Computational Topology - Mapperkbuchin/teaching/2IMA00/2018/Slides/Mapp… · I Feature selection...
Transcript of Computational Topology - Mapperkbuchin/teaching/2IMA00/2018/Slides/Mapp… · I Feature selection...
![Page 1: Computational Topology - Mapperkbuchin/teaching/2IMA00/2018/Slides/Mapp… · I Feature selection (ares, loops) Applications to Medical science data 145 patients who had diabetes,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d8bba7e708231d43ae4c6/html5/thumbnails/1.jpg)
Computational Topology - Mapper
Jiaqi Ni
Eindhoven University of Technology
June 14, 2018
![Page 2: Computational Topology - Mapperkbuchin/teaching/2IMA00/2018/Slides/Mapp… · I Feature selection (ares, loops) Applications to Medical science data 145 patients who had diabetes,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d8bba7e708231d43ae4c6/html5/thumbnails/2.jpg)
Outline
Introduction
Mapper in the continuous setting
Mapper in practice
Parameters of Mapper in practice
Applications
Summary
![Page 3: Computational Topology - Mapperkbuchin/teaching/2IMA00/2018/Slides/Mapp… · I Feature selection (ares, loops) Applications to Medical science data 145 patients who had diabetes,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d8bba7e708231d43ae4c6/html5/thumbnails/3.jpg)
Introduction
Mapper in the continuous setting
Mapper in practice
Parameters of Mapper in practice
Applications
Summary
![Page 4: Computational Topology - Mapperkbuchin/teaching/2IMA00/2018/Slides/Mapp… · I Feature selection (ares, loops) Applications to Medical science data 145 patients who had diabetes,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d8bba7e708231d43ae4c6/html5/thumbnails/4.jpg)
Introduction
I Mapper is a computational method for extracting simpledescriptions of high dimensional data sets in the form ofsimplicial complexes.
![Page 5: Computational Topology - Mapperkbuchin/teaching/2IMA00/2018/Slides/Mapp… · I Feature selection (ares, loops) Applications to Medical science data 145 patients who had diabetes,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d8bba7e708231d43ae4c6/html5/thumbnails/5.jpg)
Recap about Reeb Graph
Definition: The Reeb graph of f is the set of contours R(f).
![Page 6: Computational Topology - Mapperkbuchin/teaching/2IMA00/2018/Slides/Mapp… · I Feature selection (ares, loops) Applications to Medical science data 145 patients who had diabetes,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d8bba7e708231d43ae4c6/html5/thumbnails/6.jpg)
Recap about Reeb Graph
We can get similar result as Reeb Graph with Mapper.
![Page 7: Computational Topology - Mapperkbuchin/teaching/2IMA00/2018/Slides/Mapp… · I Feature selection (ares, loops) Applications to Medical science data 145 patients who had diabetes,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d8bba7e708231d43ae4c6/html5/thumbnails/7.jpg)
Recap about Reeb Graph
We can also get the more different results from Reeb Graph withMapper.
![Page 8: Computational Topology - Mapperkbuchin/teaching/2IMA00/2018/Slides/Mapp… · I Feature selection (ares, loops) Applications to Medical science data 145 patients who had diabetes,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d8bba7e708231d43ae4c6/html5/thumbnails/8.jpg)
Introduction
Mapper in the continuous setting
Mapper in practice
Parameters of Mapper in practice
Applications
Summary
![Page 9: Computational Topology - Mapperkbuchin/teaching/2IMA00/2018/Slides/Mapp… · I Feature selection (ares, loops) Applications to Medical science data 145 patients who had diabetes,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d8bba7e708231d43ae4c6/html5/thumbnails/9.jpg)
Cover of space
If the set X is a topological space, then a cover C of X is acollection of subsets U of X whose union is the whole spaceX. In this case we say that C covers X, or that the sets Ucover X.
Topological Space X Cover of Space X
![Page 10: Computational Topology - Mapperkbuchin/teaching/2IMA00/2018/Slides/Mapp… · I Feature selection (ares, loops) Applications to Medical science data 145 patients who had diabetes,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d8bba7e708231d43ae4c6/html5/thumbnails/10.jpg)
Cover of space
If the set X is a topological space, then a cover C of X is acollection of subsets U of X whose union is the whole spaceX. In this case we say that C covers X, or that the sets Ucover X.
