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Topological Data Analysis - PowerPoint Presentation

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Topological Data Analysis - PPT Presentation

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

points data complex dimensional data points dimensional complex simplicial creating edge vertices simplices adding start homology actual definition closeness

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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 Slide28
Slide29

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 SchaafSlide35
Slide36
Slide37
Slide38
Slide39

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 CarlssonSlide41
Slide42
Slide43
Slide44
Slide45
Slide46