1 UNC, Stat & OR Metrics in Curve Space

2 UNC, Stat & OR Metrics in Curve Quotient Space Above was Invariance for Individual Curves Now extend to:  Equivalence Classes of Curves  I.e. Orbits as Data Objects  I.e. Quotient Space

3 UNC, Stat & OR More Data Objects

4 UNC, Stat & OR More Data Objects Data Objects II ~ Kendall’s Shapes

5 UNC, Stat & OR More Data Objects Data Objects III ~ Chang’s Transfo’s

6 UNC, Stat & OR Toy Example Conventional PCA Projections Power Spread Across Spectrum

7 UNC, Stat & OR Toy Example Conventional PCA Scores Views of 1-d Curve Bending Through 4 Dim’ns’

8 UNC, Stat & OR Toy Example Conventional PCA Scores Patterns Are “Harmonics” In Scores

9 UNC, Stat & OR Toy Example Scores Plot Shows Data Are “1” Dimensional So Need Improved PCA Decomp.

10 UNC, Stat & OR Toy Example Aligned Curve PCA Projections All Var’n In 1 st Component

11 UNC, Stat & OR Toy Example Warps, PC Projections Mostly 1 st PC, But 2 nd Helps Some

12 UNC, Stat & OR Toy Example Warp Compon’ts (+ Mean) Applied to Template Mean

13 UNC, Stat & OR PNS on SRVF Sphere Toy Example Tangent Space PCA (on Horiz. Var’n) Thanks to Xiaosun Lu

14 UNC, Stat & OR PNS on SRVF Sphere Toy Example PNS Projections (Fewer Modes)

15 UNC, Stat & OR PNS on SRVF Sphere Toy Example Tangent Space PCA Note: 3 Comp’s Needed for This

16 UNC, Stat & OR PNS on SRVF Sphere Toy Example PNS Projections Only 2 for This

17 UNC, Stat & OR TIC testbed Special Feature: Answer Key of Known Peaks Goal: Find Warps To Align These

18 UNC, Stat & OR TIC testbed Fisher – Rao Alignment

19 UNC, Stat & OR Non - Euclidean Data Spaces What is “Strongly Non-Euclidean” Case? Trees as Data

20 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

21 UNC, Stat & OR Strongly Non-Euclidean Spaces Trees as Data Objects Thanks to Burcu Aydin

22 UNC, Stat & OR Strongly Non-Euclidean Spaces Trees as Data Objects From Graph Theory: Graph is set of nodes and edges Thanks to Burcu Aydin

23 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 Thanks to Burcu Aydin

24 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 Thanks to Burcu Aydin

25 UNC, Stat & OR Strongly Non-Euclidean Spaces General Graph: Thanks to Sean Skwerer

26 UNC, Stat & OR Strongly Non-Euclidean Spaces Special Case Called “Tree” Directed Acyclic

27 UNC, Stat & OR Strongly Non-Euclidean Spaces Special Case Called “Tree” Directed Acyclic Graphical note: Sometimes “grow up” Others “grow down”

28 UNC, Stat & OR Strongly Non-Euclidean Spaces Special Case Called “Tree” Directed Acyclic Terminology: Root

29 UNC, Stat & OR Strongly Non-Euclidean Spaces Special Case Called “Tree” Directed Acyclic Terminology: Children Of Parent

30 UNC, Stat & OR Strongly Non-Euclidean Spaces Motivating Example: From Dr. Elizabeth BullittDr. Elizabeth Bullitt Dept. of Neurosurgery, UNC

31 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

32 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

33 UNC, Stat & OR Blood vessel tree data Marron’s brain:  MRI (T1) view  Single Slice  From 3-d Image

34 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

35 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

36 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

37 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

38 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

39 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

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

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

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

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

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

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

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

47 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???,...,,

48 UNC, Stat & OR Blood vessel tree data Examples of Potential Specific Goals,...,,

49 UNC, Stat & OR Blood vessel tree data Examples of Potential Specific Goals (not accessible by traditional methods) Predict Stroke Tendency (Collateral Circulation),...,,

