SAMSI AOOD Opening Workshop
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11
UNC, Stat & OR
SAMSI AOOD Opening WorkshopSAMSI AOOD Opening Workshop
Tutorial
OODA of Tree Structured Objects
J. S. Marron
Dept. of Statistics and O. R., UNC
April 21, 2023
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Workshop Big PictureWorkshop Big Picture
An investment by:
• Provided Funding to Bring Us Together
• Has Specific Goal:Generating Collaborative Research
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Workshop Big PictureWorkshop Big Picture
An investment by:
Workshop Aim:Kickoff Ongoing Research
(through whole program year)
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Workshop Big PictureWorkshop 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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Working GroupsWorking 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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Working GroupsWorking Groups
Goals:
• Collaborative Research
• Among unexpected partners
Our hope:
• This group unusually well suited for
this
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Working GroupsWorking 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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Working GroupsWorking 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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Working GroupsWorking 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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Working GroupsWorking 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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Working GroupsWorking 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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Working GroupsWorking 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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Object Oriented Data Object Oriented Data AnalysisAnalysis
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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An Aside on AcronymsAn Aside on Acronyms
What is it?
OODAor
AOOD
???
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SAMSI AOOD Opening WorkshopSAMSI AOOD Opening Workshop
Tutorial
OODA of Tree Structured Objects
J. S. Marron
Dept. of Statistics and O. R., UNC
April 21, 2023
1616
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Acronym HistoryAcronym History
Original SAMSI Proposal:Object Oriented Data Analysis
(OODA)
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Acronym HistoryAcronym History
Original SAMSI Proposal:Object Oriented Data Analysis
(OODA)
SAMSI Directors’ Suggestion:Analysis of Object Oriented Data
(AOOD)
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Acronym HistoryAcronym 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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An Aside on AcronymsAn Aside on Acronyms
What is it?
OODAor
AOOD
Suggestion: Treat these as synonyms
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Object Oriented Data Object Oriented Data AnalysisAnalysis
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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Euclidean Data SpacesEuclidean Data Spaces
Data are vectors, in
Effective (and Traditional) Analysis:• Linear Methods• Mean• Covariance• Principal Component Analysis• Gaussian Distribution
d
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Euclidean Data SpacesEuclidean 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
d
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Non - Euclidean Data Non - Euclidean Data SpacesSpaces
“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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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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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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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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Non - Euclidean Data Non - Euclidean Data SpacesSpaces
What is “Strongly Non-Euclidean” Case?
Trees as Data
Special Challenge:
• No Tangent Plane
• Must Re-Invent
Data Analysis
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Strongly Non-Euclidean Strongly Non-Euclidean SpacesSpaces
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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Strongly Non-Euclidean Strongly Non-Euclidean SpacesSpaces
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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Blood vessel tree dataBlood vessel tree data
Marron’s brain:
MRI view
Single Slice
From 3-d Image
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Blood vessel tree dataBlood vessel tree data
Marron’s brain:
MRA view
“A” for
“Angiography”
Finds blood
vessels
(show up as white)
Track through 3d
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Blood vessel tree dataBlood vessel tree data
Marron’s brain:
MRA view
“A” for
“Angiography”
Finds blood
vessels
(show up as white)
Track through 3d
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Blood vessel tree dataBlood vessel tree data
Marron’s brain:
MRA view
“A” for
“Angiography”
Finds blood
vessels
(show up as white)
Track through 3d
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Blood vessel tree dataBlood vessel tree data
Marron’s brain:
MRA view
“A” for
“Angiography”
Finds blood
vessels
(show up as white)
Track through 3d
