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1 UNC, Stat & OR SAMSI AOOD Opening Workshop Tutorial OODA of Tree Structured Objects J. S. Marron Dept. of Statistics and O. R., UNC February 15, 2014

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2 UNC, Stat & OR Workshop Big Picture An investment by: Provided Funding to Bring Us Together Has Specific Goal: Generating Collaborative Research

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3 UNC, Stat & OR Workshop Big Picture An investment by: Workshop Aim: Kickoff Ongoing Research (through whole program year)

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4 UNC, Stat & OR Workshop Big Picture Thus different format: Fewer Main Talks Main Talks Aimed at Collaborations 2-Minute Madness Talks – Introductory Wed. Afternoon: Form Working Groups

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5 UNC, Stat & OR Working Groups Usual Structure Conceived of at Opening Workshop Agreed upon on Wednesday Afternoon First Meeting: Thursday or Friday Followed by weekly meetings Can Skype or WebEx in remotely

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6 UNC, Stat & OR Working Groups Goals: Collaborative Research Among unexpected partners Our hope: This group unusually well suited for this

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7 UNC, Stat & OR Working Groups Program Areas of Emphasis: Functional Data Analysis Time Dynamics Image Analysis Trees as Data Shape and Manifold Data Where are potential (new) connections?

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8 UNC, Stat & OR Working Groups Program Areas of Emphasis: Functional Data Analysis Time Dynamics Image Analysis Trees as Data Shape and Manifold Data fMRI Where are potential (new) connections?

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9 UNC, Stat & OR Working Groups Program Areas of Emphasis: Functional Data Analysis Time Dynamics Image Analysis Trees as Data DTI Shape and Manifold Data Where are potential (new) connections?

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10 UNC, Stat & OR Working Groups Program Areas of Emphasis: Functional Data Analysis Time Dynamics Image Analysis Brain Development Trees as Data Shape and Manifold Data Where are potential (new) connections?

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11 UNC, Stat & OR Working Groups Program Areas of Emphasis: Functional Data Analysis Time Dynamics Atlas of Human Body Image Analysis Trees as Data Shape and Manifold Data Where are potential (new) connections?

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12 UNC, Stat & OR Working Groups Where are potential (new) connections? Requests of you: Look for more of these Discuss with others Bring up on Wednesday Afternoon Join in on Thursday +

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13 UNC, Stat & OR Object Oriented Data Analysis What is the atom of a statistical analysis? First Course: Numbers Multivariate Analysis: Vectors Functional Data Analysis: Curves OODA: More Complicated Objects Images Movies Shapes Tree Structured Objects

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14 UNC, Stat & OR An Aside on Acronyms What is it? OODA or AOOD ???

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15 UNC, Stat & OR SAMSI AOOD Opening Workshop Tutorial OODA of Tree Structured Objects J. S. Marron Dept. of Statistics and O. R., UNC February 15, 2014

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16 UNC, Stat & OR Acronym History Original SAMSI Proposal: Object Oriented Data Analysis (OODA)

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17 UNC, Stat & OR Acronym History Original SAMSI Proposal: Object Oriented Data Analysis (OODA) SAMSI Directors Suggestion: Analysis of Object Oriented Data (AOOD)

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18 UNC, Stat & OR Acronym History Original SAMSI Proposal: Object Oriented Data Analysis (OODA) SAMSI Directors Suggestion: Analysis of Object Oriented Data (AOOD) NISS Board Suggestion: Analysis Of Object Data (AOOD)

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19 UNC, Stat & OR An Aside on Acronyms What is it? OODA or AOOD Suggestion: Treat these as synonyms

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20 UNC, Stat & OR Object Oriented Data Analysis What is the atom of a statistical analysis? First Course: Numbers Multivariate Analysis: Vectors Functional Data Analysis: Curves OODA: More Complicated Objects Images Movies Shapes Tree Structured Objects

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21 UNC, Stat & OR Euclidean Data Spaces Data are vectors, in Effective (and Traditional) Analysis: Linear Methods Mean Covariance Principal Component Analysis Gaussian Distribution

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22 UNC, Stat & OR Euclidean Data Spaces Data are vectors, in Challenges: High Dimension, Low Sample Size (Classical Methods Fail) Visualization: Find Structure (Expected & Unknown) Understand range of normal cases Find anomalies

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23 UNC, Stat & OR Non - Euclidean Data Spaces Simple Example: m-reps for shapes Data involve angles Thus lie in manifold i.e. curved feature space Typical Approach: Tangent Plane Approx. e.g. PGA Personal Terminology: Mildly non-Euclidean

