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

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

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

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

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

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

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

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

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

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

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 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 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 UNC, Stat & OR In pictures

21 UNC, Stat & OR In pictures

22 UNC, Stat & OR In pictures

23 UNC, Stat & OR In pictures

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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 UNC, Stat & OR Actual Data

64 UNC, Stat & OR Raw Brain Data - Zoomed

65 UNC, Stat & OR Raw Brain Data - Zoomed

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

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

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

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

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

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

72 UNC, Stat & OR Data Representation- Youngest

73 UNC, Stat & OR Data Representation- oldest

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

94 UNC, Stat & OR DWD

95 UNC, Stat & OR DWD

96 UNC, Stat & OR DWD

97 UNC, Stat & OR DWD

98 UNC, Stat & OR DWD

99 UNC, Stat & OR DWD

100 UNC, Stat & OR DWD

101 UNC, Stat & OR DWD

102 UNC, Stat & OR DWD

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 UNC, Stat & OR DWD/relabelling

105 UNC, Stat & OR DWD/relabelling

106 UNC, Stat & OR DWD/relabelling

107 UNC, Stat & OR DWD/relabelling

108 UNC, Stat & OR DWD/relabelling

109 UNC, Stat & OR DWD/relabelling

110 UNC, Stat & OR DWD/relabelling

111 UNC, Stat & OR DWD/relabelling

112 UNC, Stat & OR DWD/relabelling

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

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

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

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

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

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

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 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 UNC, Stat & OR ISE: in pictures

124 UNC, Stat & OR ISE in pictures

125 UNC, Stat & OR ISE in pictures

126 UNC, Stat & OR ISE in pictures

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

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 UNC, Stat & OR Other examples of tree data?

130 UNC, Stat & OR Data on actual root systems

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 UNC, Stat & OR Intuition

133 UNC, Stat & OR Random walk on data points

134 UNC, Stat & OR Folded Euclidean Approach