Restricted Linear Genomic Selection Index (RLGSI)

Implements the Restricted Linear Genomic Selection Index where genetic gains are constrained to zero for specific traits while maximizing gains for others. Uses GEBVs only (no phenotypic data required).

Usage

rlgsi(
  Gamma,
  wmat,
  wcol = 1,
  restricted_traits = NULL,
  U = NULL,
  k_I = 2.063,
  L_G = 1,
  gmat = NULL,
  GAY = NULL
)

Arguments

Gamma: GEBV variance-covariance matrix (n_traits x n_traits). This represents the variance of GEBVs, typically computed from predicted breeding values.
wmat: Economic weights matrix (n_traits x k), or vector
wcol: Weight column to use if wmat has multiple columns (default: 1)
restricted_traits: Vector of trait indices to restrict (default: NULL). Example: c(1, 3) restricts traits 1 and 3 to zero gain.
U: Constraint matrix (n_traits x n_constraints). Each column defines a restriction. Alternative to restricted_traits for custom constraints. Ignored if restricted_traits is provided.
k_I: Selection intensity (default: 2.063 for 10 percent selection)
L_G: Standardization constant (default: 1). Can be set to sqrt(w'Gw) for standardization.
gmat: Optional. True genetic variance-covariance matrix for exact accuracy calculation. If NULL, uses Gamma as approximation. Providing gmat ensures textbook-perfect accuracy metric.
GAY: Optional. Genetic advance of comparative trait for PRE calculation

Value

List with:

summary - Data frame with coefficients, response metrics
b - Vector of RLGSI coefficients ($\beta_{RG}$)
E - Named vector of expected genetic gains per trait
R - Overall selection response
U - Constraint matrix used
constrained_response - Realized gains for constrained traits (should be ~0)

Details

Mathematical Formulation (Chapter 6, Section 6.1):

The RLGSI minimizes the mean squared difference between the index I = $\beta$'$\gamma$ and the breeding objective H = w'g under the restriction: U'$\Gamma$$\beta$ = 0

Solution involves solving the augmented system: $$\begin{bmatrix} \Gamma & \Gamma U \\ U'\Gamma & 0 \end{bmatrix} \begin{bmatrix} \beta \\ v \end{bmatrix} = \begin{bmatrix} \Gamma w \\ 0 \end{bmatrix}$$

Where: - $\Gamma$ (Gamma) = Var(GEBVs) - GEBV variance-covariance matrix - U = Constraint matrix (each column is a restriction vector) - w = Economic weights - $\beta_{RG}$ = RLGSI coefficient vector - v = Lagrange multipliers

Selection response: $R_{RG} = (k_I / L_G) * sqrt(beta_RG' * Gamma * beta_RG)$

Expected gains: $E_{RG} = (k_I / L_G) * (Gamma * beta_RG) / sqrt(beta_RG' * Gamma * beta_RG)$

Examples

if (FALSE) { # \dontrun{
# Simulate GEBV variance-covariance matrix
set.seed(123)
n_traits <- 5
Gamma <- matrix(rnorm(n_traits^2), n_traits, n_traits)
Gamma <- (Gamma + t(Gamma)) / 2 # Make symmetric
diag(Gamma) <- abs(diag(Gamma)) + 2 # Ensure positive definite

# Economic weights
w <- c(10, 8, 6, 4, 2)

# Restrict traits 2 and 4 to zero gain
result <- rlgsi(Gamma, w, restricted_traits = c(2, 4))
print(result$summary)
print(result$E) # Check that traits 2 and 4 have ~0 gain
} # }