The renumbering software in BLUPF90 (renumf90) can create UPG based on year of birth or can recognize UPG in the pedigree.
For more details, check renumf90 and extra options. Independently on how UPG is included in the pedigree file, UPG in ssGBLUP can be considered for $\mathbf{A}^{-1}$, $\mathbf{A}^{-1}$ and $\mathbf{A}_{22}^{-1}$, or $\mathbf{A}^{-1}$, $\mathbf{G}^{-1}$, and $\mathbf{A}_{22}^{-1}$. For research on this topic, check Quaas 1988 Misztal et al., 2013, Tsuruta et al., 2019, Cesarani et al., 2021, and Masuda et al., 2022
UPG are considered fixed. Click here for details on how to switch to random UPG.
Note that we have un-genotyped individuals ($\mathbf{u}_1$), genotyped individuals ($\mathbf{u}_2$), and group (UPG) effects ($\mathbf{g}$). For the notation we use the order in which we put $\mathbf{g}$ at the bottom, i.e. (loosely speaking) the “inverted covariance matrix” is
$$ Var^{-1} \begin{pmatrix} \mathbf{u}_1\\ \mathbf{u}_2\\ \mathbf{g} \end{pmatrix}= \mathbf{H}^{-1} $$
This is automatically done when UPG are in the renumbered pedigree and RANDOM_TYPE is add_an_upginb. However, this is a wrong method because the assumed $\mathbf{H}$ uses $\mathbf{A}_{22}^{-1}$ pretending that that there are no UPGs, and because a UPG mean is imposed on genotyped individuals, when it should not. This corresponds to eq. [11] in Masuda et al., 2022 with $\alpha=1$, as follows.
$$ \mathbf{H}^{-1}=\mathbf{A}^{*}+ \begin{pmatrix} \mathbf{0} & \mathbf{0} & \mathbf{0}\\ \mathbf{0} & \mathbf{G}^{-1}-\mathbf{A}_{22}^{-1} & \mathbf{0}\\ \mathbf{0} & \mathbf{0} & \mathbf{0}\\ \end{pmatrix} $$
where $\mathbf{A}^{*}$ is the “traditional” A-“inverse” including UPGs Quaas 1988, $\mathbf{G}^{-1}$ is the genomic relationship matrix, inverted, and $\mathbf{A}_{22}^{-1}$ is the pedigree relationship matrix ignoring UPGs, i.e. assuming that all animals come from the same base .
This was derived in Misztal et al., 2013 and it results in the following equations, called “QP Model” in Masuda et al., 2022 where it is eq. [14]:
$$ \mathbf{H}^{-1}=\mathbf{A}^{*}+ \begin{pmatrix} \mathbf{0} & \mathbf{0} & \mathbf{0}\\ \mathbf{0} & \mathbf{G}^{-1}-\mathbf{A}_{22}^{-1} & -(\mathbf{G}^{-1}-\mathbf{A}_{22}^{-1})\mathbf{Q}_2\\ \mathbf{0} & -\mathbf{Q}_2(\mathbf{G}^{-1}-\mathbf{A}_{22}^{-1}) & \mathbf{Q}_2(\mathbf{G}^{-1}-\mathbf{A}_{22}^{-1})\mathbf{Q}_2\\ \end{pmatrix} $$
This is obtained using the option
OPTION exact_upg
which computes and multiplies the $\mathbf{Q}_2$ products. Matrix $\mathbf{Q}_2$ contains pedigree fractions of each UPG for each genotyped animal.
However, this is still not quite correct. Intuitively, matrix $\mathbf{G}^{-1}$ does not depend on pedigree and should be used without modifications.
This accounts for the previous default, and it was called “Altered QP model” in Masuda et al., 2022, with an expression as follows:
$$ \mathbf{H}^{-1}=\mathbf{A}^{*}+ \begin{pmatrix} \mathbf{0} & \mathbf{0} & \mathbf{0}\\ \mathbf{0} & \mathbf{G}^{-1}-\mathbf{A}_{22}^{-1} & -(-\mathbf{A}_{22}^{-1})\mathbf{Q}_2\\ \mathbf{0} & -\mathbf{Q}_2(-\mathbf{A}_{22}^{-1}) & \mathbf{Q}_2(-\mathbf{A}_{22}^{-1})\mathbf{Q}_2\\ \end{pmatrix} $$
It is obtained using
OPTION exact_upg OPTION TauOmegaQ2 0.0 1.0
The last OPTION TauOmegaQ2 is used to assign $\mathbf{Q}_{2}$ (the incidence matrix relating genotyped animals to unknown parent groups) only to $\mathbf{A}_{22}^{-1}$
The papers of Hsu et al 2017 Stranden et al 2022 and Belay et al. 2022 detail the use of so-called J-factors to model the difference between pedigree and genomic bases, and these J-factors have a structure similar to the UPG discussed before. In the particular case when J-factors and UPGs have exactly the same definition, their respective matrices “cancel out” and results in the previous “Altered QP model”. was correct. So we find the same inverse covariance structure as above:
$$ \mathbf{H}^{-1}=\mathbf{A}^{*}+ \begin{pmatrix} \mathbf{0} & \mathbf{0} & \mathbf{0}\\ \mathbf{0} & \mathbf{G}^{-1}-\mathbf{A}_{22}^{-1} & -(-\mathbf{A}_{22}^{-1})\mathbf{Q}_2\\ \mathbf{0} & -\mathbf{Q}_2(-\mathbf{A}_{22}^{-1}) & \mathbf{Q}_2(-\mathbf{A}_{22}^{-1})\mathbf{Q}_2\\ \end{pmatrix} $$
This is exactly specified and explained in the Appendix D of Belay et al. 2022. So the J-factor model with as many J-factors as UPGs can be run as detailed in the section above.