Linear Genomic Eigen Selection Index Method (GESIM)

Implements the GESIM by maximising the squared accuracy through the generalised eigenproblem combining phenotypic data with GEBVs (Genomic Estimated Breeding Values). No economic weights are required.

Usage

gesim(pmat, gmat, Gamma, 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). This represents Cov(y, gamma) where gamma are GEBVs.

selection_intensity

Selection intensity constant \(k_I\) (default: 2.063 for 10% selection).

Value

Object of class "gesim", 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_G

Expected genetic gains per trait.

sigma_I

Standard deviation of the index.

hI2

Index heritability (= leading eigenvalue).

rHI

Accuracy \(r_{HI}\).

R_G

Selection response.

lambda2

Leading eigenvalue.

implied_w

Implied economic weights.

selection_intensity

Selection intensity used.

Details

Eigenproblem (Section 8.2): $$(\mathbf{\Phi}^{-1}\mathbf{A} - \lambda_G^2 \mathbf{I}_{2t})\boldsymbol{\beta}_G = 0$$

where: $$\mathbf{\Phi} = \begin{bmatrix} \mathbf{P} & \mathbf{\Gamma} \\ \mathbf{\Gamma} & \mathbf{\Gamma} \end{bmatrix}$$ $$\mathbf{A} = \begin{bmatrix} \mathbf{C} & \mathbf{\Gamma} \\ \mathbf{\Gamma} & \mathbf{\Gamma} \end{bmatrix}$$

Implied economic weights: $$\mathbf{w}_G = \mathbf{A}^{-1}\mathbf{\Phi}\boldsymbol{\beta}$$

Selection response: $$R_G = k_I \sqrt{\boldsymbol{\beta}_G^{\prime}\mathbf{\Phi}\boldsymbol{\beta}_G}$$

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

References

Ceron-Rojas, J. J., & Crossa, J. (2018). Linear Selection Indices in Modern Plant Breeding. Springer International Publishing. Section 8.2.

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 (in practice, compute from genomic predictions)
Gamma <- gmat * 0.8 # Assume 80% GEBV-phenotype covariance

result <- gesim(pmat, gmat, Gamma)
print(result)
} # }