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In statistics, the '''RV coefficient'''<ref name=Robert>{{Cite journal
| author1 = Robert, P.
| author2 = Escoufier, Y.
| title = A Unifying Tool for Linear Multivariate Statistical Methods: The ''RV''-Coefficient
| journal = Applied Statistics
| year = 1976
| volume = 25
| issue = 3
| pages = 257&ndash;265
| doi = 10.2307/2347233
| jstor=2347233
}}</ref>
is a [[multivariate statistics|multivariate]] generalization of the ''squared'' [[Pearson correlation coefficient]] (because the RV coefficient takes values between 0 and 1).<ref name=HeRVé>{{Cite book
| last = Abdi
| first = Hervé
| title =  RV coefficient and congruence coefficient
| publisher = Thousand Oaks
| year = 2007
|editor1-first= Neil J
|editor1-last= Salkind
| isbn = 978-1-4129-1611-0
}}</ref>
It measures the closeness of two set of points that may each be represented in a [[matrix (mathematics)|matrix]].


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The major approaches within [[multivariate statistics|statistical multivariate data analysis]] can all be brought into a common framework in which the RV coefficient is maximised subject to relevant constraints. Specifically, these statistical methodologies include:<ref name=Robert />
:*[[principal component analysis]]
:*[[canonical correlation analysis]]
:*multivariate [[regression analysis|regression]]
:*[[statistical classification]] ([[Linear discriminant analysis|linear discrimination]]).


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One application of the RV coefficient is in [[functional neuroimaging]] where it can measure
the similarity between two subjects' series of brain scans<ref>{{Cite journal
  | title = Group analysis in functional neuroimaging: selecting subjects using similarity measures
| author1 = Ferath Kherif
| author2 = Jean-Baptiste Poline
| author3 = Sébastien Mériaux
| author4 = Habib Banali
| author5 = Guillaume Plandin
| author6 = Matthew Brett
| journal = NeuroImage
| volume = 20
| issue = 4
| year = 2003
| pages = 2197&ndash;2208
| doi = 10.1016/j.neuroimage.2003.08.018
| pmid=14683722
}}</ref>
or between different scans of a same subject.<ref>{{Cite journal
| title = How to compute reliability estimates and display confidence and tolerance intervals  for pattern classiffers using the Bootstrap and 3-way multidimensional scaling (DISTATIS)
| author1 = Herve Abdi
| author2 = Joseph P.  Dunlop
| author3 = Lynne J. Williams
| journal = NeuroImage
| volume = 45
| year = 2009
| pages = 89&ndash;95
| doi = 10.1016/j.neuroimage.2008.11.008
| pmid = 19084072
| issue = 1
}}</ref>
 
== Definitions ==
The definition of the RV-coefficient makes use of ideas<ref name=Escoufier>{{Cite journal
| author1 = Escoufier, Y.
| title = Le Traitement des Variables Vectorielles
| journal = Biometrics
| year = 1973
| volume = 29
| issue = 4
| pages = 751&ndash;760
| doi = 10.2307/2529140
| publisher = International Biometric Society
| jstor = 2529140
}}
</ref>
concerning the definition of scalar-valued quantities which are called the "variance" and "covariance" of vector-valued [[random variables]]. Note that standard usage is to have matrices for the variances and covariances of vector random variables.
Given these innovative definitions, the RV-coefficient is then just the correlation coefficient defined in the usual way.
 
Suppose that ''X'' and ''Y'' are matrices of centered random vectors (column vectors) with covariance matrix given by
:<math>\Sigma_{XY}=E( X^TY) \,,</math>
then the scalar-valued covariance (denoted by COVV) is defined by<ref name=Escoufier />
:<math>\mathrm{COVV}(X,Y)= Tr(\Sigma_{XY}\Sigma_{YX}) \, .</math>
The scalar-valued variance is defined correspondingly:
:<math>\mathrm{VAV}(X)= Tr(\Sigma_{XX}^2) \, .</math>
With these definitions, the variance and covariance have certain additive properties in relation to the formation of new vector quantities by extending an existing vector with the elements of another.<ref name=Escoufier />
 
Then the RV-coefficient is defined by<ref name=Escoufier />
:<math>\mathrm{RV}(X,Y)=\frac
  { \mathrm{COVV}(X,Y) }
  { \sqrt{ \mathrm{VAV}(X) \mathrm{VAV}(Y) } } \, .</math>
 
==See also==
*[[Congruence coefficient]]
*[[Distance correlation]]
 
== References ==
{{Reflist}}
 
[[Category:Covariance and correlation]]
[[Category:Multivariate statistics]]

Latest revision as of 15:40, 9 September 2013

In statistics, the RV coefficient[1] is a multivariate generalization of the squared Pearson correlation coefficient (because the RV coefficient takes values between 0 and 1).[2] It measures the closeness of two set of points that may each be represented in a matrix.

The major approaches within statistical multivariate data analysis can all be brought into a common framework in which the RV coefficient is maximised subject to relevant constraints. Specifically, these statistical methodologies include:[1]

One application of the RV coefficient is in functional neuroimaging where it can measure the similarity between two subjects' series of brain scans[3] or between different scans of a same subject.[4]

Definitions

The definition of the RV-coefficient makes use of ideas[5] concerning the definition of scalar-valued quantities which are called the "variance" and "covariance" of vector-valued random variables. Note that standard usage is to have matrices for the variances and covariances of vector random variables. Given these innovative definitions, the RV-coefficient is then just the correlation coefficient defined in the usual way.

Suppose that X and Y are matrices of centered random vectors (column vectors) with covariance matrix given by

then the scalar-valued covariance (denoted by COVV) is defined by[5]

The scalar-valued variance is defined correspondingly:

With these definitions, the variance and covariance have certain additive properties in relation to the formation of new vector quantities by extending an existing vector with the elements of another.[5]

Then the RV-coefficient is defined by[5]

See also

References

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