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This work was partially supported by the Joint DMS/NIGMS Initiative to Support Research in the Area of Mathematical Biology (NSF 0800285).
Isabel K. DarcyMathematics Department Applied Mathematical and Computational Sciences (AMCS)University of Iowahttp://www.math.uiowa.edu/~idarcy
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http://www.ima.umn.edu/2008-2009/ND6.15-26.09/
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Create your own homology
3 ingredients:
1.) Objects
2.) Grading
3.) Boundary map
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v2
e2e1
e3
v1 v3
2-simplex = triangle
1-simplex = edge
ev1 v2
0-simplex = vertex = v
Building blocks for a simplicial homology
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GradingGrading: Each object is assigned a unique gradeGrading = Partition of R[x]
Ex: Grade = dimension
v2
e2e1
e3v1 v3
Grade 2: 2-simplex = triangle = {v1, v2, v3}
Grade 1: 1-simplex = edge = {v1, v2} ev1 v2
Grade 0: 0-simplex = vertex = v
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Boundary Map
n : Cn Cn-1 such that 2 = 0
v2
e2e1
e3
v1 v3
ev1 v2
0
0
v1 v2
v2
e2e1
e3
v1 v3
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Cn+1 Cn Cn-1 . . . C2 C1 C0 0
Hn = Zn/Bn = (kernel of )/ (image of )
cycles
boundaries=
n+1
n+1n
n 2 1 0
v2
e2e1
e3
v1 v3
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Čech homologyGiven U Va where Va open for all a in A.
Objects = finite intersections = { V a : ai in A }
Grading = n = depth of intersection.
( V a ) = S Va
Ex: (Va) = 0, (V a Vb) = Va + Vb
(V a V b Vg) = (Va Vb) + (Va Vg) + (Vb Vg)
U
i = 1
n
i
a in A
n+1 j = 1
n
i i
U
i = 1 i ≠ j
nU
i = 1
n ( )0 1
U
U U U U U
2
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Your name homology3 ingredients:
1.) Objects
2.) Grading
3.) Boundary mapn : Cn Cn-1 such that 2 = 0
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Creating a simplicial complex from Data
Step 0.) Start by adding data points = 0-dimensional vertices (0-simplices)
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Creating a simplicial complex from Data
Step 0.) Start by adding 0-dimensional vertices (0-simplices)
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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 Rn
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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 Rn
(1, 8)
(1, 5)(2, 7)
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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 Rn
(dog, happy)
(dog, content)
(wolf, mirthful)
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Creating a simplicial complex from Data
1.) Adding 1-dimensional edges (1-simplices)Add an edge between data points that are “close”
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Creating a simplicial complex from Data
1.) Adding 1-dimensional edges (1-simplices)Add an edge between data points that are “close”
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Creating a simplicial complex from Data
1.) Adding 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 T
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Creating a simplicial complex from Data
1.) Adding 1-dimensional edges (1-simplices)Add an edge between data points that are “close”
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Creating a simplicial complex from Data
1.) Adding 1-dimensional edges (1-simplices)Add an edge between data points that are “close”
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Creating the Vietoris Rips simplicial complex
2.) Add all possible simplices of dimensional > 1.
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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 Rn
Creating the Vietoris Rips simplicial complex
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H0 counts clusters
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H0 counts clusters
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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 Rn
Creating the Vietoris Rips simplicial complex
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Cycl
es
Time
Instead of growing balls, we have a growing path (along with the cover of the path)
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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 Rn
Creating the Vietoris Rips simplicial complex
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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 fj taken at region j
Graph: 97 vertices representing 97 regions of interest edge exists between two vertices i,j if correlation between fj and fj ≥ threshold
How 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
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Vertices = Regions of Interest
Create Rips complex by growing epsilon balls (i.e. decreasing threshold) where distance between two vertices is given by
where fi = measurement at
location i
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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 fj taken at region j
Graph: 97 vertices representing 97 regions of interest edge exists between two vertices i,j if correlation between fj and fj ≥ threshold
How 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
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http://www.ima.umn.edu/videos/?id=856http://ima.umn.edu/2008-2009/ND6.15-26.09/activities/Carlsson-Gunnar/imafive-handout4up.pdf
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http://www.ima.umn.edu/videos/?id=1846http://www.ima.umn.edu/2011-2012/W3.26-30.12/activities/Carlsson-Gunnar/imamachinefinal.pdf
Application to Natural Image StatisticsWith V. de Silva, T. Ishkanov, A. Zomorodian
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An image taken by black and white digital camera can be viewed as a vector, with one coordinate for each pixel
Each 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, P
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Lee-Mumford-Pedersen [LMP] study only high contrast patches.
Collection: 4.5 x 106 high contrast patches from acollection of images obtained by van Hateren and van der Schaaf
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Eurographics Symposium on Point-Based Graphics (2004)Topological estimation using witness complexesVin de Silva and Gunnar Carlsson
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Eurographics Symposium on Point-Based Graphics (2004)Topological estimation using witness complexesVin de Silva and Gunnar Carlsson
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