I am reading a material about canonical correlation and it introduces a concept named "redundancy". I have been puzzled for one day but still could not get a understanding. The following is a screen capture of the part discussing "redundancy".enter image description here

where $\bf{Y}$ is the dependent variable vector, $\bf{X}$ is the independent variable vector, $\bf{u}$ is the canonical linear combination of $\bf{X}$ and $\bf{t}$ is the canonical linear combination of $\bf{Y}$. $\bf{u}=\bf{X}\bf{b}$ and $\bf{t}=\bf{Y}\bf{a}$.

I have several questions.

  1. Why this concept is named redundancy? Is there any intuitive interpretation?
  2. Why the first term of $Rd(\bf{t}|\bf{u})$ is just the squared canonical correlation $r^2(\bf{t},\bf{u})$?
  3. Why the second term is $\frac{\bf{g}'\bf{g}}{q}$?

Here $\bf{g}$ is the correlation between $\bf{Y}$ and $\bf{t}$. enter image description here

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    $\begingroup$ What's the name of the book? Is it available to douwnload at some place? $\endgroup$ – ttnphns May 24 '15 at 15:22
  • $\begingroup$ @ttnphns The name is Analyzing Multivariate Data by James. M. Lattin. It seems not available online. I am just taking pictures of it. I am still struggling to figure out the above problems. $\endgroup$ – Tony May 24 '15 at 21:24
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    $\begingroup$ @ttnphns I found another material which only says "redundancy of A given B is an index of of the proportions variance of A predicable from B". Does this mean "redundancy" is a measure of variance explanation? like r-squared? What is the difference? $\endgroup$ – Tony May 24 '15 at 21:27
  • $\begingroup$ I spent 4 more hours studying this problem and carefully examined the how variances are explained by principal components in PCA since the material mentions "the same notation as the variance in Y accounted for by a principal component". I now incline to say this so-called measure $Rd$ is a heuristic index rather than one with robust mathematical derivation. $\bf{Y}$ is standardized data matrix of $q$ dependent variables, so the total variance of $\bf{Y}$ is $q$ given that the variance of each dimension is normalized to $1$. $\endgroup$ – Tony May 25 '15 at 6:40
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    $\begingroup$ ...computing loadings and cross-loadings is labeled "redundancy analysis". $\endgroup$ – ttnphns May 25 '15 at 7:59

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