CovTools (version 0.5.6)

PreEst.glasso: Precision Matrix Estimation via Graphical Lasso

Description

Given a sample covariance matrix \(S\), graphical lasso aims at estimating sparse precision matrix \(X\) - inverse of covariance. It solves a following optimization problem, $$\textrm{max}_X \log\textrm{det}X - <S,X> - \lambda \|X \|_1 \textrm{ such that } X \succ 0$$ where \(\lambda\) a regularization parameter, \(<S,X>=tr(S^T X)\) , \(\|X\|_1 = \sum X_{ij}\) and \(X\succ 0\) indicates positive definiteness. We provide three modes of computations, 'fixed','confidence', or 'BIC' with respect to \(\lambda\). Please see the section below for more details.

Usage

PreEst.glasso(X, method = list(type = "fixed", param = 1), parallel = FALSE)

Value

a named list containing:

C

a \((p\times p)\) estimated precision matrix.

BIC

a dataframe containing \(\lambda\) values and corresponding BIC scores with type='BIC' method.

Arguments

X

an \((n\times p)\) data matrix where each row is an observation.

method

a list containing following parameters,

type

one of 'fixed','confidence', or 'BIC'.

param

either a numeric value or vector of values.

parallel

a logical; TRUE for using half the cores available, FALSE otherwise.

regularization parameters

We currently provide three options for solving the problem, 'fixed','confidence', or 'BIC' with respect to \(\lambda\). When the method type is 'fixed', the parameter should be a single numeric value as a user-defined \(\lambda\) value. Likewise, method type of 'confidence' requires a singule numeric value in \((0,1)\), where the value is set heuristically according to $$ \rho = \frac{t_{n-2}(\gamma) \max S_{ii}S_{jj}}{\sqrt{n-2+ t_{n-2}^2(\gamma)}} $$ for a given confidence level \(\gamma \in (0,1)\) as proposed by Banerjee et al. (2006). Finally, 'BIC' type requires a vector of \(\lambda\) values and opts for a lambda value with the lowest BIC values as proposed by Yuan and Lin (2007).

References

banerjee_convex_2006CovTools

yuan_model_2007CovTools

friedman_sparse_2008CovTools

Examples

# \donttest{
## generate data from multivariate normal with Identity precision.
pdim = 10
data = matrix(rnorm(100*pdim), ncol=pdim)

## prepare input arguments for diefferent scenarios
lbdvec <- c(0.01,0.1,1,10,100)              # a vector of regularization parameters
list1 <- list(type="fixed",param=1.0)       # single regularization parameter case
list2 <- list(type="confidence",param=0.95) # single confidence level case
list3 <- list(type="BIC",param=lbdvec)      # multiple regularizers with BIC selection

## compute with different scenarios
out1 <- PreEst.glasso(data, method=list1)
out2 <- PreEst.glasso(data, method=list2)
out3 <- PreEst.glasso(data, method=list3)

## visualize
opar <- par(no.readonly=TRUE)
par(mfrow=c(2,2), pty="s")
image(diag(pdim)[,pdim:1], main="Original Precision")
image(out1$C[,pdim:1],     main="glasso::lambda=1.0")
image(out2$C[,pdim:1],     main="glasso::Confidence=0.95")
image(out3$C[,pdim:1],     main="glasso::BIC selection")
par(opar)
# }