Covariance matrix for transformed variables

[mfb]

This sounds like a common application, but I didn't find a discussion of it.

Simple case:
I have 30 experimental values, and I have the full covariance matrix for the measurements (they are correlated). I am now interested in the sum of the first 5 measured values, the sum of the following 5 measured values, and so on. In total I want 6 values and their covariance matrix. The diagonal entries of the covariance matrix are easy - just sum the corresponding 5x5 blocks along the diagonal of the original covariance matrix. Do I get the other entries also as sum of the corresponding 5x5 blocks? I would expect so but a confirmation would be nice.

General case:
More generally, if my transformed variables are a weighted sum of the original variables (weights are not negative), how do I get the off-diagonal elements of the covariance matrix? As long as two transformed variables do not share a common measured value, I can scale everything in the covariance matrix to get back to the previous case. But what if they do? I was playing around with rotation matrices (going to a basis of transformed variables plus some dummy variables) but somehow it didn't work, and constructing 30x30 rotation matrices is ugly.

 

[fresh_42]

I haven't read it, simply relied on the source. However, it looks helpful. (If I understood you correctly and the main problem here is the diagonalization).

Spoiler: wrong language

http://www.statoek.wiso.uni-goettingen.de/veranstaltungen/Multivariate/Daten/mvsec4.pdf

May I ask you whether you try to create a risk minimal DAX portfolio made out of five paper subportfolios? If so, then there is vast literature on the issue. E.g. I've found a dissertation Risk proportion and Eigen-Risk-Portfolio.

Edit: And https://en.wikipedia.org/wiki/Principal_component_analysis provides a lot of information and links, including software tools.

 

[mfb]

I know about principal component analysis but that is not what I want to do. I could diagonalize the covariance matrix that way, but then I still have to produce a new covariance matrix for the target variables, which is yet another transformation that is very similar to the original problem, and I don't know how stable the principal component analysis would be.

The application is in physics.

 

[mfb]

I had a look at another special case interesting to me. Let X_i be the measured variables and Y_i be the transformed variables. The original covariance matrix is C, the new one is D.

Let Y₁ = X₁+X₂ and Y₂=X₂. Then $D_{11} = C_{11}+ C_{12}+ C_{21}+ C_{22}$ and $D_{22}=C_{22}$ and $D_{12}=D_{21}=C_{11}$.
I don't find a transformation that would produce this result.

 

[Stephen Tashi]

Perhaps I don't understand the question, but if you have the full covariance matrix, can't you find everything you need from properties like $COV(X_1+X_2,X_3) = COV(X_1,X_3) + COV(X_2,X_3)$ and $COV(\alpha X_1,X_3) = \alpha COV(X_1,X_3)$ ?

Or are we asking for how the consequences of those properties are implemented with block matrix manipulations ?

 

[mfb]

You are right, that is sufficient to cover all those cases. Thanks.

A nice transformation could simplify the practical side, but computers are good at adding tons of stuff anyway.