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Transcript of 1 UNC, Stat & OR SAMSI OODA Workshop SAMSI OODA Workshop Dyck path correspondence and the...
![Page 1: 1 UNC, Stat & OR SAMSI OODA Workshop SAMSI OODA Workshop Dyck path correspondence and the statistical analysis of Brain vascular networks Shankar Bhamidi,](https://reader035.fdocuments.in/reader035/viewer/2022062620/551b19085503465e7d8b6449/html5/thumbnails/1.jpg)
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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2
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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3
UNC, Stat & OR
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
![Page 4: 1 UNC, Stat & OR SAMSI OODA Workshop SAMSI OODA Workshop Dyck path correspondence and the statistical analysis of Brain vascular networks Shankar Bhamidi,](https://reader035.fdocuments.in/reader035/viewer/2022062620/551b19085503465e7d8b6449/html5/thumbnails/4.jpg)
4
UNC, Stat & OR
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
![Page 5: 1 UNC, Stat & OR SAMSI OODA Workshop SAMSI OODA Workshop Dyck path correspondence and the statistical analysis of Brain vascular networks Shankar Bhamidi,](https://reader035.fdocuments.in/reader035/viewer/2022062620/551b19085503465e7d8b6449/html5/thumbnails/5.jpg)
5
UNC, Stat & OR
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
![Page 6: 1 UNC, Stat & OR SAMSI OODA Workshop SAMSI OODA Workshop Dyck path correspondence and the statistical analysis of Brain vascular networks Shankar Bhamidi,](https://reader035.fdocuments.in/reader035/viewer/2022062620/551b19085503465e7d8b6449/html5/thumbnails/6.jpg)
6
UNC, Stat & OR
Blood vessel tree data
From MRA
Segment tree
vessel segments
Using tube tracking
Bullitt and Aylward (2002)
![Page 7: 1 UNC, Stat & OR SAMSI OODA Workshop SAMSI OODA Workshop Dyck path correspondence and the statistical analysis of Brain vascular networks Shankar Bhamidi,](https://reader035.fdocuments.in/reader035/viewer/2022062620/551b19085503465e7d8b6449/html5/thumbnails/7.jpg)
7
UNC, Stat & OR
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?
![Page 8: 1 UNC, Stat & OR SAMSI OODA Workshop SAMSI OODA Workshop Dyck path correspondence and the statistical analysis of Brain vascular networks Shankar Bhamidi,](https://reader035.fdocuments.in/reader035/viewer/2022062620/551b19085503465e7d8b6449/html5/thumbnails/8.jpg)
8
UNC, Stat & OR
Dyck path Correspondence for one tree
Tree 1
![Page 9: 1 UNC, Stat & OR SAMSI OODA Workshop SAMSI OODA Workshop Dyck path correspondence and the statistical analysis of Brain vascular networks Shankar Bhamidi,](https://reader035.fdocuments.in/reader035/viewer/2022062620/551b19085503465e7d8b6449/html5/thumbnails/9.jpg)
9
UNC, Stat & OR
Dyck path Correspondence for one tree
Tree 1
![Page 10: 1 UNC, Stat & OR SAMSI OODA Workshop SAMSI OODA Workshop Dyck path correspondence and the statistical analysis of Brain vascular networks Shankar Bhamidi,](https://reader035.fdocuments.in/reader035/viewer/2022062620/551b19085503465e7d8b6449/html5/thumbnails/10.jpg)
10
UNC, Stat & OR
Dyck path Correspondence for one tree
Tree 1
![Page 11: 1 UNC, Stat & OR SAMSI OODA Workshop SAMSI OODA Workshop Dyck path correspondence and the statistical analysis of Brain vascular networks Shankar Bhamidi,](https://reader035.fdocuments.in/reader035/viewer/2022062620/551b19085503465e7d8b6449/html5/thumbnails/11.jpg)
11
UNC, Stat & OR
Dyck path Correspondence for one tree
Tree 1
