1. 1 Phylogenetic Diversity, Similarity and Differentiation Measures based on Hill Numbers Speaker: 邱春火 清華大學統計研究所 博士後

2. 2 Outline: Species-Neutral Diversity Measures/indices (do not consider species relatedness)  Doubling property  Hill numbers Phylogenetic Diversity Measures/indices (consider phylogenetic distance between species) Philosophical Transactions of the Royal Society B. 2010 365, 3599-3609 Phylogenetic Similarity and Differentiation Measures/indices among Communities Ecology (submitted)

3. 3 The Noah’s Ark: the agony of choice The woodpecker might have to go!

4. 4 Biodiversity Measures/Indices “ the variety and variability among living organisms and the ecological complexes in which they occur ” (Wilson 1986) Two components species richness evenness among abundances

5. 5 Species: 4 More diverse Species: 3 Community1 Community2

6. 6 Species: 3 Uneven Species: 3 Even More Diverse Community1 Community2

7. 7 4 species Uneven 3 species Even Which one is more diverse? Diversity measures Monotonicity Principle of Transfer (Patil & Taillie 1982 ) Community1 Community2

8. 8 Simpson-Type Index (Gini-Simpson Index) Take two individuals, the probability that they belong to different species: (Simpson Index)

9. 9 Shannon Entropy index Measure of uncertainty uncertainty in the species identity of a randomly sampled individual

10. 10 Doubling Property MacArthur (1965), Whittaker (1972), Hill (1973) Two completely distinct (no overlapped species) communities, each with diversity measure X Combine these two, the diversity becomes 2X An essential minimum requirement for “ diversity measures ” that ecologists expect

11. 11 What kinds of measures satisfy doubling property ? Species richness? Entropy? Gini-Simpson index? Yes!! no!!

12. 12 Species richness 4 + 4 = 8 Entropy? 1.39 + 1.39 ≠ 2.08 Gini-Simpson index? 0.75 + 0.75 ≠ 0.875

13. 13 Species richness 4 + 4 = 8 Exp(entropy) 4 + 4 = 8 Inverse (Simpson) 4 + 4 = 8

14. 14 Numbers Equivalent : transform to units of “ species ” Entropy = 3 is equivalent to exp(3) = 20.8 “ species ” Simpson index = 0.1 is equivalent to 1/0.1 = 10 “ species ”

15. 15 Hill Numbers ( Hill 1973 ) q = 0, 0 D =Species richness q =1, 1 D = exponential of entropy q = 2, 2 D = inverse of Simpson index “ Doubling property ” “ Effective number of species ”, q ≥ 0

16. 16 “Effective number of species” Community1: Hill numbers = D Communty2: D species with relative abundance: 1/D, …,1/D. 

17. 17 Example 1000 species 600 species disappear + 400 species survive Gini-Simpson index 0.999 0.9983 0.9975 Inverse of Simpson index 1/(1-0.999) 1/(1-0.9975) 1/(1-0.9983) = 1000 = 600 = 400 0.9975/0.999 = 0.998 almost diversity conserved 1000 = 600 + 400 0.9983/0.999 = 0.999 almost diversity vanished

18. Short Conclusions “ Doubling property ” is essential minimum requirement for diversity measures The magnitude of changes in the measure is easily interpretable Logical consistency in conservation analyses “ Hill numbers ” are the measures that obey “ Doubling property ” and can be interpreted intuitively as “ Effective number of species ”..

19. 19 Diversity in Phylogenetic Version Community 1 Community 2 When species in each community are equally common, which community is more diverse?  ?

20. 20 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 difference between species Community 1 Community 2 <

21. 21 Previous Non-neutral diversity index (1) Branch-based measure: Phylogenetic Diversity (PD) ( Faith 1992) (generalization of species richness) sum of the branch lengths of the phylogeny connecting all species from tips to root. PD : 12 10 (1) (2)

22. 22 Previous Non-neutral diversity index (2) Abundance-Weighted measure: Quadratic entropy (Rao 1982) (generalization of Gini-Simpson index) the average of species differences weighted by their relative abundances Phylogenetic entropy (Allen et al. 2009) (generalization of Shannon entropy) A parametric class based on Tsallis entropy (Pavoine et al. 2009)

23. 23 Non-neutral diversity index Except for Faith’s PD, all indices mentioned above do NOT satisfy the “doubling property”.

