H.inverse {ASRgenomics}

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
)

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}

Value

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

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

# 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")