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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

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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?

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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

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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

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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

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

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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?

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

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

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

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

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

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

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

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

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

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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?

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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.

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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

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20 UNC, Stat & OR In pictures

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21 UNC, Stat & OR In pictures

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22 UNC, Stat & OR In pictures

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23 UNC, Stat & OR In pictures

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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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

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63 UNC, Stat & OR Actual Data

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64 UNC, Stat & OR Raw Brain Data - Zoomed

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65 UNC, Stat & OR Raw Brain Data - Zoomed

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

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

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

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

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

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

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72 UNC, Stat & OR Data Representation- Youngest

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73 UNC, Stat & OR Data Representation- oldest

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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94 UNC, Stat & OR DWD

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95 UNC, Stat & OR DWD

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96 UNC, Stat & OR DWD

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97 UNC, Stat & OR DWD

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98 UNC, Stat & OR DWD

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99 UNC, Stat & OR DWD

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100 UNC, Stat & OR DWD

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101 UNC, Stat & OR DWD

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102 UNC, Stat & OR DWD

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

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104 UNC, Stat & OR DWD/relabelling

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105 UNC, Stat & OR DWD/relabelling

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106 UNC, Stat & OR DWD/relabelling

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107 UNC, Stat & OR DWD/relabelling

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108 UNC, Stat & OR DWD/relabelling

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109 UNC, Stat & OR DWD/relabelling

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110 UNC, Stat & OR DWD/relabelling

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111 UNC, Stat & OR DWD/relabelling

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112 UNC, Stat & OR DWD/relabelling

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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?

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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)

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

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

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

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

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

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

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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)

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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)

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123 UNC, Stat & OR ISE: in pictures

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124 UNC, Stat & OR ISE in pictures

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125 UNC, Stat & OR ISE in pictures

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126 UNC, Stat & OR ISE in pictures

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

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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?

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

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130 UNC, Stat & OR Data on actual root systems

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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?

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132 UNC, Stat & OR Intuition

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133 UNC, Stat & OR Random walk on data points

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134 UNC, Stat & OR Folded Euclidean Approach

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