Extends ESIM by enforcing that genetic gains are proportional to a user-specified vector \(\mathbf{d}\): \(\Delta\mathbf{G} \propto \mathbf{d}\). A similarity transformation \(\boldsymbol{\beta}_P = \mathbf{F}\mathbf{b}_P\) aligns the eigenvector with the breeder's desired direction.
ppg_esim(pmat, gmat, d, selection_intensity = 2.063)Object of class "ppg_esim", a list with:
summaryData frame with beta (transformed b), b (raw), hI2, rHI, sigma_I, Delta_G, and lambda2.
betaNamed numeric vector of post-transformation PPG-ESIM coefficients \(\boldsymbol{\beta}_P = \mathbf{F}\mathbf{b}_P\).
bRaw eigenvector b_P before similarity transform.
Delta_GNamed vector of expected genetic gains per trait.
sigma_IIndex standard deviation.
hI2Index heritability.
rHIIndex accuracy.
lambda2Leading eigenvalue of the PPG restricted eigenproblem.
F_matDiagonal similarity transform matrix F (diag(sign(d))).
K_PPPG projection matrix (rank 1: projects onto d subspace).
D_MMallard matrix (t x t-1): orthogonal complement of d, used to construct the (t-1) restrictions.
desired_gainsInput proportional gains vector d.
selection_intensitySelection intensity used.
Phenotypic variance-covariance matrix (n_traits x n_traits).
Genotypic variance-covariance matrix (n_traits x n_traits).
Numeric vector of desired proportional gains (length n_traits). The ratios among elements define target gain proportions. Direction (positive/negative) must reflect desired improvement direction (positive = increase, negative = decrease).
Selection intensity constant (default: 2.063).
Restriction structure via the Mallard Matrix (Section 7.3):
The PPG-ESIM restricts the \((t-1)\) directions **orthogonal** to \(\mathbf{d}\), forcing the genetic gain vector to be collinear with \(\mathbf{d}\).
The Mallard matrix \(\mathbf{D}_M\) is \(t \times (t-1)\): its columns span the orthogonal complement of \(\mathbf{d}\), obtained via QR decomposition of \(\mathbf{d}/\|\mathbf{d}\|\): $$\mathbf{Q}_{QR} = [\hat{d} \mid \mathbf{D}_M], \quad \text{QR}(\hat{d}) \to \mathbf{Q}_{QR} \in \mathbb{R}^{t \times t}$$
With \(\boldsymbol{\Psi} = \mathbf{C}\) (full-trait case, \(\mathbf{U} = \mathbf{I}_t\)):
PPG projection matrix (\(t-1\) restrictions): $$\mathbf{Q}_P = \mathbf{P}^{-1}\boldsymbol{\Psi}\mathbf{D}_M (\mathbf{D}_M^{\prime}\boldsymbol{\Psi}^{\prime}\mathbf{P}^{-1}\boldsymbol{\Psi}\mathbf{D}_M)^{-1} \mathbf{D}_M^{\prime}\boldsymbol{\Psi}^{\prime}$$ $$\mathbf{K}_P = \mathbf{I}_t - \mathbf{Q}_P \quad (\text{rank 1})$$
Because \(\mathbf{K}_P\) has rank 1 (projects onto the \(\mathbf{d}\) subspace), \(\mathbf{K}_P\mathbf{P}^{-1}\mathbf{C}\) has exactly one positive eigenvalue and its eigenvector produces \(\Delta\mathbf{G} \propto \mathbf{d}\).
PPG eigenproblem (rank-1 system): $$(\mathbf{K}_P\mathbf{P}^{-1}\mathbf{C} - \lambda_P^2\mathbf{I}_t)\mathbf{b}_P = 0$$
Similarity transform: $$\boldsymbol{\beta}_P = \mathbf{F}\mathbf{b}_P$$ where \(\mathbf{F} = \text{diag}(\text{sign}(\mathbf{d}))\) aligns the eigenvector sign with the breeder's intended improvement direction.
Key response metrics: $$R_P = k_I\sqrt{\boldsymbol{\beta}_P^{\prime}\mathbf{P}\boldsymbol{\beta}_P}$$ $$\mathbf{E}_P = k_I\frac{\mathbf{C}\boldsymbol{\beta}_P}{\sqrt{\boldsymbol{\beta}_P^{\prime}\mathbf{P}\boldsymbol{\beta}_P}}$$
Ceron-Rojas, J. J., & Crossa, J. (2018). Linear Selection Indices in Modern Plant Breeding. Springer International Publishing. Section 7.3.
if (FALSE) {
gmat <- gen_varcov(seldata[, 3:9], seldata[, 2], seldata[, 1])
pmat <- phen_varcov(seldata[, 3:9], seldata[, 2], seldata[, 1])
# Desired proportional gains: increase all traits proportionally
d <- c(2, 1, 1, 1, 1, 1, 1)
result <- ppg_esim(pmat, gmat, d)
print(result)
summary(result)
}
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