Topological Space X Cover of Space X
![Page 11: Computational Topology - Mapperkbuchin/teaching/2IMA00/2018/Slides/Mapp… · I Feature selection (ares, loops) Applications to Medical science data 145 patients who had diabetes,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d8bba7e708231d43ae4c6/html5/thumbnails/11.jpg)
Cover of space
If Y is a subset of X, then a cover of Y is a collection ofsubsets of X whose union contains Y,
i.e., C is a cover of Y if Y ⊆⋃α∈C
Uα
![Page 12: Computational Topology - Mapperkbuchin/teaching/2IMA00/2018/Slides/Mapp… · I Feature selection (ares, loops) Applications to Medical science data 145 patients who had diabetes,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d8bba7e708231d43ae4c6/html5/thumbnails/12.jpg)
Cover of space
If Y is a subset of X, then a cover of Y is a collection ofsubsets of X whose union contains Y,
i.e., C is a cover of Y if Y ⊆⋃α∈C
Uα
![Page 13: Computational Topology - Mapperkbuchin/teaching/2IMA00/2018/Slides/Mapp… · I Feature selection (ares, loops) Applications to Medical science data 145 patients who had diabetes,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d8bba7e708231d43ae4c6/html5/thumbnails/13.jpg)
Cover refinement
I A refinement of a cover C of a topological space X is a newcover D of X such that every set in D is contained in someset in C.
I Formally: D = {Vβ∈B} is a refinement of C = {Uα∈A}when ∀β ∃α Vβ ⊆ Uα
Space X Cover of Space X Refinement of Cover
![Page 14: Computational Topology - Mapperkbuchin/teaching/2IMA00/2018/Slides/Mapp… · I Feature selection (ares, loops) Applications to Medical science data 145 patients who had diabetes,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d8bba7e708231d43ae4c6/html5/thumbnails/14.jpg)
Cover refinement
I A refinement of a cover C of a topological space X is a newcover D of X such that every set in D is contained in someset in C.
I Formally: D = {Vβ∈B} is a refinement of C = {Uα∈A}when ∀β ∃α Vβ ⊆ Uα
Space X Cover of Space X Refinement of Cover
![Page 15: Computational Topology - Mapperkbuchin/teaching/2IMA00/2018/Slides/Mapp… · I Feature selection (ares, loops) Applications to Medical science data 145 patients who had diabetes,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d8bba7e708231d43ae4c6/html5/thumbnails/15.jpg)
Cover refinement
I A refinement of a cover C of a topological space X is a newcover D of X such that every set in D is contained in someset in C.
I Formally: D = {Vβ∈B} is a refinement of C = {Uα∈A}when ∀β ∃α Vβ ⊆ Uα
Space X Cover of Space X Refinement of Cover
![Page 16: Computational Topology - Mapperkbuchin/teaching/2IMA00/2018/Slides/Mapp… · I Feature selection (ares, loops) Applications to Medical science data 145 patients who had diabetes,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d8bba7e708231d43ae4c6/html5/thumbnails/16.jpg)
Mapper in the continuous setting
Input:
I Continuous function(filter) f : X→ RI Cover C of im(f) by open intervals: im(f ) ⊆
⋃c∈C
c
Method:
I Compute pullback cover U of X: U = f −1(c)c∈CI Refine U by separating each of its elements into its various
connected components → connected cover VI The Mapper is the nerve of V:
I 1 vertex per element V ∈ VI 1 edge per intersection V ∪ V ′ 6= ø, V ,V ′ ∈ VI 1 k-simplex per (k + 1)-fold intersection,⋃k
i=0 Vi 6= ø,V0,V1...Vk ∈ V
![Page 17: Computational Topology - Mapperkbuchin/teaching/2IMA00/2018/Slides/Mapp… · I Feature selection (ares, loops) Applications to Medical science data 145 patients who had diabetes,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d8bba7e708231d43ae4c6/html5/thumbnails/17.jpg)
Example of Mapper in the continuous setting
![Page 18: Computational Topology - Mapperkbuchin/teaching/2IMA00/2018/Slides/Mapp… · I Feature selection (ares, loops) Applications to Medical science data 145 patients who had diabetes,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d8bba7e708231d43ae4c6/html5/thumbnails/18.jpg)
Example of Mapper in the continuous setting
