A note on the lack of symmetry in the graphical lasso

Rolfs, Benjamin T.; Rajaratnam, Bala

doi:10.1016/j.csda.2012.07.013

Cited by 5 publications

(4 citation statements)

References 7 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We define the error matrix

\Delta

\hat{\Theta}-\Theta

, and refer to its

i,{j}^{th}

element as

{\delta}_{ij}

. Although the true precision matrix is symmetric, the estimated matrix may not be; a notable example of possible asymmetry is in the graphical lasso algorithm 34 . Therefore, we considered every element of the error matrix

\Delta

rather than just upper or lower triangular components.…”

Section: Simulationmentioning

confidence: 99%

See 1 more Smart Citation

SpiderLearner: An ensemble approach to Gaussian graphical model estimation

Shutta

Balzer

Scholtens

et al. 2023

Statistics in Medicine

View full text Add to dashboard Cite

Gaussian graphical models (GGMs) are a popular form of network model in which nodes represent features in multivariate normal data and edges reflect conditional dependencies between these features. GGM estimation is an active area of research. Currently available tools for GGM estimation require investigators to make several choices regarding algorithms, scoring criteria, and tuning parameters. An estimated GGM may be highly sensitive to these choices, and the accuracy of each method can vary based on structural characteristics of the network such as topology, degree distribution, and density. Because these characteristics are a priori unknown, it is not straightforward to establish universal guidelines for choosing a GGM estimation method. We address this problem by introducing SpiderLearner, an ensemble method that constructs a consensus network from multiple estimated GGMs. Given a set of candidate methods, SpiderLearner estimates the optimal convex combination of results from each method using a likelihood-based loss function. K-fold cross-validation is applied in this process, reducing the risk of overfitting. In simulations, SpiderLearner performs better than or comparably to the best candidate methods according to a variety of metrics, including relative Frobenius norm and out-of-sample likelihood. We apply SpiderLearner to publicly available ovarian cancer gene expression data including 2013 participants from 13 diverse studies, demonstrating our tool's potential to identify biomarkers of complex disease. SpiderLearner is implemented as flexible, extensible, open-source code in the R package ensembleGGM at https://github.com/katehoffshutta/ensembleGGM.

show abstract

“…We define the error matrix

\Delta

\hat{\Theta}-\Theta

, and refer to its

i,{j}^{th}

element as

{\delta}_{ij}

\Delta

rather than just upper or lower triangular components.…”

Section: Simulationmentioning

confidence: 99%

“…Although the true precision matrix is symmetric, the estimated matrix may not be; a notable example of possible asymmetry is in the graphical lasso algorithm. 34 Therefore, we considered every element of the error matrix Δ rather than just upper or lower triangular components.…”

Section: Design Of Simulation Studymentioning

confidence: 99%

SpiderLearner: An ensemble approach to Gaussian graphical model estimation

Shutta

Balzer

Scholtens

et al. 2023

Statistics in Medicine

View full text Add to dashboard Cite

show abstract

“…Note that, although the true precision matrix is symmetric, the estimated matrix may not be: a notable example of possible asymmetry is in the graphical lasso algorithm (44). Therefore, we consider every element of the error matrix ∆ rather than just upper or lower triangular components when assessing estimation performance.…”

Section: Assessing Estimation Performancementioning

confidence: 99%

SpiderLearner: An ensemble approach to Gaussian graphical model estimation

et al. 2021

Preprint

View full text Add to dashboard Cite

Multivariate biological data are often modeled using networks in which nodes represent a biological variable (e.g., genes) and edges represent associations (e.g., coexpression). A Gaussian graphical model (GGM), or partial correlation network, is an undirected graphical model in which a weighted edge between two nodes represents the magnitude of their partial correlation, and the absence of an edge indicates zero partial correlation. A GGM provides a roadmap of direct dependencies between variables, providing a valuable systems-level perspective. Many methods exist for estimating GGMs; estimated GGMs are typically highly sensitive to choice of method, posing an outstanding statistical challenge. We address this challenge by developing SpiderLearner, a tool that combines a range of candidate GGM estimation methods to construct an ensemble estimate as a weighted average of results from each candidate. In simulation studies, SpiderLearner performs better than or comparably to the best of the candidate methods. We apply SpiderLearner to estimate a GGM for gene expression in a publicly available dataset of 260 ovarian cancer patients. Using the community structure of the GGM, we develop a network-based risk score which we validate in six independent datasets. The risk score requires only seven genes, each of which has important biological function. Our method is flexible, extensible, and has demonstrated potential to identify de novo biomarkers for complex diseases. An open-source implementation of our method is available at https://github.com/katehoffshutta/SpiderLearner.

show abstract

“…We note that the penalty ( 4 ) is slightly different from original definition 30 , expressed as When we do not assume that , the estimate of the inverse covariance matrix with ( 5 ) is not symmetric. Since the original graphical lasso algorithm does not assume that 31 , 34 , we slightly modify the penalty as in Eq. ( 4 ).…”

Section: Simulation Frameworkmentioning

confidence: 99%

Relationship between gene regulation network structure and prediction accuracy in high dimensional regression

Okinaga

Kyogoku

Kondo

et al. 2021

Sci Rep

View full text Add to dashboard Cite

The least absolute shrinkage and selection operator (lasso) and principal component regression (PCR) are popular methods of estimating traits from high-dimensional omics data, such as transcriptomes. The prediction accuracy of these estimation methods is highly dependent on the covariance structure, which is characterized by gene regulation networks. However, the manner in which the structure of a gene regulation network together with the sample size affects prediction accuracy has not yet been sufficiently investigated. In this study, Monte Carlo simulations are conducted to investigate the prediction accuracy for several network structures under various sample sizes. When the gene regulation network is a random graph, a sufficiently large number of observations are required to ensure good prediction accuracy with the lasso. The PCR provided poor prediction accuracy regardless of the sample size. However, a real gene regulation network is likely to exhibit a scale-free structure. In such cases, the simulation indicates that a relatively small number of observations, such as $$N=300$$ N = 300 , is sufficient to allow the accurate prediction of traits from a transcriptome with the lasso.

show abstract

A note on the lack of symmetry in the graphical lasso

Cited by 5 publications

References 7 publications

SpiderLearner: An ensemble approach to Gaussian graphical model estimation

SpiderLearner: An ensemble approach to Gaussian graphical model estimation

SpiderLearner: An ensemble approach to Gaussian graphical model estimation

Relationship between gene regulation network structure and prediction accuracy in high dimensional regression

Contact Info

Product

Resources

About