1. 1 UNC, Stat & OR SAMSI OODA Workshop SAMSI OODA Workshop Dyck path correspondence and the statistical analysis of Brain vascular networks Shankar Bhamidi, J.S.Marron, Dan Shen, Haipeng Shen UNC Chapel Hill September 14, 2009

2. 2 UNC, Stat & OR Overview of today’s talk  (Very) Brief introduction to the data  Dyck path or Harris correspondence between trees and functions  Modern theory of random trees  Exploratory Data Analysis and implications  Open problems: some incoherent thoughts  Modeling aspects: Natural probability models of spatial trees? (ISE)  Other datasets of trees?

3. 3 UNC, Stat & OR Basic take home messages  Last decade has witnessed an explosion in the study of Random tree models in the probability community  Many different techniques, universality results  Many interesting spatial models Probability  Large amount of data from many fields  Biology (brain networks, lung pathways); Phylogenetics; “Actual trees” (root pathways)  Amazing challenges at all levels (modeling, probabilistic analysis, statistical methodology, data analysis) Statistics

4. 4 UNC, Stat & OR Data Background Motivating Example: From Dr. Elizabeth Bullitt Dept. of Neurosurgery, UNC Blood Vessel Trees in Brains Segmented from MRAs Study population of trees Forest of Trees

5. 5 UNC, Stat & OR Strongly Non-Euclidean Spaces Trees as Data Objects From Graph Theory: Graph is set of nodes and edges Tree has root and direction and leaves

6. 6 UNC, Stat & OR Blood vessel tree data From MRA  Segment tree  vessel segments  Using tube tracking  Bullitt and Aylward (2002)

7. 7 UNC, Stat & OR Goal understand population properties:  PCA: Main sources of variation in the data?  Interpretation? (e.g. age, gender, occupation?)  Discrimination / Classification  Prediction  Models of spatial trees?

8. 8 UNC, Stat & OR Dyck path Correspondence for one tree Tree 1

9. 9 UNC, Stat & OR Dyck path Correspondence for one tree Tree 1

10. 10 UNC, Stat & OR Dyck path Correspondence for one tree Tree 1

11. 11 UNC, Stat & OR Dyck path Correspondence for one tree Tree 1

12. 12 UNC, Stat & OR Dyck path Correspondence for one tree Tree 1

13. 13 UNC, Stat & OR Dyck path Correspondence for one tree Tree 1

14. 14 UNC, Stat & OR Dyck path Correspondence for one tree Tree 1

15. 15 UNC, Stat & OR Dyck path Correspondence for one tree Tree 1

16. 16 UNC, Stat & OR Dyck path Correspondence for one tree Tree 1

17. 17 UNC, Stat & OR Dyck Path correspondence continued  One of the foremost methods in probability for analysis of random trees.  Tremendous array of random tree models arising from many different fields e.g. CS, phylogenetics, mathematics, statistical physics  Consider a “random tree” on n vertices  Rescale each edge by some factor (turns out 1/√n is the right factor)  What happens?

18. 18 UNC, Stat & OR Central Result Theorem [Aldous 90’s]: For many (most?) of the known models of random trees the Dyck path converges to standard Brownian Excursion. This also implies that the trees themselves converge to a random metric space (random fractal) called the Continuum random tree. Shall come back to this when we look at the spatial aspect.

19. 19 UNC, Stat & OR Basic intuition  Where does one get such results?  Harris: Consider a branching process with geometric (1/2) offspring  This model is “critical” (mean # of offspring=1)  Condition on size of the tree when the branching process dies out to be n.  Consider the Dyck path of this tree  Has same distribution as a simple random walk started at 0, coming back to 0 at time 2(n-1) and always above the orign otherwise

20. 20 UNC, Stat & OR In pictures

21. 21 UNC, Stat & OR In pictures

22. 22 UNC, Stat & OR In pictures

23. 23 UNC, Stat & OR In pictures

24. 24 UNC, Stat & OR Our data  Have data on a number of trees   Dyck path transformation for all of them  Exploratory Data Analysis

