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Creating a simplicial complex

MATH:7450 (22M:305) Topics in Topology: Scientific and Engineering Applications of Algebraic Topology Sept 4, 2013 Clustering Via Persistent Homology. Creating a simplicial complex. Step 0.) Start by adding 0-dimensional vertices (0-simplices). Creating a simplicial complex.

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Creating a simplicial complex

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  1. MATH:7450 (22M:305) Topics in Topology: Scientific and Engineering Applications of Algebraic Topology Sept 4, 2013 Clustering Via Persistent Homology

  2. Creating a simplicial complex Step 0.) Start by adding 0-dimensional vertices (0-simplices)

  3. Creating a simplicial complex 1.) Adding 1-dimensional edges (1-simplices) Add an edge between data points that are “close”

  4. Creating a simplicial complex 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

  5. Creating a simplicial complex 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

  6. Creating a simplicial complex 1.) Adding 1-dimensional edges (1-simplices) Add an edge between data points that are “close” Note: we only need a definition of closeness between data points. The data points do not need to be actual points in Rn

  7. Creating the Vietoris Rips simplicial complex 2.) Add all possible simplices of dimensional > 1.

  8. Creating the Vietoris Rips simplicial complex 0.) Start by adding 0-dimensional data points Use balls to measure distance (Threshold = diameter). Two points are close if their corresponding balls intersect

  9. Creating the Vietoris Rips simplicial complex 0.) Start by adding 0-dimensional data points Use balls of varying radii to determine persistence of clusters. H. Edelsbrunner, D. Letscher, and A. Zomorodian, Topological persistence and simplication, Discrete and Computational Geometry 28, 2002, 511-533.

  10. Discriminative persistent homology of brain networks, 2011 Hyekyoung LeeChung, M.K.; Hyejin Kang; Bung-Nyun Kim;Dong Soo Lee 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) and 11 pediatric control (PedCon). Data = measurement fj taken at region j Graph: 97 vertices representing 97 regions of interest edge exists between two vertices i,jif correlation between fj and fj ≥ threshold How to choose the threshold? Don’t, instead use persistent homology

  11. Creating the Vietoris Rips simplicial complex Note to determine clusters, only need vertices and edges (but we will use higher dimensional simplices Friday to look at holes in data).

  12. Creating a simplicial complex Note to determine clusters, only need vertices and edges.

  13. 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

  14. How many connected components?

  15. Counting number of connected components using homology v5 v2 e3 e4 e1 e2 v4 v6 v1 v3 e5 C0 = Z2[v1, v2, v3, v4, v5, v6] = set of 0-chains C1 = Z2[e1, e2, e3, e4, e5] = set of 1-chains

  16. Counting number of connected components using homology v5 v2 e3 e4 e1 e2 v4 v6 v1 v3 e5

  17. Counting number of connected components using homology v5 v2 e3 e4 e1 e2 v4 v6 v1 v3 Components are {v1, v2, v3} and {v4, v5, v6} (e5) = v4 + v6 (e1) = v1 + v2 (e3) = v4 + v5 (e4) = v5 + v6 e5 (e2) = v2 + v3 o o o o o

  18. Counting number of connected components using homology v5 v2 e3 e4 e1 e2 v4 v6 v1 v3 e5 C0 = Z2[v1, v2, v3, v4, v5, v6] = set of 0-chains C1 = Z2[e1, e2, e3, e4, e5] = set of 1-chains : C1  C0 o

  19. Counting number of connected components using homology v5 v2 e3 e4 e1 e2 v4 v6 v1 v3 (e5) = v4 + v6 (e4) = v5 + v6 (e1) = v1 + v2 (e3) = v4 + v5 e5 : C1  C0 (e2) = v2 + v3 o o o o o o

  20. Counting number of connected components using homology v5 v2 e3 e4 e1 e2 Extend linearly: v4 v6 v1 v3 (e4) = v5 + v6 (e3) = v4 + v5 (e5) = v4 + v6 (e1) = v1 + v2 e5 : C1  C0 (e2) = v2 + v3 o o o o o o

  21. Counting number of connected components using homology v5 v2 e3 e4 e1 e2 Extend linearly: v4 v6 v1 v3 (e4) = v5 + v6 (e3) = v4 + v5 (e5) = v4 + v6 (e1) = v1 + v2 e5 : C1  C0 (e2) = v2 + v3 o o o o o o

