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Re: st: Rescale using covariance matrix for weighted PCA?


From   "J Jones" <statafan@googlemail.com>
To   statalist@hsphsun2.harvard.edu
Subject   Re: st: Rescale using covariance matrix for weighted PCA?
Date   Wed, 21 May 2008 23:19:04 +0100

Hello--thank you for the pointer, Maarten.  I hadn't known about
different weights and it seems like a useful lesson.

I still wonder, though, about actual 'workflow' for weighting?  To
repeat:  I want to do weighted PCA, which seems like a two-step PCA in
which vars are first reduced to components and are then weighted by a
factor of teh eigenvalue of teh first component.  My reference (de
Silva 2000 - about construction of a poverty index) says:


"Factor analysis method uses the correlation structure of the
variables or the covariance
structure <epsilon>. In using the covariance matrix we can manually
rescale variables (XI_i) by dividing with
their respective standard deviations (sigma_i)."

How the heck do they use the covariance matrix to scale the variable?!

Thanks for reading!



On Wed, May 21, 2008 at 10:25 PM, J Jones <statafan@googlemail.com> wrote:
> Hello--I would like to do weighted principal-components analysis
> (weighted PCA / WPCA; initialisms added for accessibility in future
> searches); however I am not sure how to do so.  I have variables,
> which may be organized into sets if need be.  I know how to get
> covariance matrices from factor analysis or PCA and I know how to do
> simple rescaling such as dividing an original score by a variable's
> standard deviation.
>
> Is it possible to rescale a variable by manipulating its covariance
> matrix?  References say to weight variables from a particular set by
> dividing by the-reciprocal-of-the square-roo-
> of-the-eigenvalue-of-the-set's-first-principal-component (W)...  and
> also dividing by the standard deviation of the variable.
>
> The exact formula they give is:
> Rescale score=(orig score)/(st dev*W).
>
> So the general method involves doing PCA on sets of variables and
> using the weighitng factor in order to confer proper weight to each
> set.  THen you do facotr analysis on the rescaled scores.
> Can somebody help please?  Thank you.
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