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Blood vessel tree dataBlood vessel tree data
Marron’s brain:
MRA view
“A” for
“Angiography”
Finds blood
vessels
(show up as white)
Track through 3d
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Blood vessel tree dataBlood vessel tree data
Marron’s brain:
MRA view
“A” for
“Angiography”
Finds blood
vessels
(show up as white)
Track through 3d
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Blood vessel tree dataBlood vessel tree data
Marron’s brain:
From MRA
Segment tree
of vessel segments
Using tube tracking
Bullitt and Aylward (2002)
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Blood vessel tree dataBlood vessel tree data
Marron’s brain:
From MRA
Reconstruct trees
in 3d
Rotate to view
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Blood vessel tree dataBlood vessel tree data
Marron’s brain:
From MRA
Reconstruct trees
in 3d
Rotate to view
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Blood vessel tree dataBlood vessel tree data
Marron’s brain:
From MRA
Reconstruct trees
in 3d
Rotate to view
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Blood vessel tree dataBlood vessel tree data
Marron’s brain:
From MRA
Reconstruct trees
in 3d
Rotate to view
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Blood vessel tree dataBlood vessel tree data
Marron’s brain:
From MRA
Reconstruct trees
in 3d
Rotate to view
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Blood vessel tree dataBlood vessel tree data
Marron’s brain:
From MRA
Reconstruct trees
in 3d
Rotate to view
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Blood vessel tree dataBlood 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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Blood vessel tree dataBlood 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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Blood vessel tree dataBlood vessel tree data
Big Picture: 3 Approaches
1.Purely Combinatorial
2.Folded Euclidean
3.Dyck Path
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Blood vessel tree dataBlood vessel tree data
Big Picture: 3 Approaches
1.Purely Combinatorial
2.Folded Euclidean
3.Dyck Path
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Blood vessel tree dataBlood 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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Blood vessel tree dataBlood 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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Blood vessel tree dataBlood vessel tree data
Topological Representation: Each Vessel Segment (up to 1st
Split)
is a node Split Segments are child nodes Connecting lines show
relationship
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Graphical Concept: Support Graphical Concept: Support TreeTree
The union of all trees in data set T.
Consists of the nodes in any tree of T
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Support Tree ExampleSupport Tree Example
Data trees:
Support tree:
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Blood vessel tree dataBlood vessel tree data
Recall from above:
Marron’s brain:
Focus on back
Connectivity (topology) only
(also consider right & left)
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Blood vessel tree dataBlood vessel tree data
Present Focus:
Topology only
Raw data as trees
Marron’s
reduced tree
Back tree only
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Blood vessel tree dataBlood vessel tree data
Topology only
E.g. Back Trees
Full Population
Study as movie
Understand
variation?
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Strongly Non-Euclidean Strongly Non-Euclidean SpacesSpaces
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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Mildly Non-Euclidean Mildly Non-Euclidean SpacesSpaces
Useful View of Manifold Data: Tangent Space
Center:Frechét Mean
Reason forterminology“mildly nonEuclidean”
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Strongly Non-Euclidean Strongly Non-Euclidean SpacesSpaces
Mean of Population of Tree-Structured Data Objects?
Natural approach: Fréchet mean
Requires a metric (distance)
on tree space
n
ii
xxXdX
1
2,minarg
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Strongly Non-Euclidean Strongly Non-Euclidean SpacesSpaces
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 Fréchet mean
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Hamming DistanceHamming Distance
The number of nodes in the symmetric difference of two trees.
An example:
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Hamming DistanceHamming Distance
The two trees drawn on top of each other:
Common nodes: 2
Nodes only in blue tree: 4Nodes only in red tree: 2
So, distance: 4+2=6
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Strongly Non-Euclidean Strongly Non-Euclidean SpacesSpaces
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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Illust’n of PCA View: PC1 Illust’n of PCA View: PC1 ProjectionsProjections
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Illust’n of PCA View: Projections on PC1,2 Illust’n of PCA View: Projections on PC1,2 planeplane
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PCA view: Lung Cancer Microarray PCA view: Lung Cancer Microarray Data Data
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Strongly Non-Euclidean Strongly Non-Euclidean SpacesSpaces
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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PCA on Combinatorial Tree Space?
