selection.index (version 2.0.0)

genetic_genomic_varcov: Genetic-Genomic Variance-Covariance Matrix (A)

Description

Computes the genetic-genomic covariance matrix (A) as defined in Chapter 8 (Equation 8.12) for GESIM and related genomic eigen selection indices.

Structure: A = [[C, C_g-gamma], [C_gamma-g, \(\Gamma\)]] (2t x 2t, square symmetric)

where: - C = Var(g) = true genotypic variance-covariance (t x t) - \(\Gamma\) = Var(\(\gamma\)) = genomic variance-covariance (t x t) - C_g-gamma = Cov(g, \(\gamma\)) = covariance between true BVs and GEBVs (t x t) - C_gamma-g = Cov(\(\gamma\), g) = transpose of C_g-gamma (t x t)

Usage

genetic_genomic_varcov(
  gmat,
  Gamma = NULL,
  reliability = NULL,
  C_gebv_g = NULL,
  square = TRUE
)

Value

Genetic-genomic covariance matrix: - If square = TRUE: (2t × 2t) symmetric matrix for GESIM/eigen indices - If square = FALSE: (2t × t) rectangular matrix for LMSI where t is the number of traits

Arguments

gmat

Genotypic variance-covariance matrix (n_traits x n_traits)

Gamma

Genomic variance-covariance matrix (n_traits x n_traits). If NULL, assumed equal to gmat (perfect prediction).

reliability

Optional. Reliability of GEBVs (r² = squared correlation between GEBV and true BV). Can be: - Single value (applied to all traits) - Vector of length n_traits (one per trait) - NULL (default): assumes C_g,\(\gamma\) = Gamma (unbiased GEBVs with reliability = 1)

C_gebv_g

Optional. Direct specification of Cov(\(\gamma\), g) matrix (t x t). If provided, overrides reliability parameter.

square

Logical. If TRUE (default), returns (2t × 2t) square matrix as required for GESIM. If FALSE, returns (2t × t) rectangular form for LMSI.

Details

The genetic-genomic matrix relates selection on phenotypes + GEBVs to expected genetic gains.

**For GESIM (Chapter 8):** Requires the full (2t × 2t) square matrix for the eigenproblem: (\(\Phi\)^(-1) A - \(\lambda\)I)b = 0

**For LMSI/CLGSI (Chapter 4):** Can use the rectangular (2t × t) form in the equation: b = P^(-1) G w, where G is (2t × t).

When reliability is provided: - \(C_{\gamma g}\) = diag(\(\sqrt{r^2}\))

When reliability is NULL: - \(C_{\gamma g}\) = Gamma (assumes unbiased GEBVs, perfect prediction)

References

Cerón-Rojas, J. J., & Crossa, J. (2018). Linear Selection Indices in Modern Plant Breeding. Springer International Publishing. Chapters 4 & 8.

Examples

if (FALSE) {
# Generate example data
gmat <- gen_varcov(seldata[, 3:9], seldata[, 2], seldata[, 1])

# Simulate genomic covariance
Gamma <- gmat * 0.8

# For GESIM: Get square (2t × 2t) matrix
A_square <- genetic_genomic_varcov(gmat, Gamma, reliability = 0.7)
print(dim(A_square)) # Should be 14 x 14 (2t × 2t)

# For LMSI: Get rectangular (2t × t) matrix
A_rect <- genetic_genomic_varcov(gmat, Gamma, reliability = 0.7, square = FALSE)
print(dim(A_rect)) # Should be 14 x 7 (2t × t)
}