flare (version 1.2.0)

sugm: High-deimensional Sparse Undirected Graphical Models.

Description

The function "sugm" estimates sparse undirected graphical models, i.e. Gaussian precision matrix, in high dimensions. We adopt two estimation procedures based on column by column regression scheme: (1) Tuning-Insensitive Graph Estimation and Regression based on square root Lasso (tiger); (2) The Constrained L1 Minimization for Sparse Precision Matrix Estimation using either L1 penalty (clime). The optimization algorithm for all three methods are implemented based on the alternating direction method of multipliers (ADMM) with the linearization method and multi-stage screening of variables. Missing values can be tolerated for CLIME in the data matrix. The computation is memory-optimized using the sparse matrix output.

Usage

sugm(data, lambda = NULL, nlambda = NULL, lambda.min.ratio = NULL, 
     rho = NULL, method = "tiger", sym = "or", shrink=NULL, 
     prec = 1e-4, max.ite = 1e4, standardize = FALSE, 
     perturb = TRUE, verbose = TRUE)

Arguments

data

There are 2 options for "clime": (1) data is an n by d data matrix (2) a d by d sample covariance matrix. The program automatically identifies the input matrix by checking the s

lambda

A sequence of decresing positive numbers to control the regularization. Typical usage is to leave the input lambda = NULL and have the program compute its own lambda sequence based on nlambda and lambda.min.rat

nlambda

The number of values used in lambda. Default value is 5.

lambda.min.ratio

The smallest value for lambda, as a fraction of the uppperbound (MAX) of the regularization parameter. The program can automatically generate lambda as a sequence of length = nlambda starting from

rho

Penalty parameter used in the optimization algorithm for clime. The default value is $\sqrt{d}$.

method

"tiger" is applied if method = "tiger" and "clime" is applied if method="clime". Default value is "tiger".

sym

Symmetrization of output graphs. If sym = "and", the edge between node i and node j is selected ONLY when both node i and node j are selected as neighbors for each other. If sym = "or"

shrink

Shrinkage of regularization parameter based on precision of estimation. The default value is 1.5 if method = "clime" and the default value is 0 if method="tiger".

prec

Stopping criterion. The default value is 1e-4.

max.ite

The iteration limit. The default value is 1e4.

standardize

Variables are standardized to have mean zero and unit standard deviation if standardize = TRUE. The default value is FALSE.

perturb

The diagonal of Sigma is added by a positive value to guarantee that Sigma is positive definite if perturb = TRUE. User can specify a numeric value for perturbe. The default value is TRUE.

verbose

Tracing information printing is disabled if verbose = FALSE. The default value is TRUE.

Value

• An object with S3 class "sugm" is returned:
• data
  The n by d data matrix or d by d sample covariance matrix from the input.
• cov.input
  An indicator of the sample covariance.
• lambda
  The sequence of regularization parameters lambda used in the program.
• nlambda
  The number of values used in lambda.
• icov
  A list of d by d precision matrices corresponding to regularization parameters.
• sym
  The sym from the input.
• method
  The method from the input.
• path
  A list of d by d adjacency matrices of estimated graphs as a graph path corresponding to lambda.
• sparsity
  The sparsity levels of the graph path.
• ite
  If method = "clime", it is a list of two matrices where ite[[1]] is the number of external iterations and ite[[2]] is the number of internal iterations with the entry of (i,j) as the number of iteration of i-th column and j-th lambda. If method="tiger", it is a matrix of iteration with the entry of (i,j) as the number of iteration of i-th column and j-th lambda.
• df
  It is a d by nlambda matrix. Each row contains the number of nonzero coefficients along the lasso solution path.
• standardize
  The standardize from the input.
• perturb
  The perturb from the input.
• verbose
  The verbose from the input.

Details

CLIME solves the following minimization problem $$\min || \Omega ||_1 \quad \textrm{s.t. } || S \Omega - I ||_\infty \le \lambda,$$ where $||\cdot||_1$ and $||\cdot||_\infty$ are element-wise 1-norm and $\infty$-norm respectively.

"tiger" solves the following minimization problem $$\min ||X-XB||_{2,1} + \lambda ||B||_1 \quad \textrm{s.t. } B_{jj} = 0,$$ where $||\cdot||_{1}$ and $||\cdot||_{2,1}$ are element-wise 1-norm and $L_{2,1}$-norm respectively.

References

1. T. Cai, W. Liu and X. Luo. A constrained L1 minimization approach to sparse precision matrix estimation. Journal of the American Statistical Association, 2011. 2. H. Liu, L. Wang. TIGER: A tuning-insensitive approach for optimally estimating large undirected graphs. Technical Report, 2012. 3. B. He and X. Yuan. On non-ergodic convergence rate of Douglas-Rachford alternating direction method of multipliers. Technical Report, 2012.

See Also

flare-package, sugm.generator, sugm.select, sugm.plot, sugm.roc, plot.sugm, plot.select, plot.roc, plot.sim, print.sugm, print.select, print.roc and print.sim.

Examples

## load package required
library(igraph)

## generating data
n = 100
d = 100
D = sugm.generator(n=n,d=d,graph="band",g=1)
plot(D)

## sparse precision matrix estimation with method "clime"
out1 = sugm(D$data, method = "clime")
plot(out1)
sugm.plot(out1$path[[4]])

## sparse precision matrix estimation with method "tiger"
out2 = sugm(D$data, method = "tiger")
plot(out2)
sugm.plot(out2$path[[4]])