Hellinger distance

Hellinger distance

In probability theory, a branch of mathematics, given two probability measures "P" and "Q" that are absolutely continuous in respect to a third probability measure λ, the square of the Hellinger distance between "P" and "Q" is defined as the quantity

:H^2(P,Q) = frac{1}{2}displaystyle int left(sqrt{frac{dP}{dlambda - sqrt{frac{dQ}{dlambda ight)^2 dlambda.

Here, "dP" / "dλ" and "dQ" / "d"λ are the Radon-Nikodym derivatives of "P" and "Q" respectively. This definition does not depend on λ, so the Hellinger distance between "P" and "Q" does not change if λ is replaced with a different probability measure in respect to which both "P" and "Q" are absolutely continuous.

For compactness, the above formula is often written as

:H^2(P,Q) = frac{1}{2}int left(sqrt{dP} - sqrt{dQ} ight)^2.

Some authors omit the factor 1/2 in front of the integral.

The Hellinger distance "H"("P", "Q") thus defined satisfies the property

: 0le H(P,Q) le 1.

The Hellinger distance is related to the Bhattacharyya distance BC(P,Q) as it can be defined as

: H(P,Q) = frac{1}{2} sqrt{(2-2 BC(P,Q))}.

Example

The Hellinger distance "H"("P", "Q") between two normal distributions Psimmathcal{N}(x;mu_1,sigma_1^2) and Qsimmathcal{N}(x;mu_2,sigma_2^2) is: Hleft(P, Q ight)=sqrt{frac{1}{2}-sqrt{frac{sigma_1sigma_2}{2left(sigma_1^2+sigma_2^2 ight)}e^{-frac{1}{2}frac{left(mu_1-mu_2 ight)^2}{sigma_1^2+sigma_2^2

References

*
*
*


Wikimedia Foundation. 2010.

Игры ⚽ Нужно решить контрольную?

Look at other dictionaries:

  • Distance de Hellinger — En Théorie des probabilités, pour toutes mesures de probabilités P et Q absolument continues par rapport à une troisième mesure λ, le carré de la distance de Hellinger entre P et Q est donné par : où et …   Wikipédia en Français

  • Distance de Bhattacharyya — En statistiques, la distance de Bhattacharyya est une mesure de la similarité de deux distributions de probabilités discrètes. Elle est reliée au coefficient de Bhattacharyya , qui est une mesure statistique du recouvrement de deux ensembles d… …   Wikipédia en Français

  • Mahalanobis distance — In statistics, Mahalanobis distance is a distance measure introduced by P. C. Mahalanobis in 1936.[1] It is based on correlations between variables by which different patterns can be identified and analyzed. It gauges similarity of an unknown… …   Wikipedia

  • Bhattacharyya distance — In statistics, the Bhattacharyya distance measures the similarity of two discrete probability distributions. It is normally used to measure the separability of classes in classification. For discrete probability distributions p and q over the… …   Wikipedia

  • Bert Hellinger — The German psychotherapist Bert Hellinger (born 1925) is among the most well known, influential and controversial practitioners living in Europe. He is considered the principal developer of a therapeutic method best known as Family Constellations …   Wikipedia

  • Kullback–Leibler divergence — In probability theory and information theory, the Kullback–Leibler divergence[1][2][3] (also information divergence, information gain, relative entropy, or KLIC) is a non symmetric measure of the difference between two probability distributions P …   Wikipedia

  • Multivariate kernel density estimation — Kernel density estimation is a nonparametric technique for density estimation i.e., estimation of probability density functions, which is one of the fundamental questions in statistics. It can be viewed as a generalisation of histogram density… …   Wikipedia

  • List of mathematics articles (H) — NOTOC H H cobordism H derivative H index H infinity methods in control theory H relation H space H theorem H tree Haag s theorem Haagerup property Haaland equation Haar measure Haar wavelet Haboush s theorem Hackenbush Hadamard code Hadamard… …   Wikipedia

  • String metric — String metrics (also known as similarity metrics) are a class of textual based metrics resulting in a similarity or dissimilarity (distance) score between two pairs of text strings for approximate matching or comparison and in fuzzy string… …   Wikipedia

  • Projet:Mathématiques/Liste des articles de mathématiques — Cette page n est plus mise à jour depuis l arrêt de DumZiBoT. Pour demander sa remise en service, faire une requête sur WP:RBOT Cette page recense les articles relatifs aux mathématiques, qui sont liés aux portails de mathématiques, géométrie ou… …   Wikipédia en Français

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”