huge: High-dimensional undirected graph estimation

Description

The main function for high-dimensional undirected graph estimation. It provides a general interface to load huge.GECT(), huge.MBGEL() and huge.glassoM() for data analysis.

Usage

huge(L, lambda = NULL, nlambda = NULL, lambda.min.ratio = NULL, method = "MBGEL", scr = NULL, scr.num = NULL, cov.output = FALSE, sym = "or", verbose = TRUE)

Arguments

There are 2 options: (1) L is an n by d data matrix or a d by d sample covariance matrix. (2) L is a list containing L$data as an n by d d

lambda

A sequence of decresing positive numbers to control the regularization in Meinshausen & Buhlmann Graph Estimation via Lasso (MBGEL) and Graphical Lasso (GLASSO), or the thresholding in Graph Estimation via Correlation Thresholding (GECT). Typical usage is

nlambda

The number of regularization/thresholding paramters. The default value is 30 if method = "GECT" and 10 if method = "MBGEL" or method = "GLASSO".

lambda.min.ratio

If method = "MBGEL" or method = "GLASSO", it is the smallest value for lambda, as a fraction of the uppperbound (MAX) of the regularization/thresholding parameter which makes all estimates equal to

method

Graph estimation methods with 3 options: "MBGEL", "GECT" and "GLASSO". The defaulty value is "MBGEL".

scr

If scr = TRUE, Graph Sure Screening (GSS) is applied to preselect the neighborhood before MBGEL. The default value is TRUE for n and FALSE for n>=d. ONLY applicable when method = "MB

scr.num

The neighborhood size after the GSS (the number of remaining neighbors per node). ONLY applicable when scr = TRUE. The default value is n-1. An alternative value is n/log(n). ONLY applicable when scr = TRUE

cov.output

If cov.output = TRUE, the outpu will inlcude a path of estimated covariance matrices. ONLY applicable when method = "GLASSO". Since the estimated covariance matrices are generally not sparse, please use it with care, or it may ta

sym

Symmetrize the 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"

verbose

If verbose = FALSE, tracing information printing is disabled. The default value is TRUE.

Value

An object with S3 class "huge" is returned:
dataThe n by d data matrix or d by d sample covariance matrix from the input
cov.inputAn indicator of the sample covariance.
thetaThe true graph structure from the input. ONLY applicable when the input list L contains L$theta as the true graph structure.
ind.matThe scr.num by k matrix with each column correspondsing to a variable in ind.group and contains the indices of the remaining neighbors after the GSS. ONLY applicable when scr = TRUE and approx = FALSE
lambdaThe sequence of regularization parameters used in MBGEL or thresholding parameters in GECT.
symThe sym from the input. ONLY applicable when method = "MBGEL".
scrThe scr from the input. ONLY applicable when method = "MBGEL".
pathA list of k by k adjacency matrices of estimated graphs as a graph path corresponding to lambda.
sparsityThe sparsity levels of the graph path.
wiA list of d by d precision matrices as an alternative graph path (numerical path) corresponding to lambda. ONLY applicable when {method = "GLASSO"}
wA list of d by d estimated covariance matrices corresponding to lambda. ONLY applicable when cov.output = TRUE and {method = "GLASSO"}
methodThe method used in the graph estimation stage.
dfIf method = "MBGEL", it is a k by nlambda matrix. Each row corresponds to a variable in ind.group and contains the number of nonzero coefficients along the lasso solution path. If method = "GLASSO", it is a nlambda dimensional vector containing the number of nonzero coefficients along the graph path wi.
loglikA nlambda dimensional vector containing the likelihood scores along the graph path (wi). ONLY applicable when method = "GLASSO"

Details

The graph stucture is estimated using Meinshausen & Buhlmann Graph Estimation via Lasso (MBGEL) by default and it can be further accelerated by Graph SURE Screening (GSS) subroutine which preselects the graph neighborhood of each variable. In the case d >> n, the computation is memory optimized and is targeted on larger-scale problems. We also provide two alternative approaches for the graph estimation stage:(1) Graph Estimation via Correlation Thresholding (GECT) which is highly efficient and (2) A slightly modified Graphical Lasso (GLASSO) procedure in which the memory usage is optimized using sparse matrix output.

References

1.Tuo Zhao and Han Liu. HUGE: A Package for High-dimensional Undirected Graph Estimation. Technical Report, Carnegie Mellon University, 2010 2.Han Liu, John Lafferty and Larry Wasserman. The Nonparanormal: Semiparametric Estimation of High Dimensional Undirected Graphs. Journal of Machine Learning Research (JMLR), 2009 3.Jianqing Fan and Jinchi Lv. Sure independence screening for ultra-high dimensional feature space (with discussion). Journal of Royal Statistical Society B, 2008. 4.Jerome Friedman, Trevor Hastie and Rob Tibshirani. Regularization Paths for Generalized Linear Models via Coordinate Descent. Journal of Statistical Software, 2008. 5.Onureena Banerjee, Laurent El Ghaoui, Alexandre d'Aspremont: Model Selection Through Sparse Maximum Likelihood Estimation for Multivariate Gaussian or Binary Data. Journal of Machine Learning Research (JMLR), 2008. 6.Jerome Friedman, Trevor Hastie and Robert Tibshirani. Sparse inverse covariance estimation with the lasso. Biostatistics, 2007. 7.Nicolai Meinshausen and Peter Buhlmann. High-dimensional Graphs and Variable Selection with the Lasso. The Annals of Statistics, 2006.

Examples

Run this code

#generate data
L = huge.generator(n = 200, d = 80, graph = "hub")

#graph path estimation with input as a list
out1 = huge(L)
plot(out1)
plot(out1, align = TRUE)
huge.plot(out1$path[[3]])
plot(out1$lambda,out1$sparsity)

#graph path estimation using GECT
out2 = huge(L$data,method = "GECT")
plot(out2)

#graph path estimation using GLASSO
out3 = huge(L, method = "GLASSO")
plot(out3)

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