![Page 12: 1 UNC, Stat & OR SAMSI OODA Workshop SAMSI OODA Workshop Dyck path correspondence and the statistical analysis of Brain vascular networks Shankar Bhamidi,](https://reader035.fdocuments.in/reader035/viewer/2022062620/551b19085503465e7d8b6449/html5/thumbnails/12.jpg)
12
UNC, Stat & OR
Dyck path Correspondence for one tree
Tree 1
![Page 13: 1 UNC, Stat & OR SAMSI OODA Workshop SAMSI OODA Workshop Dyck path correspondence and the statistical analysis of Brain vascular networks Shankar Bhamidi,](https://reader035.fdocuments.in/reader035/viewer/2022062620/551b19085503465e7d8b6449/html5/thumbnails/13.jpg)
13
UNC, Stat & OR
Dyck path Correspondence for one tree
Tree 1
![Page 14: 1 UNC, Stat & OR SAMSI OODA Workshop SAMSI OODA Workshop Dyck path correspondence and the statistical analysis of Brain vascular networks Shankar Bhamidi,](https://reader035.fdocuments.in/reader035/viewer/2022062620/551b19085503465e7d8b6449/html5/thumbnails/14.jpg)
14
UNC, Stat & OR
Dyck path Correspondence for one tree
Tree 1
![Page 15: 1 UNC, Stat & OR SAMSI OODA Workshop SAMSI OODA Workshop Dyck path correspondence and the statistical analysis of Brain vascular networks Shankar Bhamidi,](https://reader035.fdocuments.in/reader035/viewer/2022062620/551b19085503465e7d8b6449/html5/thumbnails/15.jpg)
15
UNC, Stat & OR
Dyck path Correspondence for one tree
Tree 1
![Page 16: 1 UNC, Stat & OR SAMSI OODA Workshop SAMSI OODA Workshop Dyck path correspondence and the statistical analysis of Brain vascular networks Shankar Bhamidi,](https://reader035.fdocuments.in/reader035/viewer/2022062620/551b19085503465e7d8b6449/html5/thumbnails/16.jpg)
16
UNC, Stat & OR
Dyck path Correspondence for one tree
Tree 1
![Page 17: 1 UNC, Stat & OR SAMSI OODA Workshop SAMSI OODA Workshop Dyck path correspondence and the statistical analysis of Brain vascular networks Shankar Bhamidi,](https://reader035.fdocuments.in/reader035/viewer/2022062620/551b19085503465e7d8b6449/html5/thumbnails/17.jpg)
17
UNC, Stat & OR
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?
![Page 18: 1 UNC, Stat & OR SAMSI OODA Workshop SAMSI OODA Workshop Dyck path correspondence and the statistical analysis of Brain vascular networks Shankar Bhamidi,](https://reader035.fdocuments.in/reader035/viewer/2022062620/551b19085503465e7d8b6449/html5/thumbnails/18.jpg)
18
UNC, Stat & OR
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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19
UNC, Stat & OR
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
![Page 20: 1 UNC, Stat & OR SAMSI OODA Workshop SAMSI OODA Workshop Dyck path correspondence and the statistical analysis of Brain vascular networks Shankar Bhamidi,](https://reader035.fdocuments.in/reader035/viewer/2022062620/551b19085503465e7d8b6449/html5/thumbnails/20.jpg)
20
UNC, Stat & OR
In pictures
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21
UNC, Stat & OR
In pictures
![Page 22: 1 UNC, Stat & OR SAMSI OODA Workshop SAMSI OODA Workshop Dyck path correspondence and the statistical analysis of Brain vascular networks Shankar Bhamidi,](https://reader035.fdocuments.in/reader035/viewer/2022062620/551b19085503465e7d8b6449/html5/thumbnails/22.jpg)
22
UNC, Stat & OR
In pictures
![Page 23: 1 UNC, Stat & OR SAMSI OODA Workshop SAMSI OODA Workshop Dyck path correspondence and the statistical analysis of Brain vascular networks Shankar Bhamidi,](https://reader035.fdocuments.in/reader035/viewer/2022062620/551b19085503465e7d8b6449/html5/thumbnails/23.jpg)
23
UNC, Stat & OR
In pictures
![Page 24: 1 UNC, Stat & OR SAMSI OODA Workshop SAMSI OODA Workshop Dyck path correspondence and the statistical analysis of Brain vascular networks Shankar Bhamidi,](https://reader035.fdocuments.in/reader035/viewer/2022062620/551b19085503465e7d8b6449/html5/thumbnails/24.jpg)
