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Newsgroups: bit.listserv.stat-l
Date:         Fri, 18 Dec 1992 09:36:59 MET
From:         Joachim Benz <benz@GSF.DE>
Subject:      elimination of correlation

For a colleague of me (Anja Oetmann, email: oetmann@gsf.de) I forward
this request:
---------------------------------------------------------------------
   Our project aims to measure and to describe the present phenotypical
   variability of german indigenous populations of a wide-spread grass
   species (population in biological sense). We are interested in variation
   within and between populations as well as in the total variability of each
   character.
!   The object is to minimize the number of sites that covers the
!   over-all variance and further to preserve this variance through IN SITU
!   conservation.
   The problem is how to consider/eliminate the existing correlations
   between several characters before starting mathematical analyses.

Data:
=====

        * 100 sites, each divided in 3 subpopulations covering
          spaced heterogeneity of the sites
        * total plant number: 18 000
        * 13 variables for each plant were observed:
             5 measures:                            30 values/subpopulation
             1 measure:                             60 values/subpopulation
             7 estimations (in german: Bonituren):  60 scores/subpopulation
        * Data of 1 year and 1 experimental site (spaced plants)


STRATEGY TO SOLVE THE PROBLEM:
=============================
   1. Estimation of the univariate variability of one subpopulation
      We define the term variability of a specific variable as follows:

         VB     = [ VBL      .... VBH     ]
           i,j,l       i,j,l         i,j,l

         with:  VBL     = MEAN     -STDEV
                   i,j,l      i,j,l      i,j,l
                VBH     = MEAN     +STDEV
                   i,j,l      i,j,l      i,j,l
        (assumption: the distributions are symetric)
         i = index of variables [1, ..., 13]
         j = index of subpopulations [1,2,3]
         l = index of sites [1, ..., 100]

   2. Estimation of the univariate variability at one site

         VBS   = [ VBMIN   .... VBMAX   ]
            i,l         i,l          i,l

         with:  VBMIN     = MIN(VBL     ,VBL     ,VBL     )
                     i,l           i,1,l    i,2,l    i,3,l
                VBMAX     = MAX(VBH     ,VBH     ,VBH     )
                     i,l           i,1,l    i,2,l    i,3,l

   3. Estimation of the univariate variability over all sites

         VBG  = [ VBGMIN .... VBGMAX ]
            i           i           i

         with:  VBGMIN  = MIN(VBMIN     ,l=1,...,100)
                      i             i,l
                VBGMAX  = MAX(VBMAX     ,l=1,...,100)
                      i             i,l

    4. Selection of sites
       Select a minimal number of sites, that the cover of the total
       variabilties in all variables by the selected sites becomes a
       maximum.

But there is a serious problem we have no solution at the moment.
Between the variables the covariances are not 0. PCA we can't use as an
intermediate step to come to uncorrelated variables because the assumption
of one density function doesn't hold. On the other hand we assume if we
   ---
carry out the optimization with the original variables we will get a too
strong influence of highly correlated variables in the results.

Any suggestions and discussion are appreciated.


Thanks in advance ....

--

Mit freundlichen Gruessen


Joachim Benz

   University of Kassel
   Faculty of Agriculture (FB 20)
   Nordbahnhofstr. 1a
   D-3430 Witzenhausen
   (FRG)

   Phone: (+49)-5542-503-560
   Fax:   (+49)-5542-503-588
   email: benz@gsf.de
          C=de; A= ; P=gsf; S=benz (X.400)
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