SFSI: Sparse Family and Selection Indices

The SFSI R-package solves penalized regression problems offering tools for the solutions to penalized selection indices. In this repository we maintain the latest (developing) version.

Last update: Jun 24, 2024

Package installation

Installation of SFSI package requires a R-version ≥ 3.6.0

From CRAN (stable version) r install.packages('SFSI',repos='https://cran.r-project.org/')

From GitHub (developing version) r install.packages('remotes',repos='https://cran.r-project.org/') # 1. install remotes library(remotes) # 2. load the library install_github('MarcooLopez/SFSI') # 3. install SFSI from GitHub

Selection Indices

A selection index (SI) predicts the genetic value ($ui$) of a candidate of selection for a target trait ($yi$) as the weighted sum of $p$ measured traits $x{i1},\dots,x{ip}$ as:

$$ \color{NavyBlue}{\hat{u}_ i = \boldsymbol{x}{i}'\boldsymbol{\beta}i} $$

where $\boldsymbol{x}_ i = (x{i1},\dots,x{ip})'$ is the vector of measured traits and $\boldsymbol{\beta}_ i = (\beta{i1},\dots,\beta{ip})'$ is the vector of weights.

Standard Selection Index

The weights are derived by minimizing the optimization problem:

$$ \color{NavyBlue}{\hat{\boldsymbol{\beta}}_ i = \text{arg min}{\frac{1}{2}\mathbb{E}(ui - \boldsymbol{x}{i}'\boldsymbol{\beta}_i)}} $$

This problem is equivalent to:

$$ \color{NavyBlue}{\hat{\boldsymbol{\beta}}_ i = \text{arg min}[\frac{1}{2}\boldsymbol{\beta}'_ i\textbf{P}_ x\boldsymbol{\beta}_ i - \textbf{G}'_ {xy}\boldsymbol{\beta}_i]} $$

where $\textbf{P}_ x$ is the phenotypic variance-covariance matrix of predictors and $\textbf{G}_{xy}$ is a vector with the genetic covariances between predictors and response. Under standard assumptions, the solution to the above problem is

$$ \color{NavyBlue}{\hat{\boldsymbol{\beta}}_ i = \textbf{P}^{-1}_ x\textbf{G}_{xy}} $$

Sparse Selection Index

In the sparse selection index (SSI), the weights are derived by imposing a sparsity-inducing penalization in the above optimization function as

$$ \color{NavyBlue}{\hat{\boldsymbol{\beta}}_ i = \text{arg min}[\frac{1}{2}\boldsymbol{\beta}'_ i\textbf{P}_ x\boldsymbol{\beta}_ i - \textbf{G}'{xy}\boldsymbol{\beta}i + \lambda f(\boldsymbol{\beta}_i)]} $$

where $\lambda$ is a penalty parameter and $f(\boldsymbol{\beta}_i)$ is a penalty function on the weights. A value of $\lambda = 0$ yields the coefficients for the standard selection index. Commonly used penalty functions are based on the L1- (i.e., LASSO) and L2-norms (i.e., Ridge Regression). Elastic-Net considers a combined penalization of both norms,

$$ \color{NavyBlue}{f(\boldsymbol{\beta}_ i) = \alpha\sum^p{j=1}|\beta{ij}| + (1-\alpha)\frac{1}{2}\sum^p{j=1}\beta^2{ij}} $$

where $\alpha$ is a number between 0 and 1. The LASSO and Ridge Regression appear as special cases of the Elastic-Net when $\alpha = 1$ and $\alpha = 0$, respectively.

Functions LARS() and solveEN() can be used to obtain solutions for $\hat{\boldsymbol{\beta}}_ i$ in the above penalized optimization problem taking $\textbf{P}_ x$ and $\textbf{G}_{xy}$ as inputs. The former function provides LASSO solutions for the entire $\lambda$ path using Least Angle Regression (Efron et al., 2004), and the later finds solutions for the Elastic-Net problem for given values of $\alpha$ and $\lambda$ via the Coordinate Descent algorithm (Friedman, 2007).

Documentation (two applications)

• Application with high-throughput phenotypes: Lopez-Cruz et al. (2020). [Manuscript]. [Documentation].

• Application to Genomic Prediction: Lopez-Cruz and de los Campos (2021). [Manuscript]. [Documentation].

How to cite SFSI R-package

• Lopez-Cruz M, Olson E, Rovere G, Crossa J, Dreisigacker S, Mondal S, Singh R & de los Campos G (2020). Regularized selection indices for breeding value prediction using hyper-spectral image data. Scientific Reports, 10, 8195.

Dataset

The SFSI R-package contains a reduced version of the full data used in Lopez-Cruz et al. (2020) for the development of penalized selection indices. This full data can be found in this repository.

References

• Efron B, Hastie T, Johnstone I & Tibshirani R (2004). Least angle regression. The Annals of Statistics, 32(2), 407–499.
• Friedman J, Hastie T, Höfling H & Tibshirani R (2007). Pathwise coordinate optimization. The Annals of Applied Statistics, 1(2), 302–332.