phylogenetic diversity and functional diversity

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

https://chao.shinyapps.io/iNEXT.3D/

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


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


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


(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.

Evenness profile under Cmax = 0.996


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


Asymptotic and empirical PD profiles

Asymptotic and empirical TD profiles



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 )



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



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



(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.


Evenness profile under Cmax = 0.9997


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

phylogenetic diversity and functional diversity

Documents