Stochastic Processes for New Biology. Rice University, Houston, 21 August 2012
25 slides
Serik Sagitov
Chalmers University and University of Gothenburg
A phylogenetic confidence interval
for the optimal trait value
http://arxiv.org/abs/1207.6488
joint work with
Krzysztof Bartoszek
Comparative phylogenetics
Interspecies correlation
Yule-Brownian-Motion model
Yule-Ornstein-Uhlenbeck model
Adaptation to an optimal value
Sample mean and variance of the trait values
Phylogenetic confidence intervals
Exact and approximate formulae
Related projects
1
Comparative phylogenetics
Comparative phylogenetics: studies various trait values like
log-bodysize (X1 , . . . , Xn ) for a
family of related species
We will assume throughout that the species tree is unknown.
2
Interspecies correlation
In our previous paper Interspecies correlation for neutrally evolving
traits.J. Theor. Biol. 309 (2012) 11-19 we obtain explicit formulae for
X
1
ρ n = n
Cov [Xi , Xj ]
2 Var [X] 1≤i<j≤n
assuming that the trait evolves along a lineage as a Brownian Motion.
Aldous-Popovic (2005), Stadler (2008): unknown species trees
are modeled by branching processes conditioned on n tips.
In particular, we found for the Yule-BM-model
2
n
ρn =
−1 ,
n − 1 an
Pn
where an = i=1 (1/i) are the harmonic numbers.
3
YBM-model for the trait evolution
G.U.Yule (1924):
let us model speciation by a pure
birth process with parameter λ.
J.Felsenstein (1985):
the trait value evolves along
a lineage neutrally as a Brownian B(t) motion
with variance σ 2 and ancestral state B(0) = X0 .
Coalescent processes model the gene trees.
Conditioned branching processes model the species trees.
4
Yule-Ornstein-Uhlenbeck model
In this paper we work with the YOU-model:
(conditioned Yule tree)+(OU trait evolution)
L.Ornstein and
G.Uhlenbeck (1930)
dX(t) = −α(X(t) − θ)dt + σdB(t)
X(0) = X0
Just six parameters fully describe the YOU-model:
• number of species n
• speciation rate λ, wlog we assume λ = 1
• optimal trait value θ,
• adaptation rate α > 0
α = 0 ⇒ X(t) = B(t)
• ancestral state X0
• noise size σ > 0
σ = 0 ⇒ X(t) = θ + e−αt (X0 − θ)
5
Yule-Ornstein-Uhlenbeck model
T = time to the origin, Ti = inter-speciation times
τ = MRCA time for a random pair of species
θ = optimal trait value, X0 = ancestral trait value
(X1 , . . . , Xn ) = observed trait values identically distr. but dependent
6
Yule-Ornstein-Uhlenbeck model
OU-process: the distribution of X(t) is normal with
E [X(t)] = θ + e−αt (X0 − θ)
Var [X(t)] = σ02 (1 − e−2αt )
2
σ
σ02 =
2α
Stationary regime: E [X(t)] → θ and Var [X(t)] → σ02 as t → ∞.
