AECM Estimation for Matrix-Variate Normal-Inverse Gaussian Models

Description

This function fits MVNIG linear models for matrix-variate skew data with non-uniform data rows between subjects. Exchangeable observation row correlation and skewness structures are imposed to accommodate the varying row counts across matrices. Note that multiple restarts may be needed to account for unstable local maxima.

Usage

MVNIGmod(Y, X, theta_g = NULL, stopping = 0.001, max_iter = 50)

Arguments

Details

Fits the matrix-variate skew regression model

Y_i = X_i \Theta + E_i,

where each response Y_i is a n_i \times p matrix that indexes n_i observations and p response variables. X_i corresponds to a n_i \times q design matrix, and \Theta corresponds to a q \times p coefficient matrix. E_i corresponds to a n_i \times p error matrix, following a matrix-variate variance-gamma distribution.

The model estimates MVVG parameters \Theta, \underline{a}, r, \Psi, \tilde\gamma using the alternating expectation conditional maximization (AECM) algorithm, using the density

f(Y_i|M_i,\underline{a},r, \Psi, \tilde\gamma, n_i, p) = \dfrac{2 \exp[matlib::tr(\Sigma_i^{-1}(Y_i-M_i)\Psi^{-1}A_i^T) + \tilde\gamma]}{(2\pi)^{\frac{n_ip}{2} + 1} |\Sigma_i|^{\frac{p}{2}} |\Psi|^{\frac{n_i}{2}}} \bigg( \dfrac{\delta(Y_i; M_i, \Sigma_i, \Psi) + 1}{\rho (A_i, \Sigma_i,\Psi) + \tilde\gamma^2} \bigg)^{-\frac{(1+n_ip)}{4}} \\ \times K_{-\frac{(1+n_ip)}{2}} \big( \sqrt{[\rho(A_i, \Sigma_i, \Psi) + \tilde\gamma^2][\delta(Y_i;M_i,\Sigma_i,\Psi) + 1]} \big),

where A_i = \underline{1}_{n_i} \times \underline{a}^T, \Sigma_i = I_{n_i} + r(\underline{1}_{n_i}\underline{1}_{n_i}^T - I_{n_i}), \delta(X;M, \Sigma, \Psi) = matlib::tr(\Sigma^{-1}(X-M)\Psi^{-1}(X-M)^T), \rho(A, \Sigma, \Psi) = matlib::tr(\Sigma^{-1}A\Psi^{-1}A^T), and K_{\nu}(x) is the modified Bessel function of the second kind.

The structure of theta_g and parameter estimates returned by the function must be in the form of a list with the following named elements:

Theta:

q \times p coefficient matrix

a:

p \times 1 skewness vector

rho:

Compound symmetry parameter for row correlation matrix

Psi:

p \times p column covariance matrix

tgamma:

Univariate mixing parameter

Value

MVNIGmod returns a list with the following elements:

Iteration:

Number of iterations taken to convergence. Inf if convergence not reached.

⁠Starting Value⁠:

List of initial parameter values.

⁠Final Value⁠:

List of final parameter estimates.

⁠Stopping Criteria⁠:

Vector of |\hat\theta^{t+1} - \hat\theta^{t}|_\infty at each iteration.

AIC:

Model AIC

BIC:

Model BIC

Author(s)

Samuel Soon

Dipankar Bandyopadhyay

Qingyang Liu

Examples

MVNIGmod(Y,X,theta_mvnig)

set.seed(1234)
# num response variables
p <- ncol(gaad_res[[1]])
# num covariates
q <- ncol(gaad_cov[[1]])
# generate initial value to input, then run AECM with MVVG distribution
initial_mvnig_theta <- list(Theta = matrix(stats::rnorm(p*q), nrow = q, ncol = p),
                      A = rep(1,p),
                     rho = 0.3,
                     Psi = diag(p),
                     tgamma = 3)
MVNIGmod(gaad_res[1:30], gaad_cov[1:30], initial_mvnig_theta)

AECM Estimation for Matrix-Variate Variance Gamma (MVVG) Models

Description

This function fits MVVG linear models for matrix-variate skew data with non-uniform data rows between subjects. Exchangeable observation row correlation and skewness structures are imposed to accommodate the varying row counts across matrices. Note that multiple restarts may be needed to account for unstable local maxima.

Usage

MVVGmod(Y, X, theta_g = NULL, stopping = 0.001, max_iter = 50)

Arguments

Details

Fits the matrix-variate skew regression model

Y_i = X_i \Theta + E_i,

where each response Y_i is a n_i \times p matrix that indexes n_i observations and p response variables. X_i corresponds to a n_i \times q design matrix, and \Theta corresponds to a q \times p coefficient matrix. E_i corresponds to a n_i \times p error matrix, following a matrix-variate variance-gamma distribution.

