stenv: Fit the simultaneous envelope model

In Renvlp: Computing Envelope Estimators

Fit the simultaneous envelope model

Description

Fit the simultaneous envelope model in multivariate linear regression with dimension (q, u).

Usage

stenv(X, Y, q, u, asy = TRUE, Pinit = NULL, Ginit = NULL)

Arguments

X	Predictors. An n by p matrix, p is the number of predictors. The predictors can be univariate or multivariate, discrete or continuous.
Y	Multivariate responses. An n by r matrix, r is the number of responses and n is number of observations. The responses must be continuous variables.
q	Dimension of the X-envelope. An integer between 0 and p.
u	Dimension of the Y-envelope. An integer between 0 and r.
asy	Flag for computing the asymptotic variance of the envelope estimator. The default is TRUE. When p and r are large, computing the asymptotic variance can take much time and memory. If only the envelope estimators are needed, the flag can be set to asy = FALSE.
Pinit	The user-specified value of Phi for the X-envelope subspace. An p by q matrix. The default is the one generated by function stenvMU.
Ginit	The user-specified value of Gamma for the Y-envelope subspace. An r by u matrix. The default is the one generated by function stenvMU.

Details

This function fits the envelope model to the responses and predictors simultaneously,

Y = \mu + \beta'X+\varepsilon, \beta = \Phi\eta\Gamma', \Sigma_{Y|X}=\Gamma\Omega\Gamma'+\Gamma_{0}\Omega_{0}\Gamma'_{0}, \Sigma_{X}=\Phi\Delta\Phi'+\Phi_{0}\Delta_{0}\Phi'_{0}

using the maximum likelihood estimation. When the dimension of the Y-envelope is between 1 and r-1 and the dimension of the X-envelope is between 1 and p-1, the starting value and blockwise coordinate descent algorithm in Cook et al. (2016) is implemented. When the dimension is (p, r), then the envelope model degenerates to the standard multivariate linear regression. When the dimension of the Y-envelope is r, then the envelope model degenerates to the standard envelope model. When the dimension of X-envelope is p, then the envelope model degenerates to the envelope model in the predictor space. When the dimension is 0, it means that X and Y are uncorrelated, and the fitting is different.

Value

The output is a list that contains the following components:

beta	The envelope estimator of the regression coefficients.
SigmaYcX	The envelope estimator of the error covariance matrix.
SigmaX	The envelope estimator of the covariance matrix of X.
Gamma	An orthonormal basis of the Y-envelope subspace.
Gamma0	An orthonormal basis of the complement of the Y-envelope subspace.
eta	The coordinates of beta with respect to Gamma and Phi.
Omega	The coordinates of SigmaYcX with respect to Gamma.
Omega0	The coordinates of SigmaYcX with respect to Gamma0.
mu	The estimated intercept.
Phi	An orthonormal basis of the X-envelope subspace.
Phi0	An orthonormal basis of the complement of the X-envelope subspace.
Delta	The coordinates of SigmaX with respect to Phi.
Delta0	The coordinates of SigmaX with respect to Phi0.
loglik	The maximized log likelihood function.
covMatrix	The asymptotic covariance of vec(beta). The covariance matrix returned are asymptotic. For the actual standard errors, multiply by 1 / n.
asySE	The asymptotic standard error for elements in beta under the envelope model. The standard errors returned are asymptotic, for actual standard errors, multiply by 1 / sqrt(n).
ratio	The asymptotic standard error ratio of the standard multivariate linear regression estimator over the envelope estimator, for each element in beta.
n	The number of observations in the data.

References

Cook, R. D., Zhang, X. (2015). Simultaneous Envelopes for Multivariate Linear Regression. Technometrics 57, 11 - 25.

Cook, R. D., Forzani, L. and Su, Z. (2016) A Note on Fast Envelope Estimation. Journal of Multivariate Analysis. 150, 42-54.

Examples

data(fiberpaper)
X <- fiberpaper[, 5:7]
Y <- fiberpaper[, 1:4]
u <- u.stenv(X, Y)
u

m <- stenv(X, Y, 2, 3)
m
m$beta

Renvlp documentation built on Oct. 11, 2023, 1:06 a.m.