24. 24 Doubling Property in phylogenetic version Two completely distinct (no overlapped tree branch) communities, each with diversity measure X Combine these two, the diversity becomes 2X

25. 25 3333 3333 33333333 Faith ’ s PD 12 + 12 = 24 Phylogenetic entropy H P ? 4.16 + 4.16 ≠ 6.24 Rao ’ s Q ? 2.25 + 2.25 ≠ 2.625

26. 26 p1p1 p2p2 p3p3 p1p1 p2p2 p3p3 Phylogenetic diversity measure but also the phylogenetic relationship among species. Moreover, satisfy the essential requirement “doubling property”. Not only consider the richness and relative abundance of species,

27. 27 Basic Approach based on Hill Numbers – At any given moment t, we can find the “ species ” by slicing the tree and its relative abundance. – Obtain Hill number q D(t) at moment t. – Average over from the present time to T years ago – Call this average diversity as “ Mean Diversity of order q over T years ”, it is in units of “ species ”.

28. 28 Mean Phylogenetic Diversity of order q in the time interval [-T, 0] General Formula: B T : all branches in the time interval [-T, 0] L i : the length (duration) of Branch i in the set B T a i : the total abundance descended from Branch i When T tends to 0 or all species are maximally distinct, this measure reduce to Hill numbers.

29. 29 Generalize and unify existing measures Order q = 0 = PD / T Order q =1 Order q = 2

30. 30 3333 3333 33333333 PD/T 4 + 4 = 8 Exp(H P /T) 4 + 4 = 8 1/(1-Q/T) 4 + 4 = 8 PD 12 + 12 = 24 H P 4.16 + 4.16 ≠ 6.24 Q 2.25 + 2.25 ≠ 2.625

31. 31 Conclusion of Part I 2.Developed a unified class of PD measures based on Hill numbers and that obey the “doubling property” 1. Measures that do not obey the “doubling property” give self- contradictory results in conservation analyses 3. “the mean effective number of species over the time interval[-T, 0]” 4. Quadratic entropy and phylogenetic entropy do not satisfy “doubling porperty”, but a simple transformation converts each to our measures

32. 32 Part II Diversity in Multiple Communities Why ? Conservationist’s View Ecologist’s View

33. 33 Definition: (Whittaker 1972) Gamma diversity: dversity in total region Alpha diversity: average diversity within community Beta diversity: diversity among community Relationship: Additive : Gamma = Alpha + Beta Multiplicative: Gamma = Alpha  Beta Partition of Neutral Diversity (1)

34. 34 Partition of Neutral Diversity (2) Essential axioms (Jost 2007) Beta and Alpha should be “ mathematically independent ” ( effective number of species in the region ) ( effective number of species for individual community ) ( effective number of community )

35. 35 Neutral Similarity and Differentiation Measures among Communities Similarity measureTransform of Beta diversity Jaccard (1908) Sorenson (1948) Horn (1966) Morisita-Horn (1959) C qN (Chao et al. 2008) Differentiation measure : 1-Similarity measure Similarity measure:

36. 36 Differentiation Measures in Genetics Based on Gini-Simpson index ( Wright 1951; Nei’s G ST 1973 ), Analyses of differentiation between two populations, for a specific locus, each population has 100 equally common alleles and no shared allele. Complete differentiation!! F ST = (0.995-0.99)/0.995=0.005 Modify F ST by F ST /max(F ST ) =2 (H γ -H α )/(1-H α ) =2(0.995-0.99)/(1-0.99) =1 ( Jost’s D 2008; Chao et al’s 1-C 2N 2008 ) where and

37. 37 Non-neutral Differentiation Measures (1) Based on Rao’s Q ( Nei 1973; Nei & Li 1979 ), Example Region 1: Two communities, each contains equally abundant 500 species with difference 7 and no shared species. Region 2: Two communities, each contains 10 equally abudant species with difference 7 and 9 shared species N ST = ( 6.993-6.986 ) /6.993= 0.001 N ST = ( 6.335-6.3 ) /6.335= 0.0055 where and

38. 38 Partition of Mean Diversity (1) Based on spatial concepts of Whittaker (1972) Gamma, Alpha and Beta diversity Satisfy essential axioms ( Jost 2007 ) Beta and Alpha should be “mathematically independent” Gamma > Alpha Alpha is some type of average of diversity indices of the communities. Gamma must be completely determined by alpha and beta

39. 39 Partition of Mean Diversity (2) = When communities are completely distinct, When all communities have identical diversity K, then the average diversity (alpha) must equal to the diversity of each community. If imply Alpha diversity =K Since alpha diversity is independent of beta diversity, imply and

40. 40 Partition of Mean Diversity (3) = = ( mean effective number of species over the interval [-T, 0] ) ( mean effective number of species over the interval [-T, 0] for an individaul community ) ( The effective number of completely distinct communities in the interval [-T, 0] ) Phylogenetic Gamma diversity Phylogenetic Alpha diversity Phylogenetic Beta diversity

