1. 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

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

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

4. 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

5. 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

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

7. 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?

8. 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?

9. 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?

10. 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?

11. 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?

12. 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 +

13. 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

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

15. 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

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

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

18. 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)

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

20. 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

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

22. 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

23. 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

24. 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)

25. 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)

26. 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)

27. 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

28. 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

29. 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

30. 30 UNC, Stat & OR Blood vessel tree data Marrons brain: MRI view Single Slice From 3-d Image

31. 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

32. 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

33. 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

34. 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

35. 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

36. 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

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

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

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

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

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

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

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

44. 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???,...,,

45. 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,...,,

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

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

48. 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,...,,

49. 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

50. 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

51. 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

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

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

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

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

56. 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)

57. 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

58. 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

59. 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

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

61. 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

62. 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

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

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

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

66. 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)

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

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

69. 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?

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

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

72. 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.

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

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

75. 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

76. 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…)

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

78. 78 UNC, Stat & OR Smoothing in Tree Space

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

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

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

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

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

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

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

86. 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)

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

88. 88 UNC, Stat & OR Vessel Locations

89. 89 UNC, Stat & OR Vessel Locations

90. 90 UNC, Stat & OR Vessel Locations

91. 91 UNC, Stat & OR Vessel Locations

92. 92 UNC, Stat & OR Vessel Locations

93. 93 UNC, Stat & OR Vessel Locations

94. 94 UNC, Stat & OR Common Color

95. 95 UNC, Stat & OR Common Color

96. 96 UNC, Stat & OR Common Color

97. 97 UNC, Stat & OR Common Color

98. 98 UNC, Stat & OR Common Color

99. 99 UNC, Stat & OR Common Color

100. 100 UNC, Stat & OR Cortical Surface & Landmarks

101. 101 UNC, Stat & OR Cortical Surface & Landmarks

102. 102 UNC, Stat & OR Cortical Surface & Landmarks

103. 103 UNC, Stat & OR Cortical Surface & Landmarks

104. 104 UNC, Stat & OR Cortical Surface & Landmarks

105. 105 UNC, Stat & OR Cortical Surface & Landmarks

106. 106 UNC, Stat & OR Landmarks and Vessels

107. 107 UNC, Stat & OR Landmarks and Vessels

108. 108 UNC, Stat & OR Landmarks and Vessels

109. 109 UNC, Stat & OR Landmarks and Vessels

110. 110 UNC, Stat & OR Landmarks and Vessels

111. 111 UNC, Stat & OR Landmarks and Vessels

112. 112 UNC, Stat & OR Attach Landmarks & Subtrees

113. 113 UNC, Stat & OR Attach Landmarks & Subtrees

114. 114 UNC, Stat & OR Attach Landmarks & Subtrees

115. 115 UNC, Stat & OR Attach Landmarks & Subtrees

116. 116 UNC, Stat & OR Attach Landmarks & Subtrees

117. 117 UNC, Stat & OR Attach Landmarks & Subtrees

118. 118 UNC, Stat & OR Highlight Oprhans

119. 119 UNC, Stat & OR Highlight Oprhans

120. 120 UNC, Stat & OR Highlight Oprhans

121. 121 UNC, Stat & OR Highlight Oprhans

122. 122 UNC, Stat & OR Highlight Oprhans

123. 123 UNC, Stat & OR Highlight Oprhans

124. 124 UNC, Stat & OR Trim Oprhans

125. 125 UNC, Stat & OR Trim Oprhans

126. 126 UNC, Stat & OR Trim Oprhans

127. 127 UNC, Stat & OR Trim Oprhans

128. 128 UNC, Stat & OR Trim Oprhans

129. 129 UNC, Stat & OR Trim Oprhans

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

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

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

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

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

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

136. 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

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

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

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

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

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

142. 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

143. 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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

189. 189 UNC, Stat & OR Raw Brain Data - Zoomed

190. 190 UNC, Stat & OR Raw Brain Data - Zoomed

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

192. 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 +