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Transcript of 1 UNC, Stat & OR SAMSI OODA Workshop SAMSI OODA Workshop Dyck path correspondence and the...
1
UNC, Stat & OR
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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UNC, Stat & OR
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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UNC, Stat & OR
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