penv: Fit the partial envelope model

In Renvlp: Computing Envelope Estimators

Fit the partial envelope model

Description

Fit the partial envelope model in multivariate linear regression with dimension u. The partial envelope model focuses on the coefficients of main interest.

Usage

penv(X1, X2, Y, u, asy = TRUE, init = NULL) 

Arguments

X1	Predictors of main interest. An n by p1 matrix, n is the number of observations, and p1 is the number of main predictors. The predictors can be univariate or multivariate, discrete or continuous.
X2	Covariates, or predictors not of main interest. An n by p2 matrix, p2 is the number of covariates.
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.
u	Dimension of the partial envelope. An integer between 0 and r.
asy	Flag for computing the asymptotic variance of the partial 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 partial envelope estimators are needed, the flag can be set to asy = FALSE.
init	The user-specified value of Gamma for the partial envelope subspace. An r by u matrix. The default is the one generated by function envMU.

Details

This function fits the partial envelope model to the responses Y and predictors X1 and X2,

Y = \mu + \Gamma\eta X_{1} + \beta_{2}X_{2} +\varepsilon, \Sigma=\Gamma\Omega\Gamma'+\Gamma_{0}\Omega_{0}\Gamma'_{0}

using the maximum likelihood estimation. When the dimension of the envelope is between 1 and r - 1, we implemented the algorithm in Su and Cook (2011), but the partial envelope subspace is estimated using the blockwise coordinate descent algorithm in Cook et al. (2016). When the dimension is r, then the partial envelope model degenerates to the standard multivariate linear regression with Y as the responses and both X1 and X2 as predictors. When the dimension is 0, X1 and Y are uncorrelated, and the fitting is the standard multivariate linear regression with Y as the responses and X2 as the predictors.

Value

The output is a list that contains the following components:

beta1	The partial envelope estimator of beta1, which is the regression coefficients for X1.
beta2	The partial envelope estimator of beta2, which is the regression coefficients for X2.
Sigma	The partial envelope estimator of the error covariance matrix.
Gamma	An orthonormal basis of the partial envelope subspace.
Gamma0	An orthonormal basis of the complement of the partial envelope subspace.
eta	The coordinates of beta1 with respect to Gamma.
Omega	The coordinates of Sigma with respect to Gamma.
Omega0	The coordinates of Sigma with respect to Gamma0.
mu	The estimated intercept in the partial envelope model.
loglik	The maximized log likelihood function.
covMatrix	The asymptotic covariance of vec(beta), while beta = (beta1, beta2). The covariance matrix returned are asymptotic. For the actual standard errors, multiply by 1 / n.
asySE1	The asymptotic standard error for elements in beta1 under the partial envelope model. The standard errors returned are asymptotic, for actual standard errors, multiply by 1 / sqrt(n).
asySE2	The asymptotic standard error for elements in beta2 under the partial 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 stanard multivariate linear regression estimator over the partial envelope estimator, for each element in beta1.
n	The number of observations in the data.

References

Su, Z. and Cook, R.D. (2011). Partial envelopes for efficient estimation in multivariate linear regression. Biometrika 98, 133 - 146.

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)
X1 <- fiberpaper[, 7]
X2 <- fiberpaper[, 5:6]
Y <- fiberpaper[, 1:4]
u <- u.penv(X1, X2, Y)
u

m <- penv(X1, X2, Y, 1)
m
m$beta1

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