API

VarianceComponentModel stores the model parameters of a variance component model.

Fields

• B: p x d mean parameters
• Σ: tuple of d x d variance component parameters
• A: constraint matrix for vec(B)
• sense: vector of characters '=', '<' or '>'
• b: constraint vector for vec(B)
• lb: lower bounds for vec(B)
• ub: upper bounds for vec(B)

VarianceComponentVariate stores the data of a variance component model.

Feilds

• Y: n x d responses
• X: n x p predictors
• V: tuple of n x n covariance matrices

TwoVarCompModelRotate stores the rotated two variance component model.

Fields

• Brot: rotated mean parameters B * eigvec
• eigval: eigenvalues of eig(Σ[1], Σ[2])
• eigvec: eigenvectors of eig(Σ[1], Σ[2])
• logdetΣ2: log-determinant of Σ[2]

TwoVarCompVariateRotate stores the rotated two variance component data.

Fields

• Yrot: rotated responses eigvec * Y
• Xrot: rotated covariates eigvec * X
• eigval: eigenvalues of eig(V[1], V[2])
• eigvec: eigenvectors of eig(V[1], V[2])
• logdetV2: log-determinant of V[2]

mle_fs!(vcmodel, vcdatarot; maxiter, solver, qpsolver, verbose)

Find MLE by Fisher scoring algorithm.

Input

• vcmodel: two variane component model VarianceComponentModel, with

vcmodel.B and vcmodel.Σ used as starting point

• vcdatarot: rotated two varianec component data TwoVarCompVariateRotate

Keyword

• maxiter::Int: maximum number of iterations, default is 1000
• solver::Symbol: backend nonlinear programming solver, :Ipopt (default) or :Knitro
• qpsolver::Symbol: backend quadratic programming solver, :Ipopt (default) or :Gurobi or Mosek
• verbose::Bool: display information

Output

• maxlogl: log-likelihood at solution
• vcmodel: VarianceComponentModel with updated model parameters
• Σse=(Σse[1],Σse[2]): standard errors of estimate Σ=(Σ[1],Σ[2])
• Σcov: covariance matrix of estimate Σ=(Σ[1],Σ[2])
• Bse: standard errors of estimate B
• Bcov: covariance of estimate B

mle_mm!(vcmodel, vcdatarot; maxiter, qpsolver, verbose)

Find MLE by minorization-maximization (MM) algorithm.

Input

• vcmodel: two variane component model VarianceComponentModel, with

vcmodel.B and vcmodel.Σ used as starting point

• vcdatarot: rotated two varianec component data TwoVarCompVariateRotate

Keyword

• maxiter::Int: maximum number of iterations, default is 1000
• qpsolver::Symbol: backend quadratic programming solver, :Ipopt (default) or :Gurobi or Mosek
• verbose::Bool: display information

Output

• maxlogl: log-likelihood at solution
• vcmodel: VarianceComponentModel with updated model parameters
• Σse=(Σse[1],Σse[2]): standard errors of estimate Σ=(Σ[1],Σ[2])
• Σcov: covariance matrix of estimate Σ=(Σ[1],Σ[2])
• Bse: standard errors of estimate B
• Bcov: covariance of estimate B

Reference

• H. Zhou, L. Hu, J. Zhou, and K. Lange (2015) MM algorithms for variance components models. http://arxiv.org/abs/1509.07426

fit_mle!(vcmodel, vcdata; algo)

Find MLE of variane component model.

Input

• vcmodel: two variane component model VarianceComponentModel, with

vcmodel.B and vcmodel.Σ used as starting point

• vcdata: two varianec component data VarianceComponentVariate

Keyword

• algo::Symbol: algorithm, :FS (Fisher scoring) for :MM

(minorization-maximization algorithm)

Output

• maxlogl: log-likelihood at solution
• vcmodel: VarianceComponentModel with updated model parameters
• Σse=(Σse[1],Σse[2]): standard errors of estimate Σ=(Σ[1],Σ[2])
• Σcov: covariance matrix of estimate Σ=(Σ[1],Σ[2])
• Bse: standard errors of estimate B
• Bcov: covariance of estimate B

fit_reml!(vcmodel, vcdata; algo)

Find restricted MLE (REML) of variane component model.

Input

• vcmodel: two variane component model VarianceComponentModel, with

vcmodel.B and vcmodel.Σ used as starting point

• vcdata: two varianec component data VarianceComponentVariate

Keyword

• algo::Symbol: algorithm, :FS (Fisher scoring) for :MM

(minorization-maximization algorithm)

Output

• maxlogl: log-likelihood at solution
• vcmodel: VarianceComponentModel with updated model parameters
• Σse=(Σse[1],Σse[2]): standard errors of estimate Σ=(Σ[1],Σ[2])
• Σcov: covariance matrix of estimate Σ=(Σ[1],Σ[2])
• Bse: standard errors of estimate B
• Bcov: covariance of estimate B