On the identifiability of two tree mixtures for group-based models - - PowerPoint PPT Presentation

On the identifiability of two tree mixtures for group-based models

• E. Allman1
• S. Petrovi´

c2

• J. Rhodes1
• S. Sullivant3

1 University of Fairbanks, Alaska 2 University of Illinois Chicago 3 North Carolina State University

Phylomania 2010 – Hobart, Tasmania November 2010

On the identifiability of two tree mixtures for group-based models 1/33

Today’s talk:

◮ identifiability of 2-tree mixture models ◮ work dates from 2009 and before ◮ focus today on algebraic techniques (technical at times)

On the identifiability of two tree mixtures for group-based models 2/33

Background

Interest sparked by papers/conversations

◮ Kolaczkowski and Thornton: 2004 Nature ◮ Mossel and Vigoda; Ronquist et al: 2005, 2006, Science ◮ ˇ

Stefankoviˇ c and Vigoda: 2007, JCB, Phylogeny of Mixture Models:

Robustness of Maximum Likelihood and Non-identifiable Distributions

◮ Matsen and Steel: 2007, Sys. Bio., Phylogenetic mixtures on a

single tree can mimic a tree of another topology

◮ Matsen, Mossel, and Steel: 2008, BMB, Mixed-up trees: the

structure of phylogenetic mixtures

◮ Junhyong Kim

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Due to incomplete lineage sorting, or other biological phenonomenon, sequence data may have evolved along two or more trees.

Species Tree

Gene 1 Gene 2

Q: Is it theoretically possible to identify the two trees giving rise to expected pattern frequencies? Q’: If so, what about the numerical parameters for these trees?

On the identifiability of two tree mixtures for group-based models 4/33

Due to incomplete lineage sorting, or other biological phenonomenon, sequence data may have evolved along two or more trees.

Species Tree

Gene 1 Gene 2

Q: Is it theoretically possible to identify the two trees giving rise to expected pattern frequencies? Q’: If so, what about the numerical parameters for these trees?

On the identifiability of two tree mixtures for group-based models 4/33

Modeling sequence evolution along a tree(s)

For a fixed tree T and a model of sequence evolution (GTR,

GTR+Γ, JC, ...), the distribution of states at the leaves of T is

a function ψT of the model’s parameters.

• Eg. GTR model on a n-taxon tree T

parameterization map ψT : ST − → ∆4n−1

• π, Q, {te}
• −

→ P = (pi1··· ,in) where pi1···in is the expected frequency of pattern i = i1 · · · in at the leaves of T.

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Mixture models

Modeling sequence evolution along two or more trees requires using a mixture model.

• Eg. Suppose T1 and T2 are two n-taxon trees, then the

distribution is a point in the image of ψT1,T2 : ST1 × ST2 × [0, 1] − → ∆4n−1

• s1, s2, w
• −

→ P = (pi1··· ,in) where P = wψT1(s1) + (1 − w)ψT2(s2) is the weighted sum of the distributions for parameter choices

• n T1 and T2.

On the identifiability of two tree mixtures for group-based models 6/33

Group-based models

Today: focus on group-based models Cavender-Farris-Neyman (CFN), Jukes-Cantor (JC), Kimura 2-Parameter (K2P), Kimura 3-Parameter (K3P) These models, as well as GM, have an algebraic structure useful for analysis.

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Model parameters π, {Me} on tree T

S1 S2 M1 M2 π S3 M3 M4

pijk =

4

• l=1

4

• m=1

πlM1(l, m)M2(m, i)M3(m, j)M4(l, k) lead to a polynomial parameterization map ψT. Thus, any mixture distribution PT1,T2 ∈ ψT1,T2 is also parameterized by polynomials.

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Mixture varieties

Extending the parameterization to complex parameters, define VT1 ∗ VT2 = Im ψT1,T2, the phylogenetic mixture variety. (Point: This allows ideas from algebraic geometry to be used.)

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Algebraic geometry reminders

◮ Fundamental correspondence:

Geometry ← → Algebra V ← → IV Corresponding to any phylogenetic variety V is its ideal IV of phylogenetic invariants, the ideal of polynomials f in the pattern frequencies pi so that f (P) = 0 for any P ∈ V .

◮ Inclusion reversing correspondence:

V1 ⊆ V2 ⇐ ⇒ IV2 ⊆ IV1

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More notation

For stochastic parameter choices, denote the collection of joint distributions by MT1 ∗ MT2. Note that MT1 ∗ MT2 VT1 ∗ VT2. Though the varieties are used for proofs because of their algebraic structure (dim, good intersection properties, etc.), all results today hold for the stochastic distributions.

