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''See [[homology (mathematics)|homology]] for an introduction to the notation.''


'''Persistent homology''' is a method for computing topological features of a space at  different spatial resolutions. More persistent features are detected over a wide range of length and are deemed more likely to represent true features of the underlying space, rather than artifacts of sampling, noise, or particular choice of parameters. <ref>Carlsson, Gunnar (2009). "[http://www.ams.org/journals/bull/2009-46-02/S0273-0979-09-01249-X/ Topology and data]". ''AMS Bulletin'' '''46(2)''', 255&ndash;308.</ref>


To find the persistent homology of a space, the space must first be represented as a [[simplicial complex]]. A distance function on the underlying space corresponds to a [[Filtration (mathematics)|filtration]] of the simplicial complex, that is a nested sequence of increasing subsets.  
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Formally, consider a real-valued function on a simplicial complex <math>f:K \rightarrow \mathbb{R}</math> that is non-decreasing on increasing sequences of faces, so <math>f(\sigma) \leq f(\tau)</math> whenever <math>\sigma</math> is a face of <math>\tau</math> in <math>K</math>. Then for every <math> a \in \mathbb{R}</math> the [[sublevel set]] <math>K(a)=f^{-1}(-\infty, a]</math> is a subcomplex of K, and the ordering of the values of <math>f</math> on the simplices in <math>K</math> (which is in practice always finite) induces an ordering on the sublevel complexes that defines the filtration
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: <math> \emptyset = K_0 \subseteq K_1 \subseteq \ldots \subseteq K_n = K </math>
When <math> 0\leq i \leq j \leq n</math>, the inclusion <math>K_i \hookrightarrow K_j</math> induces a [[group homomorphism|homomorphism]] <math>f_p^{i,j}:H_p(K_i)\rightarrow H_p(K_j)</math> on the [[simplicial homology]] groups for each dimension <math>p</math>. The <math>p^{th}</math> '''persistent homology groups''' are the images of these homomorphisms, and the <math>p^{th}</math> '''persistent [[Betti numbers]]''' <math> \beta_p^{i,j}</math> are the [[rank of a group|ranks]] of those groups.<ref>Edelsbrunner, H and Harel, J (2010). ''Computational Topology: An Introduction''. American Mathematical Society.</ref>


There are various software packages for computing persistence intervals of a finite filtration, such as [http://code.google.com/p/javaplex/ javaPlex], [http://www.mrzv.org/software/dionysus/ Dionysus], [http://www.sas.upenn.edu/~vnanda/perseus/index.html Perseus], [http://phat.googlecode.com/ PHAT], and the [http://cran.r-project.org/web/packages/phom/index.html phom] R package.
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{{reflist}}


[[Category:Homology theory]]
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Latest revision as of 00:48, 24 July 2014


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