1. Persistent Homology in Topological Data Analysis Ben Fraser May 27, 2015

2. Data Analysis Suppose we start with some point cloud data, and want to extract meaningful information from it

3. Data Analysis Suppose we start with some point cloud data, and want to extract meaningful information from it We may want to visualize the data to do so, by plotting it on a graph

4. Data Analysis Suppose we start with some point cloud data, and want to extract meaningful information from it We may want to visualize the data to do so, by plotting it on a graph However, in higher dimensions, visualization becomes difficult

5. Data Analysis Suppose we start with some point cloud data, and want to extract meaningful information from it We may want to visualize the data to do so, by plotting it on a graph However, in higher dimensions, visualization becomes difficult A possible solution: dimensionality reduction

6. Principal Component Analysis Essentially, fits an ellipsoid to the data, where each of its axes corresponds to a principal component

7. Principal Component Analysis Essentially, fits an ellipsoid to the data, where each of its axes corresponds to a principal component The smaller axes are those along which the data has less variance

8. Principal Component Analysis Essentially, fits an ellipsoid to the data, where each of its axes corresponds to a principal component The smaller axes are those along which the data has less variance We could discard these less important principal components to reduce the dimensionality of the data while retaining as much of the variance as possible

9. Principal Component Analysis Essentially, fits an ellipsoid to the data, where each of its axes corresponds to a principal component The smaller axes are those along which the data has less variance We could discard these less important principal components to reduce the dimensionality of the data while retaining as much of the variance as possible Then may be easier to graph: identify clusters

10. Principal Component Analysis Done by computing the singular value decomposition of X (each row is a point, each column a dimension):

11. Principal Component Analysis Done by computing the singular value decomposition of X (each row is a point, each column a dimension): Then a truncated score matrix, where L is the number of principal components we retain:

12. Principal Component Analysis 8-dim data → 2-dim to locate clusters:

13. Principal Component Analysis 3-dim → 2-dim collapses cylinder to circle:

14. Principal Component Analysis Scale sensitive! Same transformation produces poor result on same shape/different scale data

15. Data Analysis One weakness of PCA is its sensitivity to the scale of the data

16. Data Analysis One weakness of PCA is its sensitivity to the scale of the data Also, it provides no information about the shape of our data

17. Data Analysis One weakness of PCA is its sensitivity to the scale of the data Also, it provides no information about the shape of our data We want something insensitive to scale which can identify shape (why?)

18. Data Analysis One weakness of PCA is its sensitivity to the scale of the data Also, it provides no information about the shape of our data We want something insensitive to scale which can identify shape (why?) Because “data has shape, and shape has meaning” - Ayasdi (Gunnar Carlsson)

19. Topological Data Analysis Constructs higher-dimensional structure on our point cloud via simplicial complexes

20. Topological Data Analysis Constructs higher-dimensional structure on our point cloud via simplicial complexes Then analyze this family of nested complexes with persistent homology

21. Topological Data Analysis Constructs higher-dimensional structure on our point cloud via simplicial complexes Then analyze this family of nested complexes with persistent homology Display Betti numbers in graph form

22. Topological Data Analysis Constructs higher-dimensional structure on our point cloud via simplicial complexes Then analyze this family of nested complexes with persistent homology Display Betti numbers in graph form Essentially, we approximate the shape of the data by building a graph on it and considering cliques as higher dimensional objects, and counting the cycles of such objects.

23. Algorithm Since scale doesn't matter in this analysis, we can normalize the data.

24. Algorithm Since scale doesn't matter in this analysis, we can normalize the data. Also, since we don't want to work with the entire data set (especially if it is very large), we want to choose a subset of the data to work with

25. Algorithm Since scale doesn't matter in this analysis, we can normalize the data. Also, since we don't want to work with the entire data set (especially if it is very large), we want to choose a subset of the data to work with We would ideally like this subset to be representative of the original data (but how?)