24
UNC, Stat & OR
Our data
Have data on a number of trees
Dyck path transformation for all of themExploratory Data Analysis
![Page 25: 1 UNC, Stat & OR SAMSI OODA Workshop SAMSI OODA Workshop Dyck path correspondence and the statistical analysis of Brain vascular networks Shankar Bhamidi,](https://reader035.fdocuments.in/reader035/viewer/2022062620/551b19085503465e7d8b6449/html5/thumbnails/25.jpg)
25
UNC, Stat & OR
Example 1, Assume that we have three following tree data
Tree 1 Tree 2 Tree 3
![Page 26: 1 UNC, Stat & OR SAMSI OODA Workshop SAMSI OODA Workshop Dyck path correspondence and the statistical analysis of Brain vascular networks Shankar Bhamidi,](https://reader035.fdocuments.in/reader035/viewer/2022062620/551b19085503465e7d8b6449/html5/thumbnails/26.jpg)
26
UNC, Stat & OR
Support tree: union of three tree
Tree 1 Tree 2 Tree 3
Tree 1
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27
UNC, Stat & OR
Support tree: union of three tree
Tree 1 Tree 2 Tree 3
Tree 1,2
![Page 28: 1 UNC, Stat & OR SAMSI OODA Workshop SAMSI OODA Workshop Dyck path correspondence and the statistical analysis of Brain vascular networks Shankar Bhamidi,](https://reader035.fdocuments.in/reader035/viewer/2022062620/551b19085503465e7d8b6449/html5/thumbnails/28.jpg)
28
UNC, Stat & OR
Support tree: union of three tree
Tree 1 Tree 2 Tree 3
Tree 1,2,3
![Page 29: 1 UNC, Stat & OR SAMSI OODA Workshop SAMSI OODA Workshop Dyck path correspondence and the statistical analysis of Brain vascular networks Shankar Bhamidi,](https://reader035.fdocuments.in/reader035/viewer/2022062620/551b19085503465e7d8b6449/html5/thumbnails/29.jpg)
29
UNC, Stat & OR
Now, we show how to transform the first tree as curve.
Tree 1/ Support Tree
![Page 30: 1 UNC, Stat & OR SAMSI OODA Workshop SAMSI OODA Workshop Dyck path correspondence and the statistical analysis of Brain vascular networks Shankar Bhamidi,](https://reader035.fdocuments.in/reader035/viewer/2022062620/551b19085503465e7d8b6449/html5/thumbnails/30.jpg)
30
UNC, Stat & OR
Now, we show how to transform the first tree as curve.
Tree 1/ Support Tree
![Page 31: 1 UNC, Stat & OR SAMSI OODA Workshop SAMSI OODA Workshop Dyck path correspondence and the statistical analysis of Brain vascular networks Shankar Bhamidi,](https://reader035.fdocuments.in/reader035/viewer/2022062620/551b19085503465e7d8b6449/html5/thumbnails/31.jpg)
31
UNC, Stat & OR
Now, we show how to transform the first tree as curve.
Tree 1/ Support Tree
![Page 32: 1 UNC, Stat & OR SAMSI OODA Workshop SAMSI OODA Workshop Dyck path correspondence and the statistical analysis of Brain vascular networks Shankar Bhamidi,](https://reader035.fdocuments.in/reader035/viewer/2022062620/551b19085503465e7d8b6449/html5/thumbnails/32.jpg)
32
UNC, Stat & OR
Now, we show how to transform the first tree as curve.
Tree 1/ Support Tree
![Page 33: 1 UNC, Stat & OR SAMSI OODA Workshop SAMSI OODA Workshop Dyck path correspondence and the statistical analysis of Brain vascular networks Shankar Bhamidi,](https://reader035.fdocuments.in/reader035/viewer/2022062620/551b19085503465e7d8b6449/html5/thumbnails/33.jpg)
33
UNC, Stat & OR
Now, we show how to transform the first tree as curve.
Tree 1/ Support Tree
![Page 34: 1 UNC, Stat & OR SAMSI OODA Workshop SAMSI OODA Workshop Dyck path correspondence and the statistical analysis of Brain vascular networks Shankar Bhamidi,](https://reader035.fdocuments.in/reader035/viewer/2022062620/551b19085503465e7d8b6449/html5/thumbnails/34.jpg)
34
UNC, Stat & OR
Now, we show how to transform the first tree as curve.