41. 41 Phylogenetic Similarity and Differentiation Measures (1) Normalize phylogenetic beta diversity into the phylogenetic homogeneity measures (1A) When T tends to 0 or all species are maximally distinct, this measure reduce to traditional homogeneity measure (Jost 2007). (1B) Order q=0, phylo-Jaccard measure (Lozupone & Knight 2005; Faith et al. 2009) (1C) Order q=2, phylo-Morisita Horn measure

42. 42 Phylogenetic Similarity and Differentiation Measures (2) Normalize phylogenetic beta diversity into the phylogenetic overlap measure (A) When T tends to 0 or all species are maximally distinct, this measure reduce to the classical overlap index C qN (Chao et al. 2008) (B) Order q=0, phylo-Sornson measure (Bryant et al. 2008; Ferrier et al. 2007) (C) Order q=1, phylo-Horn measure (D) Order q=2, phylo-Morisita Horn measure (E) : Phylogenetic Differentiation measures

43. 43 Example v.s. N ST ( Nei 1973 )

44. 44 Result of Example : solid N ST : dash

45. 45 Modify differentiation measure N ST (1) Beta diversity is confounded with alpha diversity N ST diversity can be standardized by its maximum value

46. 46 Region 1: Region 2: N ST /max(N ST )= 2(6.993-6.986)/(7-6.986) = 1 N ST /max(N ST )= 2(6.335-6.3)/(7-6.3) = 0.1 Modify differentiation measure N ST (2) Region 1: Two communities, each contains equally abundant 500 species with difference 7 and no shared species. Region 2:Two communities, each contains 10 equally abundant species with difference 7 and 9 shared species N ST = ( 6.993-6.986 ) /6.993=0.001 N ST = ( 6.335-6.3 ) /6.335=0.0055

47. 47 Conclusion of Part II 1. We derived a multiplicative decomposition formula for the phylogenetic mean diversity. 3. N ST (Nei & Li 1979) dose not measure genetic differentiation, and lead to counter-intuitive interpretation. 2. We derived and, the phylogenetic generalization of the Sorenson, Jaccard, Horn and Morisita-Horn measures. 4. N ST can be easily corrected, and the corrected measure is exactly our differentiation measure. 5. These measures are valid not only for phylogenetic trees but also for general trees.

48. 48 Summary Hill numbers “effective number of species” Mean Diversity “mean effective number of species over time interval [-T, 0]” Faith’s PD Allen et al.’s H P Rao’s Q Richness Shannon entropy Gini-Simpson Neutral Gamma 、 Alpha and Beta diversity measures Phylogenetic Gamma 、 Alpha and Beta diversity measures Jaccard, Sorenson, Horn, Morisita-Horn and C qN measures Phylogenetic Jaccard, Sorenson, Horn, Morisita- Horn and Overlap measures

49. 49 Overall Comments Each index measures different aspects of community Biological community (multi-dimensional structure) cannot be uniquely characterized by a single scalar index We appreciate the advantage of each index but understand its limitation

50. 50 Future work 1.Development of a unified class of phylogenetic and functional diversity measures 2. Application of these measures to resolve central questions in ecology and evolutionary genetics ecological model (neutral model, niche model,...) genetic model (finite island model, stepping stone model,…) 3. Statistical estimation of diversity and similarity/differentiation measures 4. Applications and software development G ST  1/(1+2Nm) (Crow & Kimura 1970)

51. 51 “We are all blind men (and women) trying to describe a monstrous elephant of ecological and evolutionary diversity...” (Nanney 2004 )

52. 52 THANK YOU VERY MUCH!!

53. 53 Taxonomic Diversity of Level = 3 Phylogenetic tree based on the classical Linnean taxonomic categories

54. 54 Mean Taxonomic Diversity of order q for L levels Order q = 0 = total nodes in tree / L Cladistic Diversity ( Vane-Wright et al. 1991) Order q =1 Order q = 2

55. 55 Conceptual framework for q = 0 Connect Faith’s PD to mean species richness For a fixed T, the nodes divide the tree into Segment 1, 2 and 3 with duration (length) T1, T2, and T3 In any moment of Segment 1, there are 4 species (i.e., 4 branches cut) Segment 2, there are 3 species Segment 3, there are 2 species The mean species richness over the time interval [−T, 0] is (T1/T) ×4 + (T2/T) ×3 + (T3/T) ×2 = total branch length in [-T, 0] / T (Mean Phylogenetic Diversity of order 0 over T years) If T = height of tree, then

56. 56 Conceptual framework for q > 0 When consider abundance data, imagine that There are T1 assemblages with abundance vector{p1, p2, p3, p4 }, T2 assemblages with abundance vector {p1, p2+p3, p4 } and T3 assemblages with abundance vector {p1+p2+p3, p4 }. There are a total of T1+T2+T3 = T assemblages and each is given the same weight 1/T. The “ Mean diversity of order q over T years ” is the following average (Jost 2006, 2007)