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SAMSI OODA Workshop

Dyck path correspondence and the statistical analysis of Brain

vascular networksShankar Bhamidi, J.S.Marron, Dan Shen,

Haipeng Shen

UNC Chapel Hill

September 14, 2009

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Overview of today’s talk

(Very) Brief introduction to the data Dyck path or Harris correspondence between trees

and functions Modern theory of random trees Exploratory Data Analysis and implications Open problems: some incoherent thoughts

Modeling aspects: Natural probability models of spatial trees? (ISE)

Other datasets of trees?

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Basic take home messages

Last decade has witnessed an explosion in the study of Random tree models in the probability community Many different techniques, universality results Many interesting spatial models

Probability

Large amount of data from many fields Biology (brain networks, lung pathways);

Phylogenetics; “Actual trees” (root pathways) Amazing challenges at all levels (modeling,

probabilistic analysis, statistical methodology, data analysis)

Statistics

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Data Background

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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Strongly Non-Euclidean Spaces

Trees as Data Objects

From Graph Theory:

• Graph is set of nodes and edges• Tree has root and direction and

leaves

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Blood vessel tree data

From MRA

Segment tree

vessel segments

Using tube tracking

Bullitt and Aylward (2002)

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Goal understand population properties:

PCA: Main sources of variation in the data?

Interpretation? (e.g. age, gender, occupation?)

Discrimination / Classification Prediction Models of spatial trees?

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Dyck path Correspondence for one tree

Tree 1

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Dyck path Correspondence for one tree

Tree 1

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Dyck path Correspondence for one tree

Tree 1

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Dyck path Correspondence for one tree

Tree 1

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Dyck path Correspondence for one tree

Tree 1

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Dyck path Correspondence for one tree

Tree 1

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Dyck path Correspondence for one tree

Tree 1

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Dyck path Correspondence for one tree

Tree 1

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Dyck path Correspondence for one tree

Tree 1

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Dyck Path correspondence continued

One of the foremost methods in probability for analysis of random trees.

Tremendous array of random tree models arising from many different fields e.g. CS, phylogenetics, mathematics, statistical physics

Consider a “random tree” on n vertices Rescale each edge by some factor (turns out 1/√n

is the right factor)What happens?

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Central Result

Theorem [Aldous 90’s]: For many (most?) of the known models of random trees the Dyck path converges to standard Brownian Excursion. This also implies that the trees themselves converge to a random metric space (random fractal) called the Continuum random tree.

Shall come back to this when we look at the spatial aspect.

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Basic intuition

Where does one get such results? Harris: Consider a branching process with geometric

(1/2) offspring This model is “critical” (mean # of offspring=1) Condition on size of the tree when the branching

process dies out to be n. Consider the Dyck path of this tree Has same distribution as a simple random walk

started at 0, coming back to 0 at time 2(n-1) and always above the orign otherwise

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In pictures

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In pictures

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In pictures

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In pictures

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Our data

Have data on a number of trees

Dyck path transformation for all of themExploratory Data Analysis

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Example 1, Assume that we have three following tree data

Tree 1 Tree 2 Tree 3

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Support tree: union of three tree

Tree 1 Tree 2 Tree 3

Tree 1

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Support tree: union of three tree

Tree 1 Tree 2 Tree 3

Tree 1,2

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Support tree: union of three tree

Tree 1 Tree 2 Tree 3

Tree 1,2,3

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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 first tree as curve.

Tree 1/ Support Tree

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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 first tree as curve.

Tree 1/ Support Tree

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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 first tree as curve.

Tree 1/ Support Tree

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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 first tree as curve.

Tree 1/ Support Tree

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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 first tree as curve.

Tree 1/ Support Tree

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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.

Tree 2/ Support Tree

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

Tree 2/ Support Tree

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

Tree 2/ Support Tree

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

Tree 2/ Support Tree

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

Tree 2/ Support Tree

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

Tree 2/ Support Tree

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

Tree 2/ Support Tree

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

Tree 2/ Support Tree

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

Tree 2/ Support Tree

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

Tree 2/ Support Tree

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

Tree 2/ Support Tree

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

Tree 3/ Support Tree

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

Tree 3/ Support Tree

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

Tree 3/ Support Tree

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

Tree 3/ Support Tree

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

Tree 3/ Support Tree

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

Tree 3/ Support Tree

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

Tree 3/ Support Tree

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

Tree 3/ Support Tree

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

Tree 3/ Support Tree

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

Tree 3/ Support Tree

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

Tree 3/ Support Tree

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Advantages of this encoding

If we are only interested in topological aspects then mathematically this is reasonable