Tree 1/ Support Tree
![Page 35: 1 UNC, Stat & OR SAMSI OODA Workshop SAMSI OODA Workshop Dyck path correspondence and the statistical analysis of Brain vascular networks Shankar Bhamidi,](https://reader035.fdocuments.in/reader035/viewer/2022062620/551b19085503465e7d8b6449/html5/thumbnails/35.jpg)
35
UNC, Stat & OR
Now, we show how to transform the first tree as curve.
Tree 1/ Support Tree
![Page 36: 1 UNC, Stat & OR SAMSI OODA Workshop SAMSI OODA Workshop Dyck path correspondence and the statistical analysis of Brain vascular networks Shankar Bhamidi,](https://reader035.fdocuments.in/reader035/viewer/2022062620/551b19085503465e7d8b6449/html5/thumbnails/36.jpg)
36
UNC, Stat & OR
Now, we show how to transform the first tree as curve.
Tree 1/ Support Tree
![Page 37: 1 UNC, Stat & OR SAMSI OODA Workshop SAMSI OODA Workshop Dyck path correspondence and the statistical analysis of Brain vascular networks Shankar Bhamidi,](https://reader035.fdocuments.in/reader035/viewer/2022062620/551b19085503465e7d8b6449/html5/thumbnails/37.jpg)
37
UNC, Stat & OR
Now, we show how to transform the first tree as curve.
Tree 1/ Support Tree
![Page 38: 1 UNC, Stat & OR SAMSI OODA Workshop SAMSI OODA Workshop Dyck path correspondence and the statistical analysis of Brain vascular networks Shankar Bhamidi,](https://reader035.fdocuments.in/reader035/viewer/2022062620/551b19085503465e7d8b6449/html5/thumbnails/38.jpg)
38
UNC, Stat & OR
Now, we show how to transform the first tree as curve.
Tree 1/ Support Tree
![Page 39: 1 UNC, Stat & OR SAMSI OODA Workshop SAMSI OODA Workshop Dyck path correspondence and the statistical analysis of Brain vascular networks Shankar Bhamidi,](https://reader035.fdocuments.in/reader035/viewer/2022062620/551b19085503465e7d8b6449/html5/thumbnails/39.jpg)
39
UNC, Stat & OR
Now, we show how to transform the first tree as curve.
Tree 1/ Support Tree
![Page 40: 1 UNC, Stat & OR SAMSI OODA Workshop SAMSI OODA Workshop Dyck path correspondence and the statistical analysis of Brain vascular networks Shankar Bhamidi,](https://reader035.fdocuments.in/reader035/viewer/2022062620/551b19085503465e7d8b6449/html5/thumbnails/40.jpg)
40
UNC, Stat & OR
Now, we show how to transform the second tree as curve.
Tree 2/ Support Tree
![Page 41: 1 UNC, Stat & OR SAMSI OODA Workshop SAMSI OODA Workshop Dyck path correspondence and the statistical analysis of Brain vascular networks Shankar Bhamidi,](https://reader035.fdocuments.in/reader035/viewer/2022062620/551b19085503465e7d8b6449/html5/thumbnails/41.jpg)
41
UNC, Stat & OR
Now, we show how to transform the second tree as curve.
Tree 2/ Support Tree
![Page 42: 1 UNC, Stat & OR SAMSI OODA Workshop SAMSI OODA Workshop Dyck path correspondence and the statistical analysis of Brain vascular networks Shankar Bhamidi,](https://reader035.fdocuments.in/reader035/viewer/2022062620/551b19085503465e7d8b6449/html5/thumbnails/42.jpg)
42
UNC, Stat & OR
Now, we show how to transform the second tree as curve.
Tree 2/ Support Tree
![Page 43: 1 UNC, Stat & OR SAMSI OODA Workshop SAMSI OODA Workshop Dyck path correspondence and the statistical analysis of Brain vascular networks Shankar Bhamidi,](https://reader035.fdocuments.in/reader035/viewer/2022062620/551b19085503465e7d8b6449/html5/thumbnails/43.jpg)
43
UNC, Stat & OR
Now, we show how to transform the second tree as curve.
Tree 2/ Support Tree
![Page 44: 1 UNC, Stat & OR SAMSI OODA Workshop SAMSI OODA Workshop Dyck path correspondence and the statistical analysis of Brain vascular networks Shankar Bhamidi,](https://reader035.fdocuments.in/reader035/viewer/2022062620/551b19085503465e7d8b6449/html5/thumbnails/44.jpg)
44
UNC, Stat & OR
Now, we show how to transform the second tree as curve.