In Depth Discussion Tuesday Afternoon:
Strongly Non-Euclidean Strongly Non-Euclidean SpacesSpaces
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PCA for blood vessel tree PCA for blood vessel tree datadata
Individual (each PC separately) Scores
Plot
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PCA for blood vessel tree PCA for blood vessel tree datadata
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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PCA for blood vessel tree PCA for blood vessel tree datadata
Data Analytic Goals: Age, Gender
See
these?
No…
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PCA for blood vessel tree PCA for blood vessel tree datadata
Directly study age PC scores
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PCA for blood vessel tree PCA for blood vessel tree datadata
Directly study age PC scores
• Take Deeper Look
• By Fitting Lines
• And doing Hypotest of H0: slope = 0
• Show p-values to assess significance
Compare Thickness & Descendants Corr.
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PCA for blood vessel tree PCA for blood vessel tree datadata
Directly study age PC scores
PC1
- Not Sig’t
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PCA for blood vessel tree PCA for blood vessel tree datadata
Directly study age PC scores
PC2
- Left Sig’t
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PCA for blood vessel tree PCA for blood vessel tree datadata
Directly study age PC scores
Conclusions:
- No Strong Age Connection
- Significant Connection for:
- Descendants
- Left
- PC2
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Strongly Non-Euclidean Strongly Non-Euclidean SpacesSpaces
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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Smoothing in Tree SpaceSmoothing in Tree Space
Question:
How does tree structure change with age?
Approach:
(Gaussian) Kernel Smoothing
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Smoothing in Tree SpaceSmoothing in Tree Space
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Strongly Non-Euclidean Strongly Non-Euclidean SpacesSpaces
Smoothing on Tree Space?
In Depth Discussion Tuesday Afternoon:
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Blood vessel tree dataBlood vessel tree data
Big Picture: 3 Approaches
1.Purely Combinatorial
2.Folded Euclidean
3.Dyck Path
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Folded Euclidean ApproachFolded Euclidean Approach
People:
• Scott Provan
• Sean Skwerer
• Megan Owen
• Martin Styner
• Ipek Oguz
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Folded Euclidean ApproachFolded Euclidean Approach
Setting: Connectivity & Length
Background: Phylogenetic Trees
Major Restriction: Need common
leaves
Big Payoff: Data space nearly
Euclidean
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Folded Euclidean ApproachFolded Euclidean Approach
Big Payoff: Data space nearly
Euclidean
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Folded Euclidean ApproachFolded Euclidean Approach
Big Payoff: Data space nearly
Euclidean
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Folded Euclidean ApproachFolded Euclidean Approach
Big Payoff: Data space nearly
Euclidean
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Folded Euclidean ApproachFolded 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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Blood vessel tree dataBlood vessel tree data
Marron’s brain:
From MRA
Reconstruct trees
in 3d
Rotate to view
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Vessel LocationsVessel Locations
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Vessel LocationsVessel Locations
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Vessel LocationsVessel Locations
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Vessel LocationsVessel Locations
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Vessel LocationsVessel Locations
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Vessel LocationsVessel Locations
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Common ColorCommon Color
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Common ColorCommon Color
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Common ColorCommon Color
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Common ColorCommon Color
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Common ColorCommon Color
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Common ColorCommon Color
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Cortical Surface & Cortical Surface & LandmarksLandmarks
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Cortical Surface & Cortical Surface & LandmarksLandmarks
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Cortical Surface & Cortical Surface & LandmarksLandmarks
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Cortical Surface & Cortical Surface & LandmarksLandmarks
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Cortical Surface & Cortical Surface & LandmarksLandmarks
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Cortical Surface & Cortical Surface & LandmarksLandmarks
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Landmarks and VesselsLandmarks and Vessels
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Landmarks and VesselsLandmarks and Vessels
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Landmarks and VesselsLandmarks and Vessels
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Landmarks and VesselsLandmarks and Vessels
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Landmarks and VesselsLandmarks and Vessels
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Landmarks and VesselsLandmarks and Vessels
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Attach Landmarks & Attach Landmarks & SubtreesSubtrees
