Persistent Homology in Topological Data Analysis

Ben Fraser

May 27, 2015

Data Analysis

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

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

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

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

Principal Component Analysis

Essentially, fits an ellipsoid to the data, where each of its axes corresponds to a principal component

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

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

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

Principal Component Analysis

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

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:

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

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

Principal Component Analysis Scale sensitive! Same transformation produces

poor result on same shape/different scale data

Data Analysis

One weakness of PCA is its sensitivity to the scale of the data

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

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?)

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)

Topological Data Analysis

Constructs higher-dimensional structure on our point cloud via simplicial complexes

Topological Data Analysis

Constructs higher-dimensional structure on our point cloud via simplicial complexes

Then analyze this family of nested complexes with persistent homology

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

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.

Algorithm Since scale doesn't matter in this analysis, we

can normalize the data.

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

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?)

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

Landmarking

The method used here is minMax

Landmarking

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

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

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:

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:

li = p such that dist(p,L) = max{dist(x,L) x ϵ X}∀

dist(x,L) = min{dist(x,l) l ϵ L}∀

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:

Algorithm

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

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

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

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 ϵ L, where dist(x,y) is the smallest value in D. Then replace the distance in D with 1.

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 ϵ 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 ϵ [0,1]: this is our proximity parameter, and will be important when we graph the Betti numbers later.

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 ϵ (a

0,a

1,...a

p),

and b ϵ A \ (a0,a

1,...a

p)

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 ϵ (a

0,a

1,...a

p),

and b ϵ A \ (a0,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-a0| = |x-a

1| = … = |x-a

p|.

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 ϵ (a

0,a

1,...a

p),

and b ϵ A \ (a0,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-a0| = |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.

Simplicial Complexes

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

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)

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

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

Simplicial Complexes

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

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

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

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

Boundary Matricies

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

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 4th 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.

Boundary Matricies

At each iteration, we need only add rows of zeros to the kth 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

Boundary Matricies

At each iteration, we need only add rows of zeros to the kth 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)

Betti Numbers The kth betti numbers are based on the

connectivity of the k-dimensional simplicial complexes

Betti Numbers The kth betti numbers are based on the

connectivity of the k-dimensional simplicial complexes

The kth betti number is defined as the rank of the kth homology group, H

k(X) = ker(bd

k)/im(bd

k+1)

Betti Numbers The kth betti numbers are based on the

connectivity of the k-dimensional simplicial complexes

The kth betti number is defined as the rank of the kth 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

Betti Numbers The kth betti numbers are based on the

connectivity of the k-dimensional simplicial complexes

The kth betti number is defined as the rank of the kth 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

Betti Numbers The kth betti numbers are based on the

connectivity of the k-dimensional simplicial complexes

The kth betti number is defined as the rank of the kth 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

Betti Numbers The kth betti numbers are based on the

connectivity of the k-dimensional simplicial complexes

The kth betti number is defined as the rank of the kth 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

Persistent Homology Why must we compute the betti numbers

across a range of the proximity parameter r?

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

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

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

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

Example: Circle

Choose 3200 points uniformly from the circumference of a circle

Example: Circle

Choose 3200 points uniformly from the circumference of a circle

From these, choose a landmark subset of 26 points

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

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

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

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:

Example: Cylinder

Example: Cylinder

Example: Sphere with 4 voids

Example: Sphere with 4 voids

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)

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

Trial: Lake Monitoring Data

For higher-dimensional data, may make more sense to construct higher-dimensional complexes

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

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:

Trial: Lake Monitoring Data

Trial: Lake Monitoring Data 3-complex on Sept.2,2011 data:

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

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

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

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

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

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!

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)