1. 1 UNC, Stat & OR Place Name OODA of Tree Structured Objects J. S. Marron Dept. of Statistics and Operations Research October 2, 2016

2. 2 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

3. 3 UNC, Stat & OR Euclidean Data Spaces Data are vectors, in Effective (and Traditional) Analysis: Linear Methods Mean Covariance Principal Component Analysis Gaussian Distribution

4. 4 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

5. 5 UNC, Stat & OR Euclidean Data - Visualization

6. 6 UNC, Stat & OR Euclidean Data - Visualization

7. 7 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”

8. 8 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)

9. 9 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)

10. 10 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)

11. 11 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

12. 12 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

13. 13 UNC, Stat & OR Strongly Non-Euclidean Spaces Motivating Example: From Dr. Elizabeth BullittDr. Elizabeth Bullitt Dept. of Neurosurgery, UNC Blood Vessel Trees in Brains Segmented from MRAs Study population of trees Forest of Trees

14. 14 UNC, Stat & OR Blood vessel tree data The tree team:  Very Interdsciplinary  Neurosurgery:  Bullitt, Ladha  Statistics:  Wang, Marron  Optimization:  Aydin, Pataki

15. 15 UNC, Stat & OR Blood vessel tree data Marron’s brain:  MRI view  Single Slice  From 3-d Image

16. 16 UNC, Stat & OR Blood vessel tree data Marron’s brain:  MRA view  “A” for Angiography”  Finds blood vessels (show up as white)  Track through 3d

17. 17 UNC, Stat & OR Blood vessel tree data Marron’s brain:  MRA view  “A” for Angiography”  Finds blood vessels (show up as white)  Track through 3d

18. 18 UNC, Stat & OR Blood vessel tree data Marron’s brain:  MRA view  “A” for Angiography”  Finds blood vessels (show up as white)  Track through 3d

19. 19 UNC, Stat & OR Blood vessel tree data Marron’s brain:  MRA view  “A” for Angiography”  Finds blood vessels (show up as white)  Track through 3d

20. 20 UNC, Stat & OR Blood vessel tree data Marron’s brain:  MRA view  “A” for “Angiography”  Finds blood vessels (show up as white)  Track through 3d

21. 21 UNC, Stat & OR Blood vessel tree data Marron’s brain:  MRA view  “A” for Angiography”  Finds blood vessels (show up as white)  Track through 3d

22. 22 UNC, Stat & OR Blood vessel tree data Marron’s brain:  From MRA  Segment tree  of vessel segments  Using tube tracking  Bullitt and Aylward (2002)

23. 23 UNC, Stat & OR Blood vessel tree data Marron’s brain:  From MRA  Reconstruct trees  in 3d  Rotate to view

24. 24 UNC, Stat & OR Blood vessel tree data Marron’s brain:  From MRA  Reconstruct trees  in 3d  Rotate to view

25. 25 UNC, Stat & OR Blood vessel tree data Marron’s brain:  From MRA  Reconstruct trees  in 3d  Rotate to view

26. 26 UNC, Stat & OR Blood vessel tree data Marron’s brain:  From MRA  Reconstruct trees  in 3d  Rotate to view

27. 27 UNC, Stat & OR Blood vessel tree data Marron’s brain:  From MRA  Reconstruct trees  in 3d  Rotate to view

28. 28 UNC, Stat & OR Blood vessel tree data Marron’s brain:  From MRA  Reconstruct trees  in 3d  Rotate to view

29. 29 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???,...,,

30. 30 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,...,,

31. 31 UNC, Stat & OR Blood vessel tree data Big Picture: 3 Approaches 1.Purely Combinatorial 2.Euclidean Orthant 3.Harris Correspondence

32. 32 UNC, Stat & OR Blood vessel tree data Big Picture: 3 Approaches 1.Purely Combinatorial 2.Euclidean Orthant 3.Harris Correspondence

33. 33 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,...,,

34. 34 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

35. 35 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

36. 36 UNC, Stat & OR Blood vessel tree data Correspondence (which node on left???): I.Vessel Thickness Larger Median Radius on Left

37. 37 UNC, Stat & OR Blood vessel tree data Recall from above: Marron’s brain:  Focus on back  Connectivity (topology) only (also consider right & left)

38. 38 UNC, Stat & OR Blood vessel tree data Present Focus:  Topology only  Raw data as trees  Marron’s reduced tree  Back tree only

39. 39 UNC, Stat & OR Blood vessel tree data Topology only E.g. Back Trees Full Population Study as movie Understand variation?

