Table 2:
(a) Gaussian and (b) Mixed Graphical Model estimation softwares as well as (c) recent extensions thereof. Abbreviations: AIC: Akaike information criterion, ANT: asymptotic normal thresholding, BIC: Bayesian Information Criterion, CPSS: complementary pairs stability selection, CV: cross-validation, EBIC: extended Bayesian Information Criterion, FDR: false discovery rate, mBIC: modified Bayesian Information Criterion, mVAR: mixed Vector Autoregressive, RIC: rotation information criterion, SCAD: smoothly clipped absolute deviation, STARS: stability approach to regularization selection, StEPS: Stable Edge-specific Penalty Selection.
| (a) Gaussian Graphical Models | ||||||
|---|---|---|---|---|---|---|
| Method name | Software name | Reference | Parameter estimation | Model selection | Features | Availability |
| Graphical Lasso | glasso | Friedman et al. (2008) | l1 penalized maximum likelihood inference of inverse covariance matrix | - | computationally efficient and sparse solution | R package https://CRAN.R-project.org/package=glasso |
| GGMselect | Giraud et al. (2009) | 6 different methods: C01 (Wille and Buhlmann, 2006); node-wise regression (Meinshausen et al., 2006); adaptive l1 penalty (Zou, 2006); combination of C01 and node-wise regression; combination of C01, node-wise regression, and adaptive l1 penalty; quasi-exhaustive combination of neighborhood selection with different parameter combination rules | minimization of penalized empirical risk (Giraud et al., 2008) | selection of penalization parameter(s) of any graph estimation procedure and comparison of any collection of estimation procedures possible | R package https://CRAN.R-project.org/package=GGMselect | |
| Sparse Partial Correlation Estimation | space | Peng et al. (2009) | joint sparse regression model to simultaneously perform neighborhood selection for all nodes | BIC-type criterion (Peng et al., 2009) | method specifically designed for p ≫ N scenario, particularly powerful for hub identification | R package https://CRAN.R-project.org/package=space |
| qgraph | Epskamp et al. (2012) | graphical LASSO | EBIC or local FDR | allows estimation of GGMs, graph visualization and analysis | R package https://CRAN.R-project.org/package=qgraph | |
| High-Dimensional Undirected Graph Estimation | huge | Zhao et al. (2015) | neighborhood selection (Meinshausen et al., 2006) or graphical LASSO, further acceleration by lossy screening rule preselecting neighborhood of each node via thresholding sample correlation | STARS (Liu et al., 2010), RIC, or EBIC for glasso | integrates data preprocessing, neighborhood screening, graph estimation, and model selection techniques into one pipeline | R package https://CRAN.R-project.org/package=huge |
| Covariance Shrinkage | GeneNet | Schaefer et al. (2015) | analytic shrinkage estimation of covariance and (partial) correlation matrices | parameter calibration according to (Ledoit and Wolf, 2003) and significance thresholding using the local FDR | very efficient, no parameter tuning, also suitable for dynamic (partial) correlations (Opgen-Rhein and Strimmer, 2006) | R package https://CRAN.R-project.org/package=GeneNet |
| XMRF | Wan et al. (2016) | neighborhood selection (Meinshausen et al., 2006) for GGMs | stability selection (Meinshausen and Bühlmann, 2010) and STARS (Liu et al., 2010) | allows estimation of GGMs, lsing models, and Poisson family graphical models | R package https://CRAN.R-project.org/package=XMRF | |
| FastGGM | Wang et al. (2016) | ANT algorithm (Ren et al., 2015) | - | efficient, tuning-free GGM estimation for large variable sets, supplies p-values and confidence intervals for estimated edges | R package http://www.pitt.edu/~wec47/fastGGM.html | |
| SILGGM | Zhang et al. (2018) | 4 different methods: ANT algorithm (Ren et al., 2015), de-sparsified node-wise scaled LASSO (Jankova and van de Geer, 2017), de-sparsified graphical LASSO (Jankova et al., 2015), and (scaled) LASSO GGM estimation with FDR control (Liu et al., 2013) | FDR multiple testing | provides confidence intervals, z-scores, and p-values for estimated edges, faster than FastGGM | R package https://CRAN.R-project.org/package=SILGGM | |
| GeNeCK | Zhang et al. (2019) | neighborhood selection, GeneNet, space, glasso, glasso-SF (Liu and Ihler, 2011), Bayesian-glasso (Wang et al., 2012), ESPACE, and EGLASSO for GGMs | p-value thresholding for ensemble-based network aggregation method (Zhong et al., 2014) | ensemble-based network aggregation method (Zhong et al., 2014) allows combination of networks reconstructed by different methods | web server http://lce.biohpc.swmed.edu/geneck/ | |
