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  "body": "\n## Definition\n\n*Topological data analysis* is the analysis of data by topological invariants computed through the following steps.\n\n1. The data are a finite set $X = \\set{x_1, \\dots, x_n}$ in a [[metric space|0VKD1TP7F3]] $(M, d)$.\n2. A filtration of [[simplicial complexes|V8GZ0W1ZZ5]] $(K_r)_{r \\ge 0}$ is built on $X$, with $K_r \\subseteq K_s$ for $r \\le s$.\n   A standard choice is the Vietoris–Rips complex: its simplices at scale $r$ are the nonempty subsets of $X$ of diameter at most $r$.\n3. The [[persistent homology|S8SNBHVPSP]] of the filtration, in each degree $k$ with coefficients in a field, is computed and summarised by a barcode or persistence diagram.\n4. Statistical conclusions are drawn from these summaries, for example by comparing them with the bottleneck distance.\n\n## Remarks\n\nThe stability theorem gives the summaries a statistical meaning.\nFor two finite metric spaces, the persistence diagrams of their Vietoris–Rips filtrations are at bottleneck distance at most twice their Gromov–Hausdorff distance.\n\nExample: let $X$ be the $n \\ge 4$ vertices of a regular $n$-gon in $\\R^2$, with side length $a$ and with length $b \u003e a$ of the diagonals that skip one vertex.\nFor $a \\le r \u003c b$, the Vietoris–Rips complex $K_r$ has an edge between consecutive vertices only, and no triangle, since any three vertices include two that are not consecutive.\nSo $K_r$ is the $n$-cycle graph, a circle, and its first homology has rank $1$; the bar in degree $1$ that is born at $a$ detects the circle.\n\nNon-example: computing the mean and covariance of $X$ is not topological data analysis, since it uses no filtration and no topological invariant.\n",
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