Re: A basic question on Canonical Correlation Analysis

Hi Gary -


Am 24.11.2005 07:56 schrieb ziman137:
>
> I'm glad to hear that. If you in any chance to publicize this article,
> I'd love to read it. Still to clarify what I meant for the 1st step (as
> said in previous post), When I was saying to normalize samples in
> Principal Component (PC) spaces, I really meant S^(-1/2) and R^(-1/2)
> transformations. Taking S^(-1/2) as the example, S^(-1/2) itself is
> really a 3-step procedure:
> 1. project to PC space
> 2. normalized by square-roots of PC variances
> 3. project back to original space
> Once this normalization is done, we take orthonormal projection (P' and
> Q') as the core canonical transform.

Hmm, may be I have overlooked a normalization in my procedere;
I'll give it a check; but since the comparision with some
results from data in literature was ok up to now, I think I
have only a problem to get 1.-2.-3. right in my words.

>
>
>>>Therefore, no further rotation interpretation is needed in view of
>>>this.
>>
>>This I do not understand. The canonical correlation is excatly the
>>correlation between the first principal compoent of the first
>>subspace (of the common variance only) and the first principal
>>component of the second subspace.
>>
>
>
> Yes, right - just to clarify, as we understand, we are talking about
> "principal components" in a generalized SVD manner, not in the sense of
> separated input and output.
You mean: symmetric: not one is the dependend and the other the independend?
Then there is no disagreement.

>
> This part I do not understand (my turn, ;)) -- I guess I don't quite
> get your motivation to rotate in the canonical subspace. A crucial
> point here is that, though in the above 2-step interpretation via the
> generalized SVD, the linear uncorrelation is the same as the subspace
> orthogonality, yet it's only coincidently true with the canonical
> transformation alone. Any further rotation in canonical subapce will
> only keep subspace orthogonality, while linear uncorrelation between
> different canonical components is not invariant under such rotation.
> I'm not sure of your goal for a "simple structure", but the nice
> canonical correlation property would be damaged using any further
> rotation. I'm not sure if I correctly understood your point.
>
Well I just prepared a demo, which can explain my ideas in
very fine detail.
I have a small program for factoranalyzing, rotations etc. This
program can be used to run a script performing a "Living letter"
or "demo".
I created a dataset of two 4-item-scales (Set-X and Set-Y) to
be canonical correlated. The demo-program uses this dataset
to perform the canonical correlation and then I added some
more pages, to show how I meant things like additional varimax-
rotation.

When the program works through the script to compute the
canonical correlation, every step occurs with a commenting
panel with the explanations that I just prepared yesterday.
So it works somehow like a living letter and may put things
more clearly than I could do only in formulae here.

It is a small Dos-program; no more installation needed
than to unzip and copy it into a directory.

You can download it from

http://141.51.96.22/sw/stat/CCVarimax.zip

Read the #readme.txt to see how to start

Enjoy. Also I am interested in comments.

Gottfried Helms



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