25. 25 UNC, Stat & OR Example 1, Assume that we have three following tree data Tree 1 Tree 2 Tree 3

26. 26 UNC, Stat & OR Support tree: union of three tree Tree 1 Tree 2 Tree 3 Tree 1

27. 27 UNC, Stat & OR Support tree: union of three tree Tree 1 Tree 2 Tree 3 Tree 1,2

28. 28 UNC, Stat & OR Support tree: union of three tree Tree 1 Tree 2 Tree 3 Tree 1,2,3

29. 29 UNC, Stat & OR Now, we show how to transform the first tree as curve. Tree 1/ Support Tree

30. 30 UNC, Stat & OR Now, we show how to transform the first tree as curve. Tree 1/ Support Tree

31. 31 UNC, Stat & OR Now, we show how to transform the first tree as curve. Tree 1/ Support Tree

32. 32 UNC, Stat & OR Now, we show how to transform the first tree as curve. Tree 1/ Support Tree

33. 33 UNC, Stat & OR Now, we show how to transform the first tree as curve. Tree 1/ Support Tree

34. 34 UNC, Stat & OR Now, we show how to transform the first tree as curve. Tree 1/ Support Tree

35. 35 UNC, Stat & OR Now, we show how to transform the first tree as curve. Tree 1/ Support Tree

36. 36 UNC, Stat & OR Now, we show how to transform the first tree as curve. Tree 1/ Support Tree

37. 37 UNC, Stat & OR Now, we show how to transform the first tree as curve. Tree 1/ Support Tree

38. 38 UNC, Stat & OR Now, we show how to transform the first tree as curve. Tree 1/ Support Tree

39. 39 UNC, Stat & OR Now, we show how to transform the first tree as curve. Tree 1/ Support Tree

40. 40 UNC, Stat & OR Now, we show how to transform the second tree as curve. Tree 2/ Support Tree

41. 41 UNC, Stat & OR Now, we show how to transform the second tree as curve. Tree 2/ Support Tree

42. 42 UNC, Stat & OR Now, we show how to transform the second tree as curve. Tree 2/ Support Tree

43. 43 UNC, Stat & OR Now, we show how to transform the second tree as curve. Tree 2/ Support Tree

44. 44 UNC, Stat & OR Now, we show how to transform the second tree as curve. Tree 2/ Support Tree

45. 45 UNC, Stat & OR Now, we show how to transform the second tree as curve. Tree 2/ Support Tree

46. 46 UNC, Stat & OR Now, we show how to transform the second tree as curve. Tree 2/ Support Tree

47. 47 UNC, Stat & OR Now, we show how to transform the second tree as curve. Tree 2/ Support Tree

48. 48 UNC, Stat & OR Now, we show how to transform the second tree as curve. Tree 2/ Support Tree

49. 49 UNC, Stat & OR Now, we show how to transform the second tree as curve. Tree 2/ Support Tree

50. 50 UNC, Stat & OR Now, we show how to transform the second tree as curve. Tree 2/ Support Tree

51. 51 UNC, Stat & OR Now, we show how to transform the third tree as curve. Tree 3/ Support Tree

52. 52 UNC, Stat & OR Now, we show how to transform the third tree as curve. Tree 3/ Support Tree

53. 53 UNC, Stat & OR Now, we show how to transform the third tree as curve. Tree 3/ Support Tree

54. 54 UNC, Stat & OR Now, we show how to transform the third tree as curve. Tree 3/ Support Tree

55. 55 UNC, Stat & OR Now, we show how to transform the third tree as curve. Tree 3/ Support Tree

56. 56 UNC, Stat & OR Now, we show how to transform the third tree as curve. Tree 3/ Support Tree

57. 57 UNC, Stat & OR Now, we show how to transform the third tree as curve. Tree 3/ Support Tree