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24 UNC, Stat & OR PGA for m-reps, Bladder-Prostate-Rectum Bladder – Prostate – Rectum, 1 person, 17 days PG 1 PG 2 PG 3 (analysis by Ja Yeon Jeong)

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25 UNC, Stat & OR PGA for m-reps, Bladder-Prostate-Rectum Bladder – Prostate – Rectum, 1 person, 17 days PG 1 PG 2 PG 3 (analysis by Ja Yeon Jeong)

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26 UNC, Stat & OR PGA for m-reps, Bladder-Prostate-Rectum Bladder – Prostate – Rectum, 1 person, 17 days PG 1 PG 2 PG 3 (analysis by Ja Yeon Jeong)

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27 UNC, Stat & OR Non - Euclidean Data Spaces What is Strongly Non-Euclidean Case? Trees as Data Special Challenge: No Tangent Plane Must Re-Invent Data Analysis

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28 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 Data Objects: set of trees

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29 UNC, Stat & OR Strongly Non-Euclidean Spaces 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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30 UNC, Stat & OR Blood vessel tree data Marrons brain: MRI view Single Slice From 3-d Image

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31 UNC, Stat & OR Blood vessel tree data Marrons brain: MRA view A for Angiography Finds blood vessels (show up as white) Track through 3d

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32 UNC, Stat & OR Blood vessel tree data Marrons brain: MRA view A for Angiography Finds blood vessels (show up as white) Track through 3d

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33 UNC, Stat & OR Blood vessel tree data Marrons brain: MRA view A for Angiography Finds blood vessels (show up as white) Track through 3d

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34 UNC, Stat & OR Blood vessel tree data Marrons brain: MRA view A for Angiography Finds blood vessels (show up as white) Track through 3d

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35 UNC, Stat & OR Blood vessel tree data Marrons brain: MRA view A for Angiography Finds blood vessels (show up as white) Track through 3d

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36 UNC, Stat & OR Blood vessel tree data Marrons brain: MRA view A for Angiography Finds blood vessels (show up as white) Track through 3d

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37 UNC, Stat & OR Blood vessel tree data Marrons brain: From MRA Segment tree of vessel segments Using tube tracking Bullitt and Aylward (2002)

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38 UNC, Stat & OR Blood vessel tree data Marrons brain: From MRA Reconstruct trees in 3d Rotate to view

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39 UNC, Stat & OR Blood vessel tree data Marrons brain: From MRA Reconstruct trees in 3d Rotate to view

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40 UNC, Stat & OR Blood vessel tree data Marrons brain: From MRA Reconstruct trees in 3d Rotate to view

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41 UNC, Stat & OR Blood vessel tree data Marrons brain: From MRA Reconstruct trees in 3d Rotate to view

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42 UNC, Stat & OR Blood vessel tree data Marrons brain: From MRA Reconstruct trees in 3d Rotate to view

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43 UNC, Stat & OR Blood vessel tree data Marrons brain: From MRA Reconstruct trees in 3d Rotate to view

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

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45 UNC, Stat & OR Blood vessel tree data Examples of Potential Specific Goals (not accessible by traditional methods) Predict Stroke Tendency (Collateral Circulation) Screen for Loci of Pathology Explore how age affects connectivity,...,,

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46 UNC, Stat & OR Blood vessel tree data Big Picture: 3 Approaches 1.Purely Combinatorial 2.Folded Euclidean 3.Dyck Path

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47 UNC, Stat & OR Blood vessel tree data Big Picture: 3 Approaches 1.Purely Combinatorial 2.Folded Euclidean 3.Dyck Path

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

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49 UNC, Stat & OR Blood vessel tree data Present Focus: Topology only Already challenging Later address additional challenges By adding attributes (locations, thicknesses, curvature, …) To tree nodes And extend analysis

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50 UNC, Stat & OR Blood vessel tree data Topological Representation: Each Vessel Segment (up to 1 st Split) is a node Split Segments are child nodes Connecting lines show relationship

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51 UNC, Stat & OR Graphical Concept: Support Tree The union of all trees in data set T. Consists of the nodes in any tree of T

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52 UNC, Stat & OR Support Tree Example Data trees: Support tree:

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53 UNC, Stat & OR Blood vessel tree data Recall from above: Marrons brain: Focus on back Connectivity (topology) only (also consider right & left)

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54 UNC, Stat & OR Blood vessel tree data Present Focus: Topology only Raw data as trees Marrons reduced tree Back tree only

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55 UNC, Stat & OR Blood vessel tree data Topology only E.g. Back Trees Full Population Study as movie Understand variation?

