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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#### Best Answer

A excerpt of another answer (about factor analysis), structured.

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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