58. 58 UNC, Stat & OR Now, we show how to transform the third tree as curve. Tree 3/ Support Tree

59. 59 UNC, Stat & OR Now, we show how to transform the third tree as curve. Tree 3/ Support Tree

60. 60 UNC, Stat & OR Now, we show how to transform the third tree as curve. Tree 3/ Support Tree

61. 61 UNC, Stat & OR Now, we show how to transform the third tree as curve. Tree 3/ Support Tree

62. 62 UNC, Stat & OR Advantages of this encoding  If we are only interested in topological aspects then mathematically this is reasonable  Main reason: Suppose f, g are encodings of two trees, s and t, then the sup norm between the two functions bounds the Gromov-Haussdorf distance  However a number of issues as well

63. 63 UNC, Stat & OR Actual Data

64. 64 UNC, Stat & OR Raw Brain Data - Zoomed

65. 65 UNC, Stat & OR Raw Brain Data - Zoomed

66. 66 UNC, Stat & OR Some Brain Data Points (as corresponding trees)

67. 67 UNC, Stat & OR Some Brain Data Points (as corresponding trees)

68. 68 UNC, Stat & OR Some Brain Data Points (as corresponding trees)

69. 69 UNC, Stat & OR Some Brain Data Points (as corresponding trees)

70. 70 UNC, Stat & OR Some Brain Data Points (as corresponding trees)

71. 71 UNC, Stat & OR Some Brain Data Points (as corresponding trees)

72. 72 UNC, Stat & OR Data Representation- Youngest

73. 73 UNC, Stat & OR Data Representation- oldest

74. 74 UNC, Stat & OR Average Tree-Curve and picture of the average tree

75. 75 UNC, Stat & OR Illust’n of PCA View: PC1 Projections

76. 76 UNC, Stat & OR PCA Pictures of trees that we get when we move in PC1 direction

77. 77 UNC, Stat & OR PCA Pictures of trees that we get when we move in PC1 direction

78. 78 UNC, Stat & OR PCA Pictures of trees that we get when we move in PC1 direction

79. 79 UNC, Stat & OR PCA Pictures of trees that we get when we move in PC1 direction

80. 80 UNC, Stat & OR PCA Pictures of trees that we get when we move in PC1 direction

81. 81 UNC, Stat & OR PCA Pictures of trees that we get when we move in PC1 direction

82. 82 UNC, Stat & OR PCA Pictures of trees that we get when we move in PC1 direction

83. 83 UNC, Stat & OR PCA Pictures of trees that we get when we move in PC1 direction

84. 84 UNC, Stat & OR PCA Pictures of trees that we get when we move in PC1 direction

85. 85 UNC, Stat & OR PCA Pictures of trees that we get when we move in PC2 direction

86. 86 UNC, Stat & OR PCA Pictures of trees that we get when we move in PC2 direction

87. 87 UNC, Stat & OR PCAPictures of trees that we get when we move in PC2 direction

88. 88 UNC, Stat & OR PCA Pictures of trees that we get when we move in PC2 direction

89. 89 UNC, Stat & OR PCA Pictures of trees that we get when we move in PC2 direction

90. 90 UNC, Stat & OR PCA Pictures of trees that we get when we move in PC2 direction

91. 91 UNC, Stat & OR PCA Pictures of trees that we get when we move in PC2 direction

92. 92 UNC, Stat & OR PCAPictures of trees that we get when we move in PC2 direction

93. 93 UNC, Stat & OR PCAPictures of trees that we get when we move in PC2 direction

94. 94 UNC, Stat & OR DWD

95. 95 UNC, Stat & OR DWD

96. 96 UNC, Stat & OR DWD

97. 97 UNC, Stat & OR DWD

98. 98 UNC, Stat & OR DWD

99. 99 UNC, Stat & OR DWD

100. 100 UNC, Stat & OR DWD

101. 101 UNC, Stat & OR DWD

102. 102 UNC, Stat & OR DWD

103. 103 UNC, Stat & OR DWD/relabeling  random relabeling: Suppose we randomly relabel each tree as male or female.  How does the DWD direction behave?

