MLEdag: MLE/LRT of a Gaussian directed acyclic graph
In clrdag: Constrained Likelihood Ratio Tests for a Directed Acyclic Graph

Description Usage Arguments Value Author(s) References Examples

Computes the MLE/LRT of a Gaussian directed acyclic graph using difference convex programming and alternating direction method of multipliers.

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MLEdag(X, A = NULL, Lambda = NULL, D = NULL, tau, mu, rho, 
        tol_abs = 1e-04, tol_rel = 1e-04, 
        dc_max_iter = 20, admm_max_iter = 1000, trace_obj = TRUE)

`X`	An n by p data matrix, where n is the number of observations and p is the dimension.
`A, Lambda`	Initial estimate. `A` is a p by p adjacency matrix, `Lambda` is a p by p dual matrix in acyclicity condition. `A` must be a DAG! If `A` is NULL (default), the initial estimate is provided automatically (Be careful!).
`D`	A p by p matrix indicating hypothesized edges. For the entries equal to one, no sparse penalty is imposed. If `D` is not provided, or if all off-diagonal entries of `D` are zero, no test is performed.
`tau`	A positive real number. `tau` is the threshold parameter in TLP.
`mu`	A positive real number. `mu` is the sparsity parameter.
`rho`	A positive real number. `rho` is the ADMM dual parameter.
`tol_abs, tol_rel`	Positive real. The absolute and relative tolerance.
`dc_max_iter, admm_max_iter`	Positive integer. The maximum iteration number of DC and ADMM.

trace_obj

Logical. If TRUE, the objective values are printed after each iteration.

The function returns a LIST containing the following components.

`X`	The input data matrix.
`A`	The final estimate of adjacency matrix. Returned if no test is performed.
`A.H1, A.H0`	The final estimates of adjacency matrix under alternative and null. Returned if a test is performed.
`Lambda`	The final estimate of dual variables in the acyclicity condition. Returned if no test is performed.
`Lambda.H1`	The final estimate of dual variables in the acyclicity condition under alternative. Returned if a test is performed.
`D`	A matrix indicating hypothesized edges. Returned if a test if performed.
`tau`	The input threshold parameter in TLP.
`mu`	The input sparsity parameter.
`lrt`	(2\timeslog-likelihood ratio) of alternative over null. Returned if a test is performed.
`df`	Degrees of freedom of the test. Returned if a test is performed.
`pval`	The p-value of the test. Returned if a test is performed.

Chunlin Li <li000007@umn.edu>

Li, C., Shen, X., and Pan, W. (2019). Likelihood ratio tests for a large directed acyclic graph. Journal of the American Statistical Association. Accepted. <doi:10.1080/01621459.2019.1623042>.

##
## Example: random graph
##
library(clrdag)
set.seed(2019)
p<-10
n<-1000
## random graph: randomly generate adjacency matrix A, A lower triangular
sparsity <- 2/p
A <- matrix(rbinom(p*p,1,sparsity)*sign(runif(p*p,min=-1,max=1)),p,p)
A[upper.tri(A,diag=TRUE)] <- 0
X <- matrix(rnorm(n*p),n,p) %*% t(solve(diag(p)-A))
out <- MLEdag(X=X,tau=0.3,mu=1,rho=1.2,trace_obj=FALSE) # compute the MLE
sum(abs((out$A!=0)-(A!=0))) # Hamming distance to the truth graph

# test edge 1 --> 2
D <- matrix(0,p,p)
D[2,1] <- 1
out <- MLEdag(X=X,D=D,tau=0.3,mu=1,rho=1.2,trace_obj=FALSE) # compute the MLE
out$pval