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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Persistent homology ： ウィキペディア英語版 Persistent homology ''See 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.〔Carlsson, Gunnar (2009). "(Topology and data )". ''AMS Bulletin'' 46(2), 255–308.〕 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 of the simplicial complex, that is a nested sequence of increasing subsets. Formally, consider a real-valued function on a simplicial complex $f:K\; \backslash rightarrow\; \backslash mathbb$ that is non-decreasing on increasing sequences of faces, so $f(\backslash sigma)\; \backslash leq\; f(\backslash tau)$ whenever $\backslash sigma$ is a face of $\backslash tau$ in $K$. Then for every $a\; \backslash in\; \backslash mathbb$ the sublevel set $K(a)=f^(-\backslash infty,\; a]$ is a subcomplex of K, and the ordering of the values of $f$ on the simplices in $K$ (which is in practice always finite) induces an ordering on the sublevel complexes that defines the filtration : $\backslash emptyset\; =\; K\_0\; \backslash subseteq\; K\_1\; \backslash subseteq\; \backslash ldots\; \backslash subseteq\; K\_n\; =\; K$ When $0\backslash leq\; i\; \backslash leq\; j\; \backslash leq\; n$, the inclusion $K\_i\; \backslash hookrightarrow\; K\_j$ induces a homomorphism $f\_p^:H\_p(K\_i)\backslash rightarrow\; H\_p(K\_j)$ on the simplicial homology groups for each dimension $p$. The $p^$ persistent homology groups are the images of these homomorphisms, and the $p^$ persistent Betti numbers $\backslash beta\_p^$ are the ranks of those groups.〔Edelsbrunner, H and Harer, J (2010). ''Computational Topology: An Introduction''. American Mathematical Society.〕 Persistent Betti numbers for $p=0$ coincide with the predecessor of persistence homology, i.e. the size function.〔Verri, A, Uras, C, Frosini, P and Ferri, M (1993). ''(On the use of size functions for shape analysis )'', Biological Cybernetics, 70, 99–107.〕 There are various software packages for computing persistence intervals of a finite filtration, such as (javaPlex ), (Dionysus ), (Perseus ), (PHAT ), (Gudhi ), and the (phom ) and (TDA ) R packages. ==See also== * Topological data analysis * Computational topology 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Persistent homology」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.