glassopath {glasso}

Compute the Graphical lasso along a path

Description

Estimates a sparse inverse covariance matrix using a lasso (L1) penalty, along a path of values for the regularization parameter

Usage

glassopath(s, rholist=NULL, thr=1.0e-4, maxit=1e4,  approx=FALSE, 
penalize.diagonal=TRUE, w.init=NULL,wi.init=NULL, trace=1)

Arguments

s

Covariance matrix:p by p matrix (symmetric)

rholist

Vector of non-negative regularization parameters for the lasso. Should be increasing from smallest to largest; actual path is computed from largest to smallest value of rho). If NULL, 10 values in a (hopefully reasonable) range are used. Note that the same parameter rholist[j] is used for all entries of the inverse covariance matrix; different penalties for different entries are not allowed.

thr

Threshold for convergence. Default value is 1e-4. Iterations stop when average absolute parameter change is less than thr * ave(abs(offdiag(s)))

maxit

Maximum number of iterations of outer loop. Default 10,000

approx

Approximation flag: if true, computes Meinhausen-Buhlmann(2006) approximation

penalize.diagonal

Should diagonal of inverse covariance be penalized? Dafault TRUE.

w.init

Optional starting values for estimated covariance matrix (p by p). Only needed when start="warm" is specified

wi.init

Optional starting values for estimated inverse covariance matrix (p by p) Only needed when start="warm" is specified

trace

Flag for printing out information as iterations proceed. trace=0 means no printing; trace=1 means outer level printing; trace=2 means full printing Default FALSE

Details

Estimates a sparse inverse covariance matrix using a lasso (L1) penalty, along a path of regularization paramaters, using the approach of Friedman, Hastie and Tibshirani (2007). The Meinhausen-Buhlmann (2006) approximation is also implemented. The algorithm can also be used to estimate a graph with missing edges, by specifying which edges to omit in the zero argument, and setting rho=0. Or both fixed zeroes for some elements and regularization on the other elements can be specified.

This version 1.7 uses a block diagonal screening rule to speed up computations considerably. Details are given in the paper "New insights and fast computations for the graphical lasso" by Daniela Witten, Jerry Friedman, and Noah Simon, to appear in "Journal of Computational and Graphical Statistics". The idea is as follows: it is possible to quickly check whether the solution to the graphical lasso problem will be block diagonal, for a given value of the tuning parameter. If so, then one can simply apply the graphical lasso algorithm to each block separately, leading to massive speed improvements.

Value

A list with components

w

Estimated covariance matrices, an array of dimension (nrow(s),ncol(n), length(rholist))

wi

Estimated inverse covariance matrix, an array of dimension (nrow(s),ncol(n), length(rholist))

approx

Value of input argument approx

rholist

Values of regularization parameter used

errflag

values of error flag (0 means no memory allocation error)

References

Jerome Friedman, Trevor Hastie and Robert Tibshirani (2007). Sparse inverse covariance estimation with the lasso. Biostatistics 2007. http://www-stat.stanford.edu/~tibs/ftp/graph.pdf

Meinshausen, N. and Buhlmann, P.(2006) High dimensional graphs and variable selection with the lasso. Annals of Statistics,34, p1436-1462.

Daniela Witten, Jerome Friedman, Noah Simon (2011). New insights and fast computation for the graphical lasso. To appear in Journal of Computational and Graphical Statistics.

Examples

set.seed(100)

x<-matrix(rnorm(50*20),ncol=20)
s<- var(x)
a<-glassopath(s)