![Page 19: Computational Topology - Mapperkbuchin/teaching/2IMA00/2018/Slides/Mapp… · I Feature selection (ares, loops) Applications to Medical science data 145 patients who had diabetes,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d8bba7e708231d43ae4c6/html5/thumbnails/19.jpg)
Example of Mapper in the continuous setting
![Page 20: Computational Topology - Mapperkbuchin/teaching/2IMA00/2018/Slides/Mapp… · I Feature selection (ares, loops) Applications to Medical science data 145 patients who had diabetes,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d8bba7e708231d43ae4c6/html5/thumbnails/20.jpg)
Introduction
Mapper in the continuous setting
Mapper in practice
Parameters of Mapper in practice
Applications
Summary
![Page 21: Computational Topology - Mapperkbuchin/teaching/2IMA00/2018/Slides/Mapp… · I Feature selection (ares, loops) Applications to Medical science data 145 patients who had diabetes,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d8bba7e708231d43ae4c6/html5/thumbnails/21.jpg)
Mapper in practice
Input:
I Point cloud P with distance matrix
I Continuous function(filter) f : P → RI Cover C of im(f) by open intervals: im(f ) ⊆
⋃c∈C
c
Method:
I Compute pullback cover U of X: U = f −1(c)c∈CI Refine U by applying clustering algorithm(with distance
threshold δ) → connected cover VI The Mapper is the nerve of V:
I 1 vertex per element V ∈ VI 1 edge per intersection V ∪ V ′ 6= ø, V ,V ′ ∈ VI 1 k-simplex per (k + 1)-fold intersection,⋃k
i=0 Vi 6= ø,V0,V1...Vk ∈ V
![Page 22: Computational Topology - Mapperkbuchin/teaching/2IMA00/2018/Slides/Mapp… · I Feature selection (ares, loops) Applications to Medical science data 145 patients who had diabetes,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d8bba7e708231d43ae4c6/html5/thumbnails/22.jpg)
Example of Mapper in practice
![Page 23: Computational Topology - Mapperkbuchin/teaching/2IMA00/2018/Slides/Mapp… · I Feature selection (ares, loops) Applications to Medical science data 145 patients who had diabetes,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d8bba7e708231d43ae4c6/html5/thumbnails/23.jpg)
Example of Mapper in practice
![Page 24: Computational Topology - Mapperkbuchin/teaching/2IMA00/2018/Slides/Mapp… · I Feature selection (ares, loops) Applications to Medical science data 145 patients who had diabetes,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d8bba7e708231d43ae4c6/html5/thumbnails/24.jpg)
Introduction
Mapper in the continuous setting
Mapper in practice
Parameters of Mapper in practice
Applications
Summary
![Page 25: Computational Topology - Mapperkbuchin/teaching/2IMA00/2018/Slides/Mapp… · I Feature selection (ares, loops) Applications to Medical science data 145 patients who had diabetes,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d8bba7e708231d43ae4c6/html5/thumbnails/25.jpg)
Parameters of Mapper in practice
I Filter f : P → R
I Cover C of im(f) by open intervals:
I Clustering algorithm
![Page 26: Computational Topology - Mapperkbuchin/teaching/2IMA00/2018/Slides/Mapp… · I Feature selection (ares, loops) Applications to Medical science data 145 patients who had diabetes,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d8bba7e708231d43ae4c6/html5/thumbnails/26.jpg)
Parameters of Mapper in practice
I Filter f : P → R
I Cover C of im(f) by open intervals:
I Clustering algorithm
![Page 27: Computational Topology - Mapperkbuchin/teaching/2IMA00/2018/Slides/Mapp… · I Feature selection (ares, loops) Applications to Medical science data 145 patients who had diabetes,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d8bba7e708231d43ae4c6/html5/thumbnails/27.jpg)
Parameters of Mapper in practice
I Filter f : P → R
I Cover C of im(f) by open intervals:
I Clustering algorithm
![Page 28: Computational Topology - Mapperkbuchin/teaching/2IMA00/2018/Slides/Mapp… · I Feature selection (ares, loops) Applications to Medical science data 145 patients who had diabetes,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d8bba7e708231d43ae4c6/html5/thumbnails/28.jpg)
Parameters of Mapper in practice - Filter functions
I The outcome of Mapper is highly dependent on the functionchosen to partition (filter) the data set and the choice offunctions depends mostly on the dataset.
