topological data analysis
DESCRIPTION
Topological Data Analysis. MATH 800 Fall 2011. Topological Data Analysis (TDA). An ε -chain is a finite sequence of points x 1 , ..., x n such that |x i – x i +1 | < ε for all i =1,…,n-1 x and y are ε -connected if there is an ε - chain joining them. - PowerPoint PPT PresentationTRANSCRIPT
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Topological Data Analysis
MATH 800Fall 2011
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Topological Data Analysis (TDA)
• An ε-chain is a finite sequence of points x1, ..., xn such that |xi – xi+1 | < ε for all i=1,…,n-1
• x and y are ε-connected if there is an ε-chain joining them.
• A ε-connected set is a set that any two points can be linked by an ε-chain.
• A ε-connected component is a maximal ε-connected subset.
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Topological Data Analysis (TDA)
• An ε-chain is a finite sequence of points x1, ..., xn such that |xi – xi+1 | < ε for all i=1,…,n-1
• x and y are ε-connected if there is an ε-chain joining them.
• A ε-connected set is a set that any two points can be linked by an ε-chain.
• A ε-connected component is a maximal ε-connected subset.
ε
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Topological Data Analysis (TDA)
• Cε = the number of ε-connected components
• Dε = the maximum distance of the points in ε-connected components
• Iε = the number of component consisting of a single point (isolated points)
ε
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Minimal Spanning Tree (MST)• MST is the tree of minimum total branch length that
spans the data.• To construct the MST (Prim's algorithm)
– Start with any point and its nearest neighbor,– Add the closest point– Repeat until all points are in the tree
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Minimal Spanning Tree (MST)• MST is the tree of minimum total branch length that
spans the data.• To construct the MST (Prim's algorithm)
– Start with any point and its nearest neighbor,– Add the closest point– Repeat until all points are in the tree
![Page 7: Topological Data Analysis](https://reader036.vdocuments.net/reader036/viewer/2022062316/56816707550346895ddb6ed3/html5/thumbnails/7.jpg)
Minimal Spanning Tree (MST)• MST is the tree of minimum total branch length that
spans the data.• To construct the MST (Prim's algorithm)
– Start with any point and its nearest neighbor,– Add the closest point– Repeat until all points are in the tree
ε = 6
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Topological Data Analysis (TDA)
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Topological Data Analysis (TDA)
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Topological Data Analysis (TDA)
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Topological Data Analysis (TDA)
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Applications of TDA
• Cluster analysis: is the task of assigning a set of objects into groups so that the objects in the same cluster are more similar to each other than to those in other clusters.– connected components
• Outlier: is an observation that is numerically distant from the rest of the data.
• Noise: is an unwanted value in a data– isolated points
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C
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Geometry of Data
• Consider 106 points and each point is a string of 100 numbers.– Is it one piece or more? – Is there a tunnel? Or a void?
• Point clouds in 100-D Euclidean space.
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Point Cloud
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Point Cloud
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Point Cloud
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Point Cloud
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Point Cloud
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Point Cloud
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Triangulation
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Triangulation
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Triangulation
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Triangulation
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Triangulation
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Point Cloud
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MRI
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GIS – Elevation Model
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GIS – Urban Environment