26. Algorithm Since scale doesn't matter in this analysis, we can normalize the data. Also, since we don't want to work with the entire data set (especially if it is very large), we want to choose a subset of the data to work with We would ideally like this subset to be representative of the original data (but how?) This process is called landmarking

27. Landmarking The method used here is minMax

28. Landmarking The method used here is minMax Start by computing a distance matrix D

29. Landmarking The method used here is minMax Start by computing a distance matrix D Then choose a random point l 1 to add to the subset of landmarks L

30. Landmarking The method used here is minMax Start by computing a distance matrix D Then choose a random point l 1 to add to the subset of landmarks L Then choose each subsequent i-th point to add as that which has maximum distance from the landmark it is closest to:

31. Landmarking The method used here is minMax Start by computing a distance matrix D Then choose a random point l 1 to add to the subset of landmarks L Then choose each subsequent i-th point to add as that which has maximum distance from the landmark it is closest to: l i = p such that dist(p,L) = max{dist(x,L) ∀ x X} dist(x,L) = min{dist(x,l) ∀ l L}

32. Landmarking Landmarking is not an exact science however: on certain types of data the method just used may result in a subset very unrepresentative of the original data. For example:

33. Algorithm As long as outliers are ignored, however, the method used works well to pick points as spread out as possible among the data

34. Algorithm As long as outliers are ignored, however, the method used works well to pick points as spread out as possible among the data Next we keep only the distance matrix between the landmark points, and normalize it

35. Algorithm As long as outliers are ignored, however, the method used works well to pick points as spread out as possible among the data Next we keep only the distance matrix between the landmark points, and normalize it This is all the information we need from the data: the actual position of the points is irrelevant, all we need are the distances between the landmarks, on which we will construct a neighbourhood graph

36. Neighbourhood Graph Our goal is to create a nested sequence of graphs. To be precise, by adding a single edge at a time, between points x,y L, where dist(x,y) is the smallest value in D. Then replace the distance in D with 1.

37. Neighbourhood Graph Our goal is to create a nested sequence of graphs. To be precise, by adding a single edge at a time, between points x,y L, where dist(x,y) is the smallest value in D. Then replace the distance in D with 1. At each iteration of adding an edge, we keep track of r = dist(x,y), r [0,1]: this is our proximity parameter, and will be important when we graph the Betti numbers later.

38. Witness Complex Def: A point x is a weak witness to a p-simplex (a 0,a 1,...a p ) in A if |x-a| < |x-b| ∀ a (a 0,a 1,...a p ), and b A \ (a 0,a 1,...a p )

39. Witness Complex Def: A point x is a weak witness to a p-simplex (a 0,a 1,...a p ) in A if |x-a| < |x-b| ∀ a (a 0,a 1,...a p ), and b A \ (a 0,a 1,...a p ) Def: A point x is a strong witness to a p-simplex (a 0,a 1,...a p ) in A if x is a weak witness and additionally, |x-a 0 | = |x-a 1 | = … = |x-a p |.

40. Witness Complex Def: A point x is a weak witness to a p-simplex (a 0,a 1,...a p ) in A if |x-a| < |x-b| ∀ a (a 0,a 1,...a p ), and b A \ (a 0,a 1,...a p ) Def: A point x is a strong witness to a p-simplex (a 0,a 1,...a p ) in A if x is a weak witness and additionally, |x-a 0 | = |x-a 1 | = … = |x-a p | The requirement may be added that an edge is only added between two points if there exists a weak witness to that edge.

41. Simplicial Complexes Next we want to construct higher dimensional structure on the neighbourhood graph: called a simplicial complex

42. Simplicial Complexes Next we want to construct higher dimensional structure on the neighbourhood graph: called a simplicial complex A simplex is a point, edge, triangle, tetrahedron, etc... (a k-simplex is a k+1-clique in the graph)

43. Simplicial Complexes Next we want to construct higher dimensional structure on the neighbourhood graph: called a simplicial complex A simplex is a point, edge, triangle, tetrahedron, etc... (a k-simplex is a k+1-clique in the graph) A face of a simplex is a sub-simplex of it

44. Simplicial Complexes Next we want to construct higher dimensional structure on the neighbourhood graph: called a simplicial complex A simplex is a point, edge, triangle, tetrahedron, etc... (a k-simplex is a k+1-clique in the graph) A face of a simplex is a sub-simplex of it A simplicial k-complex is a set S of simplices, each of dimension ≤ k, such that a face of any simplex in S is also in S, and the intersection of any two simplices is a face of both of them

