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Presentation transcript:

Mapper

http://www.ayasdi.com/

https://www.ayasdi.com/inquiry/academic-trial.html https://www.ayasdi.com/customers#collaborators

http://danifold.net/mapper/ https://cran.r-project.org/web/packages/TDAmapper/

http://www.nature.com/srep/2013/130207/srep01236/full/srep01236.html

Example: Point cloud data representing a hand. A) Data Set Example: Point cloud data representing a hand. B) Function f : Data Set  R Example: x-coordinate f : (x, y, z)  x Put data into overlapping bins. Example: f-1(ai, bi) Cluster each bin & create network. Vertex = a cluster of a bin. Edge = nonempty intersection between clusters http://www.nature.com/srep/2013/130207/srep01236/full/srep01236.html

Topological Methods for the Analysis of High Dimensional Data Sets and 3D Object Recognition Singh, Gurjeet; Memoli, Facundo; Carlsson, Gunnar http://diglib.eg.org/handle/10.2312/SPBG.SPBG07.091-100

http://www.nature.com/srep/2013/130207/srep01236/full/srep01236.html

Mapper Data Overlapping bins Graph http://www.nature.com/srep/2013/130207/srep01236/full/srep01236.html

Example: Point cloud data representing a hand. A) Data Set Example: Point cloud data representing a hand. http://www.nature.com/srep/2013/130207/srep01236/full/srep01236.html

Function f : Data Set  R Ex 1: x-coordinate f : (x, y, z)  x http://www.nature.com/srep/2013/130207/srep01236/full/srep01236.html

Function f : Data Set  R Ex 1: x-coordinate f : (x, y, z)  x http://www.nature.com/srep/2013/130207/srep01236/full/srep01236.html

Put data into overlapping bins. Example: f-1(ai, bi) ( ( ) ( ) ( ) ( ) ( ) ) Function f : Data Set  R Ex 1: x-coordinate f : (x, y, z)  x http://www.nature.com/srep/2013/130207/srep01236/full/srep01236.html

Put data into overlapping bins. Example: f-1(ai, bi) ( ( ) ( ) ( ) ( ) ( ) ) Function f : Data Set  R Ex 1: x-coordinate f : (x, y, z)  x http://www.nature.com/srep/2013/130207/srep01236/full/srep01236.html

D) Cluster each bin Need covering Resolution multiscale http://www.nature.com/srep/2013/130207/srep01236/full/srep01236.html

Vertex = a cluster of a bin. D) Cluster each bin Vertex = a cluster of a bin. Need covering Resolution multiscale http://www.nature.com/srep/2013/130207/srep01236/full/srep01236.html

Vertex = a cluster of a bin. Edge = nonempty intersection D) Cluster each bin & create network. Vertex = a cluster of a bin. Edge = nonempty intersection between clusters Need covering Resolution multiscale http://www.nature.com/srep/2013/130207/srep01236/full/srep01236.html

Vertex = a cluster of a bin. Edge = nonempty intersection D) Cluster each bin & create network. Vertex = a cluster of a bin. Edge = nonempty intersection between clusters Need covering Resolution multiscale http://www.nature.com/srep/2013/130207/srep01236/full/srep01236.html

Example: Point cloud data representing a hand. A) Data Set Example: Point cloud data representing a hand. B) Function f : Data Set  R Example: x-coordinate f : (x, y, z)  x Put data into overlapping bins. Example: f-1(ai, bi) Cluster each bin & create network. Vertex = a cluster of a bin. Edge = nonempty intersection between clusters http://www.nature.com/srep/2013/130207/srep01236/full/srep01236.html

To understand software, it helps to apply it to a variety of examples.

What graph do you get when you apply mapper to the ideal trefoil knot if filter = projection to x-axis? ( ( ) ( ) ( ) ( ) ( ) )

1.) Put data into overlapping bins. ( ( ) ( ) ( ) ( ) ( ) ) Example: f-1(ai, bi)

Put data into overlapping bins.

Cluster each bin.

Cluster each bin. Each cluster is represented by a vertex.

Create a graph representing the data set: Cluster each bin. Each cluster is represented by a vertex. Draw an edge between 2 vertices if their clusters have points in common.

Create a graph representing the data set: Cluster each bin. Each cluster is represented by a vertex. Draw an edge between 2 vertices if their clusters have points in common.

D C B A E

A E B C D

A B C D E 1 2 3 4

A B C D E 4 2 3 1

4 1 3 2 E D A B C

A B C D E A B C D E 1 2 3 4 A B C D E 4 2 3 1 A B C D E 4 2 3 1

A B C D E 4 2 3 1

What graph do you get when you apply mapper to the above blob if filter = projection to x-axis?

( ( ) ( ) ( ) ( ) ( ) )

( ( ) ( ) ( ) ( ) ( ) )

( ( ) ( ) ( ) ( ) ( ) )

( ( ) ( ) ( ) ( ) ( ) )

( ( ) ( ) ( ) ( ) ( ) )

( ( ) ( ) ( ) ( ) ( ) )

( ( ) ( ) ( ) ( ) ( ) )

( ( ) ( ) ( ) ( ) ( ) )

( ( ) ( ) ( ) ( ) ( ) )

( ( ) ( ) ( ) ( ) ( ) )

To understand software, it helps to apply it to a variety of examples.