Borges, J. L On exactitude in science. P. 325, In, Jorge Luis Borges, Collected Fictions (Trans. Hurley, H.) Penguin Books.


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1 ... In that Empire, the Art of Cartography attained such Perfection that the map of a single Province occupied the entirety of a City, and the map of the Empire, the entirety of a Province. In time, those Unconscionable Maps no longer satisfied, and the Cartographers Guilds struck a Map of the Empire whose size was that of the Empire, and which coincided point for point with it. The following Generations, who were not so fond of the Study of Cartography as their Forebears had been, saw that that vast Map was Useless, and not without some Pitilessness was it, that they delivered it up to the Inclemencies of Sun and Winters. In the Deserts of the West, still today, there are Tattered Ruins of that Map, inhabited by Animals and Beggars; in all the Land there is no other Relic of the Disciplines of Geography. Suárez Miranda, Viajes de varones prudentes, Libro IV, Cap. XLV, Lérida, 1658 Borges, J. L On exactitude in science. P. 325, In, Jorge Luis Borges, Collected Fictions (Trans. Hurley, H.) Penguin Books.
2 Fitting (more) Macroevolutionary Models to Data Luke J. Harmon
3 Slatkin and Pollack 2005
4 within species Slatkin and Pollack 2005
5 within species among species Slatkin and Pollack 2005
6 Hansen and Martins 1996
7 Topics Fitting models to comparative data: what do we know? Extending the set of models we can fit The future of comparative methods
8 Topics Fitting models to comparative data: what do we know? Extending the set of models we can fit The future of comparative methods
9 Example: Anolis lizards Lizards on Caribbean islands Phylogenetic and body size data for 73 species (out of ~140 total) Anolis baleatus
10
11 Brownian Motion Two parameters: starting value (zo) and rate (σ 2 ) dz(t) = σ db(t) zo t z(t)
12 Phylogeny A B
13 var(a) Phylogeny A B
14 var(a) Phylogeny A var(b) B
15 var(a) Phylogeny A cov(a,b) B var(b) A B
16 Phylogenetic variancecovariance (VCV) matrix A B A B
17 Phylogenetic variancecovariance (VCV) matrix var(a) A var(b) B A B
18 Phylogenetic variancecovariance (VCV) matrix var(a) cov(a,b) A cov(a,b) var(b) B A B
19 Phylogenetic variancecovariance (VCV) matrix var(a) cov(a,b) A σ 2 cov(a,b) var(b) B A B
20 genetic drift (co)variance + 
21 General form Tip data follow a multivariate normal distribution with mean vector zo and variancecovariance matrix where var(i) = σ 2 (di); di =distance from root to tip i cov(i,j) = σ 2 (ci,j); ci,j =shared path of tip i and j
22 We can fit Brownian motion model to comparative data using likelihood
23 16 17 σ 2 Sigma Squared 0e+00 1e 09 2e 09 3e 09 4e zo Theta
24 Quantitative genetics A quantitative genetics model of pure genetic drift also produces Brownian motion Three parameters: G, Ne, zo σ 2 = G/Ne
25 G G 1e 11 1e 10 1e 09 1e 08 1e 07 1e Ne Ne
26 G G 1e 11 1e 10 1e 09 1e 08 1e 07 1e σ 2 = G/Ne Ne Ne
27 σ 2 = G/Ne
28 σ 2 = G/Ne σ 2 / Vp = h 2 /Ne
29 Across a wide range of taxa, σ 2 /Vp is about 0.74
30 Across a wide range of taxa, σ 2 /Vp is about 0.74 That is, the average Brownian rate parameter is about 0.74 phenotypic standard deviations per million years
31 Across a wide range of taxa, σ 2 /Vp is about 0.74 That is, the average Brownian rate parameter is about 0.74 phenotypic standard deviations per million years That translates to a variance of 1.2 x 106 phenotypic sd per generation
32 Across a wide range of taxa, σ 2 /Vp is about 0.74 That is, the average Brownian rate parameter is about 0.74 phenotypic standard deviations per million years That translates to a variance of 1.2 x 106 phenotypic sd per generation About half of the time, the change from one generation to the next is phenotypic s.d.
