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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2$\begingroup$ What do you mean by "anti-image" here? $\endgroup$– gung - Reinstate MonicaCommented May 17, 2016 at 19:45
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3$\begingroup$ 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. $\endgroup$– ttnphnsCommented May 17, 2016 at 20:52
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$\begingroup$ Please edit the question to explain what an anti-image is. $\endgroup$– Sycorax ♦Commented Jun 18 at 17:44
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1 Answer
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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!! $\endgroup$– MatifouCommented Oct 17, 2018 at 21:58
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1$\begingroup$ What is the reason (if any) that image and anti-image covariance matrices both bear the label "covariance"? $\endgroup$ Commented Apr 12, 2021 at 8:45