Principal Component Analysis of Tree Topology

1. Yongdai Kim Seoul National University 2011. 6. 5 Principal Component Analysis of Tree Topology Presented by J. S. Marron, SAMSI

2. Dyck Path Challenges • Data trees not like PC projections • Branch Lengths ≥ 0 • Big flat spots

3. Brain Data: Mean – 2 σ1 PC1 Careful about values < 0

4. Interpret’n: Directions Leave Positive Orthant • (pic here)

5. Visualize Trees Important Note: Tendency Towards Large Flat Spots And Bursts of Nearby Branches

6. Dyck Path Challenges • Data trees not like PC projections • Branch Lengths ≥ 0 • Big flat spots • Alternate Approaches: • Branch Length Representation • Tree Pruning • Non-negative Matrix Factorization • Bayesian Factor Model

7. Dyck Path Challenges • Data trees not like PC projections • Branch Lengths ≥ 0 • Big flat spots • Alternate Approaches: • Branch Length Representation • Tree Pruning • Non-negative Matrix Factorization • Bayesian Factor Model Discussed by Dan

8. Dyck Path Challenges • Data trees not like PC projections • Branch Lengths ≥ 0 • Big flat spots • Alternate Approaches: • Branch Length Representation • Tree Pruning • Non-negative Matrix Factorization • Bayesian Factor Model Discussed by Dan

9. Dyck Path Challenges • Data trees not like PC projections • Branch Lengths ≥ 0 • Big flat spots • Alternate Approaches: • Branch Length Representation • Tree Pruning • Non-negative Matrix Factorization • Bayesian Factor Model Discussed by Lingsong

10. Non-neg’ve Matrix Factorization • Ideas: • Linearly Approx. Data (as in PCA) • But Stay in Positive Orthant

11. Dyck Path Challenges • Data trees not like PC projections • Branch Lengths ≥ 0 • Big flat spots • Alternate Approaches: • Branch Length Representation • Tree Pruning • Non-negative Matrix Factorization • Bayesian Factor Model

12. Contents • Introduction • Proposed Method • Bayesian Factor Model • PCA • Estimation of Projected Trees

13. Introduction • Given data , let bebranch length vectors. • Dimension p = # nodes in support (union) tree. • For tree , define tree topology vector , p-dimensional binary vector where • Goal: PCA method for

14. Visualize Trees Important Note: Tendency Towards Large Flat Spots And Bursts of Nearby Branches

15. Goal of Bayes Factor Model Model Large Flat Spots as yi = 0

16. Proposed Method • Gaussian Latent Variable Model • Est. Corr. Matrix: Bayes Factor Model • PCA on Est’ed Correlation Matrix • Interpret in Tree Space

17. Proposed Method • Gaussian Latent Variable Model

18. Proposed Method • Estimation of the correlation matrix by Bayesian factor model • Estimate and by Bayesian factor model

19. Proposed Method 3. PCA with an estimated correlation matrix • Apply the PCA to an estimated

20. Proposed Method • Estimation of projected tree • Define projected trees on PCA directions • Estimate the projected trees by MCMC algorithm

21. Bayesian Factor Model • Model • Priors • MCMC algorithm • Convergence diagnostic

22. Bayesian Factor Model • Model

23. Bayesian Factor Model • Prior • This prior has been proposed by Ghosh and Dunson(2009)

24. Bayesian Factor Model • MCMC algorithm • Notation • Step 1. generate

25. Bayesian Factor Model • MCMC algorithm • Step 2. generate where and

26. Bayesian Factor Model • MCMC algorithm • Step 3. generate where and

27. Bayesian Factor Model • MCMC algorithm • Step 4. generate where and

28. Bayesian Factor Model • MCMC algorithm • Step 5. generate

29. Bayesian Factor Model • Convergence diagnostic. • 100000 iteration of MCMC algorithm after 10000 burn-in iteration • 1000 posterior samples obtained at every 100 iteration • Trace plots, ACF (Auto Correlation functions) and histograms of the three selected s and a selected (Note ).

30. Bayesian Factor Model • Convergence diagnostic: Three s • 100000 iteration of MCMC algorithm after 10000 burn-in iteration • 1000 posterior samples obtained at every 100 iteration • Trace plot, acf functions and histograms of the three selected s

31. Bayesian Factor Model • Convergence diagnostic: A • 100000 iteration of MCMC algorithm after 10000 burn-in iteration • 1000 posterior samples obtained at every 100 iteration • Trace plot, acf functions and histograms of the three selected s(25%, 50%, 75%) and

32. PCA • Scree plot

33. Visualizing Modes of Variation

34. Visualizing Modes of Variation

35. Visualizing Modes of Variation

36. Center Point, μ

37. Approximately μ + 0.5 PC1

38. Approximately μ + 1.0 PC1

39. Approximately μ + 1.5 PC1

40. Approximately μ + 2.0 PC1

41. Center Point, μ

42. Approximately μ - 0.5 PC1

43. Approximately μ - 1.0 PC1

44. Approximately μ - 1.5 PC1

45. Approximately μ - 2.0 PC1

46. Visualizing Modes of Variation • Hard to Interpret • Scaling Issues? • Promising and Intuitive • Work in Progress … • Future goals • Improved Notion of PCA • Tune Bayes Approach for Better Interpretation • Integrate with Non-Neg. Matrix Factorization • ……..