# Help explain the “redundancy” of canonical correlation

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".

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 $$\bf{t}$$.

• What's the name of the book? Is it available to douwnload at some place? – ttnphns May 24 '15 at 15:22
• @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. – Tony May 24 '15 at 21:24
• @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? – Tony May 24 '15 at 21:27
• 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 $1$. – Tony May 25 '15 at 6:40