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56 UNC, Stat & OR 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)

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57 UNC, Stat & OR Mildly Non-Euclidean Spaces Useful View of Manifold Data: Tangent Space Center: Frech é t Mean Reason for terminology mildly non Euclidean

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58 UNC, Stat & OR 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

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59 UNC, Stat & OR 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

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60 UNC, Stat & OR Hamming Distance The number of nodes in the symmetric difference of two trees. An example:

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61 UNC, Stat & OR Hamming Distance The two trees drawn on top of each other: Common nodes: 2 Nodes only in blue tree: 4 Nodes only in red tree: 2 So, distance: 4+2=6

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62 UNC, Stat & OR 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

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63 UNC, Stat & OR Illustn of PCA View: PC1 Projections

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64 UNC, Stat & OR Illustn of PCA View: Projections on PC1,2 plane

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65 UNC, Stat & OR PCA view: Lung Cancer Microarray Data

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66 UNC, Stat & OR Strongly Non-Euclidean Spaces PCA on Tree Space? Key Idea (Jim Ramsay): 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)

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67 UNC, Stat & OR PCA on Combinatorial Tree Space? In Depth Discussion Tuesday Afternoon: Strongly Non-Euclidean Spaces

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68 UNC, Stat & OR PCA for blood vessel tree data Individual (each PC separately) Scores Plot

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69 UNC, Stat & OR 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?

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70 UNC, Stat & OR PCA for blood vessel tree data Data Analytic Goals: Age, Gender See these? No…

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71 UNC, Stat & OR PCA for blood vessel tree data Directly study age PC scores

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72 UNC, Stat & OR PCA for blood vessel tree data Directly study age PC scores Take Deeper Look By Fitting Lines And doing Hypotest of H 0 : slope = 0 Show p-values to assess significance Compare Thickness & Descendants Corr.

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73 UNC, Stat & OR PCA for blood vessel tree data Directly study age PC scores PC1 - Not Sigt

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74 UNC, Stat & OR PCA for blood vessel tree data Directly study age PC scores PC2 - Left Sigt

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75 UNC, Stat & OR PCA for blood vessel tree data Directly study age PC scores Conclusions: - No Strong Age Connection - Significant Connection for: - Descendants - Left - PC2

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76 UNC, Stat & OR Strongly Non-Euclidean Spaces Overall Impression: Interesting 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…)

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77 UNC, Stat & OR Smoothing in Tree Space Question: How does tree structure change with age? Approach: (Gaussian) Kernel Smoothing

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78 UNC, Stat & OR Smoothing in Tree Space

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79 UNC, Stat & OR Strongly Non-Euclidean Spaces Smoothing on Tree Space? In Depth Discussion Tuesday Afternoon:

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80 UNC, Stat & OR Blood vessel tree data Big Picture: 3 Approaches 1.Purely Combinatorial 2.Folded Euclidean 3.Dyck Path

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81 UNC, Stat & OR Folded Euclidean Approach People: Scott Provan Sean Skwerer Megan Owen Martin Styner Ipek Oguz

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82 UNC, Stat & OR Folded Euclidean Approach Setting: Connectivity & Length Background: Phylogenetic Trees Major Restriction: Need common leaves Big Payoff: Data space nearly Euclidean

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83 UNC, Stat & OR Folded Euclidean Approach Big Payoff: Data space nearly Euclidean

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84 UNC, Stat & OR Folded Euclidean Approach Big Payoff: Data space nearly Euclidean

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85 UNC, Stat & OR Folded Euclidean Approach Big Payoff: Data space nearly Euclidean

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86 UNC, Stat & OR Folded Euclidean Approach Major Restriction: Need common leaves Approach: Find common cortical landmarks (Oguz) corresponding across cases Treat as pseudo – leaves by projecting to points on tree (draw pic)

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87 UNC, Stat & OR Blood vessel tree data Marrons brain: From MRA Reconstruct trees in 3d Rotate to view

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88 UNC, Stat & OR Vessel Locations

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89 UNC, Stat & OR Vessel Locations

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90 UNC, Stat & OR Vessel Locations

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91 UNC, Stat & OR Vessel Locations

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92 UNC, Stat & OR Vessel Locations

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93 UNC, Stat & OR Vessel Locations

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94 UNC, Stat & OR Common Color

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95 UNC, Stat & OR Common Color

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96 UNC, Stat & OR Common Color

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97 UNC, Stat & OR Common Color

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98 UNC, Stat & OR Common Color

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99 UNC, Stat & OR Common Color

