an introduction to topological data analysis · an introduction to topological data analysis....
TRANSCRIPT
![Page 1: An Introduction to Topological Data Analysis · An Introduction to Topological Data Analysis. Motivation The classical problem of fitting data set of point in R n using linear regression](https://reader030.vdocuments.us/reader030/viewer/2022040611/5ed79e09dff44a19a36e8bb0/html5/thumbnails/1.jpg)
Mustafa Hajij
An Introduction to Topological Data Analysis
![Page 2: An Introduction to Topological Data Analysis · An Introduction to Topological Data Analysis. Motivation The classical problem of fitting data set of point in R n using linear regression](https://reader030.vdocuments.us/reader030/viewer/2022040611/5ed79e09dff44a19a36e8bb0/html5/thumbnails/2.jpg)
Motivation
The classical problem of fitting data set of point in R n using linear regression
![Page 3: An Introduction to Topological Data Analysis · An Introduction to Topological Data Analysis. Motivation The classical problem of fitting data set of point in R n using linear regression](https://reader030.vdocuments.us/reader030/viewer/2022040611/5ed79e09dff44a19a36e8bb0/html5/thumbnails/3.jpg)
Motivation
The linear shape of the data is a fundamental assumption underlying the linear regression method
The classical problem of fitting data set of point in R n using linear regression
![Page 4: An Introduction to Topological Data Analysis · An Introduction to Topological Data Analysis. Motivation The classical problem of fitting data set of point in R n using linear regression](https://reader030.vdocuments.us/reader030/viewer/2022040611/5ed79e09dff44a19a36e8bb0/html5/thumbnails/4.jpg)
Motivation
The linear shape of the data is a fundamental assumption underlying the linear regression method
Clustering algorithms assume that the data is clustered in a certain way.
The classical problem of fitting data set of point in R n using linear regression
![Page 5: An Introduction to Topological Data Analysis · An Introduction to Topological Data Analysis. Motivation The classical problem of fitting data set of point in R n using linear regression](https://reader030.vdocuments.us/reader030/viewer/2022040611/5ed79e09dff44a19a36e8bb0/html5/thumbnails/5.jpg)
Motivation
Understanding the shape of the data is a fundamental assumption underlying the analytical method
![Page 6: An Introduction to Topological Data Analysis · An Introduction to Topological Data Analysis. Motivation The classical problem of fitting data set of point in R n using linear regression](https://reader030.vdocuments.us/reader030/viewer/2022040611/5ed79e09dff44a19a36e8bb0/html5/thumbnails/6.jpg)
Concept
Space
![Page 7: An Introduction to Topological Data Analysis · An Introduction to Topological Data Analysis. Motivation The classical problem of fitting data set of point in R n using linear regression](https://reader030.vdocuments.us/reader030/viewer/2022040611/5ed79e09dff44a19a36e8bb0/html5/thumbnails/7.jpg)
Concept
Topological space
![Page 8: An Introduction to Topological Data Analysis · An Introduction to Topological Data Analysis. Motivation The classical problem of fitting data set of point in R n using linear regression](https://reader030.vdocuments.us/reader030/viewer/2022040611/5ed79e09dff44a19a36e8bb0/html5/thumbnails/8.jpg)
Concept
Metric Space
![Page 9: An Introduction to Topological Data Analysis · An Introduction to Topological Data Analysis. Motivation The classical problem of fitting data set of point in R n using linear regression](https://reader030.vdocuments.us/reader030/viewer/2022040611/5ed79e09dff44a19a36e8bb0/html5/thumbnails/9.jpg)
Concept
![Page 10: An Introduction to Topological Data Analysis · An Introduction to Topological Data Analysis. Motivation The classical problem of fitting data set of point in R n using linear regression](https://reader030.vdocuments.us/reader030/viewer/2022040611/5ed79e09dff44a19a36e8bb0/html5/thumbnails/10.jpg)
Concept
![Page 11: An Introduction to Topological Data Analysis · An Introduction to Topological Data Analysis. Motivation The classical problem of fitting data set of point in R n using linear regression](https://reader030.vdocuments.us/reader030/viewer/2022040611/5ed79e09dff44a19a36e8bb0/html5/thumbnails/11.jpg)
Roughly speaking, topology of an object studies the way this object is
connected.
Topology studies the properties of shapes that do not change under
continuous deformations.
Motivation
![Page 12: An Introduction to Topological Data Analysis · An Introduction to Topological Data Analysis. Motivation The classical problem of fitting data set of point in R n using linear regression](https://reader030.vdocuments.us/reader030/viewer/2022040611/5ed79e09dff44a19a36e8bb0/html5/thumbnails/12.jpg)
Roughly speaking, topology of an object studies the way this object is
connected.