50 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,...,,

51 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,...,,

52 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,...,,

53 UNC, Stat & OR Blood vessel tree data Big Picture: 4 Approaches 1. Purely Combinatorial 2. Euclidean Orthant (Phylogenetics) 3. Dyck Path 4. Persistent Homologies Topological DA

54 UNC, Stat & OR Blood vessel tree data Big Picture: 4 Approaches 1. Purely Combinatorial 2. Euclidean Orthant (Phylogenetics) 3. Dyck Path 4. Persistent Homologies Topological DA

55 UNC, Stat & OR Blood vessel tree data Purely Combinatorial Data Analyses (Study Connectivity Only)  Wang and Marron (2007)  Aydin et al (2009)  Wang et al (2012)  Aydin et al (2012)  Alfaro et al (2014)

56 UNC, Stat & OR D-L Visualization of Trees Challenge: Visual Display of Full Tree Structure

57 UNC, Stat & OR D-L Visualization of Trees Challenge: Visual Display of Full Tree Structure Actually Goes to Level 17 (Truncated in View)

58 UNC, Stat & OR D-L Visualization of Trees Approach: Different Coordinate System Aydin, et al (2011)

59 UNC, Stat & OR D-L Visualization of Trees Approach: Different Coordinate System Idea: Focus on Important Aspects (of Nodes) Level of Node Number of Children

60 UNC, Stat & OR D-L Visualization of Trees Approach: Different Coordinate System Idea: Focus on Important Aspects (of Nodes) Level of Node Number of Children Using These as Coordinates

61 UNC, Stat & OR D-L Visualization of Trees D-L View

62 UNC, Stat & OR D-L Visualization of Trees D-L View Nodes

63 UNC, Stat & OR D-L Visualization of Trees D-L View Level

64 UNC, Stat & OR D-L Visualization of Trees D-L View Level Much Deeper Than Early View

65 UNC, Stat & OR D-L Visualization of Trees D-L View # Des- cend- ants (log scale)

66 UNC, Stat & OR D-L Visualization of Trees D-L View Color Codes Branch Thick- ness

67 UNC, Stat & OR D-L Visualization of Trees D-L View Reveals Strange Struc- ture

68 UNC, Stat & OR D-L Visualization of Trees D-L View Reveals Strange Struc- ture Linking Errors

69 UNC, Stat & OR D-L Visualization of Trees D-L View Fixed Version

70 UNC, Stat & OR Blood vessel tree data Big Picture: 4 Approaches 1. Purely Combinatorial 2. Euclidean Orthant (Phylogenetics) 3. Dyck Path 4. Persistent Homologies Topological DA

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

72 UNC, Stat & OR Euclidean Orthant Approach Setting: Connectivity & Length

73 UNC, Stat & OR Euclidean Orthant Approach Setting: Connectivity & Length Background: Phylogenetic Trees

74 UNC, Stat & OR Euclidean Orthant Approach Setting: Connectivity & Length Background: Phylogenetic Trees Important Concept from Evolutionary Biology

75 UNC, Stat & OR Phylogenetic Trees Idea: Study “Common Ancestry” Via a tree Species are leaves thanks to Susan Holmes

76 UNC, Stat & OR Phylogenetic Trees Very Early Reference: E. Schröder (1870), Zeit. für. Math. Phys., 15, thanks to Susan Holmes

77 UNC, Stat & OR Phylogenetic Trees Important Reference: Billera L, Holmes S, & Vogtmann K (2001)

78 UNC, Stat & OR Phylogenetic Trees Important Reference: Billera L, Holmes S, & Vogtmann K (2001) Put Large Field on Firm Mathematical Basis

79 UNC, Stat & OR Euclidean Orthant Approach Setting: Connectivity & Length Background: Phylogenetic Trees Major Restriction: Need common leaves

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

81 UNC, Stat & OR Euclidean Orthant Approach Major Restriction: Need common leaves

82 UNC, Stat & OR Euclidean Orthant Approach Major Restriction: Need common leaves Approach: Find common cortical landmarks (Oguz) corresponding across cases

