0PDasy = 10.62
1PDobs = 2.29, 1PDasy = 2.29
2PDobs = 1.8, 2PDasy = 1.8
FD 0FDobs = 8.87, 0FDasy = 9.84
1FDobs = 1.82,1FDasy = 1.82
2FDobs = 1.53, 2FDasy = 1.53
Observed and asymptotic diversity estimates in fish 1998 and 2009
Diversity q = 0 q = 1 q = 2
TD 0TDobs = 32, 0TDasy = 36.9
1TDobs = 5.19, 1TDasy = 5.2
2TDobs = 3.43, 2TDasy = 3.43
PD 0PDobs = 7.72,0PDasy = 7.94
1PDobs = 2.52, 1PDasy = 2.52
2PDobs = 2.01, 2PDasy = 2.01
FD 0FDobs = 6.69, 0FDasy = 7.47
1FDobs = 2.43,1FDasy = 2.43
2FDobs = 2.02,2FDasy = 2.02
1998 2009
(1) Taxonomic Diversity
(2) Phylogenetic Diversity
(3) Functional Diversity
The coverage-based rarefaction (solid lines) and extrapolation (dashed lines) curves for TD (row 1), PD (row 2) and FD (row 3) up to the corresponding coverage value of the doubled reference sample size. Solid dots denote observed reference data points. All shaded areas denote 95% confidence bands obtained from a bootstrap method with 100 replications. The extrapolation parts for q = 1 and q = 2 for each dimension are invisible due to little increment on coverage value when reference sample size is doubled. Some confidence bands are invisible due to narrow widths.
STEP 3. Non-asymptotic coverage-based rarefaction and extrapolation analysis
Evenness profile under Cmax = 0.9997
STEP 4. Evenness profile
Application of the iNEXT.3D to time series
data 1981-2019
See Chao et al. (2021)
(a) Taxonomic diversity
(b) Phylogenetic diversity
(c) Functional diversity