R/samQL.R

In SAM: Sparse Additive Modelling

#-----------------------------------------------------------------------#
# Package: SAM                                                          #
# Method: Sparse Additive Modelling using Quadratic Loss                #
#-----------------------------------------------------------------------#

#' Training function of Sparse Additive Models
#'
#' The regression model is learned using training data.
#'
#' We adopt various computational algorithms including the block coordinate descent, fast iterative soft-thresholding algorithm, and newton method. The computation is further accelerated by "warm-start" and "active-set" tricks.
#'
#' @param X The \code{n} by \code{d} design matrix of the training set, where \code{n} is sample size and \code{d} is dimension.
#' @param y The \code{n}-dimensional response vector of the training set, where \code{n} is sample size.
#' @param p The number of basis spline functions. The default value is 3.
#' @param lambda A user supplied lambda sequence. Typical usage is to have the program compute its own lambda sequence based on nlambda and lambda.min.ratio. Supplying a value of lambda overrides this. WARNING: use with care. Do not supply a single value for lambda. Supply instead a decreasing sequence of lambda values. samQL relies on its warms starts for speed, and its often faster to fit a whole path than compute a single fit.
#' @param nlambda The number of lambda values. The default value is 30.
#' @param lambda.min.ratio Smallest value for lambda, as a fraction of lambda.max, the (data derived) entry value (i.e. the smallest value for which all coefficients are zero). The default is 5e-3.
#' @param thol Stopping precision. The default value is 1e-5.
#' @param max.ite The number of maximum iterations. The default value is 1e5.
#' @param regfunc A string indicating the regularizer. The default value is "L1". You can also assign "MCP" or "SCAD" to it.
#' @return
#' \item{p}{
#'   The number of basis spline functions used in training.
#' }
#' \item{X.min}{
#'   A vector with each entry corresponding to the minimum of each input variable. (Used for rescaling in testing)
#' }
#' \item{X.ran}{
#'   A vector with each entry corresponding to the range of each input variable. (Used for rescaling in testing)
#' }
#' \item{lambda}{
#'   A sequence of regularization parameter used in training.
#' }
#' \item{w}{
#'   The solution path matrix (\code{d*p} by length of \code{lambda}) with each column corresponding to a regularization parameter. Since we use the basis expansion, the length of each column is \code{d*p+1}.
#' }
#' \item{intercept}{
#'   The solution path of the intercept.
#' }
#' \item{df}{
#'   The degree of freedom of the solution path (The number of non-zero component function)
#' }
#' \item{knots}{
#'   The \code{p-1} by \code{d} matrix. Each column contains the knots applied to the corresponding variable.
#' }
#' \item{Boundary.knots}{
#'   The \code{2} by \code{d} matrix. Each column contains the boundary points applied to the corresponding variable.
#' }
#' \item{func_norm}{
#'   The functional norm matrix (\code{d} by length of \code{lambda}) with each column corresponds to a regularization parameter. Since we have \code{d} input variables, the length of each column is \code{d}.
#' }
#' \item{sse}{
#'   Sums of square errors of the solution path.
#' }
#' @seealso \code{\link{SAM}},\code{\link{plot.samQL},\link{print.samQL},\link{predict.samQL}}
#' @examples
#'
#' ## generating training data
#' n = 100
#' d = 500
#' X = 0.5*matrix(runif(n*d),n,d) + matrix(rep(0.5*runif(n),d),n,d)
#'
#' ## generating response
#' y = -2*sin(X[,1]) + X[,2]^2-1/3 + X[,3]-1/2 + exp(-X[,4])+exp(-1)-1
#'
#' ## Training
#' out.trn = samQL(X,y)
#' out.trn
#'
#' ## plotting solution path
#' plot(out.trn)
#'
#' ## generating testing data
#' nt = 1000
#' Xt = 0.5*matrix(runif(nt*d),nt,d) + matrix(rep(0.5*runif(nt),d),nt,d)
#'
#' yt = -2*sin(Xt[,1]) + Xt[,2]^2-1/3 + Xt[,3]-1/2 + exp(-Xt[,4])+exp(-1)-1
#'
#' ## predicting response
#' out.tst = predict(out.trn,Xt)
#' @useDynLib SAM grplasso
#' @export
samQL = function(X, y, p=3, lambda = NULL, nlambda = NULL, lambda.min.ratio = 5e-3, thol=1e-5, max.ite = 1e5, regfunc="L1"){

  gcinfo(FALSE)
  fit = list()
  fit$p = p

  fit = list()
  fit$p = p

  X = as.matrix(X)
  y = as.vector(y)

  n = nrow(X)
  d = ncol(X)
  m = d*p

  X.min = apply(X,2,min)
  X.max = apply(X,2,max)
  X.ran = X.max - X.min
  X.min.rep = matrix(rep(X.min,n),nrow=n,byrow=T)
  X.ran.rep = matrix(rep(X.ran,n),nrow=n,byrow=T)
  X = (X-X.min.rep)/X.ran.rep

  fit$X.min = X.min
  fit$X.ran = X.ran

  Z = matrix(0,n,m)
  fit$nkots = matrix(0,p-1,d)
  fit$Boundary.knots = matrix(0,2,d)
  for(j in 1:d){
    tmp = (j-1)*p + c(1:p)
    tmp0 = ns(X[,j],df=p)
    Z[,tmp] = tmp0
    fit$nkots[,j] = attr(tmp0,'knots')
    fit$Boundary.knots[,j] = attr(tmp0,'Boundary.knots')
  }