Topology studies the properties of shapes that do not change under
continuous deformations.
==
Motivation
![Page 13: An Introduction to Topological Data Analysis · An Introduction to Topological Data Analysis. Motivation The classical problem of fitting data set of point in R n using linear regression](https://reader030.vdocuments.us/reader030/viewer/2022040611/5ed79e09dff44a19a36e8bb0/html5/thumbnails/13.jpg)
Roughly speaking, topology of an object studies the way this object is
connected.
Topology studies the properties of shapes that do not change under
continuous deformations.
==
So topologically, the following objects are
equivalent because we can deform each one of
them contentiously without tearing into the other.
Motivation
![Page 14: An Introduction to Topological Data Analysis · An Introduction to Topological Data Analysis. Motivation The classical problem of fitting data set of point in R n using linear regression](https://reader030.vdocuments.us/reader030/viewer/2022040611/5ed79e09dff44a19a36e8bb0/html5/thumbnails/14.jpg)
Roughly speaking, topology of an object studies the way this object is
connected.
Topology studies the properties of shapes that do not change under
continuous deformations.
==
So topologically, the following objects are
equivalent because we can deform each one of
them contentiously without tearing into the other.
Motivation
=/=However, the sphere cannot
be continuously deformed into
the torus. Hence the sphere
and the torus are topologically
district.
![Page 15: An Introduction to Topological Data Analysis · An Introduction to Topological Data Analysis. Motivation The classical problem of fitting data set of point in R n using linear regression](https://reader030.vdocuments.us/reader030/viewer/2022040611/5ed79e09dff44a19a36e8bb0/html5/thumbnails/15.jpg)
Roughly speaking, topology of an object studies the way this object is
connected.
Topology studies the properties of shapes that do not change under
continuous deformations.
==
So topologically, the following objects are
equivalent because we can deform each one of
them contentiously without tearing into the other.
Motivation
Topology makes these notions precise.
=/=However, the sphere cannot
be continuously deformed into
the torus. Hence the sphere
and the torus are topologically
district.
![Page 16: An Introduction to Topological Data Analysis · An Introduction to Topological Data Analysis. Motivation The classical problem of fitting data set of point in R n using linear regression](https://reader030.vdocuments.us/reader030/viewer/2022040611/5ed79e09dff44a19a36e8bb0/html5/thumbnails/16.jpg)
Continuous functions
Which of the following functions is continuous ?
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Continuous functions
Which of the following functions is continuous ?
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Homeomorphism
![Page 19: An Introduction to Topological Data Analysis · An Introduction to Topological Data Analysis. Motivation The classical problem of fitting data set of point in R n using linear regression](https://reader030.vdocuments.us/reader030/viewer/2022040611/5ed79e09dff44a19a36e8bb0/html5/thumbnails/19.jpg)
Homeomorphism
![Page 20: An Introduction to Topological Data Analysis · An Introduction to Topological Data Analysis. Motivation The classical problem of fitting data set of point in R n using linear regression](https://reader030.vdocuments.us/reader030/viewer/2022040611/5ed79e09dff44a19a36e8bb0/html5/thumbnails/20.jpg)
Which of the following functions is homeomorphism? If not, which one of the three conditions is violated?
Homeomorphism
![Page 21: An Introduction to Topological Data Analysis · An Introduction to Topological Data Analysis. Motivation The classical problem of fitting data set of point in R n using linear regression](https://reader030.vdocuments.us/reader030/viewer/2022040611/5ed79e09dff44a19a36e8bb0/html5/thumbnails/21.jpg)
Homeomorphism-examples
![Page 22: An Introduction to Topological Data Analysis · An Introduction to Topological Data Analysis. Motivation The classical problem of fitting data set of point in R n using linear regression](https://reader030.vdocuments.us/reader030/viewer/2022040611/5ed79e09dff44a19a36e8bb0/html5/thumbnails/22.jpg)
How can we explain a topological space to a computer ?
![Page 23: An Introduction to Topological Data Analysis · An Introduction to Topological Data Analysis. Motivation The classical problem of fitting data set of point in R n using linear regression](https://reader030.vdocuments.us/reader030/viewer/2022040611/5ed79e09dff44a19a36e8bb0/html5/thumbnails/23.jpg)
How can we explain a topological space to a computer ?
![Page 24: An Introduction to Topological Data Analysis · An Introduction to Topological Data Analysis. Motivation The classical problem of fitting data set of point in R n using linear regression](https://reader030.vdocuments.us/reader030/viewer/2022040611/5ed79e09dff44a19a36e8bb0/html5/thumbnails/24.jpg)
How can we explain a topological space to a computer ?
