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A measure of dependence of two random variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062940/m0629401.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062940/m0629402.png" />, defined as the least upper bound of the values of the correlation coefficients between the real random variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062940/m0629403.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062940/m0629404.png" />, which are functions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062940/m0629405.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062940/m0629406.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062940/m0629407.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062940/m0629408.png" />:
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062940/m0629409.png" /></td> </tr></table>
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If this least upper bound is attained at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062940/m06294010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062940/m06294011.png" />, then the maximal correlation coefficient between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062940/m06294012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062940/m06294013.png" /> is equal to the [[Correlation coefficient|correlation coefficient]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062940/m06294014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062940/m06294015.png" />. The maximal correlation coefficient has the property: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062940/m06294016.png" /> is necessary and sufficient for the independence of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062940/m06294017.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062940/m06294018.png" />. If there is a linear correlation between the variables, then the maximal correlation coefficient coincides with the usual correlation coefficient.
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A measure of dependence of two random variables  $  X $
 +
and  $  Y $,
 +
defined as the least upper bound of the values of the correlation coefficients between the real random variables  $  \phi _ {1} ( X) $
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and  $  \phi _ {2} ( Y) $,
 +
which are functions of  $  X $
 +
and  $  Y $
 +
such that  $  {\mathsf E} \phi _ {1} ( X) = {\mathsf E} \phi _ {2} ( Y) = 0 $
 +
and  $  {\mathsf D} \phi _ {1} ( X) = {\mathsf D} \phi _ {2} ( Y) = 1 $:
 +
 
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$$
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\rho  ^ {*} ( X , Y )  = \
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\sup  {\mathsf E} [ \phi _ {1} ( X) \phi _ {2} ( Y) ] .
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$$
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If this least upper bound is attained at $  \phi _ {1} = \phi _ {1}  ^ {*} ( X) $
 +
and $  \phi _ {2} = \phi _ {2}  ^ {*} ( Y) $,  
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then the maximal correlation coefficient between $  X $
 +
and $  Y $
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is equal to the [[Correlation coefficient|correlation coefficient]] of $  \phi _ {1}  ^ {*} ( X) $
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and $  \phi _ {2}  ^ {*} ( Y) $.  
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The maximal correlation coefficient has the property: $  \rho  ^ {*} ( X , Y ) = 0 $
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is necessary and sufficient for the independence of $  X $
 +
and $  Y $.  
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If there is a linear correlation between the variables, then the maximal correlation coefficient coincides with the usual correlation coefficient.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  O.V. Sarmanov,  "The maximum correlation coefficient (symmetric case)"  ''Dokl. Akad. Nauk SSSR'' , '''120''' :  4  (1958)  pp. 715–718  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  O.V. Sarmanov,  ''Dokl. Akad. Nauk SSSR'' , '''53''' :  9  (1946)  pp. 781–784</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  Yu.V. [Yu.V. Prokhorov] Prohorov,  Yu.A. Rozanov,  "Probability theory, basic concepts. Limit theorems, random processes" , Springer  (1969)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  O.V. Sarmanov,  "The maximum correlation coefficient (symmetric case)"  ''Dokl. Akad. Nauk SSSR'' , '''120''' :  4  (1958)  pp. 715–718  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  O.V. Sarmanov,  ''Dokl. Akad. Nauk SSSR'' , '''53''' :  9  (1946)  pp. 781–784</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  Yu.V. [Yu.V. Prokhorov] Prohorov,  Yu.A. Rozanov,  "Probability theory, basic concepts. Limit theorems, random processes" , Springer  (1969)  (Translated from Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====

Latest revision as of 08:00, 6 June 2020


A measure of dependence of two random variables $ X $ and $ Y $, defined as the least upper bound of the values of the correlation coefficients between the real random variables $ \phi _ {1} ( X) $ and $ \phi _ {2} ( Y) $, which are functions of $ X $ and $ Y $ such that $ {\mathsf E} \phi _ {1} ( X) = {\mathsf E} \phi _ {2} ( Y) = 0 $ and $ {\mathsf D} \phi _ {1} ( X) = {\mathsf D} \phi _ {2} ( Y) = 1 $:

$$ \rho ^ {*} ( X , Y ) = \ \sup {\mathsf E} [ \phi _ {1} ( X) \phi _ {2} ( Y) ] . $$

If this least upper bound is attained at $ \phi _ {1} = \phi _ {1} ^ {*} ( X) $ and $ \phi _ {2} = \phi _ {2} ^ {*} ( Y) $, then the maximal correlation coefficient between $ X $ and $ Y $ is equal to the correlation coefficient of $ \phi _ {1} ^ {*} ( X) $ and $ \phi _ {2} ^ {*} ( Y) $. The maximal correlation coefficient has the property: $ \rho ^ {*} ( X , Y ) = 0 $ is necessary and sufficient for the independence of $ X $ and $ Y $. If there is a linear correlation between the variables, then the maximal correlation coefficient coincides with the usual correlation coefficient.

References

[1] O.V. Sarmanov, "The maximum correlation coefficient (symmetric case)" Dokl. Akad. Nauk SSSR , 120 : 4 (1958) pp. 715–718 (In Russian)
[2] O.V. Sarmanov, Dokl. Akad. Nauk SSSR , 53 : 9 (1946) pp. 781–784
[3] Yu.V. [Yu.V. Prokhorov] Prohorov, Yu.A. Rozanov, "Probability theory, basic concepts. Limit theorems, random processes" , Springer (1969) (Translated from Russian)

Comments

See also Canonical correlation.

References

[a1] H. Gebelein, "Das statistische Problem der Korrelation als Variations- und Eigenwertproblem und sein Zusammenhang mit der Ausgleichungrechnung" Z. Angew. Math. Mech. , 21 (1941) pp. 364–379
[a2] R. Koyak, "On measuring internal dependence in a set of random variables" Ann. Statist. , 15 (1987) pp. 1215–1229
How to Cite This Entry:
Maximal correlation coefficient. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Maximal_correlation_coefficient&oldid=47799
This article was adapted from an original article by I.O. Sarmanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article