- Topological data analysis
**Topological data analysis**is a new area of study aimed at having applications in areas such asdata mining andcomputer vision . The main problems are (1) how one infers high-dimensionalstructure from low-dimensional representations; and (2) how one assembles discrete points into global structure.The human brain can easily extract global structure from representations in a strictly lower dimension, i.e. we infer a 3D environment from a 2D image from each eye. The inference of global structure also occurs when converting

discrete data intocontinuous images. E.g. dot-matrix printers and televisions communicate images via arrays of discrete points.The main method used by topological data analysis is:

(1) replace a set of data points with a family of

simplicial complex es, indexed by a proximity parameter.(2) Analyse these topological complexes via

algebraic topology — specifically, via the new theory of**persistent homology**.(3) Encode the persistent homology of a data set in the form of a parameterized version of a

Betti number which will be called a**barcode**.**Point cloud data**Data is often represented as points in a Euclidean "n"-dimensional space E

_{"n"}. The global "shape" of the data may provide information about the phenomena that the data represent.One type of data set for which global features are certainly present is the so-called

**point cloud data**coming from physical objects in 3D. E.g. a laser can scan an object at a set of discrete points and the cloud of such points can be used in a computer representation of the object.**Point cloud data**refers to any collection of points in E_{"n"}or a (perhaps noisy) sample of points on a lower-dimensional subset.For point clouds in low-dimensional spaces there are numerous approaches for inferring features based on planar projections in the fields of

computer graphics andstatistics . Topological data analysis is needed when the spaces are high-dimensional or too twisted to allow planar projections.To convert a point cloud in a

metric space into a global object use the point cloud as thevertices of a graph whose edges are determined by proximity, then turn the graph into asimplicial complex and use algebraic topology to study it.**Persistent homology****ee also***

Dimensionality reduction

*Data mining

*Computer vision

*Computational topology

*Digital topology

*Digital Morse theory

*Shape analysis

*Structured data analysis (statistics) **References*** [

*http://www.ams.org/bull/2008-45-01/S0273-0979-07-01191-3/S0273-0979-07-01191-3.pdf BARCODES: THE PERSISTENT TOPOLOGY OF DATA*]

* [*http://www.informatics.bangor.ac.uk/%7Etporter/TDA/TDA.html Topological Data Analysis: the algebraic topology of point data clouds?*]

*

*Wikimedia Foundation.
2010.*