![Page 25: An Introduction to Topological Data Analysis · An Introduction to Topological Data Analysis. Motivation The classical problem of fitting data set of point in R n using linear regression](https://reader030.vdocuments.us/reader030/viewer/2022040611/5ed79e09dff44a19a36e8bb0/html5/thumbnails/25.jpg)
How can we explain a topological space to a computer ?
Key Idea : we use simple building blocks (called simplices)
![Page 26: An Introduction to Topological Data Analysis · An Introduction to Topological Data Analysis. Motivation The classical problem of fitting data set of point in R n using linear regression](https://reader030.vdocuments.us/reader030/viewer/2022040611/5ed79e09dff44a19a36e8bb0/html5/thumbnails/26.jpg)
How can we explain a topological space to a computer ?
Key Idea : we use simple building blocks (called simplices)
to build more complicated shape.
![Page 27: An Introduction to Topological Data Analysis · An Introduction to Topological Data Analysis. Motivation The classical problem of fitting data set of point in R n using linear regression](https://reader030.vdocuments.us/reader030/viewer/2022040611/5ed79e09dff44a19a36e8bb0/html5/thumbnails/27.jpg)
How can we explain a topological space to a computer ?
![Page 28: An Introduction to Topological Data Analysis · An Introduction to Topological Data Analysis. Motivation The classical problem of fitting data set of point in R n using linear regression](https://reader030.vdocuments.us/reader030/viewer/2022040611/5ed79e09dff44a19a36e8bb0/html5/thumbnails/28.jpg)
Graphs representation: list of edges
[ [0,1], [1,3],[1,4] [3,4], [3,2]]
• The vertices can be recovered from the edges.
• The order of the vertices is important only if the graph is directed.
![Page 29: An Introduction to Topological Data Analysis · An Introduction to Topological Data Analysis. Motivation The classical problem of fitting data set of point in R n using linear regression](https://reader030.vdocuments.us/reader030/viewer/2022040611/5ed79e09dff44a19a36e8bb0/html5/thumbnails/29.jpg)
Simplicial Complex-precise definition
![Page 30: An Introduction to Topological Data Analysis · An Introduction to Topological Data Analysis. Motivation The classical problem of fitting data set of point in R n using linear regression](https://reader030.vdocuments.us/reader030/viewer/2022040611/5ed79e09dff44a19a36e8bb0/html5/thumbnails/30.jpg)
Examples and non-examples
![Page 31: An Introduction to Topological Data Analysis · An Introduction to Topological Data Analysis. Motivation The classical problem of fitting data set of point in R n using linear regression](https://reader030.vdocuments.us/reader030/viewer/2022040611/5ed79e09dff44a19a36e8bb0/html5/thumbnails/31.jpg)
Examples and non-examples
![Page 32: An Introduction to Topological Data Analysis · An Introduction to Topological Data Analysis. Motivation The classical problem of fitting data set of point in R n using linear regression](https://reader030.vdocuments.us/reader030/viewer/2022040611/5ed79e09dff44a19a36e8bb0/html5/thumbnails/32.jpg)
Examples and non-examples
![Page 33: An Introduction to Topological Data Analysis · An Introduction to Topological Data Analysis. Motivation The classical problem of fitting data set of point in R n using linear regression](https://reader030.vdocuments.us/reader030/viewer/2022040611/5ed79e09dff44a19a36e8bb0/html5/thumbnails/33.jpg)
Examples and non-examples
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Simplicial Complex
![Page 35: An Introduction to Topological Data Analysis · An Introduction to Topological Data Analysis. Motivation The classical problem of fitting data set of point in R n using linear regression](https://reader030.vdocuments.us/reader030/viewer/2022040611/5ed79e09dff44a19a36e8bb0/html5/thumbnails/35.jpg)
Approximation of the shape
![Page 36: An Introduction to Topological Data Analysis · An Introduction to Topological Data Analysis. Motivation The classical problem of fitting data set of point in R n using linear regression](https://reader030.vdocuments.us/reader030/viewer/2022040611/5ed79e09dff44a19a36e8bb0/html5/thumbnails/36.jpg)
Cover of a space
A cover of a space X is a collection of sets U whose union is the entire space
![Page 37: An Introduction to Topological Data Analysis · An Introduction to Topological Data Analysis. Motivation The classical problem of fitting data set of point in R n using linear regression](https://reader030.vdocuments.us/reader030/viewer/2022040611/5ed79e09dff44a19a36e8bb0/html5/thumbnails/37.jpg)
