DM: Principal Components and Correlations

[C. K. Krishnadas]

Hi,

I  am having trouble with principal components and their correla-
tions with the original  variables.

Suppose I have 10 variables, many of which move together.  I have
taken  principal components.  The first principal component which
accounts for a large chunk of the variance shows a negative  cor-
relation  with  most of the variables, including the set of vari-
ables which are known to be moving together.  The  variables  are
standardized  before  computing their variance-covariance matrix.
It is also expected that the  first  principal  component  should
have  a  significant  (+ve) correlation with the set of variables
mentioned before.  But the correlations turn out to  be  negative
and  significant.  In the computation, since the eigen vectors of
the variance-covariance matrix are chosen so as to maximize vari-
ability  in their direction, with orthogonality imposed with each
other, the correlations of variables of the  variables  with  the
principal  components  can have signs contrary to common expecta-
tions.  Since the eigen vectors can be multiplied by  -1,  I  can
get  a  new  set of eigen vectors which can be used to generate a
new set of principal components which can show correlations  with
the  expected sign.  But this would involve compution of correla-
tion of the principal components with the original variables  and
a  subjective  examination depending on the nature of data or do-
main knowledge (of application).

Is there a standard method of choosing the eigen vectors or prin-
cipal components in such a way that they have correlations of the
expected (and subjectively meaningful) sign with the variables?


Thanks

  -- Krishnadas

-----------------------------------------------------------------
C. K. Krishnadas                c k krish at cyberspace dot o r g
ckkrish@cyberspace.org         http://www.cyberspace.org/~ckkrish
na.kck@na-net.ornl.gov
-----------------------------------------------------------------