  Z.mean = apply(Z,2,mean)
  Z.mean.rep = matrix(rep(Z.mean,n),nrow=n,byrow=T)
  Z = Z - Z.mean.rep
  y.mean = mean(y)
  y = y - y.mean

  lambda_input = 1
  if(is.null(lambda)){
    lambda_input = 0
    if(is.null(nlambda))
      nlambda = 30
    lambda = exp(seq(log(1),log(lambda.min.ratio),length = nlambda))
  } else nlambda = length(lambda)


  out = .C("grplasso",y = as.double(y), X = as.double(Z), lambda = as.double(lambda), nnlambda = as.integer(nlambda), nn = as.integer(n), dd = as.integer(d), pp = as.integer(p), ww = as.double(matrix(0,m,nlambda)), mmax_ite = as.integer(max.ite), tthol = as.double(thol), regfunc = as.character(regfunc), iinput = as.integer(lambda_input), df=as.integer(rep(0,nlambda)), sse=as.double(rep(0,nlambda)), func_norm = as.double(matrix(0,d,nlambda)), package="SAM")

  fit$lambda = out$lambda
  fit$w = matrix(out$w,ncol=nlambda)
  fit$df = out$df
  fit$sse = out$sse
  fit$func_norm = matrix(out$func_norm,ncol=nlambda)
  fit$intercept = rep(y.mean,nlambda) - t(Z.mean)%*%fit$w
  fit$XX = out$X

  rm(out,X,y,Z,X.min.rep,X.ran.rep,Z.mean.rep)

  class(fit) = "samQL"
  return(fit)
}

#' Printing function for S3 class \code{"samQL"}
#'
#' Summarize the information of the object with S3 class \code{samQL}.
#'
#' The output includes length and d.f. of the regularization path.
#'
#' @param x An object with S3 class \code{"samQL"}
#' @param \dots System reserved (No specific usage)
#' @seealso \code{\link{samQL}}
#' @export
print.samQL = function(x,...){
  cat("Path length:",length(x$df),"\n")
  cat("d.f.:",x$df[1],"--->",x$df[length(x$df)],"\n")
}

#' Plot function for S3 class \code{"samQL"}
#'
#' This function plots the regularization path (regularization parameter versus functional norm)
#'
#' The horizontal axis is for the regularization parameters in log scale. The vertical axis is for the functional norm of each component.
#'
#' @param x An object with S3 class \code{"samQL"}
#' @param \dots System reserved (No specific usage)
#' @seealso \code{\link{samQL}}
#' @export
plot.samQL = function(x,...){
  par = par(omi = c(0.0, 0.0, 0, 0), mai = c(1, 1, 0.1, 0.1))
  matplot(x$lambda,t(x$func_norm),type="l",xlab="Regularization Parameters",ylab = "Functional Norms",cex.lab=2,log="x",lwd=2)
}

#' Prediction function for S3 class \code{"samQL"}
#'
#' Predict the labels for testing data.
#'
#' The testing dataset is rescale to the samQLe range, and expanded by the samQLe spline basis functions as the training data.
#'
#' @param object An object with S3 class \code{"samQL"}.
#' @param newdata The testing dataset represented in a \code{n} by \code{d} matrix, where \code{n} is testing sample size and \code{d} is dimension.
#' @param \dots System reserved (No specific usage)
#' @return
#' \item{values}{
#'   Predicted values also represented in a \code{n} by the length of \code{lambda} matrix, where \code{n} is testing sample size.
#' }
#' @seealso \code{\link{samQL}}
#' @export
predict.samQL = function(object, newdata,...){
  gcinfo(FALSE)
  out = list()
  nt = nrow(newdata)
  d = ncol(newdata)
  X.min.rep = matrix(rep(object$X.min,nt),nrow=nt,byrow=T)
  X.ran.rep = matrix(rep(object$X.ran,nt),nrow=nt,byrow=T)

  newdata = (newdata-X.min.rep)/X.ran.rep
  newdata = pmax(newdata,0)
  newdata = pmin(newdata,1)

  m = object$p*d

  Zt = matrix(0,nt,m)

  for(j in 1:d){
    tmp = (j-1)*object$p + c(1:object$p)
    Zt[,tmp] = ns(newdata[,j],df=object$p,knots=object$knots[,j],Boundary.knots=object$Boundary.knots[,j])
  }

  out$values = cbind(Zt,rep(1,nt))%*%rbind(object$w,object$intercept)

  rm(Zt,newdata)

  return(out)
}