hierband: Solves main optimization problem for fixed lambda value
In hierband: Convex Banding of the Covariance Matrix

Description Usage Arguments Value See Also Examples

Solves the main optimization problem appearing in Bien, Bunea, & Xiao (2015):

min_P || Sighat - P ||_F^2 + lam * sum_l || (W_l * P)_{g_l} ||_2

where g_l are the outermost l(l+1) elements of a square matrix. and || (W_l * P)_g_l ||^2 = sum_m<=l w_lm^2 ||P_s_m||^2. If a non-NULL delta is provided, then a constraint of the form $P >= delta I_p$ is included. Problem is solved by performing blockwise coordinate descent on the dual problem. See paper for more explanation.

1 2	hierband(Sighat, lam, w = NULL, delta = NULL, maxiter = 100, tol = 1e-07)

`Sighat`	The sample covariance matrix
`lam`	Non-negative penalty parameter. Controls sparsity level.
`w`	`(p-1)`-by-`(p-1)` lower-triangular matrix (above diagonal ignored). `w[l,]` gives the `l` weights for g_l. Defaults to `w[l,m]=sqrt(2 * l)/(l - m + 1)` for `m <= l`
`delta`	Lower bound on eigenvalues. If this is NULL (which is default), then no eigenvalue constraint is included.
`maxiter`	Number of iterations of blockwise coordinate descent to perform.
`tol`	Only used when `delta` is non-NULL. When no eigenvalue changes by more than `tol` in BCD, convergence is assumed.

Returns the convex banded estimate of covariance.

hierband.path hierband.cv

set.seed(123)
p <- 100
n <- 50
K <- 10
true <- ma(p, K)
x <- matrix(rnorm(n*p), n, p) %*% true$A
Sighat <- cov(x)
fit <- hierband(Sighat, lam=0.4)
min(eigen(fit)$values)
fit2 <- hierband(Sighat, lam=0.4, delta=0.2)
min(eigen(fit2)$values)
# Use cross validation to select lambda:
path <- hierband.path(Sighat)
cv <- hierband.cv(path, x)
fit <- hierband(Sighat, lam=cv$lam.best)