Package 'varband'

Generate an autoregressive model.

Description

Generate lower triangular matrix with strict bandwidth. See, e.g., Model 1 in the paper.

Usage

ar_gen(p, phi_vec)

Arguments

p	the dimension of L
phi_vec	a K-dimensional vector for off-diagonal values

Value

a p-by-p strictly banded lower triangular matrix

Examples

true_ar <- ar_gen(p = 50, phi = c(0.5, -0.4, 0.1))

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Generate a model with block-diagonal structure

Description

Generate a model with block-diagonal structure

Usage

block_diag_gen(p)

Arguments

p	the dimension of L

Value

a p-by-p lower triangular matrix with block-diagonal structure from p/4-th row to 3p/4-th row

Examples

set.seed(123)
true_L_block_diag <- block_diag_gen(p = 50)

---

Plot the sparsity pattern of a square matrix

Description

Black, white and gray stand for positive, zero and negative respectively

Usage

matimage(Mat, main = NULL)

Arguments

Mat	A matrix to plot.
main	A plot title.

Examples

set.seed(123)
p <- 50
n <- 50
phi <- 0.4
true <- varband_gen(p = p, block = 5)
matimage(true)

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Generate random samples.

Description

Generate n random samples from multivariate Gaussian distribution N(0, (L^TL)^-1)

Usage

sample_gen(L, n)

Arguments

L	p-dimensional inverse Cholesky factor of true covariance matrix.
n	number of samples to generate.

Value

returns a n-by-p matrix with each row a random sample generated.

Examples

set.seed(123)
true <- varband_gen(p = 50, block = 5)
x <- sample_gen(L = true, n = 100)

---

Compute the varband estimate for a fixed tuning parameter value with different penalty options.

Description

Solves the main optimization problem in Yu & Bien (2017):

$min_L -2 \sum_{r=1}^p L_{rr} + tr(SLL^T) + lam * \sum_{r=2}^p P_r(L_{r.})$

where

$P_r(L_{r.}) = \sum_{\ell = 2}^{r-1} \left(\sum_{m=1}^\ell w_{\ell m}^2 L_{rm}^2\right)^{1/2}$

or

$P_r(L_{r.}) = \sum_{\ell = 1}^{r-1} |L_{r\ell}|$

Usage

varband(S, lambda, init, K = -1L, w = FALSE, lasso = FALSE)

Arguments

S	The sample covariance matrix
lambda	Non-negative tuning parameter. Controls sparsity level.
init	Initial estimate of L. Default is a closed-form diagonal estimate of L.
K	Integer between 0 and p - 1 (default), indicating the maximum bandwidth in the resulting estimate. A small value of K will result in a sparse estimate and small computing time.
w	Logical. Should we use weighted version of the penalty or not? If TRUE, we use general weight. If FALSE, use unweighted penalty. Default is FALSE.
lasso	Logical. Should we use l1 penalty instead of hierarchical group lasso penalty? Note that by using l1 penalty, we lose the banded structure in the resulting estimate. Default is FALSE.

Details

The function decomposes into p independent row problems, each of which is solved by an ADMM algorithm. see paper for more explanation.

Value

Returns the variable banding estimate of L, at most K banded, where L^TL = Omega.

See Also

varband_path varband_cv

Examples

set.seed(123)
n <- 50
true <- varband_gen(p = 50, block = 5)
x <- sample_gen(L = true, n = n)
S <- crossprod(scale(x, center = TRUE, scale = FALSE)) / n
init <- diag(1/sqrt(diag(S)))
# unweighted estimate
L_unweighted <- varband(S, lambda = 0.1, init, w = FALSE)
# at most 10-banded unweighted estimate
L_unweighted_K10 <- varband(S, lambda = 0.1, init, w = FALSE, K = 10)
# weighted estimate
L_weighted <- varband(S, lambda = 0.1, init, w = TRUE)
# lasso estimate
L_lasso <- varband(S, lambda = 0.1, init, w = TRUE, lasso = TRUE)

---

Perform nfolds-cross validation

Description

Select tuning parameter by cross validation according to the likelihood on testing data, with and without refitting.

