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Topological Data Analysis
Matteo Rucco 25/06/2015
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Topology?
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Topology
S Topology studies shapes without using distances!
S Topology is based on some special “functions” for classifying or acting over topological spaces
S We will use HOMOLOGY for classifying objects!
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Are they equivalent?
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They are TOPOLOGICALLY equivalent
Because you can define a continuous transformation without drilling, cutting or twisting the objects
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Benefits
S Topology natural provides clustering
S Topology acts well both on small and big dataset
S Topology is robust to noise: the holes will survive
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Topological Data Analysis
S Biology–Immune System dynamics
S Medicine – Cancer detection from medical images
S Optimization– Sensor network
Point Cloud Data – Timed Point Cloud Data – Networks
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DETAILS TIME
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S Simplicial Complex: is a collection of simplices – higher dimensional triangle
S Simplical complex lets you to do a finite coverage of a geometrical object
How to build a topological space
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How to build a Simplicial Complex
S From Point Cloud Data – points in Rd d-dimensions: Vietoris-Rips, Cech or with Witness complexes
S From Undirected Weighted Networks: clique-complexes
S From Directed Weighted Networks: neighborhood-complexes
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Vietoris-Rips Complexes
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Clique-Complex
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Neighborhood-Complex
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How to characterize a simplicial complex
S Using HOMOLOGY. It counts the number of n-dimensional holes. Betti Number is the output of Homology and can be used for deriving equivalence classes: “clustering”
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Persistent Homology
S It is the implementation of homology by using an incremental filtered algorithm
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Graphical representations of Betti Numbers
S persistent-barcode or persistent-diagram
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jHoles: java tool for computing PHOM (persistent homology)
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More or less at the end…
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Applications
S Immune response to vaccination
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Applications
S Time series analysis: periodicity detection
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Applications
S Cancer detection
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Applications
S Phase transition of a super glass metal
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Topology…other approaches
S Q-Analysis
Mapper
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Applications
S Mapper for Pulmonary Embolism diagnosys
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Thanks