selection.index (version 2.0.0)

rgesim: Restricted Linear Genomic Eigen Selection Index Method (RGESIM)

Description

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)

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.

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).

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) {
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
}