Cover of a space
A cover of a space X is a collection of sets U whose union is the entire space
![Page 38: An Introduction to Topological Data Analysis · An Introduction to Topological Data Analysis. Motivation The classical problem of fitting data set of point in R n using linear regression](https://reader030.vdocuments.us/reader030/viewer/2022040611/5ed79e09dff44a19a36e8bb0/html5/thumbnails/38.jpg)
Approximation of the shape
Covers can be used to obtain an approximation for the underlying data
![Page 39: An Introduction to Topological Data Analysis · An Introduction to Topological Data Analysis. Motivation The classical problem of fitting data set of point in R n using linear regression](https://reader030.vdocuments.us/reader030/viewer/2022040611/5ed79e09dff44a19a36e8bb0/html5/thumbnails/39.jpg)
Approximation of the shape
Covers can be used to obtain an approximation for the underlying data
![Page 40: An Introduction to Topological Data Analysis · An Introduction to Topological Data Analysis. Motivation The classical problem of fitting data set of point in R n using linear regression](https://reader030.vdocuments.us/reader030/viewer/2022040611/5ed79e09dff44a19a36e8bb0/html5/thumbnails/40.jpg)
Approximation of the shape
Nerve of a space
Covers can be used to obtain an approximation for the underlying data
![Page 41: An Introduction to Topological Data Analysis · An Introduction to Topological Data Analysis. Motivation The classical problem of fitting data set of point in R n using linear regression](https://reader030.vdocuments.us/reader030/viewer/2022040611/5ed79e09dff44a19a36e8bb0/html5/thumbnails/41.jpg)
Approximation of the shape
Key idea: every set is replaced by a nodeevery intersection is replaced by an edgeif we have intersection between three sets we replace them with a face and so on.
![Page 44: An Introduction to Topological Data Analysis · An Introduction to Topological Data Analysis. Motivation The classical problem of fitting data set of point in R n using linear regression](https://reader030.vdocuments.us/reader030/viewer/2022040611/5ed79e09dff44a19a36e8bb0/html5/thumbnails/44.jpg)
Mapper Construction
![Page 45: An Introduction to Topological Data Analysis · An Introduction to Topological Data Analysis. Motivation The classical problem of fitting data set of point in R n using linear regression](https://reader030.vdocuments.us/reader030/viewer/2022040611/5ed79e09dff44a19a36e8bb0/html5/thumbnails/45.jpg)
Mapper Construction
![Page 46: An Introduction to Topological Data Analysis · An Introduction to Topological Data Analysis. Motivation The classical problem of fitting data set of point in R n using linear regression](https://reader030.vdocuments.us/reader030/viewer/2022040611/5ed79e09dff44a19a36e8bb0/html5/thumbnails/46.jpg)
Mapper Construction
![Page 47: An Introduction to Topological Data Analysis · An Introduction to Topological Data Analysis. Motivation The classical problem of fitting data set of point in R n using linear regression](https://reader030.vdocuments.us/reader030/viewer/2022040611/5ed79e09dff44a19a36e8bb0/html5/thumbnails/47.jpg)
Mapper Example
Suppose that we are given a data set 𝑋 and a scalar function 𝑓: 𝑋 → [𝑎, 𝑏] defined on every point in X.
𝑋
[0,1]
0
1
![Page 48: An Introduction to Topological Data Analysis · An Introduction to Topological Data Analysis. Motivation The classical problem of fitting data set of point in R n using linear regression](https://reader030.vdocuments.us/reader030/viewer/2022040611/5ed79e09dff44a19a36e8bb0/html5/thumbnails/48.jpg)
Mapper Example
Suppose that we are given a data set 𝑋 and a scalar function 𝑓: 𝑋 → [𝑎, 𝑏] defined on every point in X.
𝑋𝑋
[0,1]
0
1
![Page 49: An Introduction to Topological Data Analysis · An Introduction to Topological Data Analysis. Motivation The classical problem of fitting data set of point in R n using linear regression](https://reader030.vdocuments.us/reader030/viewer/2022040611/5ed79e09dff44a19a36e8bb0/html5/thumbnails/49.jpg)
Mapper Example
Suppose that we are given a data set 𝑋 and a scalar function 𝑓: 𝑋 → [𝑎, 𝑏] defined on every point in X.
𝑋
[0,1]
0
1
![Page 50: An Introduction to Topological Data Analysis · An Introduction to Topological Data Analysis. Motivation The classical problem of fitting data set of point in R n using linear regression](https://reader030.vdocuments.us/reader030/viewer/2022040611/5ed79e09dff44a19a36e8bb0/html5/thumbnails/50.jpg)
Mapper Example
Suppose that we are given a data set 𝑋 and a scalar function 𝑓: 𝑋 → [𝑎, 𝑏] defined on every point in X.