Tree 2/ Support Tree
![Page 45: 1 UNC, Stat & OR SAMSI OODA Workshop SAMSI OODA Workshop Dyck path correspondence and the statistical analysis of Brain vascular networks Shankar Bhamidi,](https://reader035.fdocuments.in/reader035/viewer/2022062620/551b19085503465e7d8b6449/html5/thumbnails/45.jpg)
45
UNC, Stat & OR
Now, we show how to transform the second tree as curve.
Tree 2/ Support Tree
![Page 46: 1 UNC, Stat & OR SAMSI OODA Workshop SAMSI OODA Workshop Dyck path correspondence and the statistical analysis of Brain vascular networks Shankar Bhamidi,](https://reader035.fdocuments.in/reader035/viewer/2022062620/551b19085503465e7d8b6449/html5/thumbnails/46.jpg)
46
UNC, Stat & OR
Now, we show how to transform the second tree as curve.
Tree 2/ Support Tree
![Page 47: 1 UNC, Stat & OR SAMSI OODA Workshop SAMSI OODA Workshop Dyck path correspondence and the statistical analysis of Brain vascular networks Shankar Bhamidi,](https://reader035.fdocuments.in/reader035/viewer/2022062620/551b19085503465e7d8b6449/html5/thumbnails/47.jpg)
47
UNC, Stat & OR
Now, we show how to transform the second tree as curve.
Tree 2/ Support Tree
![Page 48: 1 UNC, Stat & OR SAMSI OODA Workshop SAMSI OODA Workshop Dyck path correspondence and the statistical analysis of Brain vascular networks Shankar Bhamidi,](https://reader035.fdocuments.in/reader035/viewer/2022062620/551b19085503465e7d8b6449/html5/thumbnails/48.jpg)
48
UNC, Stat & OR
Now, we show how to transform the second tree as curve.
Tree 2/ Support Tree
![Page 49: 1 UNC, Stat & OR SAMSI OODA Workshop SAMSI OODA Workshop Dyck path correspondence and the statistical analysis of Brain vascular networks Shankar Bhamidi,](https://reader035.fdocuments.in/reader035/viewer/2022062620/551b19085503465e7d8b6449/html5/thumbnails/49.jpg)
49
UNC, Stat & OR
Now, we show how to transform the second tree as curve.
Tree 2/ Support Tree
![Page 50: 1 UNC, Stat & OR SAMSI OODA Workshop SAMSI OODA Workshop Dyck path correspondence and the statistical analysis of Brain vascular networks Shankar Bhamidi,](https://reader035.fdocuments.in/reader035/viewer/2022062620/551b19085503465e7d8b6449/html5/thumbnails/50.jpg)
50
UNC, Stat & OR
Now, we show how to transform the second tree as curve.
Tree 2/ Support Tree
![Page 51: 1 UNC, Stat & OR SAMSI OODA Workshop SAMSI OODA Workshop Dyck path correspondence and the statistical analysis of Brain vascular networks Shankar Bhamidi,](https://reader035.fdocuments.in/reader035/viewer/2022062620/551b19085503465e7d8b6449/html5/thumbnails/51.jpg)
51
UNC, Stat & OR
Now, we show how to transform the third tree as curve.
Tree 3/ Support Tree
![Page 52: 1 UNC, Stat & OR SAMSI OODA Workshop SAMSI OODA Workshop Dyck path correspondence and the statistical analysis of Brain vascular networks Shankar Bhamidi,](https://reader035.fdocuments.in/reader035/viewer/2022062620/551b19085503465e7d8b6449/html5/thumbnails/52.jpg)
52
UNC, Stat & OR
Now, we show how to transform the third tree as curve.
Tree 3/ Support Tree
![Page 53: 1 UNC, Stat & OR SAMSI OODA Workshop SAMSI OODA Workshop Dyck path correspondence and the statistical analysis of Brain vascular networks Shankar Bhamidi,](https://reader035.fdocuments.in/reader035/viewer/2022062620/551b19085503465e7d8b6449/html5/thumbnails/53.jpg)
53
UNC, Stat & OR
Now, we show how to transform the third tree as curve.