83 UNC, Stat & OR Euclidean Orthant Approach Major Restriction: Need common leaves Approach: Find common cortical landmarks (Oguz) corresponding across cases Oguz (2009)

84 UNC, Stat & OR Euclidean Orthant 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

85 UNC, Stat & OR Labeled n-Trees 5-tree e.g. n = 5 Thanks to Sean Skwerer

86 UNC, Stat & OR Labeled n-Trees leaves - fixed a priori 5-tree

87 UNC, Stat & OR Labeled n-Trees leaves - fixed a priori root 5-tree

88 UNC, Stat & OR Labeled n-Trees leaves - fixed a priori labeled {0,1,...,n} root 5-tree

89 UNC, Stat & OR Labeled n-Trees leaves - fixed a priori labeled {0,1,...,n} Internal (nonleaf) vertices degree ≥ 3 root 5-tree

90 UNC, Stat & OR Labeled n-Trees leaves - fixed a priori labeled {0,1,...,n} Internal (nonleaf) vertices degree ≥ 3 # edges from node root 5-tree

91 UNC, Stat & OR Labeled n-Trees leaves - fixed a priori labeled {0,1,...,n} Internal (nonleaf) vertices degree ≥ 3 (note: not labelled) root 5-tree

92 UNC, Stat & OR Labeled n-Trees leaves - fixed a priori labeled {0,1,...,n} Internal (nonleaf) vertices degree ≥ 3 edge e has nonneg. length root |e 1 |=3 |e 2 |=4 |e 3 |=6 5-tree

93 UNC, Stat & OR Labeled n-Trees Note: To study ‘topology’, (i.e. tree structure) root |e 1 |=3 |e 2 |=4 |e 3 |=6 5-tree

94 UNC, Stat & OR Labeled n-Trees Note: To study ‘topology’, (i.e. tree structure) Enough to consider only lengths of internal edges root |e 1 |=3 |e 2 |=4 |e 3 |=6 5-tree

95 UNC, Stat & OR Labeled n-Trees Terminology: Leaf edges called ‘pendants’ (Care about lengths of pendants in brain arteries) root |e 1 |=3 |e 2 |=4 |e 3 |=6 5-tree

96 UNC, Stat & OR Toy Examples = Same tree, since same internal edges

97 UNC, Stat & OR Toy Examples = Different tree, since different connections

98 UNC, Stat & OR Toy Examples A valid tree, called “Star tree” or “0 tree” (since all internal edge lengths are 0)

99 UNC, Stat & OR Tree Space Examples, T-4 Set of mutually compatible splits  tree Thanks to Megan Owen

100 UNC, Stat & OR Three quadrants meeting at common axis

101 UNC, Stat & OR Three quadrants meeting at common axis Star Tree = 0 Tree

102 UNC, Stat & OR Three quadrants meeting at common axis Star Tree (point)

103 UNC, Stat & OR Three quadrants meeting at common axis Star Tree (point) 1-edge Trees (line)

104 UNC, Stat & OR Three quadrants meeting at common axis Star Tree (point) 1-edge Trees (line) 2-edge Trees (planes)

105 UNC, Stat & OR Three quadrants meeting at common axis Mathematical Construct: Manifold Stratified Space

106 UNC, Stat & OR Three quadrants meeting at common axis Mathematical Construct: Manifold Stratified Space Manifolds (planes) of Different Dimensions Glued Together

107 UNC, Stat & OR Three quadrants meeting at common axis Mathematical Construct: Manifold Stratified Space Manifolds (planes) of Different Dimensions Glued Together

108 UNC, Stat & OR Three quadrants meeting at common axis Mathematical Construct: Manifold Stratified Space Manifolds (planes) of Different Dimensions Glued Together

