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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    $\begingroup$ What do you mean by "anti-image" here? $\endgroup$ – gung - Reinstate Monica May 17 '16 at 19:45
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    $\begingroup$ I claim not to close the question: it is clear and "anti-image" is a well-defined term, although it was not explained in the question. $\endgroup$ – ttnphns May 17 '16 at 20:52

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 $\sigma_i^2-R_i^2\sigma_i^2=\sigma_i^2(1-R_i^2)$.

  • 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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  • $\begingroup$ great answer!! Any chance you could add a line about the correlation coefficient, and how it relates to these various measures? Thanks!! $\endgroup$ – Matifou Oct 17 '18 at 21:58

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