Tree 3/ Support Tree
![Page 54: 1 UNC, Stat & OR SAMSI OODA Workshop SAMSI OODA Workshop Dyck path correspondence and the statistical analysis of Brain vascular networks Shankar Bhamidi,](https://reader035.fdocuments.in/reader035/viewer/2022062620/551b19085503465e7d8b6449/html5/thumbnails/54.jpg)
54
UNC, Stat & OR
Now, we show how to transform the third tree as curve.
Tree 3/ Support Tree
![Page 55: 1 UNC, Stat & OR SAMSI OODA Workshop SAMSI OODA Workshop Dyck path correspondence and the statistical analysis of Brain vascular networks Shankar Bhamidi,](https://reader035.fdocuments.in/reader035/viewer/2022062620/551b19085503465e7d8b6449/html5/thumbnails/55.jpg)
55
UNC, Stat & OR
Now, we show how to transform the third tree as curve.
Tree 3/ Support Tree
![Page 56: 1 UNC, Stat & OR SAMSI OODA Workshop SAMSI OODA Workshop Dyck path correspondence and the statistical analysis of Brain vascular networks Shankar Bhamidi,](https://reader035.fdocuments.in/reader035/viewer/2022062620/551b19085503465e7d8b6449/html5/thumbnails/56.jpg)
56
UNC, Stat & OR
Now, we show how to transform the third tree as curve.
Tree 3/ Support Tree
![Page 57: 1 UNC, Stat & OR SAMSI OODA Workshop SAMSI OODA Workshop Dyck path correspondence and the statistical analysis of Brain vascular networks Shankar Bhamidi,](https://reader035.fdocuments.in/reader035/viewer/2022062620/551b19085503465e7d8b6449/html5/thumbnails/57.jpg)
57
UNC, Stat & OR
Now, we show how to transform the third tree as curve.
Tree 3/ Support Tree
![Page 58: 1 UNC, Stat & OR SAMSI OODA Workshop SAMSI OODA Workshop Dyck path correspondence and the statistical analysis of Brain vascular networks Shankar Bhamidi,](https://reader035.fdocuments.in/reader035/viewer/2022062620/551b19085503465e7d8b6449/html5/thumbnails/58.jpg)
58
UNC, Stat & OR
Now, we show how to transform the third tree as curve.
Tree 3/ Support Tree
![Page 59: 1 UNC, Stat & OR SAMSI OODA Workshop SAMSI OODA Workshop Dyck path correspondence and the statistical analysis of Brain vascular networks Shankar Bhamidi,](https://reader035.fdocuments.in/reader035/viewer/2022062620/551b19085503465e7d8b6449/html5/thumbnails/59.jpg)
59
UNC, Stat & OR
Now, we show how to transform the third tree as curve.
Tree 3/ Support Tree
![Page 60: 1 UNC, Stat & OR SAMSI OODA Workshop SAMSI OODA Workshop Dyck path correspondence and the statistical analysis of Brain vascular networks Shankar Bhamidi,](https://reader035.fdocuments.in/reader035/viewer/2022062620/551b19085503465e7d8b6449/html5/thumbnails/60.jpg)
60
UNC, Stat & OR
Now, we show how to transform the third tree as curve.
Tree 3/ Support Tree
![Page 61: 1 UNC, Stat & OR SAMSI OODA Workshop SAMSI OODA Workshop Dyck path correspondence and the statistical analysis of Brain vascular networks Shankar Bhamidi,](https://reader035.fdocuments.in/reader035/viewer/2022062620/551b19085503465e7d8b6449/html5/thumbnails/61.jpg)
61
UNC, Stat & OR
Now, we show how to transform the third tree as curve.