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100 UNC, Stat & OR Cortical Surface & Landmarks

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101 UNC, Stat & OR Cortical Surface & Landmarks

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102 UNC, Stat & OR Cortical Surface & Landmarks

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103 UNC, Stat & OR Cortical Surface & Landmarks

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104 UNC, Stat & OR Cortical Surface & Landmarks

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105 UNC, Stat & OR Cortical Surface & Landmarks

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106 UNC, Stat & OR Landmarks and Vessels

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107 UNC, Stat & OR Landmarks and Vessels

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108 UNC, Stat & OR Landmarks and Vessels

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109 UNC, Stat & OR Landmarks and Vessels

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110 UNC, Stat & OR Landmarks and Vessels

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111 UNC, Stat & OR Landmarks and Vessels

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112 UNC, Stat & OR Attach Landmarks & Subtrees

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113 UNC, Stat & OR Attach Landmarks & Subtrees

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114 UNC, Stat & OR Attach Landmarks & Subtrees

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115 UNC, Stat & OR Attach Landmarks & Subtrees

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116 UNC, Stat & OR Attach Landmarks & Subtrees

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117 UNC, Stat & OR Attach Landmarks & Subtrees

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118 UNC, Stat & OR Highlight Oprhans

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119 UNC, Stat & OR Highlight Oprhans

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120 UNC, Stat & OR Highlight Oprhans

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121 UNC, Stat & OR Highlight Oprhans

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122 UNC, Stat & OR Highlight Oprhans

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123 UNC, Stat & OR Highlight Oprhans

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124 UNC, Stat & OR Trim Oprhans

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125 UNC, Stat & OR Trim Oprhans

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126 UNC, Stat & OR Trim Oprhans

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127 UNC, Stat & OR Trim Oprhans

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128 UNC, Stat & OR Trim Oprhans

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129 UNC, Stat & OR Trim Oprhans

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130 UNC, Stat & OR Final Tree (common leaves)

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131 UNC, Stat & OR Final Tree (common leaves)

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132 UNC, Stat & OR Final Tree (common leaves)

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133 UNC, Stat & OR Final Tree (common leaves)

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134 UNC, Stat & OR Final Tree (common leaves)

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135 UNC, Stat & OR Final Tree (common leaves)

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136 UNC, Stat & OR Folded Euclidean Approach Next tasks: Statistical Analysis, e.g. Calculation of Mean Smoothing over time (wtd mean) PCA (Backwards approach???) Classification (linear method ???) Work in Progress Heavy & Specialized Optimization

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137 UNC, Stat & OR Strongly Non-Euclidean Spaces Statistics on Folded EuclideanTree Space? In Depth Discussion Tuesday Afternoon:

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138 UNC, Stat & OR Blood vessel tree data Big Picture: 3 Approaches 1.Purely Combinatorial 2.Euclidean Orthant 3.Dyck Path

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139 UNC, Stat & OR Dyck Path Approach People: Shankar Bhamidi Dan Shen Haipeng Shen

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140 UNC, Stat & OR Dyck Path Approach Setting: Start with connectivity only Second include lengths Should be generalizable

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141 UNC, Stat & OR Dyck Path Approach Idea: Represent trees as functions

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142 UNC, Stat & OR Dyck Path Approach Idea: Represent trees as functions Common device in probability theory Used for limiting distributions Gives access to Brownian Motion limits

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143 UNC, Stat & OR Dyck Path Approach Idea: Represent trees as functions Common device in probability theory Used for limiting distributions Gives access to Brownian Motion limits Use Functional Data Analysis Familiar, Euclidean space Many methods available

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144 UNC, Stat & OR Dyck Path Approach Idea: Represent trees as functions

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

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146 UNC, Stat & OR Support tree: union of trees Tree 1 Tree 2 Tree 3 Tree 1

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147 UNC, Stat & OR Tree 1 Tree 2 Tree 3 Tree 1,2 Support tree: union of trees

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148 UNC, Stat & OR Tree 1 Tree 2 Tree 3 Tree 1,2,3 Support tree: union of trees

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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188 UNC, Stat & OR Raw Brain Data (as curves)

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189 UNC, Stat & OR Raw Brain Data - Zoomed

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190 UNC, Stat & OR Raw Brain Data - Zoomed

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191 UNC, Stat & OR Strongly Non-Euclidean Spaces More on Dyck PathTree Space? In Depth Discussion Tuesday Afternoon:

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192 UNC, Stat & OR Working Groups Where are potential (new) connections? Requests of you: Look for more of these Discuss with others Bring up on Wednesday Afternoon Join in on Thursday +

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