1. Object Orie’d Data Analysis, Last Time Discrimination for manifold data (Sen) –Simple Tangent plane SVM –Iterated TANgent plane SVM –Manifold SVM Interesting point: Analysis done really in the manifold –Not just in projected tangent plane –Deeper than Principal Geodesic Analysis? Manifold version of DWD?

2. Mildly Non-Euclidean Spaces Useful View of Manifold Data: Tangent Space Center: Frech é t Mean Reason for terminology “ mildly non Euclidean ”

3. Strongly Non-Euclidean Spaces Trees as Data Objects From Graph Theory: Graph is set of nodes and edges Tree has root and direction Data Objects: set of trees

4. Strongly Non-Euclidean Spaces Motivating Example: Blood Vessel Trees in Brains From Dr. Elizabeth BullittDr. Elizabeth Bullitt –Dept. of Neurosurgery, UNC Segmented from MRAs Study population of trees (Forest?)

5. Blood vessel tree data Marron’s brain:  MRI view  Single Slice  From 3-d Image

6. Blood vessel tree data Marron’s brain:  MRA view  “A” for Angiography”  Finds blood vessels (show up as white)  Track through 3d

7. Blood vessel tree data Marron’s brain:  MRA view  “A” for Angiography”  Finds blood vessels (show up as white)  Track through 3d

8. Blood vessel tree data Marron’s brain:  MRA view  “A” for Angiography”  Finds blood vessels (show up as white)  Track through 3d

9. Blood vessel tree data Marron’s brain:  MRA view  “A” for Angiography”  Finds blood vessels (show up as white)  Track through 3d

10. Blood vessel tree data Marron’s brain:  MRA view  “A” for Angiography”  Finds blood vessels (show up as white)  Track through 3d

11. Blood vessel tree data Marron’s brain:  MRA view  “A” for Angiography”  Finds blood vessels (show up as white)  Track through 3d

12. Blood vessel tree data Marron’s brain:  From MRA  Segment tree  of vessel segments  Using tube tracking  Bullitt and Aylward (2002)

13. Blood vessel tree data Marron’s brain:  From MRA  Reconstruct trees  in 3d  Rotate to view

14. Blood vessel tree data Marron’s brain:  From MRA  Reconstruct trees  in 3d  Rotate to view

15. Blood vessel tree data Marron’s brain:  From MRA  Reconstruct trees  in 3d  Rotate to view

16. Blood vessel tree data Marron’s brain:  From MRA  Reconstruct trees  in 3d  Rotate to view

17. Blood vessel tree data Marron’s brain:  From MRA  Reconstruct trees  in 3d  Rotate to view

18. Blood vessel tree data Marron’s brain:  From MRA  Reconstruct trees  in 3d  Rotate to view

19. Blood vessel tree data Now look over many people (data objects) Structure of population (understand variation?) PCA in strongly non-Euclidean Space???,...,,

20. Blood vessel tree data The tree team:  Very Interdsciplinary  Neurosurgery:  Bullitt, Ladha  Statistics:  Wang, Marron  Optimization:  Aydin, Pataki

21. Blood vessel tree data Possible focus of analysis: Connectivity structure only (topology) Location, size, orientation of segments Structure within each vessel segment,...,,

22. Blood vessel tree data Present Focus: Topology only  Already challenging  Later address others  Then add attributes  To tree nodes  And extend analysis

23. Blood vessel tree data Recall from above: Marron’s brain:  Focus on back  Connectivity only

24. Blood vessel tree data Present Focus:  Topology only  Raw data as trees  Marron’s reduced tree  Back tree only

25. Blood vessel tree data Topology only E.g. Back Trees Full Population Study as movie Understand variation?

26. Strongly Non-Euclidean Spaces Statistics on Population of Tree-Structured Data Objects? Mean??? Analog of PCA??? Strongly non-Euclidean, since: Space of trees not a linear space Not even approximately linear (no tangent plane)

27. Mildly Non-Euclidean Spaces Useful View of Manifold Data: Tangent Space Center: Frech é t Mean Reason for terminology “ mildly non Euclidean ”

28. Strongly Non-Euclidean Spaces Mean of Population of Tree-Structured Data Objects? Natural approach: Fr é chet mean Requires a metric (distance) On tree space

29. Strongly Non-Euclidean Spaces Appropriate metrics on tree space: Wang and Marron (2007) Depends on: –Tree structure –And nodal attributes Won ’ t go further here But gives appropriate Fr é chet mean

30. Strongly Non-Euclidean Spaces Appropriate metrics on tree space: Wang and Marron (2007) For topology only (studied here): –Use Hamming Distance –Just number of nodes not in common Gives appropriate Fr é chet mean

31. Strongly Non-Euclidean Spaces PCA on Tree Space? Recall Conventional PCA: Directions that explain structure in data Data are points in point cloud 1-d and 2-d projections allow insights about population structure

32. Illust ’ n of PCA View: PC1 Projections

33. Illust ’ n of PCA View: Projections on PC1,2 plane

34. Source Batch Adj: PC 1-3 & DWD direction

35. Source Batch Adj: DWD Source Adjustment

36. Strongly Non-Euclidean Spaces PCA on Tree Space? Key Ideas: Replace 1-d subspace that best approximates data By 1-d representation that best approximates data Wang and Marron (2007) define notion of Treeline (in structure space)