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Attach Landmarks & Attach Landmarks & SubtreesSubtrees
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Attach Landmarks & Attach Landmarks & SubtreesSubtrees
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Attach Landmarks & Attach Landmarks & SubtreesSubtrees
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Attach Landmarks & Attach Landmarks & SubtreesSubtrees
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Attach Landmarks & Attach Landmarks & SubtreesSubtrees
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Highlight OprhansHighlight Oprhans
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Highlight OprhansHighlight Oprhans
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Highlight OprhansHighlight Oprhans
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Highlight OprhansHighlight Oprhans
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Highlight OprhansHighlight Oprhans
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Highlight OprhansHighlight Oprhans
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Trim OprhansTrim Oprhans
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Trim OprhansTrim Oprhans
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Trim OprhansTrim Oprhans
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Trim OprhansTrim Oprhans
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Trim OprhansTrim Oprhans
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Trim OprhansTrim Oprhans
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Final Tree (common Final Tree (common leaves)leaves)
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Final Tree (common Final Tree (common leaves)leaves)
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Final Tree (common Final Tree (common leaves)leaves)
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Final Tree (common Final Tree (common leaves)leaves)
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Final Tree (common Final Tree (common leaves)leaves)
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Final Tree (common Final Tree (common leaves)leaves)
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Folded Euclidean ApproachFolded Euclidean Approach
• Next tasks: Statistical Analysis,
e.g.
• Calculation of Mean
• Smoothing over time (w’td mean)
• PCA (“Backwards” approach???)
• Classification (“linear
method” ???)
• Work in Progress
• Heavy & Specialized Optimization
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Strongly Non-Euclidean Strongly Non-Euclidean SpacesSpaces
Statistics on Folded EuclideanTree Space?
In Depth Discussion Tuesday Afternoon:
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Blood vessel tree dataBlood vessel tree data
Big Picture: 3 Approaches
1.Purely Combinatorial
2.Euclidean Orthant
3.Dyck Path
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Dyck Path ApproachDyck Path Approach
People:
• Shankar Bhamidi
• Dan Shen
• Haipeng Shen
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Dyck Path ApproachDyck Path Approach
Setting:
• Start with connectivity only
• Second include lengths
• Should be generalizable
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Dyck Path ApproachDyck Path Approach
Idea:
• Represent trees as functions
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Dyck Path ApproachDyck 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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Dyck Path ApproachDyck 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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Dyck Path ApproachDyck Path Approach
Idea:
• Represent trees as functions
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Dyck Path Example
Example 1, Assume that we have three following tree data
Tree 1 Tree 2 Tree 3
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Support tree: union of trees
Tree 1 Tree 2 Tree 3
Tree 1
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Tree 1 Tree 2 Tree 3
Tree 1,2
Support tree: union of trees
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Tree 1 Tree 2 Tree 3
Tree 1,2,3
Support tree: union of trees
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Transform Tree to Curve
Now, we show how to transform the first tree as curve.
Tree 1/ Support Tree
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Transform Tree to Curve
Now, we show how to transform the first tree as curve.
Tree 1/ Support Tree
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Transform Tree to Curve
Now, we show how to transform the first tree as curve.
Tree 1/ Support Tree
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Transform Tree to Curve
Now, we show how to transform the first tree as curve.
Tree 1/ Support Tree
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Transform Tree to Curve
Now, we show how to transform the first tree as curve.
Tree 1/ Support Tree
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Transform Tree to Curve
Now, we show how to transform the first tree as curve.
Tree 1/ Support Tree
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Transform Tree to Curve
Now, we show how to transform the first tree as curve.
Tree 1/ Support Tree
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Transform Tree to Curve
Now, we show how to transform the first tree as curve.
Tree 1/ Support Tree
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Transform Tree to Curve
Now, we show how to transform the first tree as curve.
Tree 1/ Support Tree
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Transform Tree to Curve
Now, we show how to transform the first tree as curve.
Tree 1/ Support Tree
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Transform Tree to Curve
Now, we show how to transform the first tree as curve.
