Finding Clusters in Phylogenetic Trees: A Special Type of Cluster Analysis
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1 Finding lusters in Phylogenetic Trees: Special Type of luster nalysis Why try to identify clusters in phylogenetic trees? xample: origin of HIV. NUMR: Why are there so many distinct clusters? LUR04-7 SYNHRONY: Was the onset of diversification synchronized?
2 xample Observe: main features of HIV-, type M - pprox. 0 distinct subtypes - Subtypes are approx. equidistant ( sunburst ) Question: ould these features have arisen naturally? pproach: - quantitative comparison to simulated frican epidemic. Simulation details are in the models/tools: - coalescent theory, phylogenetic tree estimation, - estimate the number of subtypes, and - classical statistics: are the main features outliers with respect to our forward model? FOUS: stimate the number of subtypes
3 This talk to focus on: To choose groups, consider: Model-based clustering (Raftery et al: mclust in S+) Max likelihood + bootstrap (State of art: PHYLIP, other) Markov hain Monte arlo (M)
4 omplicated Genetic ata Structure 94Y GGTGTGGGG... 90M.4 TGGGTGGGG... xample sequence identifier: 94Y.04. : subtype 94: isolation year Y: country of origin 04: isolate : clone number nsure: global coverage, include all known subtypes widest possible span of isolation times more than one region of genome void: more than clone from same isolate Issues: genealogy implies correlation; evolution model
5 istance measures/micro evolutionary models P ij (t) = 4-by-4 transition prob. matrix P( -> in time t) = P (t), etc. For some P matrices, can define an evolutionary distance between taxa x and y each with N nucleotides (must correct for multiple substitutions) n n n G n T - aπ bπ G cπ T NF xy = n n n G n T aπ - dπ G eπ T bπ dπ - fπ T cπ eπ fπ G - n G n G n GG n Q GT ij/µ = P = e Qt n T n T n TG n TT GTR: π i P ij = π j P has 8 free parameters. ji. ommon models are special cases with fewer parameters. Use NF xy to estimate parameters. J: P ij (t) = /4 + /4e -µt for i = j, and /4 - /4e -µt for i!= j K: P ij (t) = /4 + /4e -µt + / e -µt (κ+)/, for i = j, etc. where κ is transition/transversion ratio
6 Number of subtypes: Model-based clustering nv Gag x x X x x W No. subtypes No. subtypes
7 Simulated data: 4 macro growth rates (a) N = N 0 e rt (b) N = N 0 (c) N = N 0, then N= N 0 e rt (d) N is quadratic from970 to 990
8 xample Real Trees J G H H G J nv F The ML + bootstrap approach suggests 7 clusters (subtypes) in the 9 env sequences and clusters (subtypes) in the 88 gag sequences. The data is available at hiv-web.lanl.gov and accession numbers are available upon request. NOT:, are similar and H, J are rare (omitted in this analysis) F K Gag
9 Model-based clustering as in mclust - pproximate ayes method to choose the no. of groups G. First assume: G is known and data is n cases of p-dim observations x = (x, x,, x n ) with probability density f k (x;θ) for observations from group k. Let γ = (γ,..,γ n ) be the group labels. hoose (θ,γ) to maximize L(θ;γ) = Π i f γi (x i ;θ) If f is MVN(µ k,σ k ), get a sum of squares criterion, with variations depending on assumptions on Σ k. R (99) use hierarchical agglomeration and iterative reallocation to maximize the classification likelihood: n L(x θν, ) = φ ( xi µ ν, Σν ), i= i i where φ i is MVN anfield and Raftery, iometrics 99
10 Model-based clustering as in mclust R approach: use the spectral decomposition T k k k k k Σ =λ where λ k, k, k control the volume, shape, and orientation of group k Next, to estimate p(g = r x), approximate the distribution of the ayes factor p(x G = r)/p(x G = s). llow: a noise component for new cluster cases and use heuristic method to address failure of a regularity condition in the clustering context.
11 Simulated xample x x I I VI 4 VVV V VV number of clusters valuation of emclust for a simulated data set of 0 observations from each of clusters (labeled,, in top plot) with true model VV denoting that the volume varies (V) among clusters, the shape does not vary ( for equal) among clusters, and the orientation varies (V) among clusters (model ). The I correctly chooses clusters but chooses VVV rather than the correct VV.
12 mclust suggests subtypes (tends to merge and ) G G G G G G G G x FF F F F F F F F G G G G G G G G F G F F F F F F x I I VI 4 VVV V VV 0 0 number of clusters nv ata. (Top) Hierarchical lustering; (Middle) Principle oordinate plot; (ottom) Results of mclust.
13 MM via M On different data with fewer taxa: ompare MM to ML + bootstrap in case where groups chosen in advance Probability via MM Probability via MM Probability via ML+ootstrap (c) Influenza, H gene, 9,9,94, or 9 vs 9 groups Probability via ML+ootstrap (d) Influenza, NP gene, H, S, groups
14 Summary Present method to choose the number of groups via ML + bootstrap or MM: trial and error. Usually: human eye studies tree, selects groups, then ML + bootstrap on specified groups. Similar with MM Model-based clustering offers automatic way to choose groups, but relies on pair-wise distances (less efficient than likelihood). FUTUR: consider how to automate (without human eye) cluster choices in ML + bootstrap or MM (or any other method such as weighted parsimony + bootstrap) Increasing the number of taxa: MM and ML are very slow, so currently limited to few hundred taxa onsider: identify groups, then assign new taxa to existing groups.
15 References anfield, J., & Raftery,. (99). Model-based gaussian and non-gaussian clustering. iometrics. 49, urr T., Myers G., & Hyman J. (00). The Origin of IS arwinian or Lamarkian?, Phil. Trans. R. Soc. Lond.., urr, T., Skourikhine,.N., Macken,., & runo, W. (999). onfidence measures for evolutionary trees: applications to molecular epidemiology. Proc. of the 999 I Inter. onference on Information, Intelligence and Systems, urr T., oak J., Gattiker, J., & Stanbro, W. (00a). ssessing confidence in phylogenetic trees: bootstrap versus Markov hain Monte arlo, Mathematical and ngineering Methods in Medicine and iological Sciences., urr, T., Gattiker, J., & Laerge, G. (00b). Genetic subtyping using cluster analysis, Special Interest Group on Knowledge iscovery and ata Mining xplorations., -4.
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