I Possible functions:I DensityI EccentricityI Graph LaplaciansI sum/average/max/minI x/y- axis projection
![Page 29: Computational Topology - Mapperkbuchin/teaching/2IMA00/2018/Slides/Mapp… · I Feature selection (ares, loops) Applications to Medical science data 145 patients who had diabetes,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d8bba7e708231d43ae4c6/html5/thumbnails/29.jpg)
Filter function examples
![Page 30: Computational Topology - Mapperkbuchin/teaching/2IMA00/2018/Slides/Mapp… · I Feature selection (ares, loops) Applications to Medical science data 145 patients who had diabetes,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d8bba7e708231d43ae4c6/html5/thumbnails/30.jpg)
Filter function examples
![Page 31: Computational Topology - Mapperkbuchin/teaching/2IMA00/2018/Slides/Mapp… · I Feature selection (ares, loops) Applications to Medical science data 145 patients who had diabetes,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d8bba7e708231d43ae4c6/html5/thumbnails/31.jpg)
Filter function examples
![Page 32: Computational Topology - Mapperkbuchin/teaching/2IMA00/2018/Slides/Mapp… · I Feature selection (ares, loops) Applications to Medical science data 145 patients who had diabetes,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d8bba7e708231d43ae4c6/html5/thumbnails/32.jpg)
Filter function examples
![Page 33: Computational Topology - Mapperkbuchin/teaching/2IMA00/2018/Slides/Mapp… · I Feature selection (ares, loops) Applications to Medical science data 145 patients who had diabetes,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d8bba7e708231d43ae4c6/html5/thumbnails/33.jpg)
Parameters of Mapper in practice - Cover
I Uniform cover II resolution / granularity: r (diameter of intervals)I gain: g (percentage of overlap)
I Example:
I Modification of r and g can highly effect the result.
![Page 34: Computational Topology - Mapperkbuchin/teaching/2IMA00/2018/Slides/Mapp… · I Feature selection (ares, loops) Applications to Medical science data 145 patients who had diabetes,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d8bba7e708231d43ae4c6/html5/thumbnails/34.jpg)
Parameters of Mapper in practice - Cover
I Uniform cover II resolution / granularity: r (diameter of intervals)I gain: g (percentage of overlap)
I Example:
I Modification of r and g can highly effect the result.
![Page 35: Computational Topology - Mapperkbuchin/teaching/2IMA00/2018/Slides/Mapp… · I Feature selection (ares, loops) Applications to Medical science data 145 patients who had diabetes,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d8bba7e708231d43ae4c6/html5/thumbnails/35.jpg)
Parameters of Mapper in practice - Cover
I Uniform cover II resolution / granularity: r (diameter of intervals)I gain: g (percentage of overlap)
I Example:
I Modification of r and g can highly effect the result.
![Page 36: Computational Topology - Mapperkbuchin/teaching/2IMA00/2018/Slides/Mapp… · I Feature selection (ares, loops) Applications to Medical science data 145 patients who had diabetes,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d8bba7e708231d43ae4c6/html5/thumbnails/36.jpg)
Cover examples
![Page 37: Computational Topology - Mapperkbuchin/teaching/2IMA00/2018/Slides/Mapp… · I Feature selection (ares, loops) Applications to Medical science data 145 patients who had diabetes,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d8bba7e708231d43ae4c6/html5/thumbnails/37.jpg)
Cover examples
![Page 38: Computational Topology - Mapperkbuchin/teaching/2IMA00/2018/Slides/Mapp… · I Feature selection (ares, loops) Applications to Medical science data 145 patients who had diabetes,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d8bba7e708231d43ae4c6/html5/thumbnails/38.jpg)
Cover examples
![Page 39: Computational Topology - Mapperkbuchin/teaching/2IMA00/2018/Slides/Mapp… · I Feature selection (ares, loops) Applications to Medical science data 145 patients who had diabetes,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d8bba7e708231d43ae4c6/html5/thumbnails/39.jpg)
Cover examples
![Page 40: Computational Topology - Mapperkbuchin/teaching/2IMA00/2018/Slides/Mapp… · I Feature selection (ares, loops) Applications to Medical science data 145 patients who had diabetes,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d8bba7e708231d43ae4c6/html5/thumbnails/40.jpg)
Mapper for Y-shape point cloud data
![Page 41: Computational Topology - Mapperkbuchin/teaching/2IMA00/2018/Slides/Mapp… · I Feature selection (ares, loops) Applications to Medical science data 145 patients who had diabetes,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d8bba7e708231d43ae4c6/html5/thumbnails/41.jpg)
Mapper for Y-shape point cloud data
![Page 42: Computational Topology - Mapperkbuchin/teaching/2IMA00/2018/Slides/Mapp… · I Feature selection (ares, loops) Applications to Medical science data 145 patients who had diabetes,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d8bba7e708231d43ae4c6/html5/thumbnails/42.jpg)
Parameters of uniform Cover
Parameter r:
I Small r : fine cover, Mapper close to Reeb Graph, butsensitive to δ.
