Jaro–Winkler distance: Difference between revisions
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In [[mathematics]], the '''Lévy metric''' is a [[metric (mathematics)|metric]] on the space of [[cumulative distribution function]]s of one-dimensional [[random variable]]s. It is a special case of the [[Lévy–Prokhorov metric]], and is named after the French mathematician [[Paul Lévy (mathematician)|Paul Lévy]]. | |||
==Definition== | |||
Let <math>F, G : \mathbb{R} \to [0, 1]</math> be two cumulative distribution functions. Define the '''Lévy distance''' between them to be | |||
:<math>L(F, G) := \inf \{ \varepsilon > 0 | F(x - \varepsilon) - \varepsilon \leq G(x) \leq F(x + \varepsilon) + \varepsilon \mathrm{\,for\,all\,} x \in \mathbb{R} \}.</math> | |||
Intuitively, if between the graphs of ''F'' and ''G'' one inscribes squares with sides parallel to the coordinate axes (at points of discontinuity of a graph vertical segments are added), then the side-length of the largest such square is equal to ''L''(''F'', ''G''). | |||
==See also== | |||
* [[Càdlàg]] | |||
* [[Lévy–Prokhorov metric]] | |||
* [[Wasserstein metric]] | |||
==References== | |||
* {{springer|author=V.M. Zolotarev|id=l/l058310|title=Lévy metric}} | |||
{{DEFAULTSORT:Levy metric}} | |||
[[Category:Measure theory]] | |||
[[Category:Metric geometry]] | |||
[[Category:Probability theory]] | |||
Revision as of 11:55, 20 January 2014
In mathematics, the Lévy metric is a metric on the space of cumulative distribution functions of one-dimensional random variables. It is a special case of the Lévy–Prokhorov metric, and is named after the French mathematician Paul Lévy.
Definition
Let be two cumulative distribution functions. Define the Lévy distance between them to be
Intuitively, if between the graphs of F and G one inscribes squares with sides parallel to the coordinate axes (at points of discontinuity of a graph vertical segments are added), then the side-length of the largest such square is equal to L(F, G).
See also
References
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