Tree 1/ Support Tree
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Transform Tree to Curve
Now, we show how to transform the second tree as curve.
Tree 2/ Support Tree
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Transform Tree to Curve
Now, we show how to transform the second tree as curve.
Tree 2/ Support Tree
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Transform Tree to Curve
Now, we show how to transform the second tree as curve.
Tree 2/ Support Tree
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Transform Tree to Curve
Now, we show how to transform the second tree as curve.
Tree 2/ Support Tree
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Transform Tree to Curve
Now, we show how to transform the second tree as curve.
Tree 2/ Support Tree
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Transform Tree to Curve
Now, we show how to transform the second tree as curve.
Tree 2/ Support Tree
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Transform Tree to Curve
Now, we show how to transform the second tree as curve.
Tree 2/ Support Tree
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Transform Tree to Curve
Now, we show how to transform the second tree as curve.
Tree 2/ Support Tree
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Transform Tree to Curve
Now, we show how to transform the second tree as curve.
Tree 2/ Support Tree
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Transform Tree to Curve
Now, we show how to transform the second tree as curve.
Tree 2/ Support Tree
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Transform Tree to Curve
Now, we show how to transform the second tree as curve.
Tree 2/ Support Tree
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Transform Tree to Curve
Now, we show how to transform the third tree as curve.
Tree 3/ Support Tree
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Transform Tree to Curve
Now, we show how to transform the third tree as curve.
Tree 3/ Support Tree
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Transform Tree to Curve
Now, we show how to transform the third tree as curve.
Tree 3/ Support Tree
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Transform Tree to Curve
Now, we show how to transform the third tree as curve.
Tree 3/ Support Tree
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Transform Tree to Curve
Now, we show how to transform the third tree as curve.
Tree 3/ Support Tree
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Transform Tree to Curve
Now, we show how to transform the third tree as curve.
Tree 3/ Support Tree
177177
UNC, Stat & OR
Transform Tree to Curve
Now, we show how to transform the third tree as curve.
Tree 3/ Support Tree
178178
UNC, Stat & OR
Transform Tree to Curve
Now, we show how to transform the third tree as curve.
Tree 3/ Support Tree
179179
UNC, Stat & OR
Transform Tree to Curve
Now, we show how to transform the third tree as curve.
Tree 3/ Support Tree
180180
UNC, Stat & OR
Transform Tree to Curve
Now, we show how to transform the third tree as curve.
Tree 3/ Support Tree
181181
UNC, Stat & OR
Transform Tree to Curve
Now, we show how to transform the third tree as curve.
Tree 3/ Support Tree
182182
UNC, Stat & OR
Some Brain Data PointsSome Brain Data Points(as corresponding trees)(as corresponding trees)
183183
UNC, Stat & OR
Some Brain Data PointsSome Brain Data Points(as corresponding trees)(as corresponding trees)
184184
UNC, Stat & OR
Some Brain Data PointsSome Brain Data Points(as corresponding trees)(as corresponding trees)
185185
UNC, Stat & OR
Some Brain Data PointsSome Brain Data Points(as corresponding trees)(as corresponding trees)
186186
UNC, Stat & OR
Some Brain Data PointsSome Brain Data Points(as corresponding trees)(as corresponding trees)
187187
UNC, Stat & OR
Some Brain Data PointsSome Brain Data Points(as corresponding trees)(as corresponding trees)
188188
UNC, Stat & OR
Raw Brain Data (as curves)Raw Brain Data (as curves)
189189
UNC, Stat & OR
Raw Brain Data - ZoomedRaw Brain Data - Zoomed
190190
UNC, Stat & OR
Raw Brain Data - ZoomedRaw Brain Data - Zoomed
191191
UNC, Stat & OR
Strongly Non-Euclidean Strongly Non-Euclidean SpacesSpaces
More on Dyck PathTree Space?
In Depth Discussion Tuesday Afternoon:
192192
UNC, Stat & OR
Working GroupsWorking 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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