  22. Counting number of connected components using homology v5 v2 e3 e4 e1 e2 Extend linearly: v4 v6 v1 v3 (e4) = v5 + v6 (e3) = v4 + v5 (e5) = v4 + v6 (e1) = v1 + v2 e5 : C1  C0 (e2) = v2 + v3 o o o o o o

  23. Counting number of connected components using homology v5 v2 e3 e4 e1 e2 v4 v6 v1 v3 C1  C0  0 Z0 = kernal of = {x : (x) = 0} e5 o1 o0 o0 o0

  24. Counting number of connected components using homology v5 v2 e3 e4 e1 e2 v4 v6 v1 v3 C1  C0  0 Z0 = kernal of = {x : (x) = 0} = C0 = Z2[v1, v2, v3, v4, v5, v6] e5 o1 o0 o0 o0

  25. Counting number of connected components using homology v5 v2 e3 e4 e1 e2 v4 v6 v1 v3 C1  C0  0 Z0 = Z2[v1, v2, v3, v4, v5, v6] B0 = image of (e4) = v5 + v6 (e3) = v4 + v5 (e5) = v4 + v6 (e1) = v1 + v2 e5 (e2) = v2 + v3 o o1 o0 o o o o o1

  26. Counting number of connected components using homology v5 v2 e3 e4 e1 e2 v1 + v2 = 0, v4 v6 v1 v3 H0 = Z0/B0 = <v1, v2, v3, v4, v5, v6 : e5 v2 + v3 = 0, v4 + v5 = 0, v5 + v6 = 0, v4 + v6 = 0>

  27. Counting number of connected components using homology v5 v2 e3 e4 e1 e2 v4 v6 v1 v3 Z0/B0 = <v1, v2, v3, v4, v5, v6 : v1 + v2 = 0, v2 + v3= 0, v4 + v5= 0, v5 + v6= 0, v4 + v6= 0> e5

  28. Counting number of connected components using homology v5 v2 e3 e4 e1 e2 v4 v6 v1 v3 Z0/B0 = <v1, v2, v3, v4, v5, v6 : v1 + v2 = 0, v2 + v3= 0, v4 + v5= 0, v5 + v6= 0, v4 + v6= 0> = <v1, v2, v3, v4, v5, v6 : v1 + v2 , v2 + v3 , v4 + v5 , v5 + v6 , v4 + v6 > e5

  29. Counting number of connected components using homology v5 v2 e3 e4 e1 e2 v4 v6 v1 v3 Z0/B0 = <v1, v2, v3, v4, v5, v6 : v1 + v2 = 0, v2 + v3= 0, v4 + v5= 0, v5 + v6= 0, v4 + v6= 0> e5

  30. Counting number of connected components using homology v5 v2 e3 e4 e1 e2 v4 v6 v1 v3 Z0/B0 = <v1, v2, v3, v4, v5, v6 : v1 + v2 = 0, v2 + v3 = 0, v4 + v5= 0, v5 + v6= 0, v4 + v6= 0> v1 + v2 = 0 implies v1 = v2 Z0/B0 = <v1, v3, v4, v5, v6 : v2 + v3 = 0, v4 + v5 = 0, v5 + v6 = 0, v4 + v6 = 0> e5

  31. Counting number of connected components using homology v5 v2 e3 e4 e1 e2 v4 v6 v1 v3 Z0/B0 = <v1, v3, v4, v5, v6 : v2 + v3 = 0, v4 + v5 = 0, v5 + v6 = 0, v4 + v6 = 0> Z0/B0 = <v1, v4> e5

  32. Counting number of connected components using homology v5 v2 e3 e4 e1 e2 v4 v6 v1 v3 Z0/B0 = <v1, v2, v3, v4, v5, v6 : v1 + v2 = 0, v2 + v3= 0, v4 + v5= 0, v5 + v6= 0, v4 + v6= 0> Z0/B0 = <[v1], [v4]> where [v1] = {v1, v2, v3} and [v4] = {v4, v5, v6} e5

  33. Counting number of connected components using homology Z0/B0 = <v1, v2, v3, v4, v5, v6 : v1 + v2 = 0, v2 + v3= 0, v4 + v5= 0, v5 + v6= 0, v4 + v6= 0> Use matrices: See Computing Persistent Homology by AfraZomorodian, Gunnar Carlsson

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