Package ‘glasso’
October 1, 2019
Title Graphical Lasso: Estimation of Gaussian Graphical Models
Version 1.11
Author Jerome Friedman, Trevor Hastie and Rob Tibshirani
Description Estimation of a sparse inverse covariance matrix using a lasso (L1)
penalty. Facilities are provided for estimates along a path of values
for the regularization parameter.
Maintainer Rob Tibshirani <tibs@stat.stanford.edu>
License GPL-2
URL http://www-stat.stanford.edu/~tibs/glasso
Repository CRAN
Date/Publication 2019-10-01 18:40:07 UTC
NeedsCompilation yes
R topics documented:
glasso . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
glassopath . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Index
glasso
1
4
6
Graphical lasso
Description
Estimates a sparse inverse covariance matrix using a lasso (L1) penalty
Usage
glasso(s, rho, nobs=NULL, zero=NULL, thr=1.0e-4, maxit=1e4,
penalize.diagonal=TRUE, start=c("cold","warm"),
w.init=NULL,wi.init=NULL, trace=FALSE)
1
approx=FALSE,
2
glasso
Arguments
s
Covariance matrix:p by p matrix (symmetric)
rho
(Non-negative) regularization parameter for lasso. rho=0 means no regularization. Can be a scalar (usual) or a symmetric p by p matrix, or a vector of length
p. In the latter case, the penalty matrix has jkth element sqrt(rho[j]*rho[k]).
nobs
Number of observations used in computation of the covariance matrix s. This
quantity is need to compute the value of log-likelihood. If not specified, loglik
will be returned as NA.
zero
(Optional) indices of entries of inverse covariance to be constrained to be zero.
The input should be a matrix with two columns, each row indicating the indices
of elements to be constrained to be zero. The solution must be symmetric, so
you need only specify one of (j,k) and (k,j). An entry in the zero matrix overrides
any entry in the rho matrix for a given element.
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
Approximation flag: if true, computes Meinhausen-Buhlmann(2006) approximation
penalize.diagonal
Should diagonal of inverse covariance be penalized? Dafault TRUE.
approx
start
Type of start. Cold start is default. Using Warm start, can provide starting values
for w and wi
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. Default FALSE
Details
Estimates a sparse inverse covariance matrix using a lasso (L1) penalty, 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.
3
glasso
Value
A list with components
w
Estimated covariance matrix
wi
Estimated inverse covariance matrix
loglik
Value of maximized log-likelihodo+penalty
errflag
Memory allocation error flag: 0 means no error; !=0 means memory allocation
error - no output returned
approx
Value of input argument approx
del
Change in parameter value at convergence
niter
Number of iterations of outer loop used by algorithm
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, and Noah Simon (2011). New insights and faster computations
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<-glasso(s, rho=.01)
aa<-glasso(s,rho=.02, w.init=a$w, wi.init=a$wi)
# example with structural zeros and no regularization,
# from Whittaker's Graphical models book page xxx.
s=c(10,1,5,4,10,2,6,10,3,10)
S=matrix(0,nrow=4,ncol=4)
S[row(S)>=col(S)]=s
S=(S+t(S))
diag(S)<-10
zero<-matrix(c(1,3,2,4),ncol=2,byrow=TRUE)
a<-glasso(S,0,zero=zero)
4
glassopath
glassopath
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
Approximation flag: if true, computes Meinhausen-Buhlmann(2006) approximation
penalize.diagonal
Should diagonal of inverse covariance be penalized? Dafault TRUE.
approx
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 MeinhausenBuhlmann (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.
5
glassopath
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)
Index
∗Topic graphs
glasso, 1
glassopath, 4
∗Topic models
glasso, 1
glassopath, 4
∗Topic multivariate
glasso, 1
glassopath, 4
glasso, 1
glassopath, 4
6
Jerome Friedman