# Solved – the difference between the anti-image covariance and the anti-image correlation

What is the difference between the anti-image covariance and the anti-image correlation? How are the matrices of these coefficients computed, and what is the meaning of their elements?

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Let \$bf R\$ be a correlation or covariance matrix, and \$bf D\$ be the diagonal matrix comprised of the inverses of diagonal elements of \$bf R^{-1}\$. Then

• \$bf DR^{-1}D\$ is known as anti-image covariance matrix of \$bf R\$. Its off-diagonal entries are the negatives of the partial covariance coefficients (between two variables controlled for all the other variables). The diagonal is equal to the diagonal \$bf D\$, – these diagonal values are called anti-images in \$bf R\$.

• \$bf (DR^{-1}D)-2D+R\$ is called image covariance matrix of \$bf R\$. Its diagonal entries are called images in \$bf R\$ (they are equal to the diagonal of \$bf R-D)\$. An image is \$R_i^2 sigma_i^2\$, where \$R_i^2\$ is the squared multiple correlation coefficient of dependency of variable \$i\$ on the rest variables, and \$sigma_i^2\$ is the diagonal element in \$bf R\$, the variance (or \$1\$, in case of correlation matrix).

• From the above it becomes clear that image + anti-image = \$sigma_i^2\$, and that the two are the portions of a variable's variation being, respectively, explained and unexpained (residual) by the other variables. Thus, if \$bf R\$ is correlations then image is the \$R_i^2\$ and anti-image is \$1-R_i^2\$; while if \$bf R\$ is covariances then image is \$R_i^2 sigma_i^2\$ and anti-image is \$sigma_i^2-R_i^2sigma_i^2=sigma_i^2(1-R_i^2)\$.

• Terminologic warning: the image and anti-image covariance matrices bear label "covariance" irrespective of whether \$bf R\$ is covariances or correlations.

• Anti-image correlation matrix of \$bf R\$ is computed from anti-image covariances the very usual way like we convert usual covariance into usual correlation, \$r_{ij}=cov_{ij}/(sigma_i sigma_j)\$, – i.e. here the cov and the two sigmas will be the values from an anti-image covariance matrix. Or in matrix notation: \$bf D^{-1/2} A D^{-1/2}\$, where \$bf A\$ is the anti-image covariance matrix and \$bf D^{-1/2}\$ is its diagonal, square-rooted and inversed. Equivalent formula also is \$bf D^{1/2} R^{-1} D^{1/2}\$, where \$bf D^{1/2}\$ is the \$bf R^{-1}\$'s diagonal, square-rooted and inversed. Off-diagonal elements of anti-image correlation matrix are the negatives of the partial correlation coefficients (between two variables controlled for all the other variables). And that is popular way to compute partial correlations.

• One can also convert, analogously, image covariance matrix into image correlation matrix, if needed.

• Anti-image correlation matrix will be the same – be \$bf R\$ covariances or correlations (while anti-image covariance matrix was different in the two instances).

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