GraphicalLasso.jl

glasso(s::Matrix{Float64}, obs::Int, λ::Real; penalizediag::Bool=true, γ::Real=0.0, tol::Float64=1e-05, verbose::Bool=true, maxiter::Int=100, winit::Matrix{Float64}=zeros(size(s)))

Applies the graphical lasso (glasso) algorithm to estimate a sparse inverse covariance matrix.

Arguments

• s::Matrix{Float64}: Empirical covariance matrix.
• obs::Int: Number of observations.
• λ::Real: Regularization parameter for the lasso penalty.
• penalizediag::Bool=true: Whether to penalize the diagonal entries. (optional)
• γ::Real=0.0: EBIC tuning parameter. (optional)
• tol::Float64=1e-05: Tolerance for the convergence criteria. (optional)
• verbose::Bool=true: If true, prints convergence information. (optional)
• maxiter::Int=100: Maximum number of iterations. (optional)
• winit::Matrix{Float64}=zeros(size(s)): Initial value of the precision matrix. (optional)

Returns

• NamedTuple: A named tuple with fields:
  • W::Matrix{Float64}: Estimated precision matrix.
  • θ::Matrix{Float64}: Estimated inverse covariance matrix.
  • ll::Float64: Log-likelihood of the estimate.
  • bicval::Float64: EBIC value of the estimate.

cdlasso(W11::Matrix{T}, s12::Vector{T}, λ::Real; max_iter::Int=100, tol::T=1e-5) where {T<:Real}

Solves the coordinate descent Lasso problem.

Arguments

• W11::Matrix{T}: A square matrix used in the coordinate descent update.
• s12::Vector{T}: A vector used in the coordinate descent update.
• λ::Real: Regularization parameter.
• max_iter::Int=100: Maximum number of iterations. (optional)
• tol::T=1e-5: Tolerance for the convergence criteria. (optional)

Returns

• Vector{T}: Solution vector β.

ebic(θ, ll, obs, thr, γ)

Calculates the Extended Bayesian Information Criterion (EBIC) for a given precision matrix θ. From the paper by Foygel and Drton (2010), the EBIC is defined as:

$\text{EBIC} = -2 \times \text{Log-likelihood} + \log(n) \times \mathbf{E} + 4 \times \gamma \times \mathbf{E} \times \log(p)$

where:

• n is the number of observations.
• p is the number of variables.
• gamma is a tuning parameter.
• The number of edges is calculated as the count of entries in theta that exceed a given threshold.

Arguments

• θ::AbstractMatrix: Precision matrix.
• ll::Real: Log-likelihood.
• obs::Int: Number of observations.
• thr::Real: Threshold value for counting edges.
• γ::Real: EBIC tuning parameter.

Returns

• Real: EBIC value.

critfunc(s, θ, rho; penalizediag=true)

Calculates the objective function value for the graphical lasso.

Arguments

• s::AbstractMatrix: Empirical covariance matrix.
• θ::AbstractMatrix: Precision matrix.
• rho::Real: Regularization parameter.
• penalizediag::Bool=true: Whether to penalize the diagonal entries. (optional)

Returns

• Real: Value of the objective function.

tuningselect(s::Matrix{Float64}, obs::Int, λ::AbstractVector{T}; γ::Real=0.0) where {T}

Selects the optimal regularization parameter λ for the graphical lasso using EBIC.

Arguments

• s::Matrix{Float64}: Empirical covariance matrix.
• obs::Int: Number of observations.
• λ::AbstractVector{T}: Vector of regularization parameters to be tested.
• γ::Real=0.0: EBIC tuning parameter. (optional)

Returns

• T: The optimal regularization parameter from the input vector λ.

randsparsecov(p, thr)

Generates a random sparse covariance matrix of size p x p with a specified threshold for sparsity.

Arguments

• p::Int: The dimension of the covariance matrix.
• thr::Real: Threshold value for sparsity. Values below this threshold will be set to zero.

Returns

• Hermitian{Float64}: A sparse covariance matrix.

iscov(x::AbstractMatrix{T}) where {T<:Real}

Checks if a given matrix is a valid covariance matrix. A valid covariance matrix must be square, symmetric, and positive semi-definite.

Arguments

• x::AbstractMatrix{T}: Input matrix to check.

Returns

• Bool: true if the matrix is a valid covariance matrix, false otherwise.