Sparse Covariance Estimation

Share:

Description

Provides a sparse and positive definite estimate of a covariance matrix. This function performs the majorize-minimize algorithm described in Bien & Tibshirani 2011 (see full reference below).

Usage

1
2
3
spcov(Sigma, S, lambda, step.size, nesterov = TRUE, n.outer.steps =
10000, n.inner.steps = 10000, tol.outer = 1e-04, thr.inner = 0.01,
backtracking = 0.2, trace = 0)

Arguments

Sigma

an initial guess for Sigma (suggestions: S or diag(diag(S))).

S

the empirical covariance matrix of the data. Must be positive definite (if it is not, add a small constant to the diagonal).

lambda

penalty parameter. Either a scalar or a matrix of the same dimension as Sigma. This latter choice should be used to penalize only off-diagonal elements. All elements of lambda must be non-negative.

step.size

the step size to use in generalized gradient descent. Affects speed of algorithm.

nesterov

indicates whether to use Nesterov's modification of generalized gradient descent. Default: TRUE.

n.outer.steps

maximum number of majorize-minimize steps to take (recall that MM is the outer loop).

n.inner.steps

maximum number of generalized gradient steps to take (recall that generalized gradient descent is the inner loop).

tol.outer

convergence threshold for outer (MM) loop. Stops when drop in objective between steps is less than tol.outer.

thr.inner

convergence threshold for inner (i.e. generalized gradient) loop. Stops when mean absolute change in Sigma is less than thr.inner * mean(abs(S)).

backtracking

if FALSE, then fixed step size used. If numeric and in (0,1), this is the parameter of backtracking that multiplies step.size on each step. Usually, in range of (0.1, 0.8). Default: 0.2.

trace

controls how verbose output should be.

Details

This is the R implementation of Algorithm 1 in Bien, J., and Tibshirani, R. (2011), "Sparse Estimation of a Covariance Matrix," Biometrika. 98(4). 807–820. The goal is to approximately minimize (over Sigma) the following non-convex optimization problem:

minimize logdet(Sigma) + trace(S Sigma^-1) + || lambda*Sigma ||_1 subject to Sigma positive definite.

Here, the L1 norm and matrix multiplication between lambda and Sigma are elementwise. The empirical covariance matrix must be positive definite for the optimization problem to have bounded objective (see Section 3.3 of paper). We suggest adding a small constant to the diagonal of S if it is not. Since the above problem is not convex, the returned matrix is not guaranteed to be a global minimum of the problem.

In Section 3.2 of the paper, we mention a simple modification of gradient descent due to Nesterov. The argument nesterov controls whether to use this modification (we suggest that it be used). We also strongly recommend using backtracking. This allows the algorithm to begin by taking large steps (the initial size is determined by the argument step.size) and then to gradually reduce the size of steps.

At the start of the algorithm, a lower bound (delta in the paper) on the eigenvalues of the solution is calculated. As shown in Equation (3) of the paper, the prox function for our generalized gradient descent amounts to minimizing (over a matrix X) a problem of the form

minimize (1/2)|| X-A ||_F^2 + || lambda*X ||_1 subject to X >= delta I

This is implemented using an alternating direction method of multipliers approach given in Appendix 3.

Value

Sigma

the sparse covariance estimate

n.iter

a vector giving the number of generalized gradient steps taken on each step of the MM algorithm

obj

a vector giving the objective values after each step of the MM algorithm

Author(s)

Jacob Bien and Rob Tibshirani

References

Bien, J., and Tibshirani, R. (2011), "Sparse Estimation of a Covariance Matrix," Biometrika. 98(4). 807–820.

See Also

ProxADMM

Examples

 1
 2
 3
 4
 5
 6
 7
 8
 9
10
11
12
13
14
15
16
17
18
19
20
21
22
set.seed(1)
n <- 100
p <- 20
# generate a covariance matrix:
model <- GenerateCliquesCovariance(ncliques=4, cliquesize=p / 4, 1)

# generate data matrix with x[i, ] ~ N(0, model$Sigma):
x <- matrix(rnorm(n * p), ncol=p) %*% model$A
S <- var(x)

# compute sparse, positive covariance estimator:
step.size <- 100
tol <- 1e-3
P <- matrix(1, p, p)
diag(P) <- 0
lam <- 0.06
mm <- spcov(Sigma=S, S=S, lambda=lam * P,
            step.size=step.size, n.inner.steps=200,
            thr.inner=0, tol.outer=tol, trace=1)
sqrt(mean((mm$Sigma - model$Sigma)^2))
sqrt(mean((S - model$Sigma)^2))
## Not run: image(mm$Sigma!=0)