---
title: "Stochastic Simulation of Linear Phenotypic Selection Indices"
description: >
  Stochastic simulation of recurrent plant breeding selection cycles using
  LPSI, ESIM, RLPSI, and RESIM indices. Evaluate long-term genetic gain,
  variance decay, and selection accuracy over 50+ generations with the
  selection.index R package's simulate_selection_cycles function.
Author: "Zankrut Goyani"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Stochastic Simulation of Linear Phenotypic Selection Indices}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---


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

# Introduction

Stochastic simulation provides breeders with a comprehensive method to evaluate the long-term behavior of different selection indices across numerous sequential cycles. By running *in silico* (computer simulated) models, breeders can evaluate the expected genetic gains, selection accuracy, and the eventual decay of genetic variance over an extended selection timeframe.

Chapter 10 of the `selection.index` package focuses on the performance comparison of over 50 recurrent generic population breeding cycles. The package provides the `stochastic_simulation()` function to evaluate multiple restricted and unrestricted linear phenotypic indices:

1. **LPSI** (Linear Phenotypic Selection Index - unrestricted Smith-Hazel)
2. **ESIM** (Eigen Selection Index Method - unrestricted eigen-based method)
3. **RLPSI** (Restricted Linear Phenotypic Selection Index - restricted Kempthorne-Nordskog method)
4. **RESIM** (Restricted Eigen Selection Index Method - restricted eigen-based method)

## Simulated Breeding Design

The `stochastic_simulation` function mimics a forward recurrent selection scheme. Within each cycle, the simulation performs the following biological steps:
1. Recombines parental haplotypes using Haldane's mapping function to generate double-haploid lines via crossing.
2. Simulates quantitative trait variations assuming a full pleiotropic model, applying sampled major/minor QTL effects.
3. Computes the phenotypic score of the progenies by applying realistic environmental variances based on expected heritabilities ($h^2$).
4. Evaluates the linear combinations of each chosen index to select the top-performing progenies.
5. Randomly mates the selected superior lines to act as the founders for the next evolutionary cycle.

---

# Running a Simulation

We can run a customized, small-scale simulation to demonstrate the tracking functionality. We will analyze a scenario with three traits to trace their behavior across 5 generation cycles.

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

# To ensure reproducibility
set.seed(42)
```

## Configuring the Scenario

In accordance with the textbook paradigm, let us analyze three genetically correlated target traits:
- **Trait 1:** Low heritability ($h^2 = 0.2$), negatively correlated to Trait 2.
- **Trait 2:** High heritability ($h^2 = 0.5$).
- **Trait 3:** Independent trait with high heritability ($h^2 = 0.5$).

### 1. Generating Genetic Effects

First, we generate a known initial phenotypic and genotypic covariance matrix to serve as the baseline simulation logic for our markers. 
For simplicity, we will simulate a rapid 5-cycle breeding program across a smaller subset of 50 QTL markers.

```{r define_markers}
n_traits <- 3
n_loci <- 50 # Number of segregating sites / markers

# Generate random base QTL effects for the markers across the 3 traits
# Negative correlation infused between trait 1 and 2
qtl_eff <- matrix(rnorm(n_loci * n_traits), nrow = n_loci, ncol = n_traits)
qtl_eff[, 2] <- -0.5 * qtl_eff[, 1] + 0.5 * qtl_eff[, 2]

# Define heritabilities and corresponding environmental variance
heritabilities <- c(0.2, 0.5, 0.5)

# Simulate base genetic variance to deduce correct environmental variance noise
base_gv <- apply(qtl_eff, 2, var) * n_loci
env_var <- base_gv * (1 - heritabilities) / heritabilities
```

### 2. Formulating Constraints and Weights

All the indices will apply an equal starting economic weight. However, for the Restricted indices (RLPSI and RESIM), we will force a constraint onto the first trait.

```{r define_weights}
# Equal economic trait weighting
weights <- c(1, 1, 1)

# Constraint matrix for RLPSI/RESIM: Constrain Trait 1
U_mat <- matrix(0, nrow = 3, ncol = 1)
U_mat[1, 1] <- 1
```

### 3. Executing the Simulation Routine

The `simulate_selection_cycles` routine requires various inputs covering biological configuration (individuals, cycles) and selection configurations (weights, restricted traits, selection proportion).

*Note: For performance within the vignette, we use minimal cyclic/generational values. In real breeding scenarios, you would set `n_cycles = 50` and `n_individuals = 10000` to mirror the book's 35,000 double-haploid evaluations.*

```{r run_sim}
# Run the stochastic selection (may take a moment)
sim_results <- simulate_selection_cycles(
  n_cycles = 5,
  n_individuals = 200,
  n_loci = n_loci,
  n_traits = n_traits,
  qtl_effects = qtl_eff,
  heritability = heritabilities,
  economic_weights = weights,
  selection_proportion = 0.25, # Select upper 25% progeny
  restricted_traits = 1
)
```

---

# Interpreting Results

The simulation tracks comprehensive metrics for all computed methods (LPSI, ESIM, RLPSI, RESIM) across all generated cycles. The return object provides tracking arrays for:
1. Genetic Gain (`*_gain`)
2. Genetic Variance (`*_var`)
3. Estimated Heritabilities (`*_heritability`)
4. Mean Phenotypic Value (`*_mean`)

### Genetic Gain Trajectories

We can extract the continuous genetic gain trajectories that map the success of the applied phenotypic criteria across the multiple sequential cycles. Let's observe the restricted indices:

```{r interpret_gain}
# Expected: Because Trait 1 was constrained via the U_mat for the RLPSI metric,
# its expected generational gain should stabilize at 0.
print(sim_results$rlpsi_gain)
```

As clearly demonstrated, the selection gain mapped for Trait 1 drops immediately towards effectively $0$, verifying that the restricted Kempthorne-Nordskog properties hold up systematically across stochastic evaluations. In comparison, unrestricted indices (such as LPSI) map increasing steady gains across all associated traits.

```{r interpret_lpsi_gain}
print(sim_results$lpsi_gain)
```

### Decay of Variance

Additionally, intense generational selection logically extinguishes raw genetic variability. If the selection index consistently applies extreme thresholds, the available allelic variance diminishes, making subsequent trait improvements plateau.

```{r interpret_var}
# Observe the diminishing variance arrays for the LPSI evaluations
print(sim_results$lpsi_var)
```

---

# Summary

By evaluating cyclic genetic simulations across stochastic marker environments, breeders identify crucial plateau mechanisms in multidimensional indices. Specifically:
- **Restricted indices** properly maintain and prevent target constraints over multi-generational limits, albeit slightly reducing overarching expected gains.
- **Independence of traits** limits associative "ride-along" gains obtained via pleiotropy in highly positive correlated setups.
- **Accuracies degrade linearly** as available variant variance diminishes, confirming limits built into simple additive pleiotropic models.
