1 unc, stat & or samsi aood opening workshop tutorial ooda of tree structured objects j. s....

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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 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....

Documents

data analysis slide

euclidean data spaces

data shape

shapes data

statistical analysis

traditional analysis

direction data objects

noneuclidean slide