$\ell_1$ -Penalized censored Gaussian graphical model

Abstract

Graphical lasso is one of the most used estimators for inferring genetic networks. Despite its diffusion, there are several fields in applied research where the limits of detection of modern measurement technologies make the use of this estimator theoretically unfounded, even when the assumption of a multivariate Gaussian distribution is satisfied. Typical examples are data generated by polymerase chain reactions and flow cytometer. The combination of censoring and high-dimensionality make inference of the underlying genetic networks from these data very challenging. In this article, we propose an $\ell_1$-penalized Gaussian graphical model for censored data and derive two EM-like algorithms for inference. We evaluate the computational efficiency of the proposed algorithms by an extensive simulation study and show that, when censored data are available, our proposal is superior to existing competitors both in terms of network recovery and parameter estimation. We apply the proposed method to gene expression data generated by microfluidic Reverse Transcription quantitative Polymerase Chain Reaction technology in order to make inference on the regulatory mechanisms of blood development. A software implementation of our method is available on github (https://github.com/LuigiAugugliaro/cglasso).

Joint Microbial and Metabolomic Network Estimation with the Censored Gaussian Graphical Model

“…From a methodological perspective, metaMint estimates the correlations in a marginal manner, which may not be optimal because marginal approaches ignore the fact that the correlation matrix is positive semi-definite. Augugliaro et al (2018) proposed an approximated EM algorithm that jointly estimates all entries in the correlation matrix, however their method only works well under specific settings and there is a lack of theoretical understanding about the resulting esti-mator. Obvious but non-trivial extension is to explore computationally and statistically efficient alternatives that jointly estimate all entries in the correlation matrix.…”

Joint Microbial and Metabolomic Network Estimation with the Censored Gaussian Graphical Model

“…However, direct optimization of ( 3 ) is challenging due to the integral in ( 2 ) over a potentially high-dimensional space. Augugliaro et al [ 26 ] studied a general version of ( 3 ) where variables can be left and right censored. They proposed to use the EM algorithm to optimize the expectation of the full log-likelihood with respect to the conditional distribution X c | X o .…”

“…For example, Hoffman and Johnson [ 23 ], Pesonen et al [ 24 ] and Jones et al [ 25 ] studied covariance estimation for left censored multivariate normal distribution in the classic low-dimensional setting. Recently, Augugliaro et al [ 26 ] proposed an approximated EM algorithm for inverse covariance estimation in the high-dimensional setting and applied the method to single-cell data. The work by McDavid et al [ 27 ] was also motivated by single-cell data, but the authors proposed the zero-inflated Gaussian graphical model, which treats zeros as coming from a degenerate point mass at zero instead of being censored.…”

A Computationally Efficient Correlated Mixed Probit Model for Credit Risk Inference

“…Augugliaro et al . () showed how, for a similar model, this approximate information criterion behaves well when compared with that based on the full likelihood. In contrast, considering that prediction of default is a classification problem, other criteria can also be used that are more in line with this objective of the study.…”

“…Although this method is faster than direct estimation of the moments, it is still computationally very demanding for large-scale problems. A recent strand of literature has proposed approximating the first and second moments of a multivariate truncated normal distribution through an iterative procedure within the Mstep (Guo et al, 2015;Behrouzi and Wit, 2019;Augugliaro et al, 2018), leading to a computationally much faster approach than any previous methods. Exploiting this literature, we consider a mean field approximation of the second moments, namely, for i = j and for all r = 1, 2, : : : , R, E{.y Å ir − β x ir /.y Å jr − β x jr /|y r } ≈ E{.y Å ir − β x ir /|y r }E{.y Å jr − β x jr /|y r }:…”