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Stability, akin to reproducibility, is crucial in statistical analysis. This paper examines the stability of sparse network inference in high-dimensional graphical models, where selected edges should remain consistent across different samples. Our study focuses on the Graphical Lasso and its decomposition into two steps, with the first step involving hierarchical clustering using single linkage.We provide theoretical proof that single linkage is stable, evidenced by controlled distances between two dendrograms inferred from two samples. Practical experiments further illustrate the stability of the Graphical Lasso's various steps, including dendrograms, variable clusters, and final networks. Our results, validated through both theoretical analysis and practical experiments using simulated and real datasets, demonstrate that single linkage is more stable than other methods when a modular structure is present.

Current high-throughput technologies provide a large amount of variables to describe a phenomenon. Only a few variables are generally sufficient to answer the question. Identify them in a high-dimensional Gaussian linear regression model is the one of the most-used statistical methods. In this article, we describe step-by-step the variable selection procedures built upon regularization paths. Regularization paths are obtained by combining a regularization function and an algorithm. Then, they are combined either with a model selection procedure using penalty functions or with a sampling strategy to obtain the final selected variables. We perform a comparison study by considering three simulation settings with various dependency structures on variables. %from the most classical to a most realistic one. In all the settings, we evaluate (i) the ability to discriminate between the active variables and the non-active variables along the regularization path (pROC-AUC), (ii) the prediction performance of the selected variable subset (MSE) and (iii) the relevance of the selected variables (recall, specificity, FDR). From the results, we provide recommendations on strategies to be favored depending on the characteristics of the problem at hand. We obtain that the regularization function Elastic-net provides most of the time better results than the $\ell_1$ one and the lars algorithm has to be privileged as the GD one. ESCV provides the best prediction performances. Bolasso and the knockoffs method are judicious choices to limit the selection of non-active variables while ensuring selection of enough active variables. Conversely, the data-driven penalties considered in this review are not to be favored. As for Tigress and LinSelect, they are conservative methods., Comment: 29 pages, 9 figures, 3 tables

Quantitatively predicting phenotype variables by the expression changes in a set of candidate genes is of great interest in molecular biology but it is also a challenging task for several reasons. First, the collected biological observations might be heterogeneous and correspond to different biological mechanisms. Secondly, the gene expression variables used to predict the phenotype are potentially highly correlated since genes interact though unknown regulatory networks. In this paper, we present a novel approach designed to predict quantitative trait from transcriptomic data, taking into account the heterogeneity in biological samples and the hidden gene regulatory networks underlying different biological mechanisms. The proposed model performs well on prediction but it is also fully parametric, which facilitates the downstream biological interpretation. The model provides clusters of individuals based on the relation between gene expression data and the phenotype, and also leads to infer a gene regulatory network specific for each cluster of individuals. We perform numerical simulations to demonstrate that our model is competitive with other prediction models, and we demonstrate the predictive performance and the interpretability of our model to predict alcohol sensitivity from transcriptomic data on real data from Drosophila Melanogaster Genetic Reference Panel (DGRP).

Gaussian graphical models are widely utilized to infer and visualize networks of dependencies between continuous variables. However, inferring the graph is difficult when the sample size is small compared to the number of variables. To reduce the number of parameters to estimate in the model, we propose a non-asymptotic model selection procedure supported by strong theoretical guarantees based on an oracle inequality and a minimax lower bound. The covariance matrix of the model is approximated by a block-diagonal matrix. The structure of this matrix is detected by thresholding the sample covariance matrix, where the threshold is selected using the slope heuristic. Based on the block-diagonal structure of the covariance matrix, the estimation problem is divided into several independent problems: subsequently, the network of dependencies between variables is inferred using the graphical lasso algorithm in each block. The performance of the procedure is illustrated on simulated data. An application to a real gene expression dataset with a limited sample size is also presented: the dimension reduction allows attention to be objectively focused on interactions among smaller subsets of genes, leading to a more parsimonious and interpretable modular network., Comment: Accepted in JASA