Object Orie’d Data Analysis, Last Time

Object Orie’d Data Analysis, Last Time

• Discrimination for manifold data (Sen)– Simple Tangent plane SVM

– Iterated TANgent plane SVM

– Manifold SVM

• Interesting point:

Analysis done really in the manifold– Not just in projected tangent plane

– Deeper than Principal Geodesic Analysis?

• Manifold version of DWD?

Mildly Non-Euclidean Spaces

Useful View of Manifold Data: Tangent Space

Center:Frechét Mean

Reason forterminology“mildly nonEuclidean”

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


Motivating Example:

• Blood Vessel Trees in Brains

• From Dr. Elizabeth Bullitt– Dept. of Neurosurgery, UNC

• Segmented from MRAs

• Study population of trees

(Forest?)

http://casilab.med.unc.edu/Bullitt_Home.htm

Blood vessel tree data

Marron’s brain:

MRI view

Single Slice

From 3-d Image


Marron’s brain:

MRA view

“A” for Angiography”

Finds blood vessels

(show up as white)

Track through 3d


Marron’s brain:

MRA view


Finds blood vessels

(show up as white)

Track through 3d


Marron’s brain:

MRA view


Finds blood vessels

(show up as white)

Track through 3d


Marron’s brain:

MRA view


Finds blood vessels

(show up as white)

Track through 3d


Marron’s brain:

MRA view


Finds blood vessels

(show up as white)

Track through 3d


Marron’s brain:

MRA view


Finds blood vessels

(show up as white)

Track through 3d


Marron’s brain:

From MRA

Segment tree

of vessel segments

Using tube tracking

Bullitt and Aylward (2002)


Marron’s brain:

From MRA

Reconstruct trees

in 3d

Rotate to view


Marron’s brain:

From MRA

Reconstruct trees

in 3d

Rotate to view


Marron’s brain:

From MRA

Reconstruct trees

in 3d

Rotate to view


Marron’s brain:

From MRA

Reconstruct trees

in 3d

Rotate to view


Marron’s brain:

From MRA

Reconstruct trees

in 3d

Rotate to view


Marron’s brain:

From MRA

Reconstruct trees

in 3d

Rotate to view


Now look over many people (data objects)

Structure of population (understand variation?)

PCA in strongly non-Euclidean Space???

, ... ,,


The tree team:

Very Interdsciplinary

Neurosurgery: Bullitt, Ladha

Statistics: Wang, Marron

Optimization: Aydin, Pataki


Possible focus of analysis:

• Connectivity structure only (topology)

• Location, size, orientation of segments

• Structure within each vessel segment

, ... ,,


Present Focus:

Topology only

Already challenging

Later address others

Then add attributes

To tree nodes

And extend analysis


Recall from above:

Marron’s brain:

Focus on back

Connectivity only


Present Focus:

Topology only

Raw data as trees

Marron’s

reduced tree

Back tree only


Topology only

E.g. Back Trees

Full Population

Study as movie

Understand variation?


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)

Mildly Non-Euclidean Spaces

Useful View of Manifold Data: Tangent Space

Center:Frechét Mean

Reason forterminology“mildly nonEuclidean”


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


Appropriate metrics on tree space:

Wang and Marron (2007)

• Depends on:– Tree structure

– And nodal attributes

• Won’t go further here

• But gives appropriate Fréchet mean


Appropriate metrics on tree space:


• For topology only (studied here):– Use Hamming Distance

– Just number of nodes not in common

• Gives appropriate Fréchet mean


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

Illust’n of PCA View: PC1 Projections

Illust’n of PCA View: Projections on PC1,2 plane

Source Batch Adj: PC 1-3 & DWD direction

Source Batch Adj: DWD Source Adjustment


PCA on Tree Space?

Key Ideas:

• 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)


PCA on Tree Space: Treeline

• Best 1-d representation of data

Basic idea:

• From some starting tree

• Grow only in 1 “direction”



• Best 1-d representation of data

Problem: Hard to compute

• In particular: to solve optimization problem


• Maximum 4 vessel trees

• Hard to tackle serious trees

(e.g. blood vessel trees)



Problem: Hard to compute

Solution: Burḉu Aydin & Gabor Pataki

(linear time algorithm)

(based on clever

“reformulation” of problem)

Description coming in Participant Presentation

PCA for blood vessel tree data

PCA on Tree Space: Treelines Interesting to compare:• Population of Left Trees• Population of Right Trees• Population of Back TreesAnd to study 1st, 2nd, 3rd & 4th treelines


Study

“Directions”

1, 2, 3, 4

For sub-

populations

B, L, R

(interpret

later)



Next represent data as projections

• Define as closest point in tree line

(same as Euclidean PCA)

• Have corresponding score

(length of projection along line)

• And analog of residual

(distance from data point to projection)

PCA for blood vessel tree dataRaw Data & Treelines, PC1, PC2, PC3:


Projections, Scores, Residuals


Cumulative Scores, Residuals


Now look

deeper at

“Directions”

1, 2, 3, 4

For sub-

populations

B, L, R


Notes on Treeline Directions:• PC1 always to left• BACK has most variation to right

(PC2)• LEFT has more varia’n to 2nd level

(PC2)• RIGHT has more var’n to 1st level

(PC2)See these in the data?


Notes:PC1 - leftBACK - right LEFT 2nd levRIGHT 1st levSee these??


Individual (each PC separately) Scores Plot


Identify this person


Identify this person PC Scores: 8,9,3,5


Identify this person (PC Scores: 8,9,3,5):

Red = older

0 = Female

Note:

color ~ age


Identify this person (PC Scores: 8,9,3,5):

Red = older

0 = Female

Note:

color ~ age


Identify this person (PC scores 1,10,1,1)


Identify this person (PC scores 1,10,1,1)


Explain strange (low score) correlation?


Explain strange (low score) correlation?

Revisit Treelines:

Small PC1 Score Small PC3 Score

Small PC1 Score Small PC4 Score


Individual (each PC sep’ly) Residuals Plot


Individual (each PC sep’ly) Residuals Plot

• Very strongly correlated

• Shows much variation not explained by PCs

(data are very rich)

• Note age coloring useful

Younger (bluer) more variation

Older (redder) less variation


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?


Data Analytic Goals: Age, Gender

See

these?

No…


Alternate View: Cumulative Scores


Alternate View: Cumulative Scores

• Always below 45 degree line

• Better separation of age or gender?

(doesn’t seem like it)

• This makes it easy to find:

• Best represented case

• Worst represented case


Cum. Scores: Best repr’ed case (10,19,20)


Cum. Scores: Best repr’ed case (10,19,20)

PC1 & PC2

Scores

Very Large


Cum. Scores: Worst repr’ed case (3,5,6)


Cum. Scores: Worst repr’ed case (3,5,6)

Fairly small

tree

Growth in

unusual

directions


Directly study age PC scores


Directly study age PC scores

• Graphic highlights potential connections

• But no strong correlations

• PC3 is strongest of weak lot

(less young & old for large PC3 score)


Overall Impression:

Interesting new 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…)

Documents

Object Orie’d Data Analysis, Last Time