45. Simplicial Complexes At each iteration, we add an edge: all we need to do is see if that creates any new k-simplices

46. Simplicial Complexes At each iteration, we add an edge: all we need to do is see if that creates any new k-simplices The edge itself adds a single 1-simplex to the complex

47. Simplicial Complexes At each iteration, we add an edge: all we need to do is see if that creates any new k-simplices The edge itself adds a single 1-simplex to the complex A k-simplex is formed if the intersection of neighbourhoods of a k-2 simplex contains the two points in the added edge

48. Simplicial Complexes At each iteration, we add an edge: all we need to do is see if that creates any new k-simplices The edge itself adds a single 1-simplex to the complex A k-simplex is formed if the intersection of neighbourhoods of a k-2 simplex contains the two points in the added edge In other words, if every point in a k-2 simplex is joined to the two points in the edge, then together they form a k-simplex

49. Boundary Matricies Next we compute boundary matricies. Essentially, these store the information that k-1 simplices are faces of certain k simplices

50. Boundary Matricies Next we compute boundary matricies. Essentially, these store the information that k-1 simplices are faces of certain k simplices For instance, in a simplicial complex with 100 triangles and 50 tetrahedra, the 4 th boundary matrix has 100 rows and 50 columns, with zeros everywhere except where the given triangle is a face of the given tetrahedron, where it is 1.

51. Boundary Matricies At each iteration, we need only add rows of zeros to the k th boundary matrix for each k-1 simplex that was formed, since the only k- simplices they could possibly be faces of are those new ones which were formed at this iteration

52. Boundary Matricies At each iteration, we need only add rows of zeros to the k th boundary matrix for each k-1 simplex that was formed, since the only k- simplices they could possibly be faces of are those new ones which were formed at this iteration Then add columns for each of these new k- simplices, and fill them with 0s and 1s by finding their faces (one of which is guaranteed to be one of the new k-1 simplices)

53. Betti Numbers The k th betti numbers are based on the connectivity of the k-dimensional simplicial complexes

54. Betti Numbers The k th betti numbers are based on the connectivity of the k-dimensional simplicial complexes The k th betti number is defined as the rank of the k th homology group, H k (X) = ker(bd k )/im(bd k+1 )

55. Betti Numbers The k th betti numbers are based on the connectivity of the k-dimensional simplicial complexes The k th betti number is defined as the rank of the k th homology group, H k (X) = ker(bd k )/im(bd k+1 ) In lower dimensions, can be understood as the number of k-dimensional holes

56. Betti Numbers The k th betti numbers are based on the connectivity of the k-dimensional simplicial complexes The k th betti number is defined as the rank of the k th homology group, H k (X) = ker(bd k )/im(bd k+1 ) In lower dimensions, can be understood as the number of k-dimensional holes  Betti0 – number of connected components

57. Betti Numbers The k th betti numbers are based on the connectivity of the k-dimensional simplicial complexes The k th betti number is defined as the rank of the k th homology group, H k (X) = ker(bd k )/im(bd k+1 ) In lower dimensions, can be understood as the number of k-dimensional holes  Betti0 – number of connected components  Betti1 – number of holes

58. Betti Numbers The k th betti numbers are based on the connectivity of the k-dimensional simplicial complexes The k th betti number is defined as the rank of the k th homology group, H k (X) = ker(bd k )/im(bd k+1 ) In lower dimensions, can be understood as the number of k-dimensional holes  Betti0 – number of connected components  Betti1 – number of holes  Betti2 – number of voids

59. Persistent Homology Why must we compute the betti numbers across a range of the proximity parameter r?

60. Persistent Homology Why must we compute the betti numbers across a range of the proximity parameter r? Because at low values of r, the points may be too disconnected to see any meaningful structure, and likewise at high values we are approaching a complete graph, also not useful

61. Persistent Homology However, the solution is not to “guess” an intermediate value of r whose corresponding simplicial complex best approximates the shape of the data

