H.inverse: Generates the inverse of the hybrid H matrix

Description

The single-step GBLUP approach combines the information from the pedigree relationship matrix $\boldsymbol{A}$ and the genomic relationship matrix $\boldsymbol{G}$ in one hybrid relationship matrix called $\boldsymbol{H}$. This function will calculate directly the inverse of this matrix $\boldsymbol{H}$. The user should provide the matrices $\boldsymbol{A}$ or its inverse (only one of these is required) and the inverse of the matrix $\boldsymbol{G}$ ($\boldsymbol{G_{inv}}$) in its full form. Individual names should be assigned to rownames and colnames, and individuals from $\boldsymbol{G_{inv}}$ are verified to be all a subset within individuals from $\boldsymbol{A}$ (or $\boldsymbol{A_{inv}}$).

Usage

H.inverse(
  A = NULL,
  Ainv = NULL,
  Ginv = NULL,
  lambda = NULL,
  tau = 1,
  omega = 1,
  sparseform = FALSE,
  keep.order = TRUE,
  digits = 8,
  inverse = TRUE,
  message = TRUE
)

Value

The inverse of the hybrid matrix $\boldsymbol{H}$ matrix, in full or sparse form with required attributes to be used in asreml.

Arguments

A: Input of the pedigree relationship matrix $\boldsymbol{A}$ in full form ($na \times na$) (default = NULL).
Ainv: Input of the inverse of the pedigree relationship matrix $\boldsymbol{A}^{-1}$ in full form ($na \times na$) (default = NULL).
Ginv: Input of the inverse of the genomic relationship matrix $\boldsymbol{G}^{-1}$ in full form ($ng \times ng$) (default = NULL).
lambda: The scaling factor for $(\boldsymbol{G}^{-1}-\boldsymbol{A}^{-1}_{22})$ (default = NULL).
tau: The scaling factor for $\boldsymbol{G}^{-1}$ (default = 1).
omega: The scaling factor for $\boldsymbol{A}^{-1}_{22}$ (default = 1).
sparseform: If TRUE it generates the requested matrix in sparse form to be used directly in asreml with required attributes (default = FALSE).
keep.order: If TRUE the original order of the individuals from the $\boldsymbol{A}$ or $\boldsymbol{A_{inv}}$ matrix is kept. Otherwise the non-genotyped individuals are placed first and then genotyped individuals (default = TRUE).
digits: Set up the number of digits used to round the output matrix (default = 8).
inverse: If TRUE it generates the inverse of $\boldsymbol{H}$ matrix (default = TRUE) (to be deprecated).
message: If TRUE diagnostic messages are printed on screen (default = TRUE).

Details

The generation of the $\boldsymbol{H^{-1}}$ matrix contains a few scaling factors to help with the calculation of this inverse and to allow further exploration of the combination of the information from the $\boldsymbol{A^{-1}}$ and $\boldsymbol{G^{-1}}$. We follow the specifications described by Martini et. al (2018), which is done by specifying the parameters $\lambda$, or the pair $\tau$ and $\omega$.

The general expression used is: $$\boldsymbol{H^{-1}}=\boldsymbol{A^{-1}}+\begin{bmatrix}\boldsymbol{0}&\boldsymbol{0}\\\boldsymbol{0}&(\tau\boldsymbol{G^{-1}}-\omega\boldsymbol{{A_{22}^{-1}}})\end{bmatrix}$$

and a more common representation of the above expression is found when $\tau = \omega = \lambda$, as shown below: $$\boldsymbol{H^{-1}}=\boldsymbol{A^{-1}}+\begin{bmatrix}\boldsymbol{0}&\boldsymbol{0}\\\boldsymbol{0}&\lambda(\boldsymbol{G^{-1}}-\boldsymbol{{A_{22}^{-1}}})\end{bmatrix}$$

If inverse = FALSE the $\boldsymbol{H}$ matrix is provided instead of its inverse. This option will be deprecated and it is better to use the function H.matrix.

The $\boldsymbol{H}$ matrix is obtained with the following equations: $$\boldsymbol{H}=\boldsymbol{A}+\begin{bmatrix}\boldsymbol{A}_{12}\boldsymbol{A}_{22}^{-1}(\boldsymbol{G}-\boldsymbol{A}_{22})\boldsymbol{A}_{22}^{-1}\boldsymbol{A}_{21}&\boldsymbol{A}_{12}\boldsymbol{A}_{22}^{-1}(\boldsymbol{G}-\boldsymbol{A}_{22})\\(\boldsymbol{G}-\boldsymbol{A}_{22})\boldsymbol{A}_{22}^{-1}\boldsymbol{A}_{21}&(\boldsymbol{G}-\boldsymbol{A}_{22})\end{bmatrix}$$

References

Christensen, O.F., Lund, M.S. 2010. Genomic prediction matrix when some animals are not genotyped. Gen. Sel. Evol. 42(2):1–8.

Christensen, O., Madsen, P., Nielsen, B., Ostersen, T., and Su, G. 2012. Single-step methods for genomic evaluation in pigs. Animal 6(10):1565–1571.

Legarra, A., Aguilar, I., and Misztal, I. 2009. A relationship matrix including full pedigree and genomic information. J. Dairy Sci. 92:4656-4663.

Martini, J.W.R., Schrauf, M.F., Garcia-Baccino, C.A., Pimentel, E.C.G., Munilla, S., Rogberg-Muñoz, A., Cantet, R.J.C., Reimer, C., Gao, N., Wimmer, V., and Simianer, H. 2018. The effect of the $H^{-1}$ scaling factors $\tau$ and $\omega$ on the structure of $H$ in the single-step procedure. Genet. Sel. Evol. 50:1-9.

Examples

Run this code

# \donttest{
# Get A matrix.
A <- AGHmatrix::Amatrix(data = ped.pine)
A[1:5,1:5]
dim(A)

# Read and filter genotypic data.
M.clean <- qc.filtering(
 M = geno.pine655,
 maf = 0.05,
 marker.callrate = 0.2, ind.callrate = 0.20,
 na.string = "-9",
 plots = FALSE)$M.clean

# Get G matrix.
G <- G.matrix(M.clean, method = "VanRaden", na.string = "-9")$G
G[1:5, 1:5]
dim(G)

# Match G and A.
check <- match.G2A(
 A = A, G = G,
 clean = TRUE, ord = TRUE, mism = TRUE, RMdiff = TRUE)

# Align G matrix with A.
G_align <- G.tuneup(G = check$Gclean, A = check$Aclean, align = TRUE, sparseform = FALSE)$Gb

# Get Ginverse using the G aligned.
Ginv <- G.inverse(G = G_align, sparseform = FALSE)$Ginv
Ginv[1:5, 1:5]
dim(Ginv)

# Obtain Hinv.
Hinv <- H.inverse(A = A, Ginv = Ginv, lambda = 0.90, sparseform = TRUE)
head(Hinv)
attr(Hinv, "INVERSE")
# }

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