I Large r : rough cover, less sensitive to δ, but Mapper far fromReeb Graph.
Parameter g:
I Large g(close to 1): more points inside intersections, lesssensitive to δ but far from Reeb Graph.
I Small g(close to 0): controlled Mapper dimension, close toReeb Graph.
![Page 43: Computational Topology - Mapperkbuchin/teaching/2IMA00/2018/Slides/Mapp… · I Feature selection (ares, loops) Applications to Medical science data 145 patients who had diabetes,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d8bba7e708231d43ae4c6/html5/thumbnails/43.jpg)
Parameters of Mapper in practice - Clustering algorithm
Single-linkage clustering is one of several methods of hierarchicalclustering.
I Based on grouping clusters in bottom-up fashion(agglomerative clustering).
I At each step combining two clusters that contain the closestpair of elements not yet belonging to the same cluster as eachother.
![Page 44: Computational Topology - Mapperkbuchin/teaching/2IMA00/2018/Slides/Mapp… · I Feature selection (ares, loops) Applications to Medical science data 145 patients who had diabetes,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d8bba7e708231d43ae4c6/html5/thumbnails/44.jpg)
Example of Single-linkage clustering
![Page 45: Computational Topology - Mapperkbuchin/teaching/2IMA00/2018/Slides/Mapp… · I Feature selection (ares, loops) Applications to Medical science data 145 patients who had diabetes,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d8bba7e708231d43ae4c6/html5/thumbnails/45.jpg)
Example of Single-linkage clustering
![Page 46: Computational Topology - Mapperkbuchin/teaching/2IMA00/2018/Slides/Mapp… · I Feature selection (ares, loops) Applications to Medical science data 145 patients who had diabetes,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d8bba7e708231d43ae4c6/html5/thumbnails/46.jpg)
Example of Single-linkage clustering
![Page 47: Computational Topology - Mapperkbuchin/teaching/2IMA00/2018/Slides/Mapp… · I Feature selection (ares, loops) Applications to Medical science data 145 patients who had diabetes,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d8bba7e708231d43ae4c6/html5/thumbnails/47.jpg)
Example of Single-linkage clustering
![Page 48: Computational Topology - Mapperkbuchin/teaching/2IMA00/2018/Slides/Mapp… · I Feature selection (ares, loops) Applications to Medical science data 145 patients who had diabetes,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d8bba7e708231d43ae4c6/html5/thumbnails/48.jpg)
Example of Single-linkage clustering
![Page 49: Computational Topology - Mapperkbuchin/teaching/2IMA00/2018/Slides/Mapp… · I Feature selection (ares, loops) Applications to Medical science data 145 patients who had diabetes,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d8bba7e708231d43ae4c6/html5/thumbnails/49.jpg)
Example of Single-linkage clustering
![Page 50: Computational Topology - Mapperkbuchin/teaching/2IMA00/2018/Slides/Mapp… · I Feature selection (ares, loops) Applications to Medical science data 145 patients who had diabetes,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d8bba7e708231d43ae4c6/html5/thumbnails/50.jpg)
Example of Single-linkage clustering
![Page 51: Computational Topology - Mapperkbuchin/teaching/2IMA00/2018/Slides/Mapp… · I Feature selection (ares, loops) Applications to Medical science data 145 patients who had diabetes,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d8bba7e708231d43ae4c6/html5/thumbnails/51.jpg)
Example of Clustering algorithm with different parameters
![Page 52: Computational Topology - Mapperkbuchin/teaching/2IMA00/2018/Slides/Mapp… · I Feature selection (ares, loops) Applications to Medical science data 145 patients who had diabetes,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d8bba7e708231d43ae4c6/html5/thumbnails/52.jpg)
Example of Clustering algorithm with different parameters
![Page 53: Computational Topology - Mapperkbuchin/teaching/2IMA00/2018/Slides/Mapp… · I Feature selection (ares, loops) Applications to Medical science data 145 patients who had diabetes,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d8bba7e708231d43ae4c6/html5/thumbnails/53.jpg)