Usage

varband_cv(x, w = FALSE, lasso = FALSE, K = -1, lamlist = NULL,
  nlam = 60, flmin = 0.01, folds = NULL, nfolds = 5)

Arguments

x	A n-by-p sample matrix, each row is an observation of the p-dim random vector.
w	Logical. Should we use weighted version of the penalty or not? If TRUE, we use general weight. If FALSE, use unweighted penalty. Default is FALSE.
lasso	Logical. Should we use l1 penalty instead of hierarchical group lasso penalty? Note that by using l1 penalty, we lose the banded structure in the resulting estimate. And when using l1 penalty, the becomes CSCS (Convex Sparse Cholesky Selection) introduced in Khare et al. (2016). Default value for lasso is FALSE.
K	Integer between 0 and p - 1 (default), indicating the maximum bandwidth in the resulting estimate. A small value of K will result in a sparse estimate and small computing time.
lamlist	A list of non-negative tuning parameters lambda.
nlam	If lamlist is not provided, create a lamlist with length nulam. Default is 60.
flmin	If lamlist is not provided, create a lamlist with ratio of the smallest and largest lambda in the list equal to flmin. Default is 0.01.
folds	Folds used in cross-validation
nfolds	If folds are not provided, create folds of size nfolds.

Value

A list object containing

errs_fit:

A nlam-by-nfolds matrix of negative Gaussian log-likelihood values on the CV test data sets. errs[i,j] is negative Gaussian log-likelihood values incurred in using lamlist[i] on fold j

.

errs_refit:

A nlam-by-nfolds matrix of negative Gaussian log-likelihood values of the refitting.

folds:

Folds used in cross validation.

lamlist:

lambda grid used in cross validation.

ibest_fit:

index of lamlist minimizing CV negative Gaussian log-likelihood.

ibest_refit:

index of lamlist minimizing refitting CV negative Gaussian log-likelihood.

i1se_fit:

Selected value of lambda using the one-standard-error rule.

i1se_refit:

Selected value of lambda of the refitting process using the one-standard-error rule.

L_fit:

Estimate of L corresponding to ibest_fit.

L_refit:

Refitted estimate of L corresponding to ibest_refit.

See Also

varband varband_path

Examples

set.seed(123)
p <- 50
n <- 50
true <- varband_gen(p = p, block = 5)
x <- sample_gen(L = true, n = n)
res_cv <- varband_cv(x = x, w = FALSE, nlam = 40, flmin = 0.03)

---

Generate a model with variable bandwidth.

Description

Generate lower triangular matrix with variable bandwidth. See, e.g., Model 2 and 3 in the paper.

Usage

varband_gen(p, block = 10)

Arguments

p	the dimension of L
block	the number of block diagonal structures in the resulting model, assumed to divide p

Value

a p-by-p lower triangular matrix with variable bandwidth

Examples

set.seed(123)
# small block size (big number of blocks)
true_small <- varband_gen(p = 50, block = 10)
# large block size (small number of blocks)
true_large <- varband_gen(p = 50, block = 2)

---

Solve main optimization problem along a path of lambda

Description

Compute the varband estimates along a path of tuning parameter values.

Usage

varband_path(S, w = FALSE, lasso = FALSE, K = -1, lamlist = NULL,
  nlam = 60, flmin = 0.01)

Arguments

S	The sample covariance matrix
w	Logical. Should we use weighted version of the penalty or not? If TRUE, we use general weight. If FALSE, use unweighted penalty. Default is FALSE.
lasso	Logical. Should we use l1 penalty instead of hierarchical group lasso penalty? Note that by using l1 penalty, we lose the banded structure in the resulting estimate. And when using l1 penalty, the becomes CSCS (Convex Sparse Cholesky Selection) introduced in Khare et al. (2016). Default value for lasso is FALSE.
K	Integer between 0 and p - 1 (default), indicating the maximum bandwidth in the resulting estimate. A small value of K will result in a sparse estimate and small computing time.
lamlist	A list of non-negative tuning parameters lambda.
nlam	If lamlist is not provided, create a lamlist with length node. Default is 60.
flmin	if lamlist is not provided, create a lamlist with ratio of the smallest and largest lambda in the list. Default is 0.01.

Value

A list object containing

path:

A array of dim (p, p, nlam) of estimates of L

lamlist:

a grid values of tuning parameters

See Also

varband varband_cv

Examples

set.seed(123)
n <- 50
true <- varband_gen(p = 50, block = 5)
x <- sample_gen(L = true, n = n)
S <- crossprod(scale(x, center = TRUE, scale = FALSE))/n
path_res <- varband_path(S = S, w = FALSE, nlam = 40, flmin = 0.03)

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