𝑋
[0,1]
0
1
![Page 51: An Introduction to Topological Data Analysis · An Introduction to Topological Data Analysis. Motivation The classical problem of fitting data set of point in R n using linear regression](https://reader030.vdocuments.us/reader030/viewer/2022040611/5ed79e09dff44a19a36e8bb0/html5/thumbnails/51.jpg)
Mapper Example
![Page 52: An Introduction to Topological Data Analysis · An Introduction to Topological Data Analysis. Motivation The classical problem of fitting data set of point in R n using linear regression](https://reader030.vdocuments.us/reader030/viewer/2022040611/5ed79e09dff44a19a36e8bb0/html5/thumbnails/52.jpg)
Mapper Example
![Page 53: An Introduction to Topological Data Analysis · An Introduction to Topological Data Analysis. Motivation The classical problem of fitting data set of point in R n using linear regression](https://reader030.vdocuments.us/reader030/viewer/2022040611/5ed79e09dff44a19a36e8bb0/html5/thumbnails/53.jpg)
Mapper Example
![Page 54: An Introduction to Topological Data Analysis · An Introduction to Topological Data Analysis. Motivation The classical problem of fitting data set of point in R n using linear regression](https://reader030.vdocuments.us/reader030/viewer/2022040611/5ed79e09dff44a19a36e8bb0/html5/thumbnails/54.jpg)
Mapper Example
![Page 55: An Introduction to Topological Data Analysis · An Introduction to Topological Data Analysis. Motivation The classical problem of fitting data set of point in R n using linear regression](https://reader030.vdocuments.us/reader030/viewer/2022040611/5ed79e09dff44a19a36e8bb0/html5/thumbnails/55.jpg)
The construction of Mapper on a 1d function
(a) A scalar function f : X −→ [a,b].
This gives a decomposition of the domain the domain X. The inverse image of A consists of two connected components α1 and α2, and the inverse image of B consists of three connected components β1, β3 and β3.
The connected components are represented by the nodes in the Mapper construction.
an edge is inserted whenever two connected components overlap.
![Page 56: An Introduction to Topological Data Analysis · An Introduction to Topological Data Analysis. Motivation The classical problem of fitting data set of point in R n using linear regression](https://reader030.vdocuments.us/reader030/viewer/2022040611/5ed79e09dff44a19a36e8bb0/html5/thumbnails/56.jpg)
The construction of Mapper on a 1d function
(a) A scalar function f : X −→ [a,b].
(b) This gives a decomposition of the domain the domain X. The inverse image of A consists of two connected components α1 and α2, and the inverse image of B consists of three connected components β1, β3 and β3.The connected components are represented by the nodes in the Mapper construction.
an edge is inserted whenever two connected components overlap.
Cover
(b) The range [a,b] is covered by the two intervals A,B.
![Page 57: An Introduction to Topological Data Analysis · An Introduction to Topological Data Analysis. Motivation The classical problem of fitting data set of point in R n using linear regression](https://reader030.vdocuments.us/reader030/viewer/2022040611/5ed79e09dff44a19a36e8bb0/html5/thumbnails/57.jpg)
The construction of Mapper on a 1d function
(a) A scalar function f : X −→ [a,b].
(c) This gives a decomposition of the domain the domain X. The inverse image of A consists of two connected components α1 and α2, and the inverse image of B consists of three connected components β1, β3 and β3.
(d) The connected components are represented by the nodes in the Mapper construction.
an edge is inserted whenever two connected components overlap.
Cover
(b) The range [a,b] is covered by the two intervals A,B.
![Page 58: An Introduction to Topological Data Analysis · An Introduction to Topological Data Analysis. Motivation The classical problem of fitting data set of point in R n using linear regression](https://reader030.vdocuments.us/reader030/viewer/2022040611/5ed79e09dff44a19a36e8bb0/html5/thumbnails/58.jpg)
The construction of Mapper on a 1d function
(a) A scalar function f : X −→ [a,b].
(c) This gives a decomposition of the domain the domain X. The inverse image of A consists of two connected components α1 and α2, and the inverse image of B consists of three connected components β1, β3 and β3.
(d) The connected components are represented by the nodes in the Mapper construction.
an edge is inserted whenever two connected components overlap.
Cover
(b) The range [a,b] is covered by the two intervals A,B.
![Page 59: An Introduction to Topological Data Analysis · An Introduction to Topological Data Analysis. Motivation The classical problem of fitting data set of point in R n using linear regression](https://reader030.vdocuments.us/reader030/viewer/2022040611/5ed79e09dff44a19a36e8bb0/html5/thumbnails/59.jpg)
The Mapper Algorithm
Suppose that we are given a data set 𝑋 and a scalar function 𝑓: 𝑋 → [𝑎, 𝑏] defined on every point in X.
