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.

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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 10, 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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An investment by:

Workshop Aim:Kickoff Ongoing Research

(through whole program year)

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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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Goals:

• Collaborative Research

• Among unexpected partners

Our hope:

• This group unusually well suited for

this

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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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• Functional Data Analysis• Time Dynamics• Image Analysis• Trees as Data• Shape and Manifold Data

fMRI


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• Functional Data Analysis• Time Dynamics• Image Analysis• Trees as Data DTI• Shape and Manifold Data


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• Functional Data Analysis• Time Dynamics• Image Analysis Brain

Development• Trees as Data • Shape and Manifold Data


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• Functional Data Analysis• Time Dynamics “Atlas” of Human

Body• Image Analysis• Trees as Data • Shape and Manifold Data


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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 10, 2023

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Acronym HistoryAcronym History

Original SAMSI Proposal:Object Oriented Data Analysis

(OODA)

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Original SAMSI Proposal:Object Oriented Data Analysis

(OODA)

SAMSI Directors’ Suggestion:Analysis of Object Oriented Data

(AOOD)

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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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PG 1 PG 2 PG 3


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PG 1 PG 2 PG 3


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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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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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Marron’s brain:

MRA view

“A” for

“Angiography”

Finds blood

vessels

(show up as white)

Track through 3d

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Marron’s brain:

MRA view

“A” for

“Angiography”

Finds blood

vessels

(show up as white)

Track through 3d

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Marron’s brain:

MRA view

“A” for

“Angiography”

Finds blood

vessels

(show up as white)

Track through 3d

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Marron’s brain:

MRA view

“A” for

“Angiography”

Finds blood

vessels

(show up as white)

Track through 3d

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Marron’s brain:

MRA view

“A” for

“Angiography”

Finds blood

vessels

(show up as white)

Track through 3d

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Marron’s brain:

MRA view

“A” for

“Angiography”

Finds blood

vessels

(show up as white)

Track through 3d

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Marron’s brain:

From MRA

Segment tree

of vessel segments

Using tube tracking

Bullitt and Aylward (2002)

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Marron’s brain:

From MRA

Reconstruct trees

in 3d

Rotate to view

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Marron’s brain:

From MRA

Reconstruct trees

in 3d

Rotate to view

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Marron’s brain:

From MRA

Reconstruct trees

in 3d

Rotate to view

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Marron’s brain:

From MRA

Reconstruct trees

in 3d

Rotate to view

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Marron’s brain:

From MRA

Reconstruct trees

in 3d

Rotate to view

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Marron’s brain:

From MRA

Reconstruct trees

in 3d

Rotate to view

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Now look over many people (data

objects)

Structure of population (understand

variation?)

PCA in strongly non-Euclidean Space???

, ... ,,

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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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Big Picture: 3 Approaches

1.Purely Combinatorial

2.Folded Euclidean

3.Dyck Path

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2.Folded Euclidean

3.Dyck Path

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Possible focus of analysis:

• Connectivity structure only (topology)

• Location, size, orientation of segments

• Structure within each vessel segment

, ... ,,

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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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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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Recall from above:

Marron’s brain:

Focus on back

Connectivity (topology) only

(also consider right & left)

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Present Focus:

Topology only

Raw data as trees

Marron’s

reduced tree

Back tree only

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Topology only

E.g. Back Trees

Full Population

Study as movie

Understand

variation?

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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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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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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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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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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:


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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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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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Data Analytic Goals: Age, Gender

See

these?

No…

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Directly study age PC scores

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• 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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PC1

- Not Sig’t

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PC2

- Left Sig’t

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Conclusions:

- No Strong Age Connection

- Significant Connection for:

- Descendants

- Left

- PC2

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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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Smoothing on Tree Space?


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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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Setting: Connectivity & Length

Background: Phylogenetic Trees

Major Restriction: Need common

leaves

Big Payoff: Data space nearly

Euclidean

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Euclidean

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Euclidean

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Euclidean

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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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Marron’s brain:

From MRA

Reconstruct trees

in 3d

Rotate to view

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Vessel LocationsVessel Locations

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Common ColorCommon Color

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Cortical Surface & Cortical Surface & LandmarksLandmarks

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Landmarks and VesselsLandmarks and Vessels

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Attach Landmarks & Attach Landmarks & SubtreesSubtrees

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Highlight OprhansHighlight Oprhans

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Trim OprhansTrim Oprhans

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Final Tree (common Final Tree (common leaves)leaves)

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• 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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Statistics on Folded EuclideanTree Space?


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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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Setting:

• Start with connectivity only

• Second include lengths

• Should be generalizable

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Idea:

• Represent trees as functions

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Idea:


• Common device in probability

theory

• Used for limiting distributions

• Gives access to Brownian Motion

limits

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Idea:


• 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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Idea:


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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

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Tree 1,2


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Tree 1,2,3


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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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Now, we show how to transform the second tree as curve.


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Now, we show how to transform the third tree as curve.


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Some Brain Data PointsSome Brain Data Points(as corresponding trees)(as corresponding trees)

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Raw Brain Data (as curves)Raw Brain Data (as curves)

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Raw Brain Data - ZoomedRaw Brain Data - Zoomed

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Raw Brain Data - ZoomedRaw Brain Data - Zoomed

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More on Dyck PathTree Space?


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Requests of you:• Look for more of these• Discuss with others • Bring up on Wednesday Afternoon• Join in on Thursday +

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.

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euclidean data spaces

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noneuclidean slide