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?