37. Strongly Non-Euclidean Spaces PCA on Tree Space: Treeline Best 1-d representation of data Basic idea: From some starting tree Grow only in 1 “direction”

38. Strongly Non-Euclidean Spaces PCA on Tree Space: Treeline Best 1-d representation of data Problem: Hard to compute In particular: to solve optimization problem Wang and Marron (2007) Maximum 4 vessel trees Hard to tackle serious trees (e.g. blood vessel trees)

39. Strongly Non-Euclidean Spaces PCA on Tree Space: Treeline Problem: Hard to compute Solution: Burḉu Aydin & Gabor Pataki (linear time algorithm) (based on clever “reformulation” of problem) Description coming in Participant Presentation

40. PCA for blood vessel tree data PCA on Tree Space: Treelines Interesting to compare: Population of Left Trees Population of Right Trees Population of Back Trees And to study 1 st, 2 nd, 3 rd & 4 th treelines

41. PCA for blood vessel tree data Study “Directions” 1, 2, 3, 4 For sub- populations B, L, R (interpret later)

42. Strongly Non-Euclidean Spaces PCA on Tree Space: Treeline Next represent data as projections Define as closest point in tree line (same as Euclidean PCA) Have corresponding score (length of projection along line) And analog of residual (distance from data point to projection)

43. PCA for blood vessel tree data Raw Data & Treelines, PC1, PC2, PC3:

44. PCA for blood vessel tree data Raw Data & Treelines, PC1, PC2, PC3: Projections, Scores, Residuals

45. PCA for blood vessel tree data Raw Data & Treelines, PC1, PC2, PC3: Cumulative Scores, Residuals

46. PCA for blood vessel tree data Now look deeper at “Directions” 1, 2, 3, 4 For sub- populations B, L, R

47. PCA for blood vessel tree data Notes on Treeline Directions: PC1 always to left BACK has most variation to right (PC2) LEFT has more varia’n to 2 nd level (PC2) RIGHT has more var’n to 1 st level (PC2) See these in the data?

48. PCA for blood vessel tree data Notes: PC1 - left BACK - right LEFT 2 nd lev RIGHT 1 st lev See these??

49. PCA for blood vessel tree data Individual (each PC separately) Scores Plot

50. PCA for blood vessel tree data Identify this person

51. PCA for blood vessel tree data Identify this person PC Scores: 8,9,3,5

52. PCA for blood vessel tree data Identify this person (PC Scores: 8,9,3,5): Red = older 0 = Female Note: color ~ age

53. PCA for blood vessel tree data Identify this person (PC Scores: 8,9,3,5): Red = older 0 = Female Note: color ~ age

54. PCA for blood vessel tree data Identify this person (PC scores 1,10,1,1)

55. PCA for blood vessel tree data Identify this person (PC scores 1,10,1,1)

56. PCA for blood vessel tree data Explain strange (low score) correlation?

57. PCA for blood vessel tree data Explain strange (low score) correlation? Revisit Treelines: Small PC1 Score  Small PC3 Score Small PC1 Score  Small PC4 Score

58. PCA for blood vessel tree data Individual (each PC sep’ly) Residuals Plot

59. PCA for blood vessel tree data Individual (each PC sep’ly) Residuals Plot Very strongly correlated Shows much variation not explained by PCs (data are very rich) Note age coloring useful Younger (bluer)  more variation Older (redder)  less variation

60. PCA for blood vessel tree data Important Data Analytic Goals: Understand impact of age (colors) Understand impact of gender (symbols) Understand handedness (too few) Understand ethnicity (too few) See these in PCA?

61. PCA for blood vessel tree data Data Analytic Goals: Age, Gender See these? No…

62. PCA for blood vessel tree data Alternate View: Cumulative Scores

63. PCA for blood vessel tree data Alternate View: Cumulative Scores Always below 45 degree line Better separation of age or gender? (doesn’t seem like it) This makes it easy to find: Best represented case Worst represented case

64. PCA for blood vessel tree data Cum. Scores: Best repr’ed case (10,19,20)

65. PCA for blood vessel tree data Cum. Scores: Best repr’ed case (10,19,20) PC1 & PC2 Scores Very Large

66. PCA for blood vessel tree data Cum. Scores: Worst repr’ed case (3,5,6)

67. PCA for blood vessel tree data Cum. Scores: Worst repr’ed case (3,5,6) Fairly small tree Growth in unusual directions

68. PCA for blood vessel tree data Directly study age  PC scores

69. PCA for blood vessel tree data Directly study age  PC scores Graphic highlights potential connections But no strong correlations PC3 is strongest of weak lot (less young & old for large PC3 score)

70. Strongly Non-Euclidean Spaces Overall Impression: Interesting new OODA Area Much to be to done: Refined PCA Alternate tree lines Attributes (i.e. go beyond topology) Classification / Discrimination (SVM, DWD) Other data types (e.g. lung airways…)