Example of Clustering algorithm with different parameters
![Page 54: Computational Topology - Mapperkbuchin/teaching/2IMA00/2018/Slides/Mapp… · I Feature selection (ares, loops) Applications to Medical science data 145 patients who had diabetes,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d8bba7e708231d43ae4c6/html5/thumbnails/54.jpg)
Example of Clustering algorithm with different parameters
![Page 55: Computational Topology - Mapperkbuchin/teaching/2IMA00/2018/Slides/Mapp… · I Feature selection (ares, loops) Applications to Medical science data 145 patients who had diabetes,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d8bba7e708231d43ae4c6/html5/thumbnails/55.jpg)
Parameters of graph neighborhood size
Parameter δ:
I Large δ: fewer nodes, clean Mapper but far from ReebGraph(more straight lines).
I Small δ: presence of topological structure but lots of nodes(noisy).
![Page 56: Computational Topology - Mapperkbuchin/teaching/2IMA00/2018/Slides/Mapp… · I Feature selection (ares, loops) Applications to Medical science data 145 patients who had diabetes,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d8bba7e708231d43ae4c6/html5/thumbnails/56.jpg)
Higher Dimensional Parameter Spaces
I We use 1 function and let R to be our 1-dimensionalparameter space.
I We can use M functions and let RM to be our M-dimensionalparameter space, remain to find a covering of anM-dimensional hypercube which is defined by the ranges ofthe M functions.
![Page 57: Computational Topology - Mapperkbuchin/teaching/2IMA00/2018/Slides/Mapp… · I Feature selection (ares, loops) Applications to Medical science data 145 patients who had diabetes,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d8bba7e708231d43ae4c6/html5/thumbnails/57.jpg)
Higher Dimensional Parameter Spaces
I We use 1 function and let R to be our 1-dimensionalparameter space.
I We can use M functions and let RM to be our M-dimensionalparameter space, remain to find a covering of anM-dimensional hypercube which is defined by the ranges ofthe M functions.
![Page 58: Computational Topology - Mapperkbuchin/teaching/2IMA00/2018/Slides/Mapp… · I Feature selection (ares, loops) Applications to Medical science data 145 patients who had diabetes,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d8bba7e708231d43ae4c6/html5/thumbnails/58.jpg)
Example of parameter space R2
I Assume we have a point could dataset P (2-Dim) as following.
I Assume we have two filter functions f : P → R, g : P → R,and f = f −1 and g = g−1.
![Page 59: Computational Topology - Mapperkbuchin/teaching/2IMA00/2018/Slides/Mapp… · I Feature selection (ares, loops) Applications to Medical science data 145 patients who had diabetes,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d8bba7e708231d43ae4c6/html5/thumbnails/59.jpg)
Example of parameter space R2
I Moreover, assume we have the following cover C , which isalso the cover of P since f = f −1 and g = g−1.
![Page 60: Computational Topology - Mapperkbuchin/teaching/2IMA00/2018/Slides/Mapp… · I Feature selection (ares, loops) Applications to Medical science data 145 patients who had diabetes,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d8bba7e708231d43ae4c6/html5/thumbnails/60.jpg)
Example of parameter space R2
I Moreover, assume we have the following cover C , which isalso the cover of P since f = f −1 and g = g−1.
I Assume the clustering algorithm group every points in eachrectangle as one cluster.
![Page 61: Computational Topology - Mapperkbuchin/teaching/2IMA00/2018/Slides/Mapp… · I Feature selection (ares, loops) Applications to Medical science data 145 patients who had diabetes,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d8bba7e708231d43ae4c6/html5/thumbnails/61.jpg)
Example of parameter space R2
I Moreover, assume we have the following cover C , which isalso the cover of P since f = f −1 and g = g−1.