33 G G 1e 11 1e 10 1e 09 1e 08 1e 07 1e σ 2 = G/Ne Ne Ne
34 G G 1e 11 1e 10 1e 09 1e 08 1e 07 1e Evolution is too slow for drift σ 2 = G/Ne Ne Ne
35 What about selection?
36 Brownian motion can also result if selection is random in direction and relatively weak
37
38 BM model: About half of the time, the change from one generation to the next is phenotypic s.d.
39 What happens with OU models?
40 OU Model  single optimim Three parameters: starting value (μ), rate (σ 2 ), and constraint parameter (α) i sij j T = total tree depth
41 stabilizing selection (co)variance + 
42 α = 1 / (ω+p)
43 α = 1 / (ω+p) mean α across clades: 0.34
44 α = 1 / (ω+p) mean α across clades: 0.34 Typical ω 2 = 350
45 α = 1 / (ω+p) mean α across clades: 0.34 Typical ω 2 = 350 P would have to be negative to get these alpha values
46 α = 1 / (ω+p) mean α across clades: 0.34 Typical ω 2 = 350 P would have to be negative to get these alpha values (e.g. stabilizing selection is typically stronger than what OU values suggest)
47 What does this mean? Commonly used models can fit comparative data Simplistic quantitative genetics interpretations of these models are probably not correct
48 Brownian motion is not drift OrnsteinUhlenbeck is not stabilizing selection on a single peak
49 HansenMartins environmental change model Imagine populations are subject to strong stabilizing selection on optima But the position of these optima varies according to a Brownian motion model
50 σb 2 = overall rate of drift σe 2 = rate of drift of optima Hansen and Martins 1996
51 σb 2 = overall rate of drift σe 2 = rate of drift of optima When σe 2 larger than (left term): Hansen and Martins 1996
52 σb 2 = overall rate of drift σe 2 = rate of drift of optima When σe 2 larger than (left term): σb 2 = σe 2 Hansen and Martins 1996
53 Main idea: patterns of trait means on trees might reflect the dynamics of the adaptive landscape more than they reflect processes of adaptation within populations
54 Slatkin and Pollack 2005
55 within species Slatkin and Pollack 2005
56 within species among species Slatkin and Pollack 2005
57 Possible model: The location of the optimum changes rapidly from one generation to the next Globally, optima are constrained to be within a certain range of values
58 Topics Fitting models to comparative data: what do we know? Extending the set of models we can fit The future of comparative methods
59 Extending Models Solving likelihoods for new models Using Bayesian approaches ABC
60 Extending Models Solving likelihoods for new models Using Bayesian approaches ABC
61 Early Burst Model (EB) Rate of evolution slows through time Highest rate at the root of the tree Three parameters: starting value (μ), starting rate (σ 2 o), and rate change (r) i sij j
62 Proportion of weight Proportion of weight Body size Body shape Squamates Birds Fish Insects Mammals BM CC EB NA Amphibians
63 Extending Models Solving likelihoods for new models Using Bayesian approaches ABC
64 Acceptance probability Prior =. odds Likelihood ratio Standard Bayesian MCMC
65 rjmcmc reversiblejump MCMC rjmcmc is an MCMC algorithm that can jump between models of differing complexity
66 rjmcmc moves Update parameter values (root state, rate σ 2 i) Split single rate category into two Merge two rate categories into one
67 (α1 α1 α1 α1 α1 α1 α1 α1 α1 α1) k = 1
68 (α1 α1 α1 α1 α1 α1 α1 α1 α1 α1) k = 1 split move (α1 α1 α1 α1 α1 α2 α2 α2 α2 α2) k = 2
69 (α1 α1 α1 α1 α1 α1 α1 α1 α1 α1) k = 1 split move (α1 α1 α1 α1 α1 α2 α2 α2 α2 α2) k = 2 split move (α3 α3 α3 α1 α1 α2 α2 α2 α2 α2) k = 3
70 (α1 α1 α1 α1 α1 α1 α1 α1 α1 α1) k = 1 split move (α1 α1 α1 α1 α1 α2 α2 α2 α2 α2) k = 2 split move (α3 α3 α3 α1 α1 α2 α2 α2 α2 α2) k = 3 merge move (α3 α3 α3 α1 α1 α1 α1 α1 α1 α1) k = 2
71 Acceptance probability Prior =. odds Likelihood ratio Standard Bayesian MCMC
72 Acceptance probability Prior =. odds Likelihood ratio. Hastings ratio Standard Bayesian MCMC Reversible jump MCMC
73 AUTEUR ACCOMMODATING UNCERTAINTY IN TRAIT EVOLUTION USING R Jonathan M. Eastman, Michael E. Alfaro, Paul Joyce, Andrew E. Hipp, and Luke J. Harmon
74 AUTEUR Do rates of trait evolution vary across clades in a phylogenetic tree? Are traits in some clades evolving faster than in others?
