Using PCA to reduce the number of variables split into groups First of all, sorry for the strange title, I had no idea how to describe my problem better. My issue is the following, I think it is pretty much limited to geosciences.
I have several properties for every sample, which are divided by depth.
For instance:          
$ \qquad \displaystyle \small \begin{array} {r|rrr} \hline
ID                    & 1    & 2   &3 & ...\\ \hline
\text{var1}_{0-20cm}  & 2.3  &2.0 &1.0& ...\\
\text{var1}_{20-50cm} & 2.1  &1.1 &0.0& ...\\
\text{var1}_{50-100cm}& 2.6  &1.1 &0.0& ...\\ \hline
\text{var2}_{0-20cm}  & 10.5 &5.5 &3.5& ...\\
\text{var2}_{20-50cm} & 10.9 &5.9 &1.9& ...\\
\text{var2}_{50-100cm}& 15.0 &5.0 &1.0& ...\\   \hline
  \vdots & \vdots & \vdots\\ \hline \end{array}
$
Basically these are geological layers going from surface down to 100 cm depth.
I am trying to decrease the number of variables, either with PCA or factor analysis.
The issue is, that I would like to handle properties together, no matter what the depth is.
(For instance I do not want to get rid of a layer in between the surface and the bottom layer.)
Is there any way to handle them together, or group them for PCA or whatever. I tried to find some relevant information, but I think the problem is limited to a small portion of the science (maybe I am wrong), so I could not find anything useful.
 A: What you could do is use Multiple Factor Analysis. This method allows for factor analysis in which you consider multiple groups of variables.
If you set up your analysis so that each group is a depth then it garantees that all your depth will be 'preserved'.
EDIT : Maybe explaining a bit more would be useful
In MFA, as in PCA, you have coordinates for your individuals and your variables. But what's new with MFA is the groups of variables for which you can compute coordinates too so you can extract coordinates for all of your groups (depth) on the first few dimensions, effectively reducing variable number and keeping all your depths.
If you consider your individuals, you will have several sets of coordinates, one for each of the groups of variables (a description of the individuals by each group of variable if you will) and a set of coordinates which is the centroid of all groups coordinates (partial representations), that last set of coordinates could be interpreted by how the individuals are described overall
A: If I understand you correctly, you want to use
Variable1 :=  var1_0-20cm + var1_20-50cm + var1_50-100cm
Variable2 :=  var2_0-20cm + var2_20-50cm + var2_50-100cm

(i.e. "independent of depth"; depending on how your data was generated you might want to use the mean or a weighted average instead of the sum. e.g. 0.20 times the first, 0.30 the second and 0.50 the third) instead of the full data space? What exactly is the problem with doing this? A key benefit is that this way you can control what happens quite well.
Then in the end you can e.g. use PCA on these non-divided variables. You can however try to use the PCA result to project the original data, too - do the same mapping for the "divided" attributes that you got by PCA for the non-divided variables.
A: A data driven (and thus probably not so very good) approach
Calculate four correlation matrices: One for each layer and one for the pooled data (three lines per sample). If they all look quite similar, run a PCA based on the correlation matrix of the pooled sample and go on with the first few PCs.
Instead of comparing the four correlation matrices, you could also consider the four loading matrices of the corresponding PCAs and compare the loadings of the first few PCs. This is much easier if you have lots of variables.
