---
title: "Linear Genomic Eigen Selection Index Methods"
description: >
  Genomic eigen selection index methods using GEBVs and marker data:
  MESIM, GESIM, GW-ESIM, RGESIM, and PPG-GESIM. Integrate genome-wide
  marker information into eigen-based selection frameworks to maximize
  genetic advance in plant breeding with the selection.index R package.
Author: "Zankrut Goyani"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Linear Genomic Eigen Selection Index Methods}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---


```{r, include = FALSE}
knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>"
)
```

## Introduction

This vignette provides an overview of the linear molecular and genomic eigen selection index methods, which integrate marker trait data and genome-wide marker values into the calculation. Methods covered include the Molecular Eigen Selection Index Method (MESIM), Genomic Eigen Selection Index Method (GESIM), Genome-Wide Linear Eigen Selection Index Method (GW-ESIM), Restricted Genomic Eigen Selection Index Method (RGESIM), and Predetermined Proportional Gain Genomic Eigen Selection Index Method (PPG-GESIM).

## Setup Data Matrices

```{r setup_data}
library(selection.index)

# Load standard phenotype dataset
data(maize_pheno)

# Define traits and design variables
traits <- c("Yield", "PlantHeight", "DaysToMaturity")
env_col <- "Environment"
genotype_col <- "Genotype"

# Phenotypic variance-covariance matrix (P)
pmat <- phen_varcov(maize_pheno[, traits], maize_pheno[[genotype_col]], maize_pheno[[env_col]])

# Genetic variance-covariance matrix (G)
gmat <- gen_varcov(maize_pheno[, traits], maize_pheno[[genotype_col]], maize_pheno[[env_col]])

# For the sake of demonstration within this vignette, we simulate the required molecular/genomic variance components:
set.seed(42)

# Simulate Gamma: Covariance between phenotypes and GEBVs
Gamma <- gmat * 0.85

# Molecular matrices for MESIM
S_M <- gmat * 0.75 # Covariance between phenotypic values and marker scores
S_Mg <- gmat * 0.70 # Covariance between genotypic values and marker scores
S_var <- gmat * 0.80 # Variance-covariance of marker scores

# Genomic matrices for GW-ESIM
G_M <- gmat * 0.82 # Covariance between true genotypic values and marker values
M <- gmat * 0.90 # Variance-covariance matrix of markers
```

## The Molecular Eigen Selection Index Method (MESIM)

The MESIM index integrates marker scores directly:

$$
I = \beta'_{y}\mathbf{y} + \beta'_{s}\mathbf{s} = \begin{bmatrix} \beta'_{y} & \beta'_{s} \end{bmatrix} \begin{bmatrix} \mathbf{y} \\ \mathbf{s} \end{bmatrix} = \beta'\mathbf{t}
$$

Its optimum coefficients are calculated effectively from the primary eigenvector resulting from this maximization function:

$$
(\mathbf{T}^{-1}\mathbf{\Psi} - \lambda^2_{M}\mathbf{I}_{2t})\beta_M = \mathbf{0}
$$

Using the `mesim` function:

```{r mesim_demo}
mes_index <- mesim(pmat, gmat, S_M, S_Mg, S_var)
summary(mes_index)
```

## The Linear Genomic Eigen Selection Index Method (GESIM)

The GESIM incorporates Genomic Estimated Breeding Values (GEBVs, denoted as $\gamma$):

$$
I = \beta'_{y}\mathbf{y} + \beta'_{\gamma}\mathbf{\gamma} = \begin{bmatrix} \beta'_{y} & \beta'_{\gamma} \end{bmatrix} \begin{bmatrix} \mathbf{y} \\ \mathbf{\gamma} \end{bmatrix} = \beta'\mathbf{f}
$$

The optimum index coefficients are the first eigenvector of:

$$
(\mathbf{\Phi}^{-1}\mathbf{A} - \lambda^2_{G}\mathbf{I}_{2t})\beta_G = \mathbf{0}
$$

Using the `gesim` function:

```{r gesim_demo}
ges_index <- gesim(pmat, gmat, Gamma)
summary(ges_index)
```

## The Genome-Wide Linear Eigen Selection Index Method (GW-ESIM)

When high-density markers covering the whole genome are available, the GW-ESIM formulation estimates genetic potential uniformly across the entire marker panel.

Using the `gw_esim` function:

```{r gw_esim_demo}
gw_index <- gw_esim(pmat, gmat, G_M, M)
summary(gw_index)
```

## The Restricted Linear Genomic Eigen Selection Index Method (RGESIM)

RGESIM enables breeders to impose constraints on particular traits such that their expected genetic advancements are zero, while employing the genomic capability of the GESIM.

Using the `rgesim` function to restrict improvements to `Ear_Height` (second trait):

```{r rgesim_demo}
# Restrict the second trait (PlantHeight)
U_mat <- matrix(c(0, 1, 0), nrow = 1)
rges_index <- rgesim(pmat, gmat, Gamma, U_mat)
summary(rges_index)
```

## The Predetermined Proportional Gain Linear Genomic Eigen Selection Index Method (PPG-GESIM)

This approach aims to acquire customized genetic gains according to a relative priority proportion assigned distinctly for each trait.

$$
(\mathbf{T}_{PG} - \lambda^2_{PG}\mathbf{I}_{2t})\beta_{PG} = \mathbf{0}
$$

Using the `ppg_gesim` function with relative gains heavily weighted towards `Yield` (trait 3) and `Days_To_Silking` (trait 4):

```{r ppg_gesim_demo}
# Desired genetic gain proportions: 1 for Yield, 0.5 for DaysToMaturity, 0 for others
d <- c(1, 0, 0.5)
ppg_ges_index <- ppg_gesim(pmat, gmat, Gamma, d)
summary(ppg_ges_index)
```

## Statistical Properties and Efficiency

Just as with phenotypic and standard molecular selection indices, the efficiency of Genomic Eigen Selection Indices can be evaluated via accuracy ($\rho_{HI}$), maximized selection response ($R$), and expected genetic gain per trait ($E$).

For the MESIM index, the parameters are formulated by estimating the standard deviations of the true genetic merit ($\sigma_H$) and the given index ($\sigma_I$). The maximum accuracy obtained through canonical correlation ($\lambda_M$) corresponds directly to the square root of the primary eigenvalue:

$$
\rho_{H_MI_M} = \frac{\sqrt{\beta'_{M}\mathbf{T}_M\beta_M}}{\sqrt{\beta'_{M}\mathbf{T}_M\mathbf{\Psi}_M^{-1}\mathbf{T}_M\beta_M}} = \frac{\sigma_{I_M}}{\sigma_{H_M}}
$$

The selection response is obtained using a standardized selection intensity factor ($k_I$):

$$
R_M = k_I \sqrt{\beta'_{M_1}\mathbf{T}_M\beta_{M_1}}
$$

Expected genetic gain per trait can then be calculated accordingly:

$$
\mathbf{E}_M = k_I \frac{\mathbf{\Psi}_M\beta_{M_1}}{\sqrt{\beta'_{M_1}\mathbf{T}_M\beta_{M_1}}}
$$
