Penalized Precision Matrix Estimation via ADMM

Keywords
admm, covariance-matrix, glasso, lasso, precision-matrix, ridge
CNRI-Python-GPL-Compatible/CNRI-Python-GPL-Compatible

## Overview

`ADMMsigma` is an R package that estimates a penalized precision matrix via the alternating direction method of multipliers (ADMM) algorithm. It currently supports a general elastic-net penalty that allows for both ridge and lasso-type penalties as special cases.

A (possibly incomplete) list of functions contained in the package can be found below:

• `ADMMsigma()` computes the estimated precision matrix (ridge, lasso, and elastic-net type regularization optional)

• `RIDGEsigma()` computes the estimated ridge penalized precision matrix via closed-form solution

• `plot.ADMMsigma()` produces a heat map or line graph for cross validation errors

• `plot.RIDGEsigma()` produces a heat map or line graph for cross validation errors

See package website or manual.

## Installation

```# The easiest way to install is from CRAN

# You can also install the development version from GitHub:
# install.packages("devtools")

If there are any issues/bugs, please let me know: github. You can also contact me via my website. Pull requests are welcome!

## Usage

```library(ADMMsigma)

# generate data from a sparse oracle precision matrix
# we will try to estimate this matrix from the data

# first compute the oracle covariance matrix
S = matrix(0.7, nrow = 5, ncol = 5)
for (i in 1:5){
for (j in 1:5){
S[i, j] = S[i, j]^abs(i - j)
}
}

# print oracle precision matrix
# because its sparse, some shrinkage might be useful
(Omega = round(qr.solve(S), 3))```
``````##        [,1]   [,2]   [,3]   [,4]   [,5]
## [1,]  1.961 -1.373  0.000  0.000  0.000
## [2,] -1.373  2.922 -1.373  0.000  0.000
## [3,]  0.000 -1.373  2.922 -1.373  0.000
## [4,]  0.000  0.000 -1.373  2.922 -1.373
## [5,]  0.000  0.000  0.000 -1.373  1.961
``````
```# generate 100 x 5 data matrix with rows drawn from iid N_p(0, S)
set.seed(123)
Z = matrix(rnorm(100*5), nrow = 100, ncol = 5)
out = eigen(S, symmetric = TRUE)
S.sqrt = out\$vectors %*% diag(out\$values^0.5) %*% t(out\$vectors)
X = Z %*% S.sqrt

# print sample precision matrix from the data
# this is perhaps a bad estimate (its not sparse)
Sample = (nrow(X) - 1)/nrow(X)*cov(X)
round(qr.solve(Sample), 5)```
``````##          [,1]     [,2]     [,3]     [,4]     [,5]
## [1,]  2.32976 -1.55033  0.22105 -0.08607  0.24309
## [2,] -1.55033  3.27561 -1.68026 -0.14277  0.18949
## [3,]  0.22105 -1.68026  3.19897 -1.25158 -0.11016
## [4,] -0.08607 -0.14277 -1.25158  2.76790 -1.37226
## [5,]  0.24309  0.18949 -0.11016 -1.37226  2.05377
``````
```# now use ADMMsigma to provide estimates
# elastic-net type penalty (set tolerance to 1e-8)
ADMMsigma(X, tol.abs = 1e-8, tol.rel = 1e-8)```
``````##
## Call: ADMMsigma(X = X, tol.abs = 1e-08, tol.rel = 1e-08)
##
## Iterations: 48
##
## Tuning parameters:
##       log10(lam)  alpha
## [1,]      -1.599      1
##
## Log-likelihood: -108.41003
##
## Omega:
##          [,1]     [,2]     [,3]     [,4]     [,5]
## [1,]  2.15283 -1.26902  0.00000  0.00000  0.19765
## [2,] -1.26902  2.79032 -1.32206 -0.08056  0.00925
## [3,]  0.00000 -1.32206  2.85470 -1.17072 -0.00865
## [4,]  0.00000 -0.08056 -1.17072  2.49554 -1.18959
## [5,]  0.19765  0.00925 -0.00865 -1.18959  1.88121
``````
```# lasso penalty (default tolerance)
``````##
## Call: ADMMsigma(X = X, alpha = 1)
##
## Iterations: 24
##
## Tuning parameters:
##       log10(lam)  alpha
## [1,]      -1.599      1
##
## Log-likelihood: -108.41193
##
## Omega:
##          [,1]     [,2]     [,3]     [,4]     [,5]
## [1,]  2.15308 -1.26962  0.00000  0.00000  0.19733
## [2,] -1.26962  2.79103 -1.32199 -0.08135  0.00978
## [3,]  0.00000 -1.32199  2.85361 -1.16953 -0.00921
## [4,]  0.00000 -0.08135 -1.16953  2.49459 -1.18914
## [5,]  0.19733  0.00978 -0.00921 -1.18914  1.88096
``````
```# elastic-net penalty (alpha = 0.5)
``````##
## Call: ADMMsigma(X = X, alpha = 0.5)
##
## Iterations: 20
##
## Tuning parameters:
##       log10(lam)  alpha
## [1,]      -1.821    0.5
##
## Log-likelihood: -101.13591
##
## Omega:
##          [,1]     [,2]     [,3]     [,4]     [,5]
## [1,]  2.20055 -1.32510  0.01689 -0.00350  0.21805
## [2,] -1.32510  2.90739 -1.37666 -0.19054  0.13642
## [3,]  0.01689 -1.37666  2.92556 -1.12877 -0.12032
## [4,] -0.00350 -0.19054 -1.12877  2.56559 -1.23466
## [5,]  0.21805  0.13642 -0.12032 -1.23466  1.94525
``````
```# ridge penalty
``````##
## Call: ADMMsigma(X = X, alpha = 0)
##
## Iterations: 31
##
## Tuning parameters:
##       log10(lam)  alpha
## [1,]      -1.821      0
##
## Log-likelihood: -99.19745
##
## Omega:
##          [,1]     [,2]     [,3]     [,4]     [,5]
## [1,]  2.18987 -1.31545  0.04524 -0.04093  0.23512
## [2,] -1.31545  2.90045 -1.37070 -0.22626  0.17807
## [3,]  0.04524 -1.37070  2.89459 -1.07653 -0.17369
## [4,] -0.04093 -0.22626 -1.07653  2.55028 -1.22785
## [5,]  0.23512  0.17807 -0.17369 -1.22785  1.95494
``````
```# ridge penalty (using closed-form solution)
RIDGEsigma(X, lam = 10^seq(-8, 8, 0.01))```
``````##
## Call: RIDGEsigma(X = X, lam = 10^seq(-8, 8, 0.01))
##
## Tuning parameter:
##       log10(lam)    lam
## [1,]       -2.17  0.007
##
## Log-likelihood: -109.18156
##
## Omega:
##          [,1]     [,2]     [,3]     [,4]     [,5]
## [1,]  2.15416 -1.31185  0.08499 -0.05571  0.22862
## [2,] -1.31185  2.85605 -1.36677 -0.19650  0.16880
## [3,]  0.08499 -1.36677  2.82606 -1.06325 -0.14946
## [4,] -0.05571 -0.19650 -1.06325  2.50721 -1.21935
## [5,]  0.22862  0.16880 -0.14946 -1.21935  1.92871
``````
```# produce CV heat map for ADMMsigma
ADMM = ADMMsigma(X, lam = 10^seq(-5, 5, 0.1), alpha = seq(0, 1, 0.1))
```# produce line graph for CV errors for ADMMsigma
```# produce CV heat map for RIDGEsigma
```# produce line graph for CV errors for RIDGEsigma