1- Define a cover for the interval [a,b]. Say that this cover is 𝑈 = 𝑈1, … , 𝑈𝑛2- Consider all the points x in X with f(x) in 𝑈1. Put those points in container, say 𝑉13-Consider all the points x in X with f(x) in 𝑈2. Put those points in container, say 𝑉24-Do that for every interval 𝑈𝑖 𝑖𝑛 𝑈5-Run a clustering algorithm on on 𝑉1 and store those clusters.6- Run the same clustering algorithm on every 𝑉𝑖7-Create an empty graph G. 8- For every cluster we obtain from {𝑉𝑖|1 ≤ 𝑖 ≤ 𝑛 } create a node for the graph G9- Check overlap between the clusters (nested for loop on all clusters) : whenever there is an overlap insert an edge between the corresponding nodes.10-return the graph G
![Page 60: An Introduction to Topological Data Analysis · An Introduction to Topological Data Analysis. Motivation The classical problem of fitting data set of point in R n using linear regression](https://reader030.vdocuments.us/reader030/viewer/2022040611/5ed79e09dff44a19a36e8bb0/html5/thumbnails/60.jpg)
The Mapper Algorithm
Suppose that we are given a data set 𝑋 and a scalar function 𝑓: 𝑋 → [𝑎, 𝑏] defined on every point in X.
1- Define a cover for the interval [a,b]. Say that this cover is 𝑈 = 𝑈1, … , 𝑈𝑛2- Consider all the points x in X with f(x) in 𝑈1. Put those points in container, say 𝑉13-Consider all the points x in X with f(x) in 𝑈2. Put those points in container, say 𝑉24-Do that for every interval 𝑈𝑖 𝑖𝑛 𝑈5-Run a clustering algorithm on on 𝑉1 and store those clusters.6- Run the same clustering algorithm on every 𝑉𝑖7-Create an empty graph G. 8- For every cluster we obtain from {𝑉𝑖|1 ≤ 𝑖 ≤ 𝑛 } create a node for the graph G9- Check overlap between the clusters (nested for loop on all clusters) : whenever there is an overlap insert an edge between the corresponding nodes.10-return the graph G
![Page 61: An Introduction to Topological Data Analysis · An Introduction to Topological Data Analysis. Motivation The classical problem of fitting data set of point in R n using linear regression](https://reader030.vdocuments.us/reader030/viewer/2022040611/5ed79e09dff44a19a36e8bb0/html5/thumbnails/61.jpg)
The Mapper Algorithm
Suppose that we are given a data set 𝑋 and a scalar function 𝑓: 𝑋 → [𝑎, 𝑏] defined on every point in X.
1- Define a cover for the interval [a,b]. Say that this cover is 𝑈 = 𝑈1, … , 𝑈𝑛2- Consider all the points x in X with f(x) in 𝑈1. Put those points in container, say 𝑉13-Consider all the points x in X with f(x) in 𝑈2. Put those points in container, say 𝑉24-Do that for every interval 𝑈𝑖 𝑖𝑛 𝑈5-Run a clustering algorithm on on 𝑉1 and store those clusters.6- Run the same clustering algorithm on every 𝑉𝑖7-Create an empty graph G. 8- For every cluster we obtain from {𝑉𝑖|1 ≤ 𝑖 ≤ 𝑛 } create a node for the graph G9- Check overlap between the clusters (nested for loop on all clusters) : whenever there is an overlap insert an edge between the corresponding nodes.10-return the graph G
![Page 62: An Introduction to Topological Data Analysis · An Introduction to Topological Data Analysis. Motivation The classical problem of fitting data set of point in R n using linear regression](https://reader030.vdocuments.us/reader030/viewer/2022040611/5ed79e09dff44a19a36e8bb0/html5/thumbnails/62.jpg)
The Mapper Algorithm
Suppose that we are given a data set 𝑋 and a scalar function 𝑓: 𝑋 → [𝑎, 𝑏] defined on every point in X.
1- Define a cover for the interval [a,b]. Say that this cover is 𝑈 = 𝑈1, … , 𝑈𝑛2- Consider all the points x in X with f(x) in 𝑈1. Put those points in container, say 𝑉13-Consider all the points x in X with f(x) in 𝑈2. Put those points in container, say 𝑉24-Do that for every interval 𝑈𝑖 𝑖𝑛 𝑈5-Run a clustering algorithm on on 𝑉1 and store those clusters.6- Run the same clustering algorithm on every 𝑉𝑖7-Create an empty graph G. 8- For every cluster we obtain from {𝑉𝑖|1 ≤ 𝑖 ≤ 𝑛 } create a node for the graph G9- Check overlap between the clusters (nested for loop on all clusters) : whenever there is an overlap insert an edge between the corresponding nodes.10-return the graph G
![Page 63: An Introduction to Topological Data Analysis · An Introduction to Topological Data Analysis. Motivation The classical problem of fitting data set of point in R n using linear regression](https://reader030.vdocuments.us/reader030/viewer/2022040611/5ed79e09dff44a19a36e8bb0/html5/thumbnails/63.jpg)
The Mapper Algorithm
Suppose that we are given a data set 𝑋 and a scalar function 𝑓: 𝑋 → [𝑎, 𝑏] defined on every point in X.
