From: benz@GSF.DE (Joachim Benz)
Newsgroups: bionet.population-bio
Subject: estimation of phenotypical variability
Date: 8 Jan 93 10:39:59 GMT
Distribution: bionet
Original-To: pop-bio@uk.ac.daresbury
X-Mailer: ELM [version 2.3 PL11]

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 ....


--

Sincerely,

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)