I Assume the clustering algorithm group every points in eachrectangle as one cluster.
![Page 62: Computational Topology - Mapperkbuchin/teaching/2IMA00/2018/Slides/Mapp… · I Feature selection (ares, loops) Applications to Medical science data 145 patients who had diabetes,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d8bba7e708231d43ae4c6/html5/thumbnails/62.jpg)
Example of parameter space R2
I Moreover, assume we have the following cover C , which isalso the cover of P since f = f −1 and g = g−1.
I Assume the clustering algorithm group every points in eachrectangle as one cluster.
I Whenever clusters corresponding to any n vertices have nonempty intersection, add a corresponding n-1 simplex.
![Page 63: Computational Topology - Mapperkbuchin/teaching/2IMA00/2018/Slides/Mapp… · I Feature selection (ares, loops) Applications to Medical science data 145 patients who had diabetes,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d8bba7e708231d43ae4c6/html5/thumbnails/63.jpg)
Example of parameter space R2
I Two clusters intersection = 1 edge.
![Page 64: Computational Topology - Mapperkbuchin/teaching/2IMA00/2018/Slides/Mapp… · I Feature selection (ares, loops) Applications to Medical science data 145 patients who had diabetes,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d8bba7e708231d43ae4c6/html5/thumbnails/64.jpg)
Example of parameter space R2
I Three clusters intersection = 1 triangle.
![Page 65: Computational Topology - Mapperkbuchin/teaching/2IMA00/2018/Slides/Mapp… · I Feature selection (ares, loops) Applications to Medical science data 145 patients who had diabetes,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d8bba7e708231d43ae4c6/html5/thumbnails/65.jpg)
Example of parameter space R2
I Four clusters intersection = 1 tetrahedron.
![Page 66: Computational Topology - Mapperkbuchin/teaching/2IMA00/2018/Slides/Mapp… · I Feature selection (ares, loops) Applications to Medical science data 145 patients who had diabetes,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d8bba7e708231d43ae4c6/html5/thumbnails/66.jpg)
Example of parameter space R2
I Final simplical complex.
![Page 67: Computational Topology - Mapperkbuchin/teaching/2IMA00/2018/Slides/Mapp… · I Feature selection (ares, loops) Applications to Medical science data 145 patients who had diabetes,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d8bba7e708231d43ae4c6/html5/thumbnails/67.jpg)
Higher Dimensional Parameter Spaces
Mapper to the parameter space RM can be extended in a similarfashion (by finding a covering of an M-dimensional hypercubewhich is defined by the ranges of the M functions).
![Page 68: Computational Topology - Mapperkbuchin/teaching/2IMA00/2018/Slides/Mapp… · I Feature selection (ares, loops) Applications to Medical science data 145 patients who had diabetes,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d8bba7e708231d43ae4c6/html5/thumbnails/68.jpg)
Introduction
Mapper in the continuous setting
Mapper in practice
Parameters of Mapper in practice
Applications
Summary
![Page 69: Computational Topology - Mapperkbuchin/teaching/2IMA00/2018/Slides/Mapp… · I Feature selection (ares, loops) Applications to Medical science data 145 patients who had diabetes,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d8bba7e708231d43ae4c6/html5/thumbnails/69.jpg)
Mapper in Applications
Most commonly used in:
I Clustering
I Feature selection (flares, loops)
![Page 70: Computational Topology - Mapperkbuchin/teaching/2IMA00/2018/Slides/Mapp… · I Feature selection (ares, loops) Applications to Medical science data 145 patients who had diabetes,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d8bba7e708231d43ae4c6/html5/thumbnails/70.jpg)
Applications to Medical science data
145 patients who had diabetes, for each patient, six quantitieswere measured:
I Age
I Relative weight
I Fasting plasma glucose
I Area under the plasma glucose curve for the three hourglucose tolerance test (OGTT)
I Aarea under the plasma insulin curve for the (OGTT)
I Steady state plasma glucose response
This creates a 6 dimensional data set.
![Page 71: Computational Topology - Mapperkbuchin/teaching/2IMA00/2018/Slides/Mapp… · I Feature selection (ares, loops) Applications to Medical science data 145 patients who had diabetes,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d8bba7e708231d43ae4c6/html5/thumbnails/71.jpg)
Applications to Medical science data
I Applying projection pursuit methods to obtain a projectioninto three dimensional Euclidean space
We want to use Mapper as an automatic tool for detectingsuch flares in the data.
