DiffPrecTest.jl

Inference for the difference Δ = Ωx - Ωy of the precision matrices of two independent Gaussian samples.

• Confidence intervals for one entry of Δ: estimate with
  • SymmetricNormal: estimate Δ by the weighted ℓ1 penalized direct difference estimator, estimate the corresponding row of the inverse Hessian, and refit on the union of the two supports; AsymmetricNormal and ReducedNormal are variants
  • SeparateNormal: estimate Ωx and Ωy separately by node-wise lasso regressions (Xia, Cai and Cai, 2015)
  • refits on a given support: SymmetricOracleNormal, AsymmetricOracleNormal, ReducedOracleNormal, and with bootstrap standard errors SymmetricOracleBoot, AsymmetricOracleBoot
• Simultaneous confidence intervals for all entries by the bootstrap: bootstrap, simul_ci, simul_ci_studentized
• Support recovery: support_estimate with ANTSupport, BootStdSupport or DTraceValidationSupport

DiffPrecTest is one of five Julia packages for high-dimensional statistics, three for estimation and inference and two for the optimization beneath them:

Package
HighDimensionalModels.jl	the lasso and its paths; the group, square-root, scaled and heteroscedastic lasso; varying-coefficient models
CovarianceSelection.jl	Gaussian graphical models: the graphical lasso, CLIME and ACLIME, precision matrices by the scaled lasso, the fused graphical lasso, and the difference of two precision matrices
DiffPrecTest.jl (this package)	confidence intervals and tests for the difference of two precision matrices
ProximalSolvers.jl	the solvers: coordinate descent with penalty paths, screening and active sets; proximal gradient methods, proximal Newton and ADMM
ProximalBase.jl	the building blocks: losses, penalties and constraints with their gradients and proximal operators, and sparse iterates; documentation with the mathematics

Most users need one of the first three. To solve a problem of your own, use ProximalSolvers; to write your own algorithm, ProximalBase.

Usage

using DiffPrecTest, LinearAlgebra

p, n = 10, 500
Ωx = Matrix(SymTridiagonal(fill(1., p), fill(0.4, p - 1)))
Ωy = Matrix(SymTridiagonal(fill(1., p), fill(0.2, p - 1)))
X = randn(n, p) / cholesky(Ωx).U'     # rows from N(0, inv(Ωx))
Y = randn(n, p) / cholesky(Ωy).U'

# 95% confidence interval for Δ[2, 1] = 0.2
res, = estimate(SymmetricNormal(), X, Y, 2, 1)
res.p .+ [-1.96, 1.96] .* res.std

# simultaneous 95% confidence intervals for the entries of the lower triangle of Δ;
# the off-diagonal entries are multiplied by √2
straps, = bootstrap(X, Y; boot_samples=500)
CI = simul_ci_studentized(straps, 0.95)

Distributions also exports estimate; when both packages are loaded, call DiffPrecTest.estimate.

The exper/ folder holds the simulation scripts, with their own environment; see exper/README.md.

The penalized estimators of the first and second stages are solved by the coordinate descent of ProximalSolvers.jl, on losses of ProximalBase.jl: the direct difference loss of CovarianceSelection.jl and the inverse Hessian losses are its KronQuadratic. The functions of the two stages (diff_estimation, inv_q_sym_hessian, inv_asym_hessian_estimation, reduced_diff_estimation, inv_hessian_reduced) take the solver options as a ProximalSolvers CDSolver; so are the node-wise lasso regressions of SeparateNormal.

Naming migration

Types retain UpperCamelCase; descriptive functions, fields and keyword arguments use snake_case. Mathematical identifiers remain available.

Previous spelling	Canonical spelling
diffEstimation	diff_estimation
invAsymHessianEstimation, invQSymHessian, invHessianReduced	inv_asym_hessian_estimation, inv_q_sym_hessian, inv_hessian_reduced
supportEstimate	support_estimate
simulCI, simulCIstudentized	simul_ci, simul_ci_studentized
computeSimulationResult	compute_simulation_result
bootSamples, estimSupport	boot_samples, estim_support
SimulationResult.lenCoverage	SimulationResult.len_coverage

Previous public spellings are deprecated aliases for a transition period. Supplying both spellings of a keyword raises an error. Package versions are unchanged. Active environments use sibling master checkouts of CovarianceSelection, ProximalBase and ProximalSolvers through local sources. The pinned benchmark/compare/old environment remains a historical baseline.

Development tests

From the repository root, run:

julia scripts/test_master.jl

The script prepares an isolated test environment using dependency master checkouts and test extras on Julia 1.10 and newer. See CONTRIBUTING.md for the naming policy and environment setup.