| (b) Mixed Graphical Models | ||||||
| Method name | Software name | Reference | Parameter estimation | Model selection | Features | Availability |
| Graphical Random Forests | (Fellinghauer et al., 2013) | individual nonlinear regressions with Random Forests | stability selection (Meinshausen and Bühlmann, 2010) | appropriate edge ranking among mixed data types based on Random Forest’s variable importance measure | R code https://ars.els-cdn.com/content/image/1-s2.0-S0167947313000789-mmc1.zip | |
| Chen et al. (2014) | node-wise penalized conditional likelihood | BIC | MGM estimation for Gaussian, Bernoulli, and Poisson variables | R code on github https://github.com/ChenShizhe/MixedGraphicalModels | ||
| Lee and Hastie (2015) | maximum pseudo-log-likelihood with calibrated weighting scheme for penalization | MGM estimation for p ≫ N scenario with individually weighted penalization for each edge type | Matlab code https://jasondlee88.github.io/learningmgm.html | |||
| mgm | Haslbeck and Waldorp (2016) | node-wise neighborhood selection by penalized (default: l1, also supports elastic net penalty (Zou and Hastie, 2005)) multinomial logistic regression in case of discrete response node and linear regression in case of Gaussian response node | EBIC or CV | estimation of k-order MGM and mVAR models in high-dimensional data, Gaussian, categorical, and Poisson data, also time-varying MGMs and mVAR models, allows to compute predictions and node-wise errors from these models and to assess model stability via resampling | R package https://CRAN.R-project.org/package=mgm | |
| (c) Extensions of GGMs and MGMs | ||||||
| Method name | Software name | Reference | Parameter estimation | Model selection | Features | Availability |
| Sparse Time Series Chain Graphical Models | SparseTSCGM | Abegaz and Wit (2013) | penalized maximum likelihood inference with SCAD penalty | BIC or CV | estimation of time series chain graphical models | R package https://CRAN.R-project.org/package=SparseTSCGM |
| Sparse Inverse Covariance Estimation for Ecological Association Inference | SpiecEasi | Kurtz et al. (2015) | neighborhood selection or glasso | STARS | GGM estimation for compositional data | R package https://github.com/zdkl23/SpiecEasi |
| prior Lasso | pLasso | Wang et al. (2013) | neighborhood selection | mBIC or pBIC (Wang et al., 2013) | incorporation of prior knowledge in GGM estimation | Matlab code https://nba.uth.tmc.edu/homepage/liu/pLasso/ |
| weighted graphical lasso | wglasso | Li and Jackson (2015) | graphical Lasso | BIC | incorporation of prior knowledge in GGM estimation | R code on github https://github.com/bioops/wglasso |
| differentially weighted graphical lasso | dwglasso | Zuo et al. (2017) | glasso | CV | wglasso for two groups and subsequent differential network score calculation for each variable | R code on github https://github.com/Hurricanerl989/dwgLASS0-R-codes |
| ESPACE/EGLASSO | espace | Yu et al. (2017) | extension of SPACE/graphical Lasso with additional tuning parameter to individually change penalization of hub gene edges | GIC (Yu et al., 2015) | allows incorporation of prior biological knowledge about hub genes to improve model estimation | R package for ESPACE https://sites.google.com/site/dhyeonyu/software |
| Joint Graphical Lasso | JGL | Danaher et al. (2014) | graphical Lasso with two penalty functions: Fused Graphical Lasso (FGL), employs fused penalty to encourage inverse covariance matrices to be similar across classes, and Group Graphical Lasso (GGL), which encourages similar network structure between classes | AIC | jointly estimates multiple graphical models corresponding to distinct but related conditions (multi-class GGMs) | R package https://CRAN.R-project.org/package=JGL |
| CausalMGM | Sedgewick et al. (2018) | penalized maximum pseudo-likelihood method of (Lee and Hastie, 2015) with different sparsity penalties for each edge type (Sedgewick et al., 2016), PC- and CPC-algorithm (Colombo and Maathuis, 2014) for directionality search | StEPS (Sedgewick et al., 2016) and CPSS (Shah and Sam worth, 2013) | estimation of both undirected and directed MGMs | R package https://CRAN.R-project.org/package=causalMGM | |