Applications II This work was partially supported by the Joint DMSNIGMS Initiative to Support Research in the Area of Mathematical Biology NSF 0800285 Isabel K Darcy Mathematics Department ID: 615640
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Slide1
Topological Data Analysis
Applications II
This work was partially supported by the Joint DMS/NIGMS Initiative to Support Research in the Area of Mathematical Biology (NSF 0800285).
Isabel K. Darcy
Mathematics Department
Applied Mathematical and
Computational
Sciences (AMCS)
University of Iowa
http://www.math.uiowa.edu/~idarcy Slide2
http://www.ima.umn.edu/2008-2009/ND6.15-26.09
/Slide3
Create your own
h
omology
3 ingredients:
1.) Objects
2.) Grading
3.) Boundary mapSlide4
v
2
e
2
e
1
e
3
v
1
v
3
2-simplex = triangle
1
-simplex = edge
e
v
1
v
2
0-simplex = vertex
= v
Building blocks for a
simplicial
homologySlide5
Grading
Grading: Each object is assigned a unique grade
Grading = Partition of R[x
]
Ex: Grade =
dimension
v
2
e
2
e
1
e
3
v
1
v
3
Grade
2:
2-simplex = triangle
=
{v
1
, v
2
, v
3
}
Grade
1:
1-simplex = edge
=
{v
1
, v
2
}
e
v
1
v
2
Grade 0:
0-simplex = vertex
= vSlide6
Boundary Map
n
:
C
n
C
n-1
such that
2
= 0
v
2
e
2
e
1
e
3
v
1
v
3
e
v
1
v
2
0
0
v
1
v
2
v
2
e
2
e
1
e
3
v
1
v
3Slide7
Cn+1
Cn Cn-1 . . .
C2
C1 C0 0
H
n
= Z
n
/
B
n
= (kernel of )/ (image of )
cycles
boundaries
=
n
+1
n+1
n
n
2
1
0
v
2
e
2
e
1
e
3
v
1
v
3Slide8
Čech
homology
Given U
V
a
where
V
a
open for all
a
in
A.
Objects = finite intersections = {
V
a
:
a
i
in A }
Grading = n = depth of intersection.
(
V
a
) = S
V
a
Ex: (
Va)
= 0, (Va
Vb
) = V
a + V
b (
Va V
b Vg
) = (V
a Vb)
+ (V
a Vg
)
+ (V
b V
g)
U
i = 1
n
ia
in A
n
+1
j
= 1
n
i
i
U
i
= 1
i
≠ j
n
U
i
= 1
n
( )
0
1
U
U
U
U
U
U
2Slide9
Your name homology
3 ingredients:
1.) Objects
2.) Grading
3.) Boundary map
n
:
C
n
C
n-1
such that
2
= 0Slide10
Creating a
simplicial
complex from Data
Step 0.) Start by adding data points
= 0-dimensional vertices (0-simplices) Slide11
Creating a
simplicial
complex from Data
Step 0.) Start by adding 0-dimensional vertices
(0-simplices)Slide12
Creating a
simplicial
complex from Data
0.) Start by adding 0-dimensional data points
Note: we only need a definition of closeness between data points. The data points do not need to be actual points in
R
nSlide13
Creating a
simplicial
complex from Data
0.) Start by adding 0-dimensional data points
Note: we only need a definition of closeness between data points. The data points do not need to be actual points in
R
n
(1, 8)
(1, 5)
(2, 7)Slide14
Creating a
simplicial
complex from Data
0.) Start by adding 0-dimensional data points
Note: we only need a definition of closeness between data points. The data points do not need to be actual points in
R
n
(dog, happy)
(dog, content)
(wolf, mirthful)Slide15
Creating a
simplicial
complex from Data
1
.)
A
dding
1
-dimensional edges (1-simplices)
Add an edge between data points that are “close”Slide16
Creating a
simplicial
complex from Data
1
.)
A
dding
1
-dimensional edges (1-simplices)
Add an edge between data points that are “close”Slide17
Creating a
simplicial
complex from Data
1
.)
A
dding
1
-dimensional edges (1-simplices)
Let T = Threshold =
Connect vertices v and w with an edge
iff
the distance between v and w is less than TSlide18
Creating a
simplicial
complex from Data
1
.)
