rlgsi: Restricted Linear Genomic Selection Index (RLGSI)

In selection.index: Analysis of Selection Index in Plant Breeding

Restricted Linear Genomic Selection Index (RLGSI)

Description

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

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)

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)

Examples

## Not run: 
# 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

## End(Not run)

selection.index documentation built on March 9, 2026, 1:06 a.m.