109 UNC, Stat & OR ‘Connectivity’ of T-4

110 UNC, Stat & OR ‘Connectivity’ of T-4  Cone Structure (Reflects edge lengths > 0)

111 UNC, Stat & OR ‘Connectivity’ of T-4  Cone Structure  Star (0) Tree (At Origin)

112 UNC, Stat & OR ‘Connectivity’ of T-4  Cone Structure  Star (0) Tree  Single Edge Trees (On Boundary Lines)

113 UNC, Stat & OR ‘Connectivity’ of T-4  Cone Structure  Star (0) Tree  Single Edge Trees  Full (2 Edge) Trees (On Planes)

114 UNC, Stat & OR ‘Connectivity’ of T-4 Each Line Connects to 3 planes (# compatible edges)

115 UNC, Stat & OR ‘Connectivity’ of T-4 Each Line Connects to 3 planes (# compatible edges)

116 UNC, Stat & OR Geodesic Paths Given 2 trees, Shortest path Some math: Can show unique in this space Both geodesics & shortest paths

117 UNC, Stat & OR Geodesic Examples, T-4 Thanks to Megan Owen Very Interesting Geometry

118 UNC, Stat & OR Geodesic Examples, T-4 Thanks to Megan Owen Recall: Manifold Stratified Space

119 UNC, Stat & OR Geodesic Examples, T-4 Thanks to Megan Owen Recall: Manifold Stratified Space 2-d Strata (Orthants)

120 UNC, Stat & OR Geodesic Examples, T-4 Thanks to Megan Owen Recall: Manifold Stratified Space 2-d Strata (Orthants) Glued With 1-d Strata (Lines)

121 UNC, Stat & OR Geodesic Examples, T-4 Thanks to Megan Owen Recall: Manifold Stratified Space 0-d Stratum (Origin = Star Tree)

122 UNC, Stat & OR Geodesic Examples, T-4 Thanks to Megan Owen 3-Orthant Geodesic (angle < 180 o )

123 UNC, Stat & OR Geodesic Examples, T-4 Thanks to Megan Owen Cone Path Geodesic (angle > 180 o )

124 UNC, Stat & OR Geodesic Examples, T-4 Thanks to Megan Owen Endpoints In Same Orthants

125 UNC, Stat & OR Geodesic Paths Given 2 trees, Shortest path between is called the geodesic Fast Computation (polynomial time): Owen & Provan (2011)

126 UNC, Stat & OR The Geodesic Path Algorithm Owen & Provan (2011)’s polynomial algorithm for finding geodesics between n-trees

127 UNC, Stat & OR The Geodesic Path Algorithm Owen & Provan (2011)’s polynomial algorithm for finding geodesics between n-trees:  Start with the cone path connecting the two trees through the origin (“star tree”).

128 UNC, Stat & OR Geodesic Paths in Tree Space Thanks to Sean Skwerer

129 UNC, Stat & OR The Geodesic Path Algorithm Owen & Provan (2011)’s polynomial algorithm for finding geodesics between n-trees:  Start with the cone path connecting the two trees through the origin (“star tree”).  Successively “slide” the path into successively larger sets of orthants, each time decreasing the length of the path.

130 UNC, Stat & OR Geodesic Paths in Tree Space

131 UNC, Stat & OR Geodesic Paths in Tree Space

132 UNC, Stat & OR The Geodesic Path Algorithm Owen & Provan (2011)’s polynomial algorithm for finding geodesics between n-trees:  Start with the cone path connecting the two trees through the origin (“star tree”).  Successively “slide” the path into successively larger sets of orthants, each time decreasing the length of the path.  Stop when shortest path is found.

133 UNC, Stat & OR Geodesics for Artery Trees,...,,

134 UNC, Stat & OR Geodesics for Artery Trees To illustrate geodesics Study trees along geodesic, From Case 2 To Case 3 Common Edges

135 UNC, Stat & OR March Along 2 to 3 Geodesic Thanks to Sean Skwerer

136 UNC, Stat & OR March Along 2 to 3 Geodesic Thanks to Sean Skwerer Leaf Node Numbers

137 UNC, Stat & OR March Along 2 to 3 Geodesic Thanks to Sean Skwerer Interior Edge Lengths

138 UNC, Stat & OR March Along 2 to 3 Geodesic Thanks to Sean Skwerer Distance From Root

139 UNC, Stat & OR March Along 2 to 3 Geodesic Thanks to Sean Skwerer Show Interior Edges Midway Between Children

140 UNC, Stat & OR March Along 2 to 3 Geodesic Thanks to Sean Skwerer Unfortunate Consequence: Crossing Branches General Problem Embedding 3-d Trees in 2-d