Tree 3/ Support Tree
![Page 62: 1 UNC, Stat & OR SAMSI OODA Workshop SAMSI OODA Workshop Dyck path correspondence and the statistical analysis of Brain vascular networks Shankar Bhamidi,](https://reader035.fdocuments.in/reader035/viewer/2022062620/551b19085503465e7d8b6449/html5/thumbnails/62.jpg)
62
UNC, Stat & OR
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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UNC, Stat & OR
Actual Data
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64
UNC, Stat & OR
Raw Brain Data - Zoomed
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65
UNC, Stat & OR
Raw Brain Data - Zoomed
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66
UNC, Stat & OR
Some Brain Data Points(as corresponding trees)
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67
UNC, Stat & OR
Some Brain Data Points(as corresponding trees)
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68
UNC, Stat & OR
Some Brain Data Points(as corresponding trees)
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69
UNC, Stat & OR
Some Brain Data Points(as corresponding trees)
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70
UNC, Stat & OR
Some Brain Data Points(as corresponding trees)
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71
UNC, Stat & OR
Some Brain Data Points(as corresponding trees)
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72
UNC, Stat & OR
Data Representation- Youngest
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73
UNC, Stat & OR
Data Representation- oldest
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74
UNC, Stat & OR
Average Tree-Curve and picture of the average tree
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75
UNC, Stat & OR
Illust’n of PCA View: PC1 Projections
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76
UNC, Stat & OR
PCA Pictures of trees that we get when we move in PC1 direction
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77
UNC, Stat & OR
PCA Pictures of trees that we get when we move in PC1 direction
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78
UNC, Stat & OR
PCA Pictures of trees that we get when we move in PC1 direction
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79
UNC, Stat & OR
PCA Pictures of trees that we get when we move in PC1 direction
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80
UNC, Stat & OR
PCA Pictures of trees that we get when we move in PC1 direction
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81
UNC, Stat & OR
PCA Pictures of trees that we get when we move in PC1 direction
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82
UNC, Stat & OR
PCA Pictures of trees that we get when we move in PC1 direction
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UNC, Stat & OR
PCA Pictures of trees that we get when we move in PC1 direction
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UNC, Stat & OR
PCA Pictures of trees that we get when we move in PC1 direction
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UNC, Stat & OR
PCA Pictures of trees that we get when we move in PC2 direction
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UNC, Stat & OR
PCA Pictures of trees that we get when we move in PC2 direction
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87
UNC, Stat & OR
PCAPictures of trees that we get when we move in PC2 direction
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UNC, Stat & OR
PCA Pictures of trees that we get when we move in PC2 direction
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UNC, Stat & OR
PCA Pictures of trees that we get when we move in PC2 direction
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90
UNC, Stat & OR
PCA Pictures of trees that we get when we move in PC2 direction
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91
UNC, Stat & OR
PCA Pictures of trees that we get when we move in PC2 direction
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92
UNC, Stat & OR
PCAPictures of trees that we get when we move in PC2 direction
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UNC, Stat & OR
PCAPictures of trees that we get when we move in PC2 direction
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UNC, Stat & OR
DWD
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UNC, Stat & OR
DWD
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UNC, Stat & OR
DWD
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UNC, Stat & OR
DWD
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UNC, Stat & OR
DWD
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UNC, Stat & OR
DWD
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UNC, Stat & OR
DWD
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UNC, Stat & OR
DWD
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UNC, Stat & OR
DWD
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103
UNC, Stat & OR
DWD/relabeling
random relabeling: Suppose we randomly relabel each tree as male or female.
How does the DWD direction behave?
![Page 104: 1 UNC, Stat & OR SAMSI OODA Workshop SAMSI OODA Workshop Dyck path correspondence and the statistical analysis of Brain vascular networks Shankar Bhamidi,](https://reader035.fdocuments.in/reader035/viewer/2022062620/551b19085503465e7d8b6449/html5/thumbnails/104.jpg)
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UNC, Stat & OR
DWD/relabelling
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UNC, Stat & OR
DWD/relabelling
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UNC, Stat & OR
DWD/relabelling
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UNC, Stat & OR
DWD/relabelling
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UNC, Stat & OR
DWD/relabelling
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UNC, Stat & OR
DWD/relabelling
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UNC, Stat & OR
DWD/relabelling
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UNC, Stat & OR
DWD/relabelling
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UNC, Stat & OR
DWD/relabelling
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UNC, Stat & OR
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?