40. 40 UNC, Stat & OR Blood vessel tree data Correspondence (which node on left???): I.Vessel Thickness Larger Median Radius on Left II.Descendants Most Descendants on Left

41. 41 UNC, Stat & OR Blood vessel tree data Marron’s Back Thickness Descendant Trees Look Similar

42. 42 UNC, Stat & OR Blood vessel tree data Case 34 Back Thickness Descendant Trees Quite Different

43. 43 UNC, Stat & OR Blood vessel tree data Whole Population – Thickness Correspondence

44. 44 UNC, Stat & OR Blood vessel tree data Whole Population – Descendant Correspondence

45. 45 UNC, Stat & OR Graphical Concept: Support Tree The union of all trees in data set T. Consists of the nodes that exist in T and their “weights”. Weight of a node, w(v,T) is the number of trees it occurs in the data set. Use to compare Thickness & Descendant Correspences

46. 46 UNC, Stat & OR Support Tree Example Data trees: Support tree:

47. 47 UNC, Stat & OR Graphical Concept: Support Tree Thickness: More Spread Descendant: More Compact

48. 48 UNC, Stat & OR Graphical Concept: Support Tree Comparison of Correspondences:  Thickness Corr. gives bushier trees (shows more population structure)  Descendant Corr. suggests compact rep’n (easier to decompose, e.g. “PCA”?)

49. 49 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)

50. 50 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 ”

51. 51 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

52. 52 UNC, Stat & OR 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

53. 53 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

54. 54 UNC, Stat & OR Hamming Distance The number of nodes in the symmetric difference of two trees. An example:

55. 55 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

56. 56 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

57. 57 UNC, Stat & OR Illust’n of PCA View: PC1 Projections

58. 58 UNC, Stat & OR Illust’n of PCA View: Projections on PC1,2 plane

59. 59 UNC, Stat & OR PCA view: Lung Cancer Microarray Data

60. 60 UNC, Stat & OR Source Batch Adj: PC 1-3 & DWD direction

61. 61 UNC, Stat & OR Source Batch Adj: DWD Source Adjustment

62. 62 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)

63. 63 UNC, Stat & OR 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”

64. 64 UNC, Stat & OR Definition of Tree-Line A set of trees, L, such that: L={u 0,u 1,u 2,…,u m } u i can be obtained by adding a single node (v i ) to u i-1 v i+1 is a child of vi Also, we will call the set of nodes hanged to u 0 as Path(L), path of treeline L: Path(L) = u m /u 0

65. 65 UNC, Stat & OR Definition of Projection of a data tree t i onto treeline L is the closest point on L to the data tree:

66. 66 UNC, Stat & OR Suppose we have the data tree: And the treeline: Example of projection Which point on the treeline is the closest to the data tree?

67. 67 UNC, Stat & OR Example of projection u 0 : d(u 0,t) = 6

68. 68 UNC, Stat & OR Example of projection u 0 : d(u 0,t) = 6 u 1 : d(u 1,t)=5

69. 69 UNC, Stat & OR Example of projection u 0 : d(u 0,t) = 6 u 1 : d(u 1,t)=5 u 2 : d(u 2,t)=4

70. 70 UNC, Stat & OR Example of projection u 0 : d(u 0,t) = 6 u 1 : d(u 1,t)=5 u 2 : d(u 2,t)=4 u 3 : d(u 3,t)=5

71. 71 UNC, Stat & OR Example of projection u 0 : d(u 0,t) = 6 u 1 : d(u 1,t)=5 u 2 : d(u 2,t)=4 u 3 : d(u 3,t)=5 Closest!

72. 72 UNC, Stat & OR Example of projection So, P L (t)=u 2 in this case Observation: Projection onto the treeline is u 0 combined with the members of P(L) that are in the data tree: The data tree: Its projection:

73. 73 UNC, Stat & OR Example of projection Useful statistical summary: Score Length of Projection (analog of PC in PCA) Above Example: Score (~PCA Coeff.) for this tree = 3 The data tree:Its projection:

74. 74 UNC, Stat & OR Best Tree Line The treeline that gives the smallest sum of distances:

75. 75 UNC, Stat & OR 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)

76. 76 UNC, Stat & OR Strongly Non-Euclidean Spaces PCA on Tree Space: Treeline Problem: Hard to compute Solution: Burcu Aydin & Gabor Pataki (linear time algorithm) (based on clever “reformulation” of problem)

77. 77 UNC, Stat & OR 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

78. 78 UNC, Stat & OR PCA for blood vessel tree data Study “Directions” 1, 2, 3, 4 For sub- populations B, L, R (Thickness)

79. 79 UNC, Stat & OR 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?

