lecture 6: creating a simplicial complex from data. in a series of preparatory lectures for the fall...
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![Page 1: Lecture 6: Creating a simplicial complex from data. in a series of preparatory lectures for the Fall 2013 online course MATH:7450 (22M:305) Topics in Topology:](https://reader035.vdocuments.us/reader035/viewer/2022062511/5519d0d455034649768b48c7/html5/thumbnails/1.jpg)
Lecture 6: Creating a simplicial complex from data. in a series of preparatory lectures for the Fall 2013 online course MATH:7450 (22M:305) Topics in Topology: Scientific and Engineering Applications of Algebraic Topology
Target Audience: Anyone interested in topological data analysis including graduate students, faculty, industrial researchers in bioinformatics, biology, business, computer science, cosmology, engineering, imaging, mathematics, neurology, physics, statistics, etc.
Isabel K. Darcy Mathematics Department/Applied Mathematical & Computational Sciences
University of Iowa
http://www.math.uiowa.edu/~idarcy/AppliedTopology.html
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Creating a simplicial complex
Step 0.) Start by adding 0-dimensional vertices (0-simplices)
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Creating a simplicial complex
Step 0.) Start by adding data points = 0-dimensional vertices (0-simplices)
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Creating a simplicial complex
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
![Page 5: Lecture 6: Creating a simplicial complex from data. in a series of preparatory lectures for the Fall 2013 online course MATH:7450 (22M:305) Topics in Topology:](https://reader035.vdocuments.us/reader035/viewer/2022062511/5519d0d455034649768b48c7/html5/thumbnails/5.jpg)
Creating a simplicial complex
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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(1, 5)(2, 7)
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Creating a simplicial complex
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
1.) Adding 1-dimensional edges (1-simplices)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
1.) Adding 1-dimensional edges (1-simplices)Add an edge between data points that are “close”
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Creating a simplicial complex
1.) Adding 1-dimensional edges (1-simplices)Add an edge between data points that are “close”
![Page 10: Lecture 6: Creating a simplicial complex from data. in a series of preparatory lectures for the Fall 2013 online course MATH:7450 (22M:305) Topics in Topology:](https://reader035.vdocuments.us/reader035/viewer/2022062511/5519d0d455034649768b48c7/html5/thumbnails/10.jpg)
Creating a simplicial complex
1.) Adding 1-dimensional edges (1-simplices)Let T = ThresholdConnect 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
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
![Page 12: Lecture 6: Creating a simplicial complex from data. in a series of preparatory lectures for the Fall 2013 online course MATH:7450 (22M:305) Topics in Topology:](https://reader035.vdocuments.us/reader035/viewer/2022062511/5519d0d455034649768b48c7/html5/thumbnails/12.jpg)
Creating a simplicial complex
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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Creating the Vietoris Rips simplicial complex
2.) Add all possible simplices of dimensional > 1.
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Step 0.) Start by adding data points = 0-dimensional vertices (0-simplices)
Creating the Vietoris Rips simplicial complex
![Page 16: Lecture 6: Creating a simplicial complex from data. in a series of preparatory lectures for the Fall 2013 online course MATH:7450 (22M:305) Topics in Topology:](https://reader035.vdocuments.us/reader035/viewer/2022062511/5519d0d455034649768b48c7/html5/thumbnails/16.jpg)
1.) Adding 1-dimensional edges (1-simplices)Add an edge between data points that are “close”
Creating the Vietoris Rips simplicial complex
![Page 17: Lecture 6: Creating a simplicial complex from data. in a series of preparatory lectures for the Fall 2013 online course MATH:7450 (22M:305) Topics in Topology:](https://reader035.vdocuments.us/reader035/viewer/2022062511/5519d0d455034649768b48c7/html5/thumbnails/17.jpg)
2.) Add 2-dimensional triangles (2-simplices)
Add all possible 2-simplices.
Creating the Vietoris Rips simplicial complex
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3.) Add 3-dimensional tetrahedrons (3-simplices)
Add all possible 3-simplices.
Creating the Vietoris Rips simplicial complex
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4.) Add 4-simplices
Add all possible 4-simplices.
Creating the Vietoris Rips simplicial complex
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Step 0.) Start by adding data points = 0-dimensional vertices (0-simplices)
Creating the Vietoris Rips simplicial complex
![Page 21: Lecture 6: Creating a simplicial complex from data. in a series of preparatory lectures for the Fall 2013 online course MATH:7450 (22M:305) Topics in Topology:](https://reader035.vdocuments.us/reader035/viewer/2022062511/5519d0d455034649768b48c7/html5/thumbnails/21.jpg)
1.) Adding 1-dimensional edges (1-simplices)Add an edge between data points that are “close”
Creating the Vietoris Rips simplicial complex
![Page 22: Lecture 6: Creating a simplicial complex from data. in a series of preparatory lectures for the Fall 2013 online course MATH:7450 (22M:305) Topics in Topology:](https://reader035.vdocuments.us/reader035/viewer/2022062511/5519d0d455034649768b48c7/html5/thumbnails/22.jpg)
Creating the Vietoris Rips simplicial complex
2.) Add all possible simplices of dimensional > 1.
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Creating the Vietoris Rips simplicial complex
Let T = Threshold =Connect vertices v and w with an edge iff the distance between v and w is less than T
![Page 24: Lecture 6: Creating a simplicial complex from data. in a series of preparatory lectures for the Fall 2013 online course MATH:7450 (22M:305) Topics in Topology:](https://reader035.vdocuments.us/reader035/viewer/2022062511/5519d0d455034649768b48c7/html5/thumbnails/24.jpg)
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
![Page 25: Lecture 6: Creating a simplicial complex from data. in a series of preparatory lectures for the Fall 2013 online course MATH:7450 (22M:305) Topics in Topology:](https://reader035.vdocuments.us/reader035/viewer/2022062511/5519d0d455034649768b48c7/html5/thumbnails/25.jpg)
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
![Page 26: Lecture 6: Creating a simplicial complex from data. in a series of preparatory lectures for the Fall 2013 online course MATH:7450 (22M:305) Topics in Topology:](https://reader035.vdocuments.us/reader035/viewer/2022062511/5519d0d455034649768b48c7/html5/thumbnails/26.jpg)
Creating the Vietoris Rips simplicial complex
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Creating the Vietoris Rips simplicial complex
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Creating the Vietoris Rips simplicial complex
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Creating the Vietoris Rips simplicial complex
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Creating the Vietoris Rips simplicial complex
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Creating the Vietoris Rips simplicial complex
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Creating the Vietoris Rips simplicial complex