GeneNet
GeneNet is a linear shrinkage estimator for a covariance matrix followed by a Gaussian graphical
model (GGM) selection based on the partial correlation obtained from the shrinkage estimator. With a multiple
testing procedure using the local false discovery rate, the GGM selection controls the false discovery rate
under a pre-determined level $\alpha$.
One of the most commonly used linear shrinkage estimators \(S^{*}\) for the covariance matrix \(\Sigma\) is
$$S^{*} = \lambda^{*}T + (1 - \lambda^{*})S$$
where
\( S = (s_{ij})_{1 \leq i,j \leq p}\)
is the sample covariance matrix,
\( T = diag(s_{11}, s_{22}, ..., s_{pp}) \)
is the shrinkage target matrix, and
$\lambda^{*}=\sum_{i\neq j}\hat{Var}(s_{ij})/(\sum_{i\neq j}s_{ij}^{2})$
is the optimal shrinkage intensity. With this estimator \(S^{*}\), the matrix of the partial correlations
$P = (\hat{\rho}^{ij})_{1\leq i, j\leq p}$
is defined as
$\hat{\rho}^{ij} = -\hat{\omega}_{ij}/ \sqrt{\hat{\omega}_{ii}\hat{\omega}_{jj}}$,
where
$\hat{\Omega} = (\hat{\omega}_{ij})_{1\leq i, j\leq p} = (S^{*})^{-1}$.
To identify the significant edges, the distribution of the partial correlations is supposed to be as the mixture
$$f(\rho) = \eta_{0}f_{0}(\rho, \nu) + (1-\eta_{0})f_{1}(\rho)$$
where $f_0$ is the null distribution, $f_1$ is the alternative distribution corresponding to the true edges,
and $η_0$ is the unknown mixing parameter. GeneNet identifies significant edges that have the
local false positive rate
$$fdr(\rho) = \frac{\eta_{0}f_{0}(\rho, \hat{\nu})}{\hat{f}(\rho)}$$
smaller than the pre-determined level $\alpha$, where
$f_0(\rho, \upsilon) = |\rho|Be(\rho^2; 0.5, (\upsilon - 1)/2)$, $Be(x;a,b)$
is the density of the Beta distribution and $\upsilon$ is the reciprocal variance of the null $\rho$.
Reference:
1. Donghyeon Yu, Johan Lim, Xinlei Wang, Faming Liang, and Guanghua Xiao. "Enhanced construction of gene regulatory networks using hub gene information." BMC bioinformatics 18.1 (2017): 186.
2. Schäfer, Juliane, and Korbinian Strimmer. "A shrinkage approach to large-scale covariance matrix estimation and implications for functional genomics." Statistical applications in genetics and molecular biology 4.1 (2005): 32.
Note:
Change the $\alpha$ level (false positive rate) to control the sparsity of network $(0 < \alpha < 1)$.
A larger $\alpha$ will give you more estimated edges, but with lower confidence.