80. 80 UNC, Stat & OR PCA for blood vessel tree data (Thickness) Notes: PC1 - left BACK - right LEFT 2 nd lev RIGHT 1 st lev See these??

81. 81 UNC, Stat & OR PCA for blood vessel tree data Revisit PC “Directions” (Thickness Corr.) Compare to Descendant?

82. 82 UNC, Stat & OR PCA for blood vessel tree data Descendant Corr. - Similar Lessons - But Effects Stronger - Better Repres’n?

83. 83 UNC, Stat & OR PCA for blood vessel tree data (Descendant) Notes: PC1 - left BACK - right LEFT 2 nd lev RIGHT 1 st lev See stronger Lessons?

84. 84 UNC, Stat & OR 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)

85. 85 UNC, Stat & OR PCA for blood vessel tree data Raw Data & Treelines, PC1, PC2, PC3:

86. 86 UNC, Stat & OR PCA for blood vessel tree data Raw Data & Treelines, PC1, PC2, PC3: Projections, Scores, Residuals

87. 87 UNC, Stat & OR PCA for blood vessel tree data Raw Data & Treelines, PC1, PC2, PC3: Cumulative Scores, Residuals

88. 88 UNC, Stat & OR PCA for blood vessel tree data Individual (each PC separately) Scores Plot

89. 89 UNC, Stat & OR PCA for blood vessel tree data Identify this person

90. 90 UNC, Stat & OR PCA for blood vessel tree data Identify this person PC Scores: 8,9,3,5

91. 91 UNC, Stat & OR PCA for blood vessel tree data Identify this person (PC Scores: 8,9,3,5): Red = older 0 = Female Note: color ~ age

92. 92 UNC, Stat & OR PCA for blood vessel tree data Identify this person (PC Scores: 8,9,3,5): Red = older 0 = Female Note: color ~ age

93. 93 UNC, Stat & OR PCA for blood vessel tree data Identify this person (PC scores 1,10,1,1)

94. 94 UNC, Stat & OR PCA for blood vessel tree data Identify this person (PC scores 1,10,1,1)

95. 95 UNC, Stat & OR PCA for blood vessel tree data Explain strange (low score) correlation?

96. 96 UNC, Stat & OR 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

97. 97 UNC, Stat & OR PCA for blood vessel tree data Individual (each PC sep’ly) Residuals Plot

98. 98 UNC, Stat & OR 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

99. 99 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?

100. 100 UNC, Stat & OR PCA for blood vessel tree data Data Analytic Goals: Age, Gender See these? No…

101. 101 UNC, Stat & OR PCA for blood vessel tree data Alternate View: Cumulative Scores

102. 102 UNC, Stat & OR 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

103. 103 UNC, Stat & OR PCA for blood vessel tree data Cum. Scores: Best repr’ed case (10,19,20)

104. 104 UNC, Stat & OR PCA for blood vessel tree data Cum. Scores: Best repr’ed case (10,19,20) PC1 & PC2 Scores Very Large

105. 105 UNC, Stat & OR PCA for blood vessel tree data Cum. Scores: Worst repr’ed case (3,5,6)

106. 106 UNC, Stat & OR PCA for blood vessel tree data Cum. Scores: Worst repr’ed case (3,5,6) Fairly small tree Growth in unusual directions

107. 107 UNC, Stat & OR PCA for blood vessel tree data Directly study age  PC scores

108. 108 UNC, Stat & OR PCA for blood vessel tree data Directly study age  PC scores Thickness Corr. does not show much Could look deeper, with linear fits What about Descendants Corr.???

109. 109 UNC, Stat & OR PCA for blood vessel tree data Directly study age  PC scores - Descendants - PC1 all 11s - PC2 spread - PC3 4 or 11 - Corr. w/ Age?

110. 110 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.

111. 111 UNC, Stat & OR PCA for blood vessel tree data Directly study age  PC scores PC1 Only - Thickness Not Sig’t - Descendants Not Diverse

112. 112 UNC, Stat & OR PCA for blood vessel tree data Directly study age  PC scores PC1 + PC2 - Thickness Not Sig’t - Descendants Left Sig’t

113. 113 UNC, Stat & OR PCA for blood vessel tree data Directly study age  PC scores PC1 + 2 + 3 - Thickness Not Sig’t - Descendants Left Sig’t (less so)

114. 114 UNC, Stat & OR PCA for blood vessel tree data Directly study age  PC scores PC1+2+3+4 - Thickness Not Sig’t - Descendants Left Sig’t (again less so)

115. 115 UNC, Stat & OR PCA for blood vessel tree data Directly study age  PC scores PC2 only - Thickness Not Sig’t - Descendants Left Sig’t (again strong)

116. 116 UNC, Stat & OR PCA for blood vessel tree data Directly study age  PC scores PC3 only - Thickness Not Sig’t - Descendants Not Sig’t (Only PC2?)