A
dding
1
-dimensional edges (1-simplices)
Add an edge between data points that are “close”Slide19
Creating a
simplicial
complex from Data
1
.)
A
dding
1
-dimensional edges (1-simplices)
Add an edge between data points that are “close”Slide20
Creating the
Vietoris
Rips
simplicial complex
2
.) Add all possible simplices of dimensional > 1.Slide21
0.) Start by adding 0-dimensional data points
Note: we only need a definition of closeness between data points. The data points do not need to be actual points in
R
n
Creating the
Vietoris
Rips simplicial complexSlide22
H
0
counts clustersSlide23
H
0
counts clustersSlide24
0.) Start by adding 0-dimensional data points
Note: we only need a definition of closeness between data points. The data points do not need to be actual points in
R
n
Creating the
Vietoris
Rips simplicial complexSlide25
Cycles
Time
Instead of growing balls, we have a growing path (along with the cover of the path)Slide26
0.) Start by adding 0-dimensional data points
Note: we only need a definition of closeness between data points. The data points do not need to be actual points in
R
n
Creating the
Vietoris
Rips simplicial complexSlide27
Constructing functional brain
networks with 97 regions of interest (ROIs) extracted from FDG-PET data for 24 attention-deficit hyperactivity disorder (
ADHD),26 autism spectrum disorder (ASD) and11 pediatric
control (PedCon).Data = measurement f
j
taken at region j
Graph: 97 vertices representing 97 regions of interest
edge exists between two vertices
i,j
if correlation
between
f
j and fj ≥ thresholdHow to choose the threshold?
Don’t, instead use persistent homology
Discriminative persistent homology of brain networks, 2011
Hyekyoung Lee
Chung, M.K.
;
Hyejin Kang
;
Bung-Nyun
Kim
;
Dong Soo Lee Slide28Slide29
Vertices = Regions of Interest
Create Rips complex by growing epsilon balls (i.e. decreasing threshold) where distance between two vertices is given bywhere
fi = measurement at location iSlide30
Constructing functional brain
networks with 97 regions of interest (ROIs) extracted from FDG-PET data for 24 attention-deficit hyperactivity disorder (
ADHD),26 autism spectrum disorder (ASD) and11 pediatric
control (PedCon).Data = measurement f
j
taken at region j
Graph: 97 vertices representing 97 regions of interest
edge exists between two vertices
i,j
if correlation
between
f
j and fj ≥ thresholdHow to choose the threshold?
Don’t, instead use persistent homology
Discriminative persistent homology of brain networks, 2011
Hyekyoung Lee
Chung, M.K.
;
Hyejin Kang
;
Bung-Nyun
Kim
;
Dong Soo Lee Slide31
http://www.ima.umn.edu/videos/?id=856
http://ima.umn.edu/2008-2009/ND6.15-26.09/activities/Carlsson-Gunnar/imafive-handout4up.pdfSlide32
http://www.ima.umn.edu/videos/?id=1846
http://www.ima.umn.edu/2011-2012/W3.26-30.12/activities/Carlsson-Gunnar/imamachinefinal.pdf
Application to Natural Image Statistics
With V. de Silva, T. Ishkanov, A. ZomorodianSlide33
An
image taken by black and white digital camera can be viewed as a vector, with one coordinate for each pixelEach pixel has a “gray scale” value, can be thought of as a
real number (in reality, takes one of 255 values)Typical camera uses tens of thousands of pixels, so images lie
in a very high dimensional space, call it pixel space, PSlide34
Lee
-Mumford-Pedersen [LMP] study only high
contrast patches.Collection: 4.5 x 10
6 high contrast patches from acollection of images obtained by van Hateren and van der SchaafSlide35Slide36Slide37Slide38Slide39
Eurographics
Symposium on Point-Based Graphics (2004)Topological estimation using witness complexesVin de Silva and Gunnar CarlssonSlide40
Eurographics
Symposium on Point-Based Graphics (2004)Topological estimation using witness complexesVin de Silva and Gunnar CarlssonSlide41Slide42Slide43Slide44Slide45Slide46