![Page 72: Computational Topology - Mapperkbuchin/teaching/2IMA00/2018/Slides/Mapp… · I Feature selection (ares, loops) Applications to Medical science data 145 patients who had diabetes,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d8bba7e708231d43ae4c6/html5/thumbnails/72.jpg)
Applications to Medical science data
I Applying projection pursuit methods to obtain a projectioninto three dimensional Euclidean space
We want to use Mapper as an automatic tool for detectingsuch flares in the data.
![Page 73: Computational Topology - Mapperkbuchin/teaching/2IMA00/2018/Slides/Mapp… · I Feature selection (ares, loops) Applications to Medical science data 145 patients who had diabetes,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d8bba7e708231d43ae4c6/html5/thumbnails/73.jpg)
Applications to Medical science data
I Left: 3 intervals, 50% overlap.
I Right: 4 intervals, 50% overlap.I For each output:
I Left flare: adult onset Right flare: juvenile onsetI Distance function: L2-distanceI Filter function: density kernel with e=130,000
![Page 74: Computational Topology - Mapperkbuchin/teaching/2IMA00/2018/Slides/Mapp… · I Feature selection (ares, loops) Applications to Medical science data 145 patients who had diabetes,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d8bba7e708231d43ae4c6/html5/thumbnails/74.jpg)
Mapper in Applications
I Innate and adaptive T cells in asthmatic patients:Relationship to severity and disease mechanisms, Hinks et al.,J. Allergy Clinical Immunology, 2015
I Topological Data Analysis for Discovery in Preclinical SpinalCord Injury and Traumatic Brain Injury, Nielson et al., Nature,2015
I Using Topological Data Analysis for Diagnosis PulmonaryEmbolism, Rucco et al., arXiv preprint, 2014
I CD8 T-cell reactivity to islet antigens is unique to type 1while CD4 T-cell reactivity exists in both type 1 and type 2diabetes, Sarikonda et al., J. Autoimmunity, 2013
I Extracting insights from the shape of complex data usingtopology, Lum et al., Nature, 2013
I Topological Methods for Exploring Low-density States inBiomolecular Folding Pathways, Yao et al., J. ChemicalPhysics, 2009
![Page 75: Computational Topology - Mapperkbuchin/teaching/2IMA00/2018/Slides/Mapp… · I Feature selection (ares, loops) Applications to Medical science data 145 patients who had diabetes,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d8bba7e708231d43ae4c6/html5/thumbnails/75.jpg)
Introduction
Mapper in the continuous setting
Mapper in practice
Parameters of Mapper in practice
Applications
Summary
![Page 76: Computational Topology - Mapperkbuchin/teaching/2IMA00/2018/Slides/Mapp… · I Feature selection (ares, loops) Applications to Medical science data 145 patients who had diabetes,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d8bba7e708231d43ae4c6/html5/thumbnails/76.jpg)
Summary
I Mapper: a computational method which retrieves ahigher-level understanding of the structure of data.
I Mapper in continuous setting.
I Mapper in practiceI Parameters of Mapper in practice
I filter function.I covering algorithm.I clustering algorithm.
I Applications
![Page 77: Computational Topology - Mapperkbuchin/teaching/2IMA00/2018/Slides/Mapp… · I Feature selection (ares, loops) Applications to Medical science data 145 patients who had diabetes,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d8bba7e708231d43ae4c6/html5/thumbnails/77.jpg)
Sources
I [SMG07] G. Singh, F. M’emoli, G. Carlsson, TopologicalMethods for the Analysis of High Dimensional Data Sets and3D Object Recognition, Eurographics Symposium onPoint-Based Graphics 2007.
I Examples and images from Tutorial of topological dataanalysis part 3(Mapper algorithm):https://www.slideshare.net/Eniod/tutorial-of-topological-data-analysis-part-3mapper-algorithm
I Examples and images from Introduction to Topological DataAnalysis:https://www.slideshare.net/hendrikarisma/introduction-to-topological-data-analysis-59759836
I Examples and images from KeplerMapper:https://mlwave.github.io/kepler-mapper/