117. 117 UNC, Stat & OR PCA for blood vessel tree data Directly study age  PC scores PC4 only - Thickness Back Sig’t? - Descendants Not Sig’t (Only PC2?)

118. 118 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

119. 119 UNC, Stat & OR PCA for blood vessel tree data Directly study age  PC scores Explanation? - Difference in age is here? - Only for Descendants?

120. 120 UNC, Stat & OR PCA for blood vessel tree data Overall Assessment of Tree-Line PCA Somewhat finds desired aspects (e.g. age effects) Descendant corr. better than thickness But representation seems too sparse (PCs not explaining enough variation) Need to improve notion of 1-d representation

121. 121 UNC, Stat & OR Upcoming New Approach Replace Tree-Lines by Tree-Curves: Goal: Explain more variation Approach: Start at a tree (root for now) Sequentially add nodes (now can add anywhere) End at Support Tree Find: Best Fitting Tree Curve

122. 122 UNC, Stat & OR Upcoming New Approach Replace Tree-Lines by Tree-Curves: Challenge: Solve Optimization problem Currently Open Problem Np complete??? Current Approach: Heuristics based approximate solution

123. 123 UNC, Stat & OR Upcoming New Approach Replace Tree-Lines by Tree-Curves:

124. 124 UNC, Stat & OR Upcoming New Approach Replace Tree-Lines by Tree-Curves: Notes: Back bushier on right (consistent) Right has much longer curve This was for Thickness Correspondence

125. 125 UNC, Stat & OR Upcoming New Approach Projections on Tree-Curves (Thickness): Display: Show: Tree ⋂ Projection Show: Tree \ Projection Show: Projection \ Tree

126. 126 UNC, Stat & OR Upcoming New Approach Projections on Tree-Curves (Thickness):

127. 127 UNC, Stat & OR Upcoming New Approach Scores for Thickness Tree-Curves: Skewed Distributions Too many small scores Poor Repres’n?

128. 128 UNC, Stat & OR Upcoming New Approach Scores for Descendants Tree-Curves: “Gaussian” Distributions Good spread of scores Better Repres’n?

129. 129 UNC, Stat & OR Preliminary Tree-Curve Results Relate Tree Curve Scores to Age: Fit line to highlight dependence Test Slope Hypothesis Assess with p-value

130. 130 UNC, Stat & OR Preliminary Tree-Curve Results Relate Tree Curve Scores to Age (Thickness): Back: No Sign’t Slope Don’t Find Relat’n

131. 131 UNC, Stat & OR Preliminary Tree-Curve Results Relate Tree Curve Scores to Age (Descendants): Back: Sign’t Slope Now Find Relat’n Better Corr.?

132. 132 UNC, Stat & OR Preliminary Tree-Curve Results Relate Tree Curve Scores to Age (Thickness): Left: No Sign’t Slope Don’t Find Relat’n

133. 133 UNC, Stat & OR Preliminary Tree-Curve Results Relate Tree Curve Scores to Age (Descendants): Left: Sign’t Slope Now Find Relat’n Again: Better Corr.?

134. 134 UNC, Stat & OR Preliminary Tree-Curve Results Relate Tree Curve Scores to Age (Thickness): Right: Sign’t Slope Now Find Relat’n?!?

135. 135 UNC, Stat & OR Preliminary Tree-Curve Results Relate Tree Curve Scores to Age (Descendants): Right: No Sign’t Slope No Relat’n?!? Really have: Better Corr.?

