Predetermined Proportional Gain Genomic Eigen Selection Index (PPG-GESIM)

Implements the PPG-GESIM which extends GESIM to enforce that genetic gains are proportional to a user-specified vector d. Combines eigen approach with predetermined gain proportions.

Usage

ppg_gesim(pmat, gmat, Gamma, d, 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).
d: Numeric vector of desired proportional gains (length n_traits). The ratios among elements define target gain proportions.
selection_intensity: Selection intensity constant $k_I$ (default: 2.063 for 10% selection).

Value

Object of class "ppg_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_PG: Expected genetic gains per trait.
gain_ratios: Ratios of actual to desired gains (should be constant).
d: Original desired proportional gains (length t).
d_PG: Extended proportional gains (length 2t, includes GEBV targets).
sigma_I: Standard deviation of the index.
hI2: Index heritability.
rHI: Accuracy.
R_PG: Selection response.
lambda2: Leading eigenvalue.
implied_w: Implied economic weights.
U_PG: Restriction matrix ((2t-1) x 2t).
selection_intensity: Selection intensity used.

Details

Eigenproblem (Section 8.5): $$(\mathbf{T}_{PG} - \lambda_{PG}^2 \mathbf{I}_{2t})\boldsymbol{\beta}_{PG} = 0$$

where: $$\mathbf{T}_{PG} = \mathbf{K}_{RG}\mathbf{\Phi}^{-1}\mathbf{A} + \mathbf{B}$$ $$\mathbf{B} = \boldsymbol{\delta}\boldsymbol{\varphi}^{\prime}$$

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

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

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

References

Ceron-Rojas, J. J., & Crossa, J. (2018). Linear Selection Indices in Modern Plant Breeding. Springer International Publishing. Section 8.5.

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

# Desired proportional gains (e.g., 2:1:3 ratio for first 3 traits)
d <- c(2, 1, 3, 1, 1, 1, 1)

result <- ppg_gesim(pmat, gmat, Gamma, d)
print(result)
print(result$gain_ratios) # Should be approximately constant
} # }