104. 104 UNC, Stat & OR DWD/relabelling

105. 105 UNC, Stat & OR DWD/relabelling

106. 106 UNC, Stat & OR DWD/relabelling

107. 107 UNC, Stat & OR DWD/relabelling

108. 108 UNC, Stat & OR DWD/relabelling

109. 109 UNC, Stat & OR DWD/relabelling

110. 110 UNC, Stat & OR DWD/relabelling

111. 111 UNC, Stat & OR DWD/relabelling

112. 112 UNC, Stat & OR DWD/relabelling

113. 113 UNC, Stat & OR Implications  “Eyeballing” the data, the PC1 directions (and PC2) do not seem to be capturing variation in the data  Because of encoding all the trees to form a support tree?  Perhaps because inherently PCA works well in the Euclidean regime?  Path of Dyck paths a weird subset of function space?  Any math theory that can be developed about families of large trees?  Modeling of these trees?

114. 114 UNC, Stat & OR Blood vessel tree data Marron’s brain:  From MRA  Segment tree  of vessel segments  Using tube tracking  Bullitt and Aylward (2002)

115. 115 UNC, Stat & OR Blood vessel tree data Marron’s brain:  From MRA  Reconstruct trees  in 3d  Rotate to view

116. 116 UNC, Stat & OR Blood vessel tree data Marron’s brain:  From MRA  Reconstruct trees  in 3d  Rotate to view

117. 117 UNC, Stat & OR Blood vessel tree data Marron’s brain:  From MRA  Reconstruct trees  in 3d  Rotate to view

118. 118 UNC, Stat & OR Blood vessel tree data Marron’s brain:  From MRA  Reconstruct trees  in 3d  Rotate to view

119. 119 UNC, Stat & OR Blood vessel tree data Marron’s brain:  From MRA  Reconstruct trees  in 3d  Rotate to view

120. 120 UNC, Stat & OR Blood vessel tree data Marron’s brain:  From MRA  Reconstruct trees  in 3d  Rotate to view

121. 121 UNC, Stat & OR Thoughts I: Probabilistic models of spatial trees?  What are natural models of spatial trees such as those in this talk?  At least two natural directions to proceed in  ISE (Integrated Superbrownian Excursion): Arising from modelling of critical random systems in euclidean space  Engineering and biological principles of flow distribution: (Constructal theory)

122. 122 UNC, Stat & OR ISE: Integrated Superbrownian excursion  Formulated in the late 90s by Aldous  Has now come to be one of the standard models of spatial trees  Arises as the scaling limit of many different systems  Example: Random trees on the integer lattice  Critical contact process in high dimensions etc  Thought to be the scaling limit of many systems at criticality  Use Standard Brownian excursion and Brownian motion to construct a random tree in 3 (or higher dimensions)

123. 123 UNC, Stat & OR ISE: in pictures

124. 124 UNC, Stat & OR ISE in pictures

125. 125 UNC, Stat & OR ISE in pictures

126. 126 UNC, Stat & OR ISE in pictures

127. 127 UNC, Stat & OR ISE  Any notion of data driven ISE?

128. 128 UNC, Stat & OR Blood vessel tree data  Notion of ISE on the sphere?  Notion of ISE where the Brownian motion has some sort of drift?  How does one estimate drift from data?  Model of thickness on edges to the data?

129. 129 UNC, Stat & OR Other examples of tree data?

130. 130 UNC, Stat & OR Data on actual root systems

131. 131 UNC, Stat & OR PCA and Random Walks on Tree space?  In this study we tried usual notion of PCA  Ok when data are “Gaussian in nature”  Tree space intuitively very non-linear  Can one use random walks to explore this space?

132. 132 UNC, Stat & OR Intuition

133. 133 UNC, Stat & OR Random walk on data points

134. 134 UNC, Stat & OR Folded Euclidean Approach