In stochastics , the correlation matrix is a symmetrical and positive semidefinite matrix that records the correlation between the components of a random vector . The correlation matrix can be obtained from the variance-covariance matrix and vice versa.
definition
The correlation matrix, as a matrix of all paired correlation coefficients of the elements of a random vector, contains information about the correlations between its components. Analogous to the variance-covariance matrix , the correlation matrix is defined as
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{\ displaystyle \ mathbf {X} = (X_ {1}, X_ {2}, \ dotsc, X_ {n}) ^ {\ top}}
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{\ displaystyle \ mathbf {\ Sigma}}
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{\ displaystyle \ mathbf {P} \ equiv \ operatorname {Corr} (\ mathbf {X}) = {\ begin {pmatrix} \ rho _ {11} & \ rho _ {12} & \ cdots & \ rho _ { 1n} \\\\\ rho _ {21} & \ rho _ {22} & \ cdots & \ rho _ {2n} \\\\\ vdots & \ vdots & \ ddots & \ vdots \\\\\ rho _ {n1} & \ rho _ {n2} & \ cdots & \ rho _ {nn} \ end {pmatrix}} = {\ begin {pmatrix} 1 & \ rho _ {12} & \ cdots & \ rho _ {1n } \\\\\ rho _ {21} & 1 & \ cdots & \ rho _ {2n} \\\\\ vdots & \ vdots & \ ddots & \ vdots \\\\\ rho _ {n1} & \ rho _ {n2} & \ cdots & 1 \ end {pmatrix}}}
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where the correlation coefficient is between and .
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{\ displaystyle \ rho _ {ij} = \ operatorname {Cov} (X_ {i}, X_ {j}) / {\ sqrt {\ operatorname {Var} (X_ {i}) \ operatorname {Var} (X_ { j})}} = \ sigma _ {ij} / \ sigma _ {i} \ sigma _ {j}}
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{\ displaystyle X_ {i}}
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{\ displaystyle X_ {j}}
For example, the second line of includes the correlation of with every other variable. The population correlation matrix is called or and the sample correlation matrix is called. If one defines the diagonal matrix , one obtains by and vice versa:
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{\ displaystyle X_ {2}}
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{\ displaystyle \ mathbf {P} _ {\ rho}}
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{\ displaystyle \ mathbf {D} = \ left (\ operatorname {diag} ({\ varvec {\ Sigma}}) \ right) ^ {1/2} = \ operatorname {diag} (\ sigma _ {1}, \ sigma _ {2}, \ dotsc, \ sigma _ {n})}
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{\ displaystyle {\ boldsymbol {\ Sigma}}}
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{\ displaystyle \ mathbf {P} = \ mathbf {D} ^ {- 1} \, {\ varvec {\ Sigma}} \, \ mathbf {D} ^ {- 1}}
or equivalent
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{\ displaystyle {\ boldsymbol {\ Sigma}} = \ mathbf {D} \, \ mathbf {P} \, \ mathbf {D}}
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properties
Sample Correlation Matrix
An estimate of the population correlation matrix is obtained by replacing the population correlation coefficients with the empirical correlation coefficients (their empirical counterparts) . This leads to the sample correlation matrix
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{\ displaystyle {\ widehat {\ mathbf {P}}}}
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{\ displaystyle \ rho _ {ij}}
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{\ displaystyle {\ begin {aligned} \ mathbf {R} = {\ widehat {\ mathbf {P}}} = {\ widehat {\ operatorname {Corr} (\ mathbf {X})}} & = {\ begin {pmatrix} 1 & r_ {12} & \ cdots & r_ {1k} \\\\ r_ {21} & 1 & \ cdots & r_ {2k} \\\\\ vdots & \ vdots & \ ddots & \ vdots \\\\ r_ { k1} & r_ {k2} & \ cdots & 1 \ end {pmatrix}} \ end {aligned}}}
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See also
Individual evidence
^ Ludwig Fahrmeir , Thomas Kneib , Stefan Lang, Brian Marx: Regression: models, methods and applications. Springer Science & Business Media, 2013, ISBN 978-3-642-34332-2 , p. 646.ff.
^ Rencher, Alvin C., and G. Bruce Schaalje: Linear models in statistics. , John Wiley & Sons, 2008., p. 77.
^ Rencher, Alvin C., and G. Bruce Schaalje: Linear models in statistics. , John Wiley & Sons, 2008., p. 247.
Special matrices in statistics
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