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

In selection.index: Analysis of Selection Index in Plant Breeding

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
)

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 - \lambdaI)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)

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

References

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

Examples

## Not run: 
# 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)

## End(Not run)

selection.index documentation built on March 9, 2026, 1:06 a.m.