Construct an APY Relationship Precision Matrix

Constructs the Algorithm for Proven and Young (APY) approximation for a positive-definite relationship matrix and a user-specified core set. In the genomic evaluation literature, the input is the genomic relationship matrix \(G\) for genotyped animals, usually calculated from SNP marker data. The core and non-core partition is within that genotyped set; it is not a pedigree-versus-genomic partition. The non-core conditional covariance is approximated by its diagonal. The returned matrix is a sparse precision matrix suitable for the Gu argument of mmes() in a genomic BLUP model.

Usage

APY(G, core, tol = 1e-10, return.details = FALSE)

Arguments

G: A finite, symmetric, positive-definite numeric relationship matrix. For the standard APY genetic evaluation use, this is the genomic relationship matrix among genotyped animals, commonly obtained from marker genotypes with A.mat(). Row and column names, when present, must agree exactly. This implementation receives G as a dense matrix.
core: Unique one-based row indices or row names selecting the core individuals.
tol: Positive tolerance below one, used for symmetry validation and the minimum acceptable conditional variance relative to the relationship scale. Small or non-positive conditional variances cause an error; they are not silently regularized.
return.details: If TRUE, return a list containing the sparse precision matrix as Gu, core and non-core indices, regression matrix B, and non-core conditional variances.

Value

A sparse symmetric APY precision matrix with individual dimnames and attr(x, "inverse") = TRUE, ready for Gu. Metadata describing the core and conditional variances is stored in attr(x, "APY"). If return.details = TRUE, returns the diagnostic list described above.

Details

Partition \(G\) into core and non-core blocks. With \(B = G_{nc}G_{cc}^{-1}\) and \(D = \operatorname{diag}(G_{nn} - BG_{cn})\), APY uses the precision \(\left[\begin{smallmatrix} G_{cc}^{-1} + B' D^{-1} B & -B' D^{-1} \\ -D^{-1} B & D^{-1} \end{smallmatrix}\right]\). It preserves the core block and core-to-non-core relationships while replacing non-core conditional correlations with zero. Thus APY is exact when the non-core conditional covariance is diagonal; otherwise it is an approximation to the supplied relationship model.

APY was developed to provide a computationally economical approximation to the inverse genomic relationship matrix for large genomic evaluations (Misztal, 2016; Fragomeni et al., 2015). The algorithm partitions genotyped animals into core and non-core individuals. The core contains the relationships that are modeled jointly; each non-core animal is regressed on the core, with its remaining conditional variance treated as independent of other non-core animals. Thus non-core animals are not independent of the core: their core-to-non-core relationships are retained.

Pedigree relationships are a distinct input. For pedigree-only models, the pedigree relationship matrix \(A\) and its sparse inverse \(A^{-1}\) are normally used directly; APY is not required. In single-step genomic BLUP, pedigree and genomic information are combined, conventionally through \(H^{-1} = A^{-1} + \left[\begin{smallmatrix}0&0\\0&G^{-1}-A_{22}^{-1}\end{smallmatrix}\right]\), where genotyped animals are ordered last and \(A_{22}\) is their pedigree relationship block. APY approximates the genomic term \(G^{-1}\) in that correction; it does not replace the pedigree term or construct \(H^{-1}\). This function returns only the APY genomic precision. Do not use it as the complete single-step precision for a population that also contains non-genotyped animals.

This function constructs APY from an already available relationship matrix. It does not avoid the cost of constructing a full dense relationship matrix from markers, and it does not change the solver algorithm. Marker-based block construction and single-step matrix assembly are not implemented here.

References

Misztal I (2016). Inexpensive Computation of the Inverse of the Genomic Relationship Matrix in Populations with Small Effective Population Size. Genetics, 202(2), 401–409. doi:10.1534/genetics.115.182089.

Fragomeni BO, Lourenco DA, Tsuruta S, Masuda Y, Aguilar I, Legarra A, Lawlor TJ, Misztal I (2015). Use of genomic recursions and algorithm for proven and young animals for single-step genomic BLUP analyses–a simulation study. Journal of Animal Breeding and Genetics, 132(5), 340–345. doi:10.1111/jbg.12161.

Examples

## Marker-derived GBLUP for genotyped animals.
set.seed(2)
ids <- paste0("animal", seq_len(12))
markers <- matrix(sample(c(-1, 0, 1), 12 * 80, replace = TRUE), nrow = 12,
                  dimnames = list(ids, NULL))
G <- A.mat(markers)

## Blend slightly with the identity so this small illustrative G is positive
## definite, then construct APY precision for a six-animal core.
G <- 0.95 * G + 0.05 * diag(nrow(G))
dimnames(G) <- list(ids, ids)
Ginv.apy <- APY(G, core = ids[seq_len(6)])

phenotypes <- data.frame(
  id = factor(rep(ids, each = 3), levels = ids),
  y = rnorm(length(ids) * 3)
)
fit <- mmes(y ~ 1, random = ~vsm(ism(id), Gu = Ginv.apy),
            rcov = ~units, data = phenotypes, verbose = FALSE)

## In a single-step evaluation, this APY approximation supplies G^{-1} for
## the genotyped block correction; it is not itself the complete H^{-1}.