136. 136 UNC, Stat & OR Improved Correspondence Overall Impressions: Tree Curves better than Tree Lines Explain more variation Found more age structure Descendant Corr. better than Thickness Better Scores Distribution Found more age structure

137. 137 UNC, Stat & OR 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…)

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

139. 139 UNC, Stat & OR Euclidean Orthant Approach People: Scott Provan Sean Skwerer Megan Owen Martin Styner Ipek Oguz

140. 140 UNC, Stat & OR Euclidean Orthant Approach Setting: Connectivity & Length Background: Phylogenetic Trees Major Restriction: Need common leaves Big Payoff: Data space nearly Euclidean

141. 141 UNC, Stat & OR Euclidean Orthant Approach Big Payoff: Data space nearly Euclidean

142. 142 UNC, Stat & OR Euclidean Orthant Approach Big Payoff: Data space nearly Euclidean

143. 143 UNC, Stat & OR Euclidean Orthant Approach Big Payoff: Data space nearly Euclidean

144. 144 UNC, Stat & OR Euclidean Orthant Approach Major Restriction: Need common leaves Approach: Find common cortical landmarks corresponding across cases Treat as pseudo – leaves by projecting to points on tree (draw pic)

145. 145 UNC, Stat & OR Blood vessel tree data Marron’s brain:  From MRA  Reconstruct trees  in 3d  Rotate to view

146. 146 UNC, Stat & OR Vessel Locations

147. 147 UNC, Stat & OR Vessel Locations

148. 148 UNC, Stat & OR Vessel Locations

149. 149 UNC, Stat & OR Vessel Locations

150. 150 UNC, Stat & OR Vessel Locations

151. 151 UNC, Stat & OR Vessel Locations

152. 152 UNC, Stat & OR Common Color

153. 153 UNC, Stat & OR Common Color

154. 154 UNC, Stat & OR Common Color

155. 155 UNC, Stat & OR Common Color

156. 156 UNC, Stat & OR Common Color

157. 157 UNC, Stat & OR Common Color

158. 158 UNC, Stat & OR Cortical Surface & Landmarks

159. 159 UNC, Stat & OR Cortical Surface & Landmarks

160. 160 UNC, Stat & OR Cortical Surface & Landmarks

161. 161 UNC, Stat & OR Cortical Surface & Landmarks

162. 162 UNC, Stat & OR Cortical Surface & Landmarks

163. 163 UNC, Stat & OR Cortical Surface & Landmarks

164. 164 UNC, Stat & OR Landmarks and Vessels

165. 165 UNC, Stat & OR Landmarks and Vessels

166. 166 UNC, Stat & OR Landmarks and Vessels

167. 167 UNC, Stat & OR Landmarks and Vessels

168. 168 UNC, Stat & OR Landmarks and Vessels

169. 169 UNC, Stat & OR Landmarks and Vessels

170. 170 UNC, Stat & OR Attach Landmarks & Subtrees

171. 171 UNC, Stat & OR Attach Landmarks & Subtrees

172. 172 UNC, Stat & OR Attach Landmarks & Subtrees

173. 173 UNC, Stat & OR Attach Landmarks & Subtrees

174. 174 UNC, Stat & OR Attach Landmarks & Subtrees

175. 175 UNC, Stat & OR Attach Landmarks & Subtrees

176. 176 UNC, Stat & OR Highlight Oprhans

177. 177 UNC, Stat & OR Highlight Oprhans

178. 178 UNC, Stat & OR Highlight Oprhans

179. 179 UNC, Stat & OR Highlight Oprhans

180. 180 UNC, Stat & OR Highlight Oprhans

181. 181 UNC, Stat & OR Highlight Oprhans

182. 182 UNC, Stat & OR Trim Oprhans

183. 183 UNC, Stat & OR Trim Oprhans

184. 184 UNC, Stat & OR Trim Oprhans

185. 185 UNC, Stat & OR Trim Oprhans

186. 186 UNC, Stat & OR Trim Oprhans

187. 187 UNC, Stat & OR Trim Oprhans

188. 188 UNC, Stat & OR Final Tree (common leaves)

189. 189 UNC, Stat & OR Final Tree (common leaves)

190. 190 UNC, Stat & OR Final Tree (common leaves)

191. 191 UNC, Stat & OR Final Tree (common leaves)

192. 192 UNC, Stat & OR Final Tree (common leaves)

193. 193 UNC, Stat & OR Final Tree (common leaves)

194. 194 UNC, Stat & OR Blood vessel tree data Big Picture: 3 Approaches 1.Purely Combinatorial 2.Euclidean Orthant 3.Harris Correspondence

195. 195 UNC, Stat & OR Harris Correspondence Approach People: Shankar Bhamidi Dan Shen

196. 196 UNC, Stat & OR Harris Correspondence Approach Setting: Start with connectivity only Should be generalizable

197. 197 UNC, Stat & OR Harris Correspondence Approach Idea: Represent trees as functions (draw some pics)

198. 198 UNC, Stat & OR Harris Correspondence 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

199. 199 UNC, Stat & OR Harris Correspondence Approach Very new idea, no results yet.