grasps

Groupwise Regularized Adaptive Sparse Precision Solution

The goal of grasps is to provide a collection of statistical methods that incorporate both element-wise and group-wise penalties to estimate a precision matrix, making them user-friendly and useful for researchers and practitioners.

$$\hat{\Omega}(\lambda,\alpha,\gamma) = {\arg\min}_{\Omega \succ 0} { -\log\det(\Omega) + \text{tr}(S\Omega) + \lambda P_{\alpha,\gamma}(\Omega) },$$

$$P_{\alpha,\gamma}(\Omega) = \alpha P^\text{idv}_\gamma(\Omega) + (1-\alpha) P^\text{grp}_\gamma(\Omega),$$

$$P^\text{idv}_\gamma(\Omega) = \sum_{i,j} p_\gamma(\vert\omega_{ij}\vert),$$

$$P^\text{grp}_\gamma(\Omega) = \sum_{g,g^\prime} p_\gamma(\Vert\Omega_{gg^\prime}\Vert_F).$$

For more details, see the vignette Penalized Precision Matrix Estimation in grasps.

Penalties

The package grasps provides functions to estimate precision matrices using the following penalties:

Penalty	Reference
Lasso (penalty = "lasso")	Tibshirani (1996); Friedman et al. (2008)
Adaptive lasso (penalty = "adapt")	Zou (2006); Fan et al. (2009)
Atan (penalty = "atan")	Wang and Zhu (2016)
Exp (penalty = "exp")	Wang et al. (2018)
Lq (penalty = "lq")	Frank and Friedman (1993); Fu (1998); Fan and Li (2001)
LSP (penalty = "lsp")	Candès et al. (2008)
MCP (penalty = "mcp")	Zhang (2010)
SCAD (penalty = "scad")	Fan and Li (2001); Fan et al. (2009)

See the vignette Penalized Precision Matrix Estimation in grasps for more details.

Installation

You can install the development version of grasps from GitHub with:

# install.packages("devtools")
devtools::install_github("Carol-seven/grasps")

Example

library(grasps)

## reproducibility for everything
set.seed(1234)

## block-structured precision matrix based on SBM
sim <- gen_prec_sbm(d = 30, K = 3,
                    within.prob = 0.25, between.prob = 0.05,
                    weight.dists = list("gamma", "unif"),
                    weight.paras = list(c(shape = 20, rate = 10),
                                        c(min = 0, max = 5)),
                    cond.target = 100)

## synthetic data
library(MASS)
X <- mvrnorm(n = 20, mu = rep(0, 30), Sigma = sim$Sigma)

## solution
res <- grasps(X = X, membership = sim$membership, penalty = "adapt", crit = "HBIC")

## visualization
plot(res)

## performance
performance(hatOmega = res$hatOmega, Omega = sim$Omega)
#>      measure    value
#> 1   sparsity   0.9103
#> 2  Frobenius  24.6796
#> 3         KL   7.2063
#> 4  quadratic  54.1949
#> 5   spectral  13.1336
#> 6         TP  22.0000
#> 7         TN 370.0000
#> 8         FP  17.0000
#> 9         FN  26.0000
#> 10       TPR   0.4583
#> 11       FPR   0.0439
#> 12        F1   0.5057
#> 13       MCC   0.4545

Reference

Candès, Emmanuel J., Michael B. Wakin, and Stephen P. Boyd. 2008. “Enhancing Sparsity by Reweighted $\ell_1$ Minimization.” Journal of Fourier Analysis and Applications 14 (5): 877–905. https://doi.org/10.1007/s00041-008-9045-x.

Fan, Jianqing, Yang Feng, and Yichao Wu. 2009. “Network Exploration via the Adaptive LASSO and SCAD Penalties.” The Annals of Applied Statistics 3 (2): 521–41. https://doi.org/10.1214/08-aoas215.

Fan, Jianqing, and Runze Li. 2001. “Variable Selection via Nonconcave Penalized Likelihood and Its Oracle Properties.” Journal of the American Statistical Association 96 (456): 1348–60. https://doi.org/10.1198/016214501753382273.

Frank, Lldiko E., and Jerome H. Friedman. 1993. “A Statistical View of Some Chemometrics Regression Tools.” Technometrics 35 (2): 109–35. https://doi.org/10.1080/00401706.1993.10485033.

Friedman, Jerome, Trevor Hastie, and Robert Tibshirani. 2008. “Sparse Inverse Covariance Estimation with the Graphical Lasso.” Biostatistics 9 (3): 432–41. https://doi.org/10.1093/biostatistics/kxm045.

Fu, Wenjiang J. 1998. “Penalized Regressions: The Bridge Versus the Lasso.” Journal of Computational and Graphical Statistics 7 (3): 397–416. https://doi.org/10.1080/10618600.1998.10474784.

Tibshirani, Robert. 1996. “Regression Shrinkage and Selection via the Lasso.” Journal of the Royal Statistical Society: Series B (Methodological) 58 (1): 267–88. https://doi.org/10.1111/j.2517-6161.1996.tb02080.x.

Wang, Yanxin, Qibin Fan, and Li Zhu. 2018. “Variable Selection and Estimation Using a Continuous Approximation to the $L_0$ Penalty.” Annals of the Institute of Statistical Mathematics 70 (1): 191–214. https://doi.org/10.1007/s10463-016-0588-3.

Wang, Yanxin, and Li Zhu. 2016. “Variable Selection and Parameter Estimation with the Atan Regularization Method.” Journal of Probability and Statistics 2016: 6495417. https://doi.org/10.1155/2016/6495417.

Zhang, Cun-Hui. 2010. “Nearly Unbiased Variable Selection Under Minimax Concave Penalty.” The Annals of Statistics 38 (2): 894–942. https://doi.org/10.1214/09-AOS729.

Zou, Hui. 2006. “The Adaptive Lasso and Its Oracle Properties.” Journal of the American Statistical Association 101 (476): 1418–29. https://doi.org/10.1198/016214506000000735.