\name{envnr}
\alias{envnr}
\title{Fit the envelope model when the number of reponses is smaller than the number of samples.}
\description{
  This function fits the envelope model to the responses and predictors,
  using the maximum likelihood estimation. When the dimension of the envelope is between 1 and r-1, we implemented the algorithm in Z. Su, G. Zhu, X. Chen and Y. Yang(2016). We use the algorithm in Rothman, A. J., Bickel, P. J., Levina, E., and Zhu, J. (2008) to estimate sigY and sigRes.
  }
\usage{
       ModelOutput = envnr(X, Y, u)
}

\arguments{
  \item{X}{Predictors. An n by p matrix, p is the number of predictors. The predictors can be univariate or multivariate, discrete or continuous.}
  \item{Y}{Multivariate responses. An n by r matrix, r is the number of responses and n is the number of observations. The responses must be continuous variables.}
  \item{u}{Dimension of the envelope. An integer between 0 and r.}
  \item{maxiter}{Maximum number of iterations.  Default value: 100.}
  \item{eps}{Relative convergence tolerance.  Default value: 1e-1. } 
  \item{init}{The initial value for the envelope subspace. An r by u matrix. Default value is the one generated by function initial_value.}
  \item{verbose}{Flag for print out model fitting process, logical 0 or 1. Default value: 0.}
}
%\details{}

\value{
  \item{alpha}{The estimated intercept in the envelope model.  An r by 1 vector.}
  \item{beta}{The envelope estimator of the regression coefficients. An r by p matrix.}
  \item{Gamma}{The orthogonal basis of the envelope subspace. An r by u semi-orthogonal matrix.}
  \item{Gamma0}{The orthogonal basis of the complement of the envelope subspace.  An r by r-u semi-orthogonal matrix.}
  \item{eta}{The coordinates of beta with respect to Gamma. A u by p matrix.} 
  \item{paramNum}{The number of parameters in the envelope model.  A positive integer.}
  \item{n}{The number of observations in the data.  A positive integer.}
  \item{r}{The number of response. A nonnegative integer.}
  \item{u}{Dimension of the envelope. An integer between 0 and r.}
  \item{p}{The number of predictors. A positive integer.}    
  \item{fit_time}{The time costs for fitting the envelope model.}
}
  
\references{
Su, Z., Zhu, G., Chen, X. and Yang, Y. (2016), Sparse Envelope Model: Efficient Estimation and Response Variable Selection in Multivariate Linear Regression. \emph{Biometrika}. 103, 579-593. 
}

\author{Zhihua Su\cr Maintainer: Guangyu Zhu \email{gzhu22@ufl.edu}}