constrained {GGMncv}

Precision Matrix with Known Graph

Description

Compute the maximum likelihood estimate of the precision matrix, given a known graphical structure (i.e., an adjacency matrix). This approach was originally described in "The Elements of Statistical Learning" (see pg. 631, Hastie et al. 2009).

Usage

constrained(Sigma, adj)

mle_known_graph(Sigma, adj)

Arguments

Sigma

Covariance matrix

adj

Adjacency matrix that encodes the constraints, where a zero indicates that element should be zero.

Value

A list containing the following:

• Theta: Inverse of the covariance matrix (precision matrix)

• Sigma: Covariance matrix.

• wadj: Weighted adjacency matrix, corresponding to the partial correlation network.

Note

The algorithm is written in c++, and should scale to high dimensions nicely.

Note there are a variety of algorithms for this purpose. Simulation studies indicated that this approach is both accurate and computationally efficient (HFT therein, Emmert-Streib et al. 2019)

References

Emmert-Streib F, Tripathi S, Dehmer M (2019). “Constrained covariance matrices with a biologically realistic structure: Comparison of methods for generating high-dimensional Gaussian graphical models.” Frontiers in Applied Mathematics and Statistics, 5, 17.

Hastie T, Tibshirani R, Friedman J (2009). The elements of statistical learning: data mining, inference, and prediction. Springer Science \& Business Media.

Examples

# data
y <- ptsd

# fit model
fit <- ggmncv(cor(y), n = nrow(y),
              penalty = "lasso",
              progress = FALSE)

# set negatives to zero (sign restriction)
adj_new <- ifelse( fit$P <= 0, 0, 1)

check_zeros <- TRUE

# track trys
iter <- 0

# iterate until all positive
while(check_zeros){
  iter <- iter + 1
  fit_new <- constrained(cor(y), adj = adj_new)
  check_zeros <- any(fit_new$wadj < 0)
  adj_new <- ifelse( fit_new$wadj <= 0, 0, 1)
}



# alias

# data
y <- ptsd

# nonreg (lambda = 0)
fit <- ggmncv(cor(y), n = nrow(y),
              lambda = 0,
              progress = FALSE)

# set values less than |0.1| to zero
adj_new <- ifelse( abs(fit$P) <= 0.1, 0, 1)

# mle given the graph
mle_known_graph(cor(y), adj_new)