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Monomial parameterization

Hendy, Penny, Sz´ ekely, Erd¨

• s, Evans, Speed, Sturmfels, Sullivant:

Group-based models can be diagonalized by means of the discrete Fourier transform over G (Hadamard transform). In the Fourier coordinates, group-based models give rise to toric varieties. (In this setting, ψT is parameterized by monomials.) Moreover, the discrete Fourier transform is a linear change of variables, so it behaves well with respect to taking mixtures of group-based models. F(MT1) ∗ F(MT2) = F(MT1 ∗ MT2)

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Fourier coordinates

For each split A|B in T, introduce a set of Fourier parameters {aA|B

g

: g ∈ G}.

Theorem (Hendy-Penny)

In the Fourier coordinates, a group-based phylogenetic model is given parameterically by: qg1,...,gn =

A|B∈Σ(T) aA|B P

a∈A ga

if g1 + · · · + gn = 0 if g1 + · · · + gn = 0 ‘Coordinates’ in this parameterization are called q-coordinates.

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Fourier coordinates

For JC, K2P, we take G = Z2 × Z2 = {A, C, G, T}.

◮ For K2P model, we have aA|B G

= aA|B

T

for all A|B

◮ For JC model, we have aA|B C

= aA|B

G

= aA|B

T

for all A|B.

On the identifiability of two tree mixtures for group-based models 14/33

Tree parameter identifiability (stochastic version)

Definition

The tree parameters T1, . . . , Tk in a k-class phylogenetic mixture model are identifiable, if for all P ∈ MT1 ∗ · · · ∗ MTk there does not exist another set of k trees T ′

1, . . . , T ′ k such that

P ∈ MT ′

1 ∗ · · · ∗ MT ′ k. On the identifiability of two tree mixtures for group-based models 15/33

Tree parameter identifiability (geometric version)

VT1*VT2 VT3*VTi

Definition

The tree parameters in a k-class phylogenetic mixture model are generically identifiable if for all non-equal multisets {T1, . . . , Tk}, and {T ′

1, . . . , T ′ k},

dim(VT1 ∗ · · · ∗ VTk ∩ VT ′

1 ∗ · · · ∗ VT ′ k) < dim(VT1 ∗ · · · ∗ VTk). On the identifiability of two tree mixtures for group-based models 16/33

Generic identifiability of tree parameters

An immediate consequence of the geometric definition: dim(VT1 ∗ VT2 ∩ VT ′

1 ∗ VT ′ 2) < dim(VT1 ∗ VT2)

is that tree parameters are generically identifiable for stochastic parameter choices too. That is, the trees giving rise to MT1 ∗ MT2 are identifiable, except on a non-generic set E of stochastic parameters (s1, s2, π) of Lebesque measure zero where ψT1,T2(s1, s2, π) = ψT ′

1,T ′ 2(s′

1, s′ 2, π′).

(E is the set of bad parameters.) On the identifiability of two tree mixtures for group-based models 17/33

Algebraic methods for proofs

Use

◮ dimension counts for phylogenetic varieties ◮ all phylogenetic mixture varieties are irreducible, since they

are parameterized

◮ two irreducible varieties of the same dimension either

coincide or intersect in a sub-variety of lower dimension Analogy with linear spaces. = ⇒ if two phylogenetic varieties are distinct, then parameters will be generically identifiable

◮ two varieties V1 and V2 are distinct if IV1 = IV2

and V1 V2 if there exists an invariant f2 ∈ IV2 \ IV1

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VT1*VT2 VT3*VTi

IVT1∗VT2 = IVT3∗VT4

VT1* VT2 VT3

∃f ∈ IVT1∗VT2 \ IVT3

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Algebraic methods for proofs

Use

◮ group-based models (JC and K2P) have linear invariants

which can be used to construct invariants for 2-tree mixtures

◮ computational algebra packages like Singular

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Main theorem (tree parameters)

Theorem

The tree parameters of the 2-tree mixture model MT1 ∗ MT2 are generically identifiable under the Jukes-Cantor and Kimura 2-parameter models if T1, T2 are binary with n ≥ 4 leaves. Strategy: Prove theorem for quartets n = 4, then lift to larger trees.

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Identifiability of quartet trees

Proposition

Let T1 = 12|34, T2 = 14|23, T3 = 13|24. Then ℓ(q) = qGGGG + qGTGT − qGGTT − qGTTG satisfies ℓ(q) = 0 for all q ∈ MT1 ∗ MT2, but ℓ(q) = 0 for some q ∈ MT3 for the JC and K2P models.

Corollary

Generic identifiabiliity of tree parameters holds for n = 4. a few details ....

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If q = wq1 + (1 − w)q2, then since ℓ is linear = ⇒ ℓ(q) = ℓ(wq1 + (1 − w)q2) = wℓ(q1) + (1 − w)ℓ(q2)

qGGGG qGTGT qGGTT qGTTG

+

• T1

1 2 4 3 1 2 4 3 1 2 4 3 1 2 4 3 +

• T2

1 4 3 2 1 4 3 2 1 4 3 2 1 4 3 2 +

• T3

1 3 4 2 1 3 4 2 1 3 4 2 1 3 4 2 +

• = 0

= 0 = 0

l(q) =

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Generic identifiabiliity of tree parameters holds for n = 4.

