Transcript Powerpoint Presentation

Structural EM for Phylogentic
Inference
Nir Friedman
Matan Ninio
Computer Science & Engineering
Hebrew University
Computer Science & Engineering
Hebrew University
Itsik Pe’er
Tal Pupko
Computer Science
Tel-Aviv University
Inst. of Statistical Mathematics
Tokyo
Copyright N. Friedman, M. Ninio. I. Pe’er, and T. Pupko. 2001
RECOMB, April 2001
Introduction
 Phylogentic
inference:
“reconstruction of the tree of evolution based on
DNA/Protein sequences of current day species”
 Maximum Likelihood inference
 Model evolution as a stochastic process
 Use likelihood of observed sequences to
evaluate different trees
 Computational task
 Construct the maximum likelihood tree
 We describe a new procedure that use a variant of
EM to efficiently learn better trees
Copyright N. Friedman, M. Ninio. I. Pe’er, and T. Pupko. 2001
RECOMB, April 2001
Evolution of a Character over Time
 Probability
of change: Pab(t)
Assumptions:
 Lack of memory:

Pa c (t  t ')   Pa b (t )Pb c (t ')
b
 Reversibility:

A
C
G
T
Exist stationary probabilities
{Pa} s.t. Pa Pa b (t )  PbPb a (t )
Copyright N. Friedman, M. Ninio. I. Pe’er, and T. Pupko. 2001
RECOMB, April 2001
Phylogenetic Tree
branch
internal node
leaf
 Topology:


bifurcating
Leaves - 1…N
Internal nodes N+1…2N-2
t = {ti,j} for each branch (i,j)
 Phylogenetic tree = (Topology, Lengths)
 Lengths
Copyright N. Friedman, M. Ninio. I. Pe’er, and T. Pupko. 2001
RECOMB, April 2001
Joint Distribution
Random variables X1... X2N-2 for all nodes.
 x[1...N] - Observed values of X[1...N].
Joint distribution:
P (x[1,2N 2] |T ,t )   pxi
i
pxi x j (ti , j )

px j
(i , j )T
Marginal distribution:
P (x[1,,N ] |T ,t ) 
 p 
x[ N 1,...,2 N 2 ]
i
xi
(i , j )T
pxi x j (ti , j )
px j
 Computation
of marginal distribution:
by dynamic programming
Copyright N. Friedman, M. Ninio. I. Pe’er, and T. Pupko. 2001
RECOMB, April 2001
Maximum Likelihood Reconstruction
Observed data: (D )
 N sequences of length M
 Each position: an independent sample from the
marginal distribution over N current day taxa
Likelihood:
 Given a tree (T,t) :
l (T ,t : D )  log P (D |T ,t )
  log P (x[m1,,N ] |T ,t )
M
Goal:
 Find a tree (T,t) that maximizes l(T,t:D) .
Copyright N. Friedman, M. Ninio. I. Pe’er, and T. Pupko. 2001
RECOMB, April 2001
Current Approaches
 Perform
search over possible topologies
T1
T2
T3
Parameter space
Parametric
optimization
(EM)
Local Maxima
T4
Copyright N. Friedman, M. Ninio. I. Pe’er, and T. Pupko. 2001
Tn
RECOMB, April 2001
Computational Problem
 Such
procedures are computationally expansive!
 Computation of optimal parameters, per candidate,
requires non-trivial optimization step.
 Spend non-negligible computation on a candidate,
even if it is a low scoring one.
 In practice, such learning procedures can only
consider small sets of candidate structures
Copyright N. Friedman, M. Ninio. I. Pe’er, and T. Pupko. 2001
RECOMB, April 2001
Structural EM
Idea:
Use parameters found for current topology to help
evaluate new topologies.
Outline:
 Perform search in (T, t) space.
 Use EM-like iterations:
 E-step: use current solution to compute
expected sufficient statistics for all topologies
 M-step: select new topology based on these
expected sufficient statistics
Copyright N. Friedman, M. Ninio. I. Pe’er, and T. Pupko. 2001
RECOMB, April 2001
The Complete-Data Scenario
Suppose we observe H, the ancestral sequences.
l complete T ,t : D, H    log P (x 1...2N 2m  |T ,t )
m
  log pxi m  
i
 log
m
 const 
(i , j )T
 F (t
(i , j )T
i ,j
F is a linear function of Si,j
Si,j is a matrix of # of
co-occurrences for each
pair (a,b) in the taxa i,j
Find: topology T that maximizes
w
(i , j )T
Copyright N. Friedman, M. Ninio. I. Pe’er, and T. Pupko. 2001
px j m 
m
, Si , j )
Define: wi , j  maxt F (ti , j , Si , j )
i ,j
pxi m x j m  (ti , j )
i ,j
RECOMB, April 2001
Expected Likelihood
with a tree (T0,t0)
 Compute
 Start
E [S(i , j ) (a , b )]   P (Xi m  a , X jm  b | x[m1,,N ] ,T 0 ,t 0 )
m
Formal justification:
complete
0
0
Q
(
T
,
t
)

E
[
l
(
D
,
H
:
T
,
t
)
|
T
,
t
]
 Define:

F (t
(i , j )T
i,j
, E [Si , j ])  const
0
0
0
0
Q
(
T
,
t
)