The model estimates MVVG parameters \Theta, \underline{a}, r, \Psi, \gamma using the alternating expectation conditional maximization (AECM) algorithm, using the density

f(Y_i| X_i\Theta,\underline{a},r, \Psi, \gamma, n_i, p) = \dfrac{2\gamma^\gamma \exp[matlib::tr(\Sigma_i^{-1}(Y_i- X_i\Theta)\Psi^{-1}A_i^T)]}{(2\pi)^{n_ip/2} |\Sigma_i|^{p/2} |\Psi|^{n_i/2} \Gamma(\gamma)} \bigg( \dfrac{\delta(Y_i; X_i\Theta, \Sigma_i, \Psi)}{\rho (A_i, \Sigma_i,\Psi) + 2\gamma} \bigg)^{(\gamma - n_ip/2)/2} \\ \times K_{(\gamma - n_ip/2)} \big( \sqrt{[\rho(A_i, \Sigma_i, \Psi) + 2\gamma][\delta(Y_i; X_i\Theta,\Sigma_i,\Psi)]} \big),

where A_i = \underline{1}_{n_i} \times \underline{a}^T, \Sigma_i = I_{n_i} + r(\underline{1}_{n_i}\underline{1}_{n_i}^T - I_{n_i}), \delta(X;M, \Sigma, \Psi) = matlib::tr(\Sigma^{-1}(X-M)\Psi^{-1}(X-M)^T), \rho(A, \Sigma, \Psi) = matlib::tr(\Sigma^{-1}A\Psi^{-1}A^T), and K_{\nu}(x) is the modified Bessel function of the second kind.

The structure of theta_g and parameter estimates returned by the function must be in the form of a list with the following named elements:

Theta:

q \times p coefficient matrix

a:

p \times 1 skewness vector

rho:

Compound symmetry parameter for row correlation matrix

Psi:

p \times p column covariance matrix

gamma:

Univariate mixing parameter

Value

MVVGmod returns a list with the following elements:

Iteration:

Number of iterations taken to convergence. Inf if convergence not reached.

⁠Starting Value⁠:

List of initial parameter values.

⁠Final Value⁠:

List of final parameter estimates.

⁠Stopping Criteria⁠:

Vector of |\hat\theta^{t+1} - \hat\theta^{t}|_\infty at each iteration.

AIC:

Model AIC

BIC:

Model BIC

Author(s)

Samuel Soon

Dipankar Bandyopadhyay

Qingyang Liu

Examples

MVVGmod(Y,X,theta_mvvg)

set.seed(1234)
# num response variables
p <- ncol(gaad_res[[1]])
# num covariates
q <- ncol(gaad_cov[[1]])
# generate initial value to input, then run AECM with MVVG distribution
initial_gaad_theta_mvvg <- list(Theta = matrix(stats::rnorm(p*q), nrow = q, ncol = p),
                      A = rep(1,p),
                     rho = 0.3,
                     Psi = diag(p),
                     gamma = 4)
MVVGmod(gaad_res[1:50], gaad_cov[1:50], initial_gaad_theta_mvvg)

Toy Covariate Matrices

Description

Part of toy dataset for examples.

Usage

X

Format

List of covariate matrices for individual subjects

Examples

X

Toy Response Matrices

Description

Part of toy dataset for examples.

Usage

Y

Format

List of response matrices for individual subjects

Examples

Y

GAAD Data

Description

These data sets describe periodontal measurements performed on members of the Gullah-Speaking African American community.

Usage

gaad_cov

Format

Each is a list of matrices, with rows denoting tooth sites and columns denoting CAL/PPD response.

Details

gaad_res and gaad_cov contain the response and covariate matrices of the GAAD data.

Examples

gaad_cov
gaad_res

GAAD Data

Description

These data sets describe periodontal measurements performed on members of the Gullah-Speaking African American community.

Usage

gaad_res

Format

Each is a list of matrices, with rows denoting tooth sites and columns denoting CAL/PPD response.

Details

gaad_res and gaad_cov contain the response and covariate matrices of the GAAD data.

Examples

gaad_cov
gaad_res

MVVG Parameter Format

Description

This is an example of the format of input parameter list theta.

Usage

gaad_theta_mvvg

Format

List of model parameters.

Examples

gaad_theta_mvvg

MVSK Model Prediction

Description

Predicts response values given a list of covariate matrices and a model output from either MVVGmod or MVNIGmod.

Usage

predict(mod, X)

Arguments

Value

Returns a list of predicted response matrices

Author(s)

Samuel Soon

Dipankar Bandyopadhyay

Qingyang Liu

Examples

set.seed(1234)
# num response variables
p <- ncol(gaad_res[[1]])
# num covariates
q <- ncol(gaad_cov[[1]])
# generate initial value to input, then run AECM with MVVG distribution
initial_mvnig_theta <- list(Theta = matrix(stats::rnorm(p*q), nrow = q, ncol = p),
                      A = rep(1,p),
                     rho = 0.3,
                     Psi = diag(p),
                     tgamma = 4)
mvnig_mod <- MVNIGmod(gaad_res[1:50], gaad_cov[1:50], initial_mvnig_theta)

predict(mvnig_mod, gaad_cov[1:50])

Toy Response Initial Parameter (MVNIG)

Description

Part of toy dataset for examples.

Usage

theta_mvnig

Format

List of parameters for input to MVNIGmod function

Examples

theta_mvvg

Toy Response Initial Parameter (MVVG)

Description

Part of toy dataset for examples.

Usage

theta_mvvg

Format

List of parameters for input to MVVGmod function

Examples

theta_mvvg