62. Persistent Homology However, the solution is not to “guess” an intermediate value of r whose corresponding simplicial complex best approximates the shape of the data Indeed, as seen in the previous example, features may briefly appear at some value of r only to disappear within a few edge-adding iterations

63. Persistent Homology However, the solution is not to “guess” an intermediate value of r whose corresponding simplicial complex best approximates the shape of the data Indeed, as seen in the previous example, features may briefly appear at some value of r only to disappear within a few edge-adding iterations So, the idea is to see which features “persist”, as they are more likely to accurately represent the shape of the data

64. Example: Circle Choose 3200 points uniformly from the circumference of a circle

65. Example: Circle Choose 3200 points uniformly from the circumference of a circle From these, choose a landmark subset of 26 points

66. Example: Circle Choose 3200 points uniformly from the circumference of a circle From these, choose a landmark subset of 26 points Iteratively add one edge, compute the simplicial 2-complex, boundary matrices, and betti numbers

67. Example: Circle Choose 3200 points uniformly from the circumference of a circle From these, choose a landmark subset of 26 points Iteratively add one edge, compute the simplicial 2-complex, boundary matrices, and betti numbers Plot the betti numbers against the proximity parameter

68. Example: Circle As expected, we find a single hole in the data, and it persists across a wide range of r values. The graph has 1 component

69. Example: Circle The important information is the lifetime of a feature, which can be displayed in a persistence diagram/interval graph/barcode, as shown below:

70. Example: Cylinder

72. Example: Sphere with 4 voids

74. Trial: Lake Monitoring Data Data was collected from buoys on Lake Nipissing:  Temperature  Specific conductivity  Dissolved oxygen concentration  pH  Chlorophyll (RFU – relative fluorescence units)  Total Algae (RFU)

75. Trial: Lake Monitoring Data Sept.4,2011, 3-complex, all 6 dimensions:

76. Trial: Lake Monitoring Data For higher-dimensional data, may make more sense to construct higher-dimensional complexes

77. Trial: Lake Monitoring Data For higher-dimensional data, may make more sense to construct higher-dimensional complexes Also, to focus our attention to dimensions that we expect to be more strongly correlated

78. Trial: Lake Monitoring Data For higher-dimensional data, may make more sense to construct higher-dimensional complexes Also, to focus our attention to dimensions that we expect to be more strongly correlated The next trial constructs a 2-complex on DO concentration, pH, and algae, using a larger set of data from Sept.4,2011:

79. Trial: Lake Monitoring Data

80. 3-complex on Sept.2,2011 data:

81. Trial: Lake Monitoring Data Each combination of dimension of the data and dimension of complex being built has so far failed to recognize any significant features in shape of the data

82. Trial: Lake Monitoring Data Each combination of dimension of the data and dimension of complex being built has so far failed to recognize any significant features in shape of the data Combining data sets from different times of year might result in greater variation in the data, and a greater chance of patterns being found

83. Summary Construct a filtration of a simplicial complex on our data by building a sequence of neighbourhood graphs across an interval of the proximity parameter

84. Summary Construct a filtration of a simplicial complex on our data by building a sequence of neighbourhood graphs across an interval of the proximity parameter Plot betti numbers against this proximity parameter

85. Summary Construct a filtration of a simplicial complex on our data by building a sequence of neighbourhood graphs across an interval of the proximity parameter Plot betti numbers against this proximity parameter Features which persist longer more likely represent the shape of the data

86. Summary Construct a filtration of a simplicial complex on our data by building a sequence of neighbourhood graphs across an interval of the proximity parameter Plot betti numbers against this proximity parameter Features which persist longer more likely represent the shape of the data Shape is important!

87. Acknowledgments Mark Wachowiak (supervisor, artificial data sets) Renata Smolikova-Wachowiak (lake monitoring data) Gunnar Carlsson (see “on the shape of data”: https://www.youtube.com/watch?v=kctyag2Xi8o) Adam Cutbill (author of original program) Afra Zomorodian (fast construction of the Vietoris-Rips complex) Vin de Silva (topological estimation using witness complexes)