\name{spenvnr}
\alias{spenvnr}
\title{Fit the sparse envelope model when the number of reponses is greater than the number of samples.}
\description{
  This function fits the sparese envelope model to the responses
  using the maximum likelihood estimation.  When the dimension of the
  envelope is between 1 and r-1, we implemented the algorithm in Su Z, Zhu G, Chen X, Yang Y. Sparse envelope model: efficient estimation and response variable selection in multivariate linear regression. Biometrika. 2016 Sep 1;103(3):579-93. We use the algorithm in Rothman, A. J., Bickel, P. J., Levina, E., and Zhu, J. (2008) to estimate sigY and sigRes.
  }
\usage{
		ModelOutput = spenvnr(X, Y, u, lambda)
		
}

\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 number of observations. The responses must be continuous variables.}
  \item{u}{Dimension of the envelope. An integer between 0 and r.}
  \item{lambda}{Tuning parameter.}
  \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 predictor. A positive integer.} 
  \item{q}{The number of active response. A positive integer.}   
  \item{where_1}{The active responses chosen by the sparse envelope model.}
  \item{where_0}{The inactive responses chosen by the sparse envelope model.}
  \item{lambda}{The tuning parameter choosed by BIC.}
  \item{fit_time}{The time costs for fitting the envelope model.}
}
  
\references{
The codes are implemented based on the algorithm in 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}}


\keyword{models}
\keyword{regression}