The Yule tree conditioned on n tips:
run forward a linear birth process from a single branch and
stop at the n-th speciation disregarding the nascent branch.
Interspeciation times
Ti = time duration with exactly i branches in the tree
Ti are independent exponential with rates i = 1, . . . , n.
Backward description reminds a stretched Kingman’s coalescent.
7
Sample mean and variance
Classical summary statistics: sample mean and sample variance
X1 + . . . + Xn
X̄n =
n
n
1 X
2
and Sn =
(Xi − X̄)2
n − 1 i=1
Denoting X, Y a random pair of sampled trait values we get
E X̄n = E [X]
ρn = Cov [X, Y ] / Var [X]
Var X̄n = Var [X] − (n − 1)n−1 (1 − ρn ) Var [X]
2
E Sn = (1 − ρn ) Var [X]
and it becomes crucial to know the Laplace transforms of (T, τ ):
−αT E [X] = θ + (X0 − θ) E e
−2αT −αT 2
2
Var [X] = σ0 (1 − E e
) + (X0 − θ) Var e
−2ατ −2αT −αT 2
2
Cov [X, Y ] = σ0 (E e
−E e
) + (X0 − θ) Var e
8
Laplace transforms of T and τ
A Yule tree conditioned on n tips has n − 1 splittings.
Let κn be the splitting number (counted forward in time)
corresponding to the coalescent of two randomly sampled tips.
Key Lemma. We claim that
P(κn = k) =
2(n + 1)
, k = 1, . . . , n − 1
(n − 1)(k + 2)(k + 1)
and that for any x > −1 and y > −1
h
E e−x(T −τ )−yτ
i
n−1
bk,x
2(n + 1)bn,y X
=
(n − 1)
(k + 2)(k + 1)bk,y
k=1
bn,x =
1
2
n
·
···
1+x 2+x
n+x
9
Moments of T and τ
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E[τ]
6
Using the Key Lemma we derive recursively all the joint moments
k m
for k ≥ 0 and m ≥ 0.
E (T − τ ) τ
2
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true
simulated
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Number of tips
Simulated and true values of E [T ] and E [τ ] .
Each point comes from 104 simulated Yule trees.
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true
simulated
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Each point comes from 104 simulated Yule trees.
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Exact formulae for the moments of X̄n and Sn2
Using the Key Lemma we compute
1
2
n
·
···
1+α 2+α
n+α
−ατ 2 − (n + 1)(y + 1)bn,α
E e
=
(n − 1)(α − 1)
−αT E e
= bn,α =
which imply
E X̄n = bn,α X0 + (1 − bn,α )θ
2 2α + 1 − (4αn + 2α + 1)bn,2α
Var X̄n = σ0 ·
+ (X0 − θ)2 (bn,2α − b2n,α )
(2α − 1)n
2
(2α + 1)(n + 1)bn,2α − 2
E Sn = σ02 1 +
(2α − 1)(n − 1)
12
For example, with α = 1 we get
Phylogenetic point estimates
n
1
X0 + n+1
θ.
E X̄n = n+1
X̄n and Sn2 are asymptotically unbiased estimates of the
optimum θ and the stationary OU-variance σ02 respectively.
Concistency of the estimate X̄n

