Microarray Statistics

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DETECTING RECOMBINATION IN THE FELSENSTEIN ZONE
Alexander V. Mantzaris Wolfgang Lehrach Dirk
Husmeier
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Inference of a phylogenetic tree
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Nucleotide substitution model
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Maximum likelihood tree
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Probabilistic phylogenetic model
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Steps of recombination
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Steps of recombination
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Steps of recombination
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Steps of recombination
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Steps of recombination
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Steps of recombination
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Sequence alignment changes from recombination
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Sequence alignment changes from recombination
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Multiple change point model (MCP)

• Suchard 2003 the MCP uses Reversible Jump MCMC
  (RJMCMC) to place break points modelling the
  change in topology, rate of evolution and
  substitution parameters

s-topology r-rate t-substitution parameters
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Dual multiple change point model (dual MCP)

• Decoupling the topology from the substitution
  parameters removes the prior belief that these
  changes must coincide Minin 2005

s-topology r-rate t-substitution parameters
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Analytic integration of branch lengths

• Following Suchard 2003 both the MCP and Dual MCP
  models analytically integrate the branch lengths
  from the model
• The approximation assumes independence of the
  branch lengths to make the model more tractable

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Introduction of the prior into the model
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Felsenstein parameters

• Branch lengths are in groups d2 and d3 which all
  share the same length

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Data

• Sequence alignment data produced synthetically
• seq-gen Rambaut, A. and Grassly, N. C. (1996)

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Dualbrothers
d2-0.25 d3-0.25
Posterior probabilities per topology vs. position
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Dualbrothers
d2-0.45 d3-0.65
Posterior probabilities per topology vs. position
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Dualbrothers long branch attraction

• d2-0.05 d3-0.85

Posterior probabilities per topology vs. position
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Dualbrothers long branch attraction

• d2-0.15 d3-0.85

Posterior probabilities per topology vs. position
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Phylo-Factorial Hidden Markov model

• Two independent chains for rate and topology
  inference
• Husmeier, Bioinformatics 2005

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Sampling in phylo-FHMM
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FHMM

• d2-0.25 d3-0.25

Posterior probabilities per topology vs. position
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FHMM
d2-0.45 d3-0.65
Posterior probabilities per topology vs. position
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FHMM long branch attraction

• d2-0.05 d3-0.85

Posterior probabilities per topology vs. position
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FHMM long branch attraction

• d2-0.15 d3-0.85

Posterior probabilities per topology vs. position
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Long branch attraction

• Long branches can produce a statistically
  consistent wrong tree that converges towards a
  parsimonious tree
• Parsimony in a likelihood method

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Felsenstein zone

• Joseph Felsenstein, Systematic Zoology 1978

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Recpars on synthetic data set

• Viewing most parsimonious tree with no
  recombination

score of correct topology-White low score of
correct topology-Black
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Model integrating out the branch lengths

• The data is dependent on the topology and rate
  with a hyper-parameter for the rate distribution

Model 1
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Removing the branch length independence assumption

• Model incorporates the branch length sampling

Model 2
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Contrast to model without analytic integration

• This new model incorporates branch lengths

Model 1
Model 2
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Intra-model explorationAnnealed Importance
Sampling (AIS)
Objective P(DM)
Inter-model exploration Simultaneous MCMC
sampling of model topologies and parameters
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Annealed importance sampling

• Radford Neal 1998, based on importance sampling
  with annealing transitions

The first sample is produced from the prior
-prior
-posterior
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Annealed importance sampling
Each annealing transition produces importance
weights
The weights allow computing the marginal
likelihood with respect to the sampling
distribution (prior here).
The ratio of the marginal likelihood with the
prior normalisation factor
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AIS analytic integration

• 200 samples, 5 MCMC steps, 5 transitions

Model 1
High posterior probability of correct
topology-White low posterior probability of
correct topology-Black
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Annealed Importance Sampler

• Average posterior probabilities of correct
  topology over 8 independent simulations

Model 2
High posterior probability of correct
topology-White low posterior probability of
correct topology-Black
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Model 1/Model 2 comparison with AIS
Model 2
Model 1
Model 2
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Remarks on Annealed importance sampling

• Sporadic wrong inferences deep in the Felsenstein
  zone assumed to be due to the sampler
  occasionally missing the narrow high density
  regions as noted in Neal 1998
• Longer simulations alleviate this weakness, which
  is unsuitable for large batch jobs
• The method is particularly good where the large
  density region is less narrow

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Intra-model explorationAnnealed Importance
Sampling (AIS)
Objective P(DM)
Inter-model exploration Simultaneous MCMC
sampling of model topologies and parameters
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MCMC procedure without branch lengths

• The peeling algorithm is used to calculate the
  probability of the data

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Region of long branch attraction
Model 1

• Long branch attraction where expected from the
  theory

High posterior probability of correct
topology-White low posterior probability of
correct topology-Black
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MCMC procedure including branch lengths

• The peeling algorithm is used to calculate the
  probability of the data

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MCMC procedure including branch lengths

• Symmetric proposal distributions centred on the
  current value-Cauchy used

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Convergence Diagnostics

• Gelman-Rubin tests performed as a heuristic test
  for convergence
• PSR factor below 1.3 required 15K iterations for
  burn-in and 5K sampling

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Avoiding the Felsenstein zone
Model 2

• Metropolis-Hastings with Cauchy distribution
  escapes long branch attraction

High posterior probability of correct
topology-White low posterior probability of
correct topology-Black
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Integration of branch length sampling into
phylo-FHMM
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Contrast to model without analytic integration

• This new model incorporates branch lengths

Model 1
Model 2
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Felsenstein parameters

• Branch lengths are in groups d2 and d3 which all
  share the same length

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FHMM long branch attraction

• d2-0.15 d3-0.85

Posterior probabilities per topology vs. position
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Integration of branch length sampling into
phylo-FHMM

• Previous incorrect inference corrected D2-0.15
  D3-0.85

Posterior probabilities per topology vs. position
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Mosaic sequence alignment Model2

• D2-0.25 D3-0.25 flanking 500bp
• D2-0.15 D3-0.85 center

Model 2
Model 1
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Conclusion

• Long branch attraction is evident in the region
  of the Felsenstein zone for models using the
  independence assumption for branch lengths as in
  Suchard 2003
• The more complex model escapes false positives
• Future work on real biological DNA sequence
  alignments

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Appendix
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Reliability of sampling scheme

• For problems outside the Felsenstein zone the
  scheme is reliable
• Within the Felsenstein zone where the region of
  high density is narrow there can be occasions
  with wrong inference from a lack of convergence
• Sufficient iterations are needed in any case