AlphaPart update paper

[janaobsteter]

@gregorgorjanc and @ros, I am opening this discussion to write down the epifanies from this week. Gregor, if maybe you can write the math?

There is three layers to the output:

1. Whether we use pedigree or genomic data/model

2. The assumptions we make in the estimation vs partitioning (mainly regarding the intercept and the founder population)

3. Centering of the data

4. Pedigree data. When we run PBLUP, centering is implicit in the A matrix. BUT when we do the estimation with UPGs, we also need to include UPGs in the AlphaPart.

5. With genomic model, centering is all about the allele frequencies and that has implications for the model intercept. We can partition the deviations from the intercept (where the intercept is one / many UPGs), or partition the whole value on the original scale.
   ...

[janaobsteter]

TODO:

1. run BLUP without mean, because we can't estimate three UPGs plus a mean (4 unknowns) with the data (after UPGs, there is nothing left to estimate the mean)
2. Show the "correct" thing - run BLUP with UPGs and partition with UPGs (plus the mean should get us back to the original "whole" value)
3. Show a couple of the "wrong" things - estimate with UPGs and then partition without UPGs, use a different centering in estimation/partitioning ...