−2α

, if 0 < α < 0.5,
C
·
n
α,δ


Var X̄n ∼ σ 2 ·
2n−1 ln n,
if α = 0.5,



2α+1
if α > 0.5,
2α(2α−1)n ,
where δ =
|X0 −θ|
σ
Cα,δ
is the normalized deviation of X0 from θ, and
2
=
Γ(2α + 1) + δ 2 Γ(2α + 1) − δ 2 Γ2 (α + 1).
1 − 2α
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−1.5
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0.0
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Sampling distributions
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n = 110
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n = 10
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0.0
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n = 140
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Histograms of X̄n for α = 1, X0 = 1, θ = 0, σ = 1.
Each histogram is based on 104 independently simulated samples.
14
Phylogenetic confidence interval
For sufficiently strong adaptation when α > 0.5 as n → ∞ we have
E X̄n = θ + O(δn−α )
2α + 1 σ02
Var X̄n ∼
·
2α − 1 n
2
E Sn → σ02
σ02 = σ 2 /(2α)
Let α > 0.5 and δ = |X0σ−θ| stay bounded as n → ∞. We propose a
convenient approximate formula for a 95% confidence interval for θ:
r
2α + 1
Kα
Kα =
X̄n ± 1.96 · Sn · √
2α − 1
n
√
Compared to (X̄n ± 1.96 · Sn / n) it has an extra factor Kα > 1
reflecting a positive correlation among the sample observations.
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n
Simulations results for α = 1, σ = 1, X0 = 1, θ = 0.
Left: variance of the sample mean, simulated and theoretical values.
Center: each dot gives the coverage proportion of 104 confidence
intervals. The horizontal line: predicted coverage of 95%.
2
Right: simulated values of Var Sn decrease toward zero with n
indicating that Sn2 is a consistent estimate of the stationary variance.
16
Asymptotic normality
A strict mathematical justification of this formula requires two
results which will be addressed elsewhere.
On one hand, have we have to prove that
2
Var Sn → 0, n → ∞.
On the other hand, the following Central Limit Theorem has to be
verified
Conjecture. As n → ∞, the standardized sample mean
−1/2
Zn = (X̄n − E X̄n )(Var X̄n )
is asymptotically normally distributed with parameters (0,1).
17
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Left: p–value of Shapiro–Wilk test for normality of Zn .
Right: A more thorough 0.975 level quantile check of normality for
Zn is given by a hybrid Monte–Carlo–numerical approach.
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If α = 0 and σ = 1, then E X̄n = X0 with Var X̄n → 2 as n → ∞.
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X0 estimate for the YMB-model
For YBM-model X̄n is an unbiased but not consistent estimate X0 .
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0.975 quantile
2.0
2.10
Assuming σ is known one can think of an approximate 95% CI for X0
σ √
X̄n ± qn √
2n − an
n
●
0
50
100
true
simulated
●
150
0
Number of tips
50
simulated quantile of standarized sample
t with n−1 df quantile
100
n
20
150
200
●
1.2
●
2.0
1.4
X0 estimate for the YMB-model
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0.2
0.5
0.4
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2.025
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zoom
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2.015
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50
100
150
200
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50
100
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150
200
n
0.0
0
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0.0
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2.005
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0.8
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97.5% quantile
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1.5
●
1.0
●
●
97.5% quantile
1.0
●
Excess kurtosis of Xn
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
0
50
100
150
200
n
n
Asymptotic distribution of X̄n for the Brownian motion model with
X0 = 0 and σ = 1 is symmetric but not normal.
21
Interspecies correlation
From
σ02 (1 − E e−2ατ )
E [e−2αT ]) + (X0 − θ)2
ρn = 1 −
σ02 (1 −
we obtain putting δ =
ρn = 1 −
Var [e−αT ]
|X0 −θ|
σ
2α(n − 1) + (n + 1)((2α + 1)bn,2α − 1)
(n − 1)(2α − 1)(1 + (2αδ 2 − 1)bn,2α ) − 2αδ 2 b2n,α )
As n → ∞

−1

2(ln
n)
,




 c n−2α ,
α,δ
ρn ∼
−1

2n
ln n,




 2 n−1 ,
2α−1
if α = 0
if 0 < α < 0.5
if α = 0.5
if α > 0.5
22
ρn =
E [T − τ ]
E [T ]
Interspecies correlation
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Simulation results for n = 30, σ = 1, X0 = 0, θ = 1. Thick lines fitted
to simulated data are indistinguishable from the predicted (dashed)
lines y = ρ30 x + (1 − b30,α )(1 − ρ30 ). Left: α = 0.05. Right: α = 5.
23
Related projects
Ongoing project 1 (with K.Bartoszek)
YOU-model with jumps at speciation events.
Ongoing project 2 (with K.Bartoszek, G.Jones, B.Oxelman):
suppose a tetraploid comes as a hybrid of a pair in a set of n diploids.
The time to the hybridization event is asymptotically exponential.
Assumption 1: conditioned Yule tree for n diploid species
with parameter λ. Assumption 2: Poissonian clock for
hybridization events with rate β per a pair of diploids.
Acknowledgement. This work was supported by the Swedish Research Council
grant 621-2010-5623.
24
References
1. Sagitov S. and Bartoszek K.
Interspecies correlation for neutrally evolving traits. J. Theor.
Biol. 309 (2012) 11-19
2. Bartoszek K. and Sagitov S.
A phylogenetic confidence interval for the optimal trait value.
Submitted to J. Math. Biol. http://arxiv.org/abs/1207.6488
3. Graham J., Sagitov S., and Oxelman B.
Statistical inference of allopolyploid species networks in the
presence of incomplete lineage sorting.
Submitted to Syst. Biol. http://arxiv.org/abs/1208.3606
4. Bartoszek K., Jones G., Oxelman B., and Sagitov S.
Time to a single hybridization event in a group of species with
unknown ancestral history. In progress.
25