rrenv: Fit the reduced-rank envelope model

In Renvlp: Computing Envelope Estimators

Fit the reduced-rank envelope model

Description

Fit the reduced-rank envelope model with rank d and fixed envelope dimension u.

Usage

rrenv(X, Y, u, d, asy = TRUE)

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.
u	Dimension of the envelope. An integer between 0 and r.
d	The rank of the coefficient matrix. An integer between 0 and u.
asy	Flag for computing the asymptotic variance of the reduced rank 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.

Details

This function fits the reduced rank envelope model to the responses and predictors,

Y_{i} = \alpha + \Gamma\eta BX_{i}+\varepsilon_{i}, \Sigma=\Gamma\Omega\Gamma'+\Gamma_{0}\Omega_{0}\Gamma'_{0}, i=1, ..., n,

using the maximum likelihood estimation. The errors \varepsilon_{i} follow a normal distribution. When 0 < d < u < r, the estimation procedure in Cook et al. (2015) is implemented. When d < u = r, then the model is equivalent to a reduced rank regression model. When d = u, or d = p < r, then B can be taken as the identity matrix and the model reduces to a response envelope model. When the dimension is d = u = r, then the envelope model degenerates to the standard multivariate linear regression. When the u = 0, it means that X and Y are uncorrelated, and the fitting is different. If the error covariance matrix is nonconstant, see the function rrenv.apweights.

Value

The output is a list that contains the following components:

Gamma	An orthogonal basis of the envelope subspace.
Gamma0	An orthogonal basis of the complement of the envelope subspace.
mu	The estimated intercept.
beta	The envelope estimator of the regression coefficients.
Sigma	The envelope estimator of the error covariance matrix.
eta	The eta matrix in the coefficient matrix.
B	The B matrix in the coefficient matrix.
Omega	The coordinates of Sigma with respect to Gamma.
Omega0	The coordinates of Sigma with respect to Gamma0.
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 reduced rank 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 (with consideration of nonconstant variance) over the envelope estimator, for each element in beta.
n	The number of observations in the data.

References

Cook, R. D., Forzani, L. and Zhang, X. (2015). Envelopes and reduced-rank regression. Biometrika 102, 439-456.

Forzani, L. and Su, Z. (2021). Envelopes for elliptical multivariate linear regression. Statist. Sinica 31, 301-332.

Examples

data(vehicles)
X <- vehicles[, 1:11]  
Y <- vehicles[, 12:15]
X <- scale(X)
Y <- scale(Y)  # The scales of Y are vastly different, so scaling is reasonable here
d <- d.select(X, Y, 0.01)
d

## Not run: u <- u.rrenv(X, Y, 2)
## Not run: u

m <- rrenv(X, Y, 4, 2)
m
m$beta

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