Case: two different trees in mixture

◮ The linear invariant ℓ ∈ IVT1 ∗VT2 \ IVT3 . Thus, VT3 VT1 ∗ VT2. ◮ Since VT3 ⊂ VT3 ∗ VTi , we have IVT3 ∗VTi ⊂ IVT3 and thus,

ℓ ∈ IVT1 ∗VT2 \ IVT3 ∗VTi .

◮ Since VT1 ∗ VT2 and VT3 ∗ VTi are irreducible of the same

dimension with different ideals, they are distinct.

VT1* VT2 VT3

∃ℓ ∈ IVT1∗VT2 \ IVT3

VT1*VT2 VT3*VTi

IVT1∗VT2 = IVT3∗VTi

On the identifiability of two tree mixtures for group-based models 24/33

Generic identifiability of continuous parameters

Definition

The continuous parameters of a 2-tree mixture model are generically identifiable if for generic choices of (s1, s2, w), ψT1,T2(s1, s2, w) = ψT1,T2(s′

1, s′ 2, w′)

implies (s1, s2, w) = (s′

1, s′ 2, w′)

• r, in the case where T1 = T2, that

(s1, s2, w) = (s′

2, s′ 1, 1 − w′).

On the identifiability of two tree mixtures for group-based models 25/33

Main theorem (continuous parameters)

Theorem*

The continuous parameters of the 2-tree mixture model MT1 ∗ MT2 are generically identifiable under the Jukes-Cantor and Kimura 2-parameter models if T1, T2 are binary with n ≥ 5 leaves.

Definition

Theorem* means that the result holds with high probability. Note: If T1 = T2, no ∗ needed.

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Theorem* ?

Proposition:

Let ψ : Cd → Cm be a polynomial (or rational)

• map. Then for some k ∈ {1, 2, 3, . . . , ∞},

ψ is generically k − to − 1. That is, except for some exceptional set E of parameter space, the map will be k − to − 1. Moreover, E is of Lebesgue measure 0 within the parameter space. For example, ψ : C → C given by ψ(z) = z2 and ψ(z) = 1 z2 Taking E = {0}, then k = 2.

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Theorem* ?

Proposition:

Let ψ : Cd → Cm be a polynomial (or rational)

• map. Then for some k ∈ {1, 2, 3, . . . , ∞},

ψ is generically k − to − 1. That is, except for some exceptional set E of parameter space, the map will be k − to − 1. Moreover, E is of Lebesgue measure 0 within the parameter space. For example, ψ : C → C given by ψ(z) = z2 and ψ(z) = 1 z2 Taking E = {0}, then k = 2.

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Polynomial maps are generically k − to − 1

• 1. To prove* the Theorem* for a particular tree, repeatedly

generate random rational parameter choices θ and then symbolically solve the simultaneous polynomial system ψ(t) = ψ(θ) and hope for one solution.

(One solution means that parameters are ‘probably’ generically identifiable.)

• 2. We check this using software Singular, for JC and K2P
• n 4 and 5-taxon trees.
• 3. Recovering parameters uniquely on quartets =

⇒ recover parameters on arbitrarily sized trees.

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Why n = 5 in Theorem*?

Proposition*

For T a 4-taxon tree under the Jukes-Cantor model, the continuous parameters in MT ∗ MT are not generically

• identifiable. The map ψT,T is generically 6-to-1 (up to label

swapping). For biologically relevant parameters, we observed between 1 and 4 biologically relevant preimages.

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Another Mathematical Surprise

1 2 3 4 5 1 2 4 5 3 1 4 2 3

5 T1 T2 T3

Theorem

For the Jukes-Cantor model VT2 ⊆ VT1 ∗ VT3. Q: Is M2 ⊆ M1 ∗ M3? A*: No, unless you allow 0 and/or infinite branch lengths in T1 and T3.

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Another Mathematical Surprise

1 2 3 4 5 1 2 4 5 3 1 4 2 3

5 T1 T2 T3

Theorem

For the Jukes-Cantor model VT2 ⊆ VT1 ∗ VT3. Q: Is M2 ⊆ M1 ∗ M3? A*: No, unless you allow 0 and/or infinite branch lengths in T1 and T3.

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Mixtures of many trees

Recent work of Rhodes and Sullivant has advanced these results:

Theorem

◮ Under the general Markov model of sequence evolution,

the tree parameter and continuous parameters are generically identifiable for a k-class mixture on the same tree, provided k < 4⌈n/4⌉−1.

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Open problems

◮ Develop methods to remove * from Theorem* ◮ Beyond group-based models: GTR, rate variation ◮ Arbitrary k-tree mixtures

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Thank you.

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