1- Define a cover for the interval [a,b]. Say that this cover is 𝑈 = 𝑈1, … , 𝑈𝑛2- Consider all the points x in X with f(x) in 𝑈1. Put those points in container, say 𝑉13-Consider all the points x in X with f(x) in 𝑈2. Put those points in container, say 𝑉24-Do that for every interval 𝑈𝑖 𝑖𝑛 𝑈5-Run a clustering algorithm on on 𝑉1 and store those clusters.6- Run the same clustering algorithm on every 𝑉𝑖7-Create an empty graph G. 8- For every cluster we obtain from {𝑉𝑖|1 ≤ 𝑖 ≤ 𝑛 } create a node for the graph G9- Check overlap between the clusters (nested for loop on all clusters) : whenever there is an overlap insert an edge between the corresponding nodes.10-return the graph G
![Page 64: An Introduction to Topological Data Analysis · An Introduction to Topological Data Analysis. Motivation The classical problem of fitting data set of point in R n using linear regression](https://reader030.vdocuments.us/reader030/viewer/2022040611/5ed79e09dff44a19a36e8bb0/html5/thumbnails/64.jpg)
The Mapper Algorithm
Suppose that we are given a data set 𝑋 and a scalar function 𝑓: 𝑋 → [𝑎, 𝑏] defined on every point in X.
1- Define a cover for the interval [a,b]. Say that this cover is 𝑈 = 𝑈1, … , 𝑈𝑛2- Consider all the points x in X with f(x) in 𝑈1. Put those points in container, say 𝑉13-Consider all the points x in X with f(x) in 𝑈2. Put those points in container, say 𝑉24-Do that for every interval 𝑈𝑖 𝑖𝑛 𝑈5-Run a clustering algorithm on on 𝑉1 and store those clusters.6- Run the same clustering algorithm on every 𝑉𝑖7-Create an empty graph G. 8- For every cluster we obtain from {𝑉𝑖|1 ≤ 𝑖 ≤ 𝑛 } create a node for the graph G9- Check overlap between the clusters (nested for loop on all clusters) : whenever there is an overlap insert an edge between the corresponding nodes.10-return the graph G
![Page 65: An Introduction to Topological Data Analysis · An Introduction to Topological Data Analysis. Motivation The classical problem of fitting data set of point in R n using linear regression](https://reader030.vdocuments.us/reader030/viewer/2022040611/5ed79e09dff44a19a36e8bb0/html5/thumbnails/65.jpg)
The Mapper Algorithm
Suppose that we are given a data set 𝑋 and a scalar function 𝑓: 𝑋 → [𝑎, 𝑏] defined on every point in X.
1- Define a cover for the interval [a,b]. Say that this cover is 𝑈 = 𝑈1, … , 𝑈𝑛2- Consider all the points x in X with f(x) in 𝑈1. Put those points in container, say 𝑉13-Consider all the points x in X with f(x) in 𝑈2. Put those points in container, say 𝑉24-Do that for every interval 𝑈𝑖 𝑖𝑛 𝑈5-Run a clustering algorithm on on 𝑉1 and store those clusters.6- Run the same clustering algorithm on every 𝑉𝑖7-Create an empty graph G. 8- For every cluster we obtain from {𝑉𝑖|1 ≤ 𝑖 ≤ 𝑛 } create a node for the graph G9- Check overlap between the clusters (nested for loop on all clusters) : whenever there is an overlap insert an edge between the corresponding nodes.10-return the graph G
![Page 66: An Introduction to Topological Data Analysis · An Introduction to Topological Data Analysis. Motivation The classical problem of fitting data set of point in R n using linear regression](https://reader030.vdocuments.us/reader030/viewer/2022040611/5ed79e09dff44a19a36e8bb0/html5/thumbnails/66.jpg)
The Mapper Algorithm
Suppose that we are given a data set 𝑋 and a scalar function 𝑓: 𝑋 → [𝑎, 𝑏] defined on every point in X.