141 UNC, Stat & OR March Along 2 to 3 Geodesic

142 UNC, Stat & OR March Along 2 to 3 Geodesic

143 UNC, Stat & OR March Along 2 to 3 Geodesic

144 UNC, Stat & OR March Along 2 to 3 Geodesic

145 UNC, Stat & OR March Along 2 to 3 Geodesic

146 UNC, Stat & OR March Along 2 to 3 Geodesic

147 UNC, Stat & OR March Along 2 to 3 Geodesic

148 UNC, Stat & OR March Along 2 to 3 Geodesic

149 UNC, Stat & OR March Along 2 to 3 Geodesic

150 UNC, Stat & OR March Along 2 to 3 Geodesic

151 UNC, Stat & OR March Along 2 to 3 Geodesic

152 UNC, Stat & OR March Along 2 to 3 Geodesic

153 UNC, Stat & OR March Along 2 to 3 Geodesic

154 UNC, Stat & OR March Along 2 to 3 Geodesic

155 UNC, Stat & OR March Along 2 to 3 Geodesic

156 UNC, Stat & OR March Along 2 to 3 Geodesic

157 UNC, Stat & OR March Along 2 to 3 Geodesic

158 UNC, Stat & OR March Along 2 to 3 Geodesic

159 UNC, Stat & OR March Along 2 to 3 Geodesic

160 UNC, Stat & OR March Along 2 to 3 Geodesic

161 UNC, Stat & OR March Along 2 to 3 Geodesic

162 UNC, Stat & OR March Along 2 to 3 Geodesic

163 UNC, Stat & OR March Along 2 to 3 Geodesic

164 UNC, Stat & OR March Along 2 to 3 Geodesic

165 UNC, Stat & OR March Along 2 to 3 Geodesic

166 UNC, Stat & OR March Along 2 to 3 Geodesic

167 UNC, Stat & OR March Along 2 to 3 Geodesic

168 UNC, Stat & OR March Along 2 to 3 Geodesic

169 UNC, Stat & OR March Along 2 to 3 Geodesic

170 UNC, Stat & OR March Along 2 to 3 Geodesic Count of # of Nodes

171 UNC, Stat & OR March Along 2 to 3 Geodesic Count of # of Nodes, as Function of Geodesic Step

172 UNC, Stat & OR March Along 2 to 3 Geodesic Also Total Branch Length

173 UNC, Stat & OR Geodesics for Artery Trees Summarize Lessons: Very few Common Edges (only 5) Most edges swap out

174 UNC, Stat & OR Geodesics for Artery Trees Summarize Lessons: Very few Common Edges (only 5) Most edges swap out Edges get much shorter (bottom plot) Recall this Later

175 UNC, Stat & OR Geodesics for Artery Trees Summarize Lessons: Very few Common Edges (only 5) Most edges swap out Edges get much shorter (bottom plot) Recall this Later # of edges roughly constant (middle plot)

176 UNC, Stat & OR Euclidean Orthant Approach Reference for More: Skwerer et al (2013) Some Related Probability Theory: Hotz et al (2013)

177 UNC, Stat & OR Blood vessel tree data Big Picture: 4 Approaches 1. Purely Combinatorial 2. Euclidean Orthant (Phylogenetics) 3. Dyck Path 4. Persistent Homologies

178 UNC, Stat & OR Blood vessel tree data Persistent Homology Approach Topological Data Analysis Bendich et al (2014) Gave Deepest Results to Date

179 UNC, Stat & OR Carry Away Concept OODA is more than a “framework” It Provides a Focal Point Highlights Pivotal Choices: What should be the Data Objects? How should they be Represented?