Q
(
T
,
t
)

l
(
D
:
T
,
t
)

l
(
D
:
T
,
t
)
Theorem:
Consequence: improvement in expected score 
Copyright N. Friedman, M. Ninio. I. Pe’er, and T. Pupko. 2001
improvement in likelihood
RECOMB, April 2001
Algorithm Outline
0
0
Compute: E [S(i , j ) (a , b ) | D,T ,t ]
Weights: wi , j  maxt F (t , E [Si , j ])
Original Tree (T0,t0)
Unlike standard EM for trees,
we compute all possible pairwise
statistics
Time: O(N2M)
Copyright N. Friedman, M. Ninio. I. Pe’er, and T. Pupko. 2001
RECOMB, April 2001
Algorithm Outline
0
0
Compute: E [S(i , j ) (a , b ) | D,T ,t ]
Weights: wi , j  maxt F (t , E [Si , j ])
Find: T ' arg maxT
w
(i , j )T
i,j
Pairwise weights
This stage also computes the
branch length for each pair (i,j)
Copyright N. Friedman, M. Ninio. I. Pe’er, and T. Pupko. 2001
RECOMB, April 2001
Algorithm Outline
0
0
Compute: E [S(i , j ) (a , b ) | D,T ,t ]
Weights: wi , j  maxt F (t , E [Si , j ])
Find: T ' arg maxT
w
(i , j )T
i,j
Construct bifurcation T1
Max. Spanning Tree
Fast greedy procedure to find tree
Copyright N. Friedman, M. Ninio. I. Pe’er, and T. Pupko. 2001
By construction:
Q(T’,t’)  Q(T0,t0)
Thus,
April )
l(T’,t’)  RECOMB,
l(T0,t
0 2001
Algorithm Outline
0
0
Compute: E [S(i , j ) (a , b ) | D,T ,t ]
Weights: wi , j  maxt F (t , E [Si , j ])
Find: T ' arg maxT
w
(i , j )T
i,j
Construct bifurcation T1
Fix Tree
Remove redundant nodes
Add nodes to break large degree
This operation preserves likelihood
l(T1,t’) =l(T’,t’)  l(T0,t0)
Copyright N. Friedman, M. Ninio. I. Pe’er, and T. Pupko. 2001
RECOMB, April 2001
Algorithm Outline
0
0
Compute: E [S(i , j ) (a , b ) | D,T ,t ]
Weights: wi , j  maxt F (t , E [Si , j ])
Find: T ' arg maxT
w
(i , j )T
i,j
Construct bifurcation T1
Thm: l(T1,t1)  l(T0,t0)
New Tree
These steps are then repeated until convergence
Copyright N. Friedman, M. Ninio. I. Pe’er, and T. Pupko. 2001
RECOMB, April 2001
Evaluation
 Comparison
to MOLPHY (PROTML):
Evaluation on
 Synthetic data sets
 Sampled from a tree we generated
 Allows us to control # taxa and #positions
 Can compare to “true” generating tree
 Real-life
data
Copyright N. Friedman, M. Ninio. I. Pe’er, and T. Pupko. 2001
RECOMB, April 2001
Number of Positions (48 taxa)
-2
-4
-6
-8
-10
-12
10
Log-probability (per position)
relative to original
0
2
SEMPHY
MOLPHY
Original
Original
(no training)
1.5
1
0.5
0
100
1000
-0.5 on test data
Performance
10
100
relative to original model
Number of Positions
Copyright N. Friedman, M. Ninio. I. Pe’er, and T. Pupko. 2001
RECOMB, April 2001
1000
Number of taxa (100 pos)
0
-0.4
-0.6
-0.8
-1
-1.2
-1.4
-1.6
-1.8
0
20
Log-probability (per position)
relative to original
-0.2
SEMPHY
MOLPHY
Original
Original
(no training)
1
0.8
0.6
0.4
0.2
0
40-0.2
60
80
100
Performance
-0.4 on test data
0
20
40
60
80
relative to original
model
Number of Taxa
Copyright N. Friedman, M. Ninio. I. Pe’er, and T. Pupko. 2001
RECOMB, April 2001
100
Run times
2000
SEMPHY
MOLPHY
1800
Time in seconds
1600
1400
1200
1000
800
600
400
200
0
0
10
20
30
40
50
60
70
80
90
100
Number of taxa
Copyright N. Friedman, M. Ninio. I. Pe’er, and T. Pupko. 2001
RECOMB, April 2001
Real life data
Lysozyme
Mitochondrial
# taxa
43
34
# pos
122
3,578
-2,916.2
-74,227.9
-2,892.1
-70,533.5
0.19
1.03
MOLPHY
likelihood
SEMPHY
likelihood
Diff per
position
Copyright N. Friedman, M. Ninio. I. Pe’er, and T. Pupko. 2001
RECOMB, April 2001
Discussion
 New
algorithmic approach for optimizing the
likelihood of models
 SEMPHY: an implementation for protein sequences
 Incorporates standard models for Pab(t)
 Early results shows that it outperforms current
programs for ML reconstruction
 In terms of running time & solution quality
Work in progress
 Escaping “local” maxima
 More elaborate models of evolution
 Variable rate
 Co-evolution
Copyright N. Friedman, M. Ninio. I. Pe’er, and T. Pupko. 2001
RECOMB, April 2001
Preliminary Results:
Annealed Structural EM
log-likelihood relative to
optimized original
Original
SEMPHY
Anneal SEMPHY
0.02
0
-0.02
-0.04
-0.06
-0.08
-0.1
-0.12
10
20
30
40
50
60
70
80
90 100
Number of Taxa
Copyright N. Friedman, M. Ninio. I. Pe’er, and T. Pupko. 2001
RECOMB, April 2001
THE END
Copyright N. Friedman, M. Ninio. I. Pe’er, and T. Pupko. 2001
RECOMB, April 2001