CovTools (version 0.4.0)

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)

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.

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.

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

# NOT RUN {
## generate data from multivariate normal with Identity precision.
data = mvtnorm::rmvnorm(100, sigma=diag(10))

## 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
par(mfrow=c(2,2), pty="s")
image(pracma::flipud(diag(10)),main="Original Precision")
image(pracma::flipud(out1$C), main="glasso::lambda=1.0")
image(pracma::flipud(out2$C), main="glasso::Confidence=0.95")
image(pracma::flipud(out3$C), main="glasso::BIC selection")

# }