Package 'gif'

Synthetic multivariate Gaussian data

Description

A synthetic dataset includes 200 samples under multivariate Gaussian distribution with 100 variables.

Format

ar1$x

A numeric matrix with 200 rows and 100 variables where each row represents a sample.

ar1$Omega

The corresponding inverse covariance matrix of the Gaussian graphical model.

Details

This synthetic dataset contains 200 samples, each of them is a vector following multivariate Gaussian distribution with 100 variables. The inverse covariance matrix of the distribution is as follows,

Omega[i, i] = 1.

Omega[i, i + 1] = Omega[i, i - 1] = 0.5.

Otherwise: Omega[i, j] = 0.

The corresponding graph structure is the so-called AR(1).

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Simulate Data from Gaussian Graphical Model

Description

Produce one or more samples from the specified Gaussian graphical model.

Usage

ggm.generator(n, Omega)

Arguments

n	The number of samples required.
Omega	The inverse covariance matrix of the specified Gaussian graphical model.

Value

A numeric matrix with n rows and p variables where p corresponds to the dimension of Omega.

Note

Omega should be positive definite.

Examples

library(gif)

set.seed(1)
n <- 200
p <- 100
Omega <- diag(1, p, p)
for(i in 1:(p - 1)) {
  Omega[i, i + 1] <- 0.5
  Omega[i + 1, i] <- 0.5
}
x <- ggm.generator(n, Omega)

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Hard Graphical Thresholding Algorithm

Description

Estimates sparse inverse covariance matrix.

Usage

hgt(x, size, active.entry = NULL, bcd.opt = list(max.iter = 10, eps = 0.001))

Arguments

x	There are 2 options: (1) x is an $n$ by $p$ data matrix; (2) a $p$ by $p$ sample covariance matrix. The program automatically identifies the input matrix by checking the symmetry. ($n$ is the sample size and $p$ is the dimension.)
size	A non-negative integer for determining the model size, i.e., the number of non-zero off-diagonal entries in the upper-triangular precision matrix, which is also the number of edges in the graph. size must range from 0 to $(p^2 - p) / 2$.
active.entry	Pre-determined non-zero off-diagonal entries positions of the precision matrix. Default: active.entry = NULL.
bcd.opt	A list of options that control details of the block coordinate descent algorithm.

Details

Hard Graphical Thresholding (HGT) algorithm proceeds by thresholding the sample correlation matrix and estimating the inverse covariance matrix with block coordinate descent algorithm. HGT algorithm could recover the inverse covariance matrix given model size or given active entries. When active entries are given directly, model fitting is the so-called covariance selection.

Value

A list with following components:

Omega	Estimated inverse covariance matrix.
active.entry	The position of the non-zero off-diagonal entries of Omega in the upper-triangular part.

Note

Either size or active.entry should be specified when function hgt is called. If both arguments are given, size would be omitted and the inverse covariance matrix would be estimated based on the given active.entry.

If arguments active.entry is specified, only one of the entries in symmetric positions should be given.

References

Luo, Shikai, Rui Song, and Daniela Witten (2014). Sure Screening for Gaussian Graphical Models. arXiv preprint arXiv:1407.7819. URL https://arxiv.org/abs/1407.7819.

Dempster, A.P. (1972). Covariance Selection. Biometrics, 28(1), 157-175. doi:10.2307/2528966

Examples

library(gif)

data("ar1")
p <- 100
non_zero_num <- sum(ar1[["Omega"]] != 0) - p
res <- hgt(ar1[["x"]], size = non_zero_num / 2)

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Soft Graphical Thresholding Algorithm

Description

Estimates a sparse inverse covariance matrix using the closed form solution of graphical lasso under acyclic graph structure.

Usage

sgt(x, lambda, size = NULL)

Arguments

x	There are 2 options: (1) x is an $n$ by $p$ data matrix; (2) a $p$ by $p$ sample covariance matrix. The program automatically identifies the input matrix by checking the symmetry. ($n$ is the sample size and $p$ is the dimension.)
lambda	The regularization parameter for graphical lasso.
size	A non-negative integer for determining the model size, i.e., the number of non-zero off-diagonal entries in the upper-triangular precision matrix, which is also the number of edges in the graph. size must range from 0 to $(p^2 - p) / 2$.

Details

Soft Graphical Thresholding (SGT) algorithm proceeds by thresholding the sample covariance matrix and estimating the inverse covariance matrix with a closed-form formula. If the graph structure detected by the thresholding procedure is acyclic, then the estimation is equivalent to the solution of graphical lasso.

Value

A list with following components:

Omega	Estimated inverse covariance matrix.
active.entry	The position of the non-zero entries of Omega.
is.acyclic	The boolean flag of whether the detected graph structure is acyclic or not.

Note

Either lambda or size should specified when function sgt is called. If both arguments are given, only lambda would be considered.

References

Fattahi, Salar, and Somayeh Sojoudi. Graphical Lasso and Thresholding: Equivalence and Closed-form Solutions. Journal of Machine Learning Research 20.10 (2019): 1-44. doi: 10.5555/3322706.3322716

Examples

library(gif)

data("ar1")
res <- sgt(ar1[["x"]], lambda = 0.01)

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