1- Define a cover for the interval [a,b]. Say that this cover is 𝑈 = 𝑈1, … , 𝑈𝑛2- Consider all the points x in X with f(x) in 𝑈1. Put those points in container, say 𝑉13-Consider all the points x in X with f(x) in 𝑈2. Put those points in container, say 𝑉24-Do that for every interval 𝑈𝑖 𝑖𝑛 𝑈5-Run a clustering algorithm on on 𝑉1 and store those clusters.6- Run the same clustering algorithm on every 𝑉𝑖7-Create an empty graph G. 8- For every cluster we obtain from {𝑉𝑖|1 ≤ 𝑖 ≤ 𝑛 } create a node for the graph G9- Check overlap between the clusters (nested for loop on all clusters) : whenever there is an overlap insert an edge between the corresponding nodes.10-return the graph G
![Page 67: An Introduction to Topological Data Analysis · An Introduction to Topological Data Analysis. Motivation The classical problem of fitting data set of point in R n using linear regression](https://reader030.vdocuments.us/reader030/viewer/2022040611/5ed79e09dff44a19a36e8bb0/html5/thumbnails/67.jpg)
The Mapper Algorithm
Suppose that we are given a data set 𝑋 and a scalar function 𝑓: 𝑋 → [𝑎, 𝑏] defined on every point in X.
1- Define a cover for the interval [a,b]. Say that this cover is 𝑈 = 𝑈1, … , 𝑈𝑛2- Consider all the points x in X with f(x) in 𝑈1. Put those points in container, say 𝑉13-Consider all the points x in X with f(x) in 𝑈2. Put those points in container, say 𝑉24-Do that for every interval 𝑈𝑖 𝑖𝑛 𝑈5-Run a clustering algorithm on on 𝑉1 and store those clusters.6- Run the same clustering algorithm on every 𝑉𝑖7-Create an empty graph G. 8- For every cluster we obtain from {𝑉𝑖|1 ≤ 𝑖 ≤ 𝑛 } create a node for the graph G9- Check overlap between the clusters (nested for loop on all clusters) : whenever there is an overlap insert an edge between the corresponding nodes.10-return the graph G
![Page 68: An Introduction to Topological Data Analysis · An Introduction to Topological Data Analysis. Motivation The classical problem of fitting data set of point in R n using linear regression](https://reader030.vdocuments.us/reader030/viewer/2022040611/5ed79e09dff44a19a36e8bb0/html5/thumbnails/68.jpg)
The Mapper Algorithm
Suppose that we are given a data set 𝑋 and a scalar function 𝑓: 𝑋 → [𝑎, 𝑏] defined on every point in X.
1- Define a cover for the interval [a,b]. Say that this cover is 𝑈 = 𝑈1, … , 𝑈𝑛2- Consider all the points x in X with f(x) in 𝑈1. Put those points in container, say 𝑉13-Consider all the points x in X with f(x) in 𝑈2. Put those points in container, say 𝑉24-Do that for every interval 𝑈𝑖 𝑖𝑛 𝑈5-Run a clustering algorithm on on 𝑉1 and store those clusters.6- Run the same clustering algorithm on every 𝑉𝑖7-Create an empty graph G. 8- For every cluster we obtain from {𝑉𝑖|1 ≤ 𝑖 ≤ 𝑛 } create a node for the graph G9- Check overlap between the clusters (nested for loop on all clusters) : whenever there is an overlap insert an edge between the corresponding nodes.10-return the graph G
![Page 69: An Introduction to Topological Data Analysis · An Introduction to Topological Data Analysis. Motivation The classical problem of fitting data set of point in R n using linear regression](https://reader030.vdocuments.us/reader030/viewer/2022040611/5ed79e09dff44a19a36e8bb0/html5/thumbnails/69.jpg)
The Mapper Algorithm
Suppose that we are given a data set 𝑋 and a scalar function 𝑓: 𝑋 → [𝑎, 𝑏] defined on every point in X.
1- Define a cover for the interval [a,b]. Say that this cover is 𝑈 = 𝑈1, … , 𝑈𝑛2- Consider all the points x in X with f(x) in 𝑈1. Put those points in container, say 𝑉13-Consider all the points x in X with f(x) in 𝑈2. Put those points in container, say 𝑉24-Do that for every interval 𝑈𝑖 𝑖𝑛 𝑈5-Run a clustering algorithm on on 𝑉1 and store those clusters.6- Run the same clustering algorithm on every 𝑉𝑖7-Create an empty graph G. 8- For every cluster we obtain from {𝑉𝑖|1 ≤ 𝑖 ≤ 𝑛 } create a node for the graph G9- Check overlap between the clusters (nested for loop on all clusters) : whenever there is an overlap insert an edge between the corresponding nodes.10-return the graph G