Restricted Linear Genomic Eigen Selection Index Method (RGESIM)

Implements the RGESIM which extends GESIM to allow restrictions on genetic gains of certain traits. Uses the eigen approach with Lagrange multipliers.

Usage

rgesim(pmat, gmat, Gamma, U_mat, selection_intensity = 2.063)

Arguments

pmat: Phenotypic variance-covariance matrix (n_traits x n_traits).
gmat: Genotypic variance-covariance matrix (n_traits x n_traits).
Gamma: Covariance between phenotypes and GEBVs (n_traits x n_traits).
U_mat: Restriction matrix (r x n_traits) where r is number of restrictions. Each row specifies a linear combination of traits to be held at zero gain.
selection_intensity: Selection intensity constant $k_I$ (default: 2.063 for 10% selection).

Value

Object of class "rgesim", a list with:

summary: Data frame with coefficients and metrics.
b_y: Coefficients for phenotypic data.
b_gamma: Coefficients for GEBVs.
b_combined: Combined coefficient vector.
E_RG: Expected genetic gains per trait.
constrained_response: U' * E (should be near zero).
sigma_I: Standard deviation of the index.
hI2: Index heritability.
rHI: Accuracy.
R_RG: Selection response.
lambda2: Leading eigenvalue.
implied_w: Implied economic weights.
K_RG: Projection matrix.
Q_RG: Constraint projection matrix.
selection_intensity: Selection intensity used.

Details

Eigenproblem (Section 8.4): $$(\mathbf{K}_{RG}\mathbf{\Phi}^{-1}\mathbf{A} - \lambda_{RG}^2 \mathbf{I}_{2t})\boldsymbol{\beta}_{RG} = 0$$

where: $$\mathbf{K}_{RG} = \mathbf{I}_{2t} - \mathbf{Q}_{RG}$$ $$\mathbf{Q}_{RG} = \mathbf{\Phi}^{-1}\mathbf{A}\mathbf{U}_G(\mathbf{U}_G^{\prime}\mathbf{A}\mathbf{\Phi}^{-1}\mathbf{A}\mathbf{U}_G)^{-1}\mathbf{U}_G^{\prime}\mathbf{A}$$

Implied economic weights: $$\mathbf{w}_{RG} = \mathbf{A}^{-1}[\mathbf{\Phi} + \mathbf{Q}_{RG}^{\prime}\mathbf{A}]\boldsymbol{\beta}_{RG}$$

Selection response: $$R_{RG} = k_I \sqrt{\boldsymbol{\beta}_{RG}^{\prime}\mathbf{\Phi}\boldsymbol{\beta}_{RG}}$$

Expected genetic gain per trait: $$\mathbf{E}_{RG} = k_I \frac{\mathbf{A}\boldsymbol{\beta}_{RG}}{\sqrt{\boldsymbol{\beta}_{RG}^{\prime}\mathbf{\Phi}\boldsymbol{\beta}_{RG}}}$$

References

Ceron-Rojas, J. J., & Crossa, J. (2018). Linear Selection Indices in Modern Plant Breeding. Springer International Publishing. Section 8.4.

Examples

if (FALSE) { # \dontrun{
gmat <- gen_varcov(seldata[, 3:9], seldata[, 2], seldata[, 1])
pmat <- phen_varcov(seldata[, 3:9], seldata[, 2], seldata[, 1])

# Simulate GEBV covariance
Gamma <- gmat * 0.8

# Restrict first trait to zero gain
# U_mat must be (n_traits x n_restrictions)
n_traits <- nrow(gmat)
U_mat <- matrix(0, n_traits, 1)
U_mat[1, 1] <- 1 # Restrict trait 1

result <- rgesim(pmat, gmat, Gamma, U_mat)
print(result)
print(result$constrained_response) # Should be near zero
} # }