Matrix-Free Pedigree Relationship Products

Computes products of the form $A x$, $A X$, $A^{-1} x$, or $A^{-1} X$ directly from an ordered pedigree, without constructing the full additive relationship matrix $A$. This is useful for large pedigrees where explicitly storing the dense $n \times n$ matrix is infeasible.

Usage

pedprod(ped, x, method = c("A", "Ainv"))

Arguments

ped: A complete pedigree accepted by tidyped, preferably an existing tidyped object.
x: A numeric or logical vector, or a numeric or logical matrix. An unnamed vector must have one value per pedigree individual in IndNum order. A named vector may contain a subset of individuals; omitted individuals receive value zero. For a matrix, the same rules apply to its rows and rownames.
method: Character scalar. "A" (default) computes products with the additive relationship matrix. "Ainv" computes products with its inverse.

Value

For vector input, a named numeric vector in pedigree order. For matrix input, a numeric matrix whose rows are in pedigree order; input column names are preserved.

Details

For an ordered pedigree, the additive relationship matrix can be written as $$A = T D T^\prime,$$ where $T$ is the transmission matrix and $D$ contains Mendelian sampling variances. pedprod() applies these factors through backward and forward pedigree traversals. For an $n \times p$ right-hand side, computation requires $O(np)$ time and working memory rather than materializing the $O(n^2)$ relationship matrix.

Named inputs are aligned by individual ID. Every supplied name must occur in ped$Ind, and duplicate, missing, or empty names are rejected. Missing values and non-finite input values are also rejected to avoid propagating an invalid value through all descendants or relatives.

References

Colleau, J. J. (2002). An indirect approach to the extensive calculation of relationship coefficients. Genetics Selection Evolution, 34, 409–421. doi:10.1051/gse:2002015

Colleau, J. J., Palhiere, I., Rodriguez-Ramilo, S. T., & Legarra, A. (2017). A fast indirect method to compute functions of genomic relationships concerning genotyped and ungenotyped individuals, for diversity management. Genetics Selection Evolution, 49, 87. doi:10.1186/s12711-017-0363-9

Examples

tped <- tidyped(small_ped)

# Equal contributions from two candidates; unspecified individuals are zero.
contributions <- c(Z1 = 0.5, Z2 = 0.5)
relationship_to_group <- pedprod(tped, contributions)

# Average additive relationship among the weighted candidates: c' A c.
sum(contributions * relationship_to_group[names(contributions)])
#> [1] 0.7910156

# Multiple contribution schemes can be evaluated in one call.
schemes <- cbind(
  Equal = c(A = 0.5, B = 0.5),
  SireA = c(A = 1.0, B = 0.0)
)
pedprod(tped, schemes)
#>      Equal   SireA
#> A  0.50000 1.00000
#> B  0.50000 0.00000
#> F  0.00000 0.00000
#> I  0.00000 0.00000
#> J1 0.00000 0.00000
#> J2 0.00000 0.00000
#> N  0.00000 0.00000
#> O  0.00000 0.00000
#> R  0.00000 0.00000
#> C  0.50000 0.50000
#> D  0.50000 0.50000
#> E  0.50000 0.50000
#> P  0.00000 0.00000
#> Q  0.00000 0.00000
#> G  0.25000 0.25000
#> H  0.25000 0.25000
#> K  0.25000 0.25000
#> L  0.25000 0.25000
#> M  0.25000 0.25000
#> S  0.00000 0.00000
#> T  0.00000 0.00000
#> U  0.12500 0.12500
#> V  0.25000 0.25000
#> W  0.12500 0.12500
#> X  0.18750 0.18750
#> Y  0.06250 0.06250
#> Z1 0.09375 0.09375
#> Z2 0.09375 0.09375