75
76 Em
77 Em Graptemys Pseudemys
78 Extending Models Solving likelihoods for new models Using Bayesian approaches ABC
79 Standard Bayes MCMC Acceptance probability Prior =. odds Likelihood ratio
80
81 Prior Density θ
82 Prior Density θ
83 Prior compute likelihood of data under model M with θ Density θ
84 Prior compute likelihood of data under model M with θ Density Likelihood ratio θ
85 Prior compute likelihood of data under model M with θ accept proposal with probability h Posterior Density Likelihood ratio Density θ θ
86 Prior compute likelihood of data under model M with θ accept proposal with probability h Posterior Density Likelihood ratio Density θ θ
87 Prior compute likelihood of data under model M with θ accept proposal with probability h Posterior Density Likelihood ratio Density θ θ otherwise reject
88 Prior compute likelihood of data under model M with θ accept proposal with probability h Posterior Density Likelihood ratio Density θ θ otherwise reject
89 Standard Bayes MCMC Acceptance probability Prior =. odds Likelihood ratio
90 Standard Bayes MCMC Acceptance probability Prior =. odds Likelihood ratio But what if we can t compute the likelihood ratio?
91 Standard Bayes MCMC Acceptance probability Prior =. odds Likelihood ratio But what if we can t compute the likelihood ratio? Use Approximate Bayesian Computation (ABC)
92 ABC
93 ABC Prior Density θ
94 ABC Prior Density θ
95 ABC Prior simulate data under model M with θ Density θ
96 ABC Prior simulate data under model M with θ Density θ
97 ABC Prior simulate data under model M with θ Density θ
98 ABC Prior simulate data under model M with θ Density θ
99 ABC Prior simulate data under model M with θ Density θ
100 ABC Prior simulate data under model M with θ Density θ
101 ABC Prior simulate data under model M with θ Posterior Density Density θ θ
102 ABC Prior simulate data under model M with θ Posterior Density Density θ θ
103 MECCA Simulation Algorithm
104 n
105 1. Draw ancestral character state Θ n 3 5 Θ
106 1. Draw ancestral character state Θ n 3 5 Θ Simulate characters along branches under BM
107 3. Simulate characters in unresolved clades using a branching diffusion process (birthdeath plus Brownian motion)
108 ABC summary statistics
109 ABC summary statistics
110 ABC summary statistics a
111 ABC summary statistics σ 2 a
112 ABC summary statistics σ 2 a a σ 2
113 λ θμ μ a
114 λ λ μ θμ a
115 λ λ μ θμ a
116 λ λ μ θμ a σ 2 σ 2
117 λ λ μ θμ a a σ 2 σ 2
118 λ λ μ θμ a a σ 2 σ 2
119 λ λ μ θμ a a σ 2 σ 2
120 Diversification rates
121 Diversification rates Lambda Mu density Lambda Mu Index
122 Diversification rates Lambda Mu density Lambda Mu λ = 0.19 ( ) μ = 0.10 ( ) Index
123 Character evolution Density Density σ 2 sigmasq lnθ Root State
124 Character evolution Density Density σ 2 sigmasq ( ) lnθ Root State 7.1 kg ( )
125 ACME C MECCA Modeling the Evolution of Continuous Characters using ABC Graham Slater, Luke Harmon, Paul Joyce, Liam Revell, and Michael Alfaro
126 Topics Fitting models to comparative data: what do we know? Extending the set of models we can fit The future of comparative methods
127 Fitting the data people actually have Incomplete data where sampling is nonrandom Combinations of different types of data, like within and among species
128 Adding to the set of models Moving beyond Brownian motion Estimating meaningful parameters Adding complexity when possible
129 New Statistical Tools Dealing with all forms of uncertainty More flexible Bayesian analyses ABC
130 Topics Fitting models to comparative data: what do we know? Extending the set of models we can fit The future of comparative methods
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