ppg_esim: Predetermined Proportional Gain Eigen Selection Index...

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

Predetermined Proportional Gain Eigen Selection Index (PPG-ESIM)

Description

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.

Usage

ppg_esim(pmat, gmat, 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).
d	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	Selection intensity constant (default: 2.063).

Details

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}}

Value

Object of class "ppg_esim", a list with:

summary

Data frame with beta (transformed b), b (raw), hI2, rHI, sigma_I, Delta_G, and lambda2.

beta

Named numeric vector of post-transformation PPG-ESIM coefficients \boldsymbol{\beta}_P = \mathbf{F}\mathbf{b}_P.

b

Raw eigenvector b_P before similarity transform.

Delta_G

Named vector of expected genetic gains per trait.

sigma_I

Index standard deviation.

hI2

Index heritability.

rHI

Index accuracy.

lambda2

Leading eigenvalue of the PPG restricted eigenproblem.

F_mat

Diagonal similarity transform matrix F (diag(sign(d))).

K_P

PPG projection matrix (rank 1: projects onto d subspace).

D_M

Mallard matrix (t x t-1): orthogonal complement of d, used to construct the (t-1) restrictions.

desired_gains

Input proportional gains vector d.

selection_intensity

Selection intensity used.

References

Ceron-Rojas, J. J., & Crossa, J. (2018). Linear Selection Indices in Modern Plant Breeding. Springer International Publishing. Section 7.3.

Examples

## Not run: 
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)

## End(Not run)

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