\name{spxenvnp}
\alias{spxenvnp}
\title{Fit the sparse envelope model when the number of predictors is greater than the number of samples.}
\description{
  This function fits the sparese envelope model to the predictors
  using the maximum likelihood estimation.  When the dimension of the
  envelope is between 1 and p-1, we implemented the algorithm in Guangyu Zhu and Zhihua Su (2017). 'Envelope-based Sparse Partial Least Squares', Manuscript submitted for publication. We use the algorithm in Rothman, A. J., Bickel, P. J., Levina, E., and Zhu, J. (2008) to estimate sigX and sigXcY.
  }
\usage{
		ModelOutput = spxenvnp(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{mu}{The estimated intercept in the envelope model.  An r by 1 vector.}
  \item{beta}{The envelope estimator of the regression coefficients beta. A p by r matrix.} 
   \item{Gamma}{The orthogonal basis of the envelope subspace. A p by u semi-orthogonal matrix.}
  \item{Gamma0}{The orthogonal basis of the complement of the envelope subspace.  A p by p-u semi-orthogonal matrix.}
  \item{eta}{The coordinates of beta with respect to Gamma. A u by r 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 Guangyu Zhu and Zhihua Su (2017). 'Envelope-based Sparse Partial Least Squares', Manuscript submitted for publication. 
}

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


\keyword{models}
\keyword{regression}