![Page 114: 1 UNC, Stat & OR SAMSI OODA Workshop SAMSI OODA Workshop Dyck path correspondence and the statistical analysis of Brain vascular networks Shankar Bhamidi,](https://reader035.fdocuments.in/reader035/viewer/2022062620/551b19085503465e7d8b6449/html5/thumbnails/114.jpg)
114
UNC, Stat & OR
Blood vessel tree data
Marron’s brain:
From MRA
Segment tree
of vessel segments
Using tube tracking
Bullitt and Aylward (2002)
![Page 115: 1 UNC, Stat & OR SAMSI OODA Workshop SAMSI OODA Workshop Dyck path correspondence and the statistical analysis of Brain vascular networks Shankar Bhamidi,](https://reader035.fdocuments.in/reader035/viewer/2022062620/551b19085503465e7d8b6449/html5/thumbnails/115.jpg)
115
UNC, Stat & OR
Blood vessel tree data
Marron’s brain:
From MRA
Reconstruct trees
in 3d
Rotate to view
![Page 116: 1 UNC, Stat & OR SAMSI OODA Workshop SAMSI OODA Workshop Dyck path correspondence and the statistical analysis of Brain vascular networks Shankar Bhamidi,](https://reader035.fdocuments.in/reader035/viewer/2022062620/551b19085503465e7d8b6449/html5/thumbnails/116.jpg)
116
UNC, Stat & OR
Blood vessel tree data
Marron’s brain:
From MRA
Reconstruct trees
in 3d
Rotate to view
![Page 117: 1 UNC, Stat & OR SAMSI OODA Workshop SAMSI OODA Workshop Dyck path correspondence and the statistical analysis of Brain vascular networks Shankar Bhamidi,](https://reader035.fdocuments.in/reader035/viewer/2022062620/551b19085503465e7d8b6449/html5/thumbnails/117.jpg)
117
UNC, Stat & OR
Blood vessel tree data
Marron’s brain:
From MRA
Reconstruct trees
in 3d
Rotate to view
![Page 118: 1 UNC, Stat & OR SAMSI OODA Workshop SAMSI OODA Workshop Dyck path correspondence and the statistical analysis of Brain vascular networks Shankar Bhamidi,](https://reader035.fdocuments.in/reader035/viewer/2022062620/551b19085503465e7d8b6449/html5/thumbnails/118.jpg)
118
UNC, Stat & OR
Blood vessel tree data
Marron’s brain:
From MRA
Reconstruct trees
in 3d
Rotate to view
![Page 119: 1 UNC, Stat & OR SAMSI OODA Workshop SAMSI OODA Workshop Dyck path correspondence and the statistical analysis of Brain vascular networks Shankar Bhamidi,](https://reader035.fdocuments.in/reader035/viewer/2022062620/551b19085503465e7d8b6449/html5/thumbnails/119.jpg)
119
UNC, Stat & OR
Blood vessel tree data
Marron’s brain:
From MRA
Reconstruct trees
in 3d
Rotate to view
![Page 120: 1 UNC, Stat & OR SAMSI OODA Workshop SAMSI OODA Workshop Dyck path correspondence and the statistical analysis of Brain vascular networks Shankar Bhamidi,](https://reader035.fdocuments.in/reader035/viewer/2022062620/551b19085503465e7d8b6449/html5/thumbnails/120.jpg)
120
UNC, Stat & OR
Blood vessel tree data
Marron’s brain:
From MRA
Reconstruct trees
in 3d
Rotate to view
![Page 121: 1 UNC, Stat & OR SAMSI OODA Workshop SAMSI OODA Workshop Dyck path correspondence and the statistical analysis of Brain vascular networks Shankar Bhamidi,](https://reader035.fdocuments.in/reader035/viewer/2022062620/551b19085503465e7d8b6449/html5/thumbnails/121.jpg)
121
UNC, Stat & OR
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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122
UNC, Stat & OR
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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UNC, Stat & OR
ISE: in pictures
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124
UNC, Stat & OR
ISE in pictures
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125
UNC, Stat & OR
ISE in pictures
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UNC, Stat & OR
ISE in pictures
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127
UNC, Stat & OR
ISE
Any notion of data driven ISE?
![Page 128: 1 UNC, Stat & OR SAMSI OODA Workshop SAMSI OODA Workshop Dyck path correspondence and the statistical analysis of Brain vascular networks Shankar Bhamidi,](https://reader035.fdocuments.in/reader035/viewer/2022062620/551b19085503465e7d8b6449/html5/thumbnails/128.jpg)
128
UNC, Stat & OR
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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UNC, Stat & OR
Other examples of tree data?
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UNC, Stat & OR
Data on actual root systems
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131
UNC, Stat & OR
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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UNC, Stat & OR
Intuition
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UNC, Stat & OR
Random walk on data points
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134
UNC, Stat & OR
Folded Euclidean Approach