Correlation matrix

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 ${\displaystyle \mathbf {X} =(X_{1},X_{2},\dotsc ,X_{n})^{\top }}$ ${\displaystyle \mathbf {\Sigma } }$

${\displaystyle \mathbf {P} \equiv \operatorname {Corr} (\mathbf {X} )={\begin{pmatrix}\rho _{11}&\rho _{12}&\cdots &\rho _{1n}\\\\\rho _{21}&\rho _{22}&\cdots &\rho _{2n}\\\\\vdots &\vdots &\ddots &\vdots \\\\\rho _{n1}&\rho _{n2}&\cdots &\rho _{nn}\end{pmatrix}}={\begin{pmatrix}1&\rho _{12}&\cdots &\rho _{1n}\\\\\rho _{21}&1&\cdots &\rho _{2n}\\\\\vdots &\vdots &\ddots &\vdots \\\\\rho _{n1}&\rho _{n2}&\cdots &1\end{pmatrix}}}$,

where the correlation coefficient is between and . ${\displaystyle \rho _{ij}=\operatorname {Cov} (X_{i},X_{j})/{\sqrt {\operatorname {Var} (X_{i})\operatorname {Var} (X_{j})}}=\sigma _{ij}/\sigma _{i}\sigma _{j}}$ ${\displaystyle X_{i}}$ ${\displaystyle X_{j}}$

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: ${\displaystyle \mathbf {P} }$ ${\displaystyle X_{2}}$ ${\displaystyle X}$ ${\displaystyle \mathbf {P} _{\rho }}$ ${\displaystyle \mathbf {P} }$ ${\displaystyle \mathbf {R} }$ ${\displaystyle \mathbf {D} =\left(\operatorname {diag} ({\boldsymbol {\Sigma }})\right)^{1/2}=\operatorname {diag} (\sigma _{1},\sigma _{2},\dotsc ,\sigma _{n})}$ ${\displaystyle \mathbf {P} }$ ${\displaystyle {\boldsymbol {\Sigma }}}$

${\displaystyle \mathbf {P} =\mathbf {D} ^{-1}\,{\boldsymbol {\Sigma }}\,\mathbf {D} ^{-1}}$

or equivalent

${\displaystyle {\boldsymbol {\Sigma }}=\mathbf {D} \,\mathbf {P} \,\mathbf {D} }$.

properties

• If all components of the random vector are linearly independent , then positive is definite . ${\displaystyle \mathbf {X} }$ ${\displaystyle \mathbf {R} }$
• The correlation of the sizes with themselves is calculated on the main diagonal . Since the relationship between the sizes is strictly linear, the correlation on the main diagonal is always one.
• When sampling from a multidimensional normal distribution , the sample correlation matrix is ​​the maximum likelihood estimator of the correlation matrix in the population . ${\displaystyle \mathbf {R} }$ ${\displaystyle \mathbf {P} }$

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 ${\displaystyle {\widehat {\mathbf {P} }}}$ ${\displaystyle \rho _{ij}}$ ${\displaystyle r_{ij}}$ ${\displaystyle \mathbf {R} }$

${\displaystyle {\begin{aligned}\mathbf {R} ={\widehat {\mathbf {P} }}={\widehat {\operatorname {Corr} (\mathbf {X} )}}&={\begin{pmatrix}1&r_{12}&\cdots &r_{1k}\\\\r_{21}&1&\cdots &r_{2k}\\\\\vdots &\vdots &\ddots &\vdots \\\\r_{k1}&r_{k2}&\cdots &1\end{pmatrix}}\end{aligned}}}$.

See also

• Covariance matrix

Individual evidence

1. ^ 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.
2. ^ Rencher, Alvin C., and G. Bruce Schaalje: Linear models in statistics. , John Wiley & Sons, 2008., p. 77.
3. ^ Rencher, Alvin C., and G. Bruce Schaalje: Linear models in statistics. , John Wiley & Sons, 2008., p. 247.