Main reason: Suppose f, g are encodings of two trees, s and t, then the sup norm between the two functions bounds the Gromov-Haussdorf distance

However a number of issues as well

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Actual Data

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Raw Brain Data - Zoomed

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Raw Brain Data - Zoomed

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Some Brain Data Points(as corresponding trees)

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Some Brain Data Points(as corresponding trees)

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Some Brain Data Points(as corresponding trees)

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Some Brain Data Points(as corresponding trees)

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Some Brain Data Points(as corresponding trees)

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Some Brain Data Points(as corresponding trees)

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Data Representation- Youngest

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Data Representation- oldest

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Average Tree-Curve and picture of the average tree

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Illust’n of PCA View: PC1 Projections

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PCA Pictures of trees that we get when we move in PC1 direction

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PCA Pictures of trees that we get when we move in PC1 direction

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PCA Pictures of trees that we get when we move in PC1 direction

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PCA Pictures of trees that we get when we move in PC1 direction

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PCA Pictures of trees that we get when we move in PC1 direction

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PCA Pictures of trees that we get when we move in PC1 direction

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PCA Pictures of trees that we get when we move in PC1 direction

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PCA Pictures of trees that we get when we move in PC1 direction

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PCA Pictures of trees that we get when we move in PC1 direction

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PCA Pictures of trees that we get when we move in PC2 direction

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PCA Pictures of trees that we get when we move in PC2 direction

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PCAPictures of trees that we get when we move in PC2 direction

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PCA Pictures of trees that we get when we move in PC2 direction

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PCA Pictures of trees that we get when we move in PC2 direction

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PCA Pictures of trees that we get when we move in PC2 direction

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PCA Pictures of trees that we get when we move in PC2 direction

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PCAPictures of trees that we get when we move in PC2 direction

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PCAPictures of trees that we get when we move in PC2 direction

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DWD

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DWD

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DWD

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DWD

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DWD

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DWD

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DWD

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DWD

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DWD

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DWD/relabeling

random relabeling: Suppose we randomly relabel each tree as male or female.

How does the DWD direction behave?

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DWD/relabelling

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DWD/relabelling

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DWD/relabelling

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DWD/relabelling

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DWD/relabelling

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DWD/relabelling

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DWD/relabelling

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DWD/relabelling

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DWD/relabelling

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Implications

“Eyeballing” the data, the PC1 directions (and PC2) do not seem to be capturing variation in the data

Because of encoding all the trees to form a support tree?

Perhaps because inherently PCA works well in the Euclidean regime?

Path of Dyck paths a weird subset of function space? Any math theory that can be developed about

families of large trees? Modeling of these trees?

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Blood 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 data

Marron’s brain:

From MRA

Reconstruct trees

in 3d

Rotate to view

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Blood vessel tree data

Marron’s brain:

From MRA

Reconstruct trees

in 3d

Rotate to view

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Blood vessel tree data

Marron’s brain:

From MRA

Reconstruct trees

in 3d

Rotate to view

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Blood vessel tree data

Marron’s brain:

From MRA

Reconstruct trees

in 3d

Rotate to view

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Blood vessel tree data

Marron’s brain:

From MRA

Reconstruct trees

in 3d

Rotate to view

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Blood vessel tree data

Marron’s brain:

From MRA

Reconstruct trees

in 3d

Rotate to view

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Thoughts I: Probabilistic models of spatial trees?

What are natural models of spatial trees such as those in this talk?

At least two natural directions to proceed in ISE (Integrated Superbrownian Excursion): Arising

from modelling of critical random systems in euclidean space

Engineering and biological principles of flow distribution: (Constructal theory)

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ISE: Integrated Superbrownian excursion

Formulated in the late 90s by Aldous Has now come to be one of the standard models of

spatial trees Arises as the scaling limit of many different systems Example: Random trees on the integer lattice Critical contact process in high dimensions etc Thought to be the scaling limit of many systems at

criticality

Use Standard Brownian excursion and Brownian motion to construct a random tree in 3 (or higher dimensions)

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ISE: in pictures

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ISE in pictures

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ISE in pictures

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ISE in pictures

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ISE

Any notion of data driven ISE?

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Blood vessel tree data

Notion of ISE on the

sphere?

Notion of ISE where the

Brownian motion has

some sort of drift?

How does one estimate

drift from data?

Model of thickness on

edges to the data?

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Other examples of tree data?

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Data on actual root systems

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PCA and Random Walks on Tree space?

In this study we tried usual notion of PCA

Ok when data are “Gaussian in nature”

Tree space intuitively very non-linear

Can one use random walks to explore this space?

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Intuition

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Random walk on data points

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Folded Euclidean Approach