APY is a technique that allows representing the inverse of a genomic relationship matrix in a sparse format by choosing a "core" of animals (here and here). The authoritative guide to APY is Bermann et al. 2022: this paper .
One of the key aspects of genomic models is the ability of estimating SNP effects and then apply them to newly genotyped animals, what is know as Indirect Predictions. Matias and I found out that it's easier than we thought as shown here.
Among many other things Bermann et al. show that one can write indirect predictions of "non-core" animals as (I use the original equation numbering)
eq. (10) : \mathbf{{u}_n}= \mathbf{Z}_n \mathbf{Z}'_c (\mathbf{Z}_c \mathbf{Z}'_c)^{-1} \mathbf{Z}_c \mathbf{{a}} + \boldsymbol{\xi}
where \mathbf{{a}} are SNP effects and \boldsymbol{\xi} is an error term that does not depend on \mathbf{{u}_c} .
we obtain SNP effects from eq. 21 in Bermann et al:
\mathbf{\hat{a}}=k \mathbf{Z}'_c \mathbf{G}_{cc}^{-1} \mathbf{\hat{u}}_{c}
we plug that into (10) and expand:
\mathbf{\hat{u}_n}= \mathbf{Z}_n \mathbf{Z}'_c (\mathbf{Z}_c \mathbf{Z}'_c)^{-1} \mathbf{Z}_c \mathbf{\hat{a}} =k \mathbf{Z}_n \mathbf{Z}'_c (k \mathbf{Z}_c \mathbf{Z}'_c)^{-1} \mathbf{Z}_c \mathbf{\hat{a}}
we substitute for \mathbf{\hat{a}} :
\mathbf{\hat{u}_n} =k \mathbf{Z}_n \mathbf{Z}'_c (k \mathbf{Z}_c \mathbf{Z}'_c)^{-1} \mathbf{Z}_c k \mathbf{Z}'_c \mathbf{G}_{cc}^{-1} \mathbf{\hat{u}}_{c}=k \mathbf{Z}_n \mathbf{Z}'_c \mathbf{G}_{cc}^{-1} \mathbf{\hat{u}}_{c} = \mathbf{Z}_n \mathbf{\hat{a}}
which is the original eq. 21. This is because \hat{a} is part of the column space of \mathbf{Z}'_c , then \mathcal{P}\mathbf{\hat{a}} = \mathbf{\hat{a}} .
Finally, obtaining Indirect Predictions from APY is deadly simple and intuitive:
- \mathbf{\hat{a}}=k \mathbf{Z}'_c \mathbf{G}_{cc}^{-1} \mathbf{\hat{u}}_{c}
- \mathbf{\hat{u}_n} = \mathbf{Z}_n \mathbf{\hat{a}}