nnls: The Lawson-Hanson NNLS implemention of non-negative least squares

Description

An R interface to the Lawson-Hanson NNLS implementation of an algorithm for non-negative linear least squares that solves \(\min{\parallel A x - b \parallel_2}\) with the constraint \(x \ge 0\), where \(x \in R^n, b \in R^m\) and \(A\) is an \(m \times n\) matrix.

Usage

nnls(A, b)

Value

nnls returns an object of class "nnls".

The generic accessor functions coefficients,

fitted.values, deviance and residuals extract various useful features of the value returned by nnls.

An object of class "nnls" is a list containing the following components:

x: the parameter estimates.
deviance: the residual sum-of-squares.
residuals: the residuals, that is response minus fitted values.
fitted: the fitted values.
mode: a character vector containing a message regarding why termination occured.
passive: vector of the indices of x that are not bound at zero.
bound: vector of the indices of x that are bound at zero.
nsetp: the number of elements of x that are not bound at zero.

Arguments

A: numeric matrix with m rows and n columns
b: numeric vector of length m

References

Lawson CL, Hanson RJ (1974). Solving Least Squares Problems. Prentice Hall, Englewood Cliffs, NJ.

Lawson CL, Hanson RJ (1995). Solving Least Squares Problems. Classics in Applied Mathematics. SIAM, Philadelphia.

Examples

Run this code

## simulate a matrix A
## with 3 columns, each containing an exponential decay 
t <- seq(0, 2, by = .04)
k <- c(.5, .6, 1)
A <- matrix(nrow = 51, ncol = 3)
Acolfunc <- function(k, t) exp(-k*t)
for(i in 1:3) A[,i] <- Acolfunc(k[i],t)

## simulate a matrix X
## with 3 columns, each containing a Gaussian shape 
## the Gaussian shapes are non-negative
X <- matrix(nrow = 51, ncol = 3)
wavenum <- seq(18000,28000, by=200)
location <- c(25000, 22000, 20000) 
delta <- c(3000,3000,3000)
Xcolfunc <- function(wavenum, location, delta)
  exp( - log(2) * (2 * (wavenum - location)/delta)^2)
for(i in 1:3) X[,i] <- Xcolfunc(wavenum, location[i], delta[i])

## set seed for reproducibility
set.seed(3300)

## simulated data is the product of A and X with some
## spherical Gaussian noise added 
matdat <- A %*% t(X) + .005 * rnorm(nrow(A) * nrow(X))

## estimate the rows of X using NNLS criteria 
nnls_sol <- function(matdat, A) {
  X <- matrix(0, nrow = 51, ncol = 3)
  for(i in 1:ncol(matdat)) 
     X[i,] <- coef(nnls(A,matdat[,i]))
  X
}
X_nnls <- nnls_sol(matdat,A) 

matplot(X_nnls,type="b",pch=20)
abline(0,0, col=grey(.6))

if (FALSE) {
## can solve the same problem with L-BFGS-B algorithm
## but need starting values for x 
bfgs_sol <- function(matdat, A) {
  startval <- rep(0, ncol(A))
  fn1 <- function(par1, b, A) sum( ( b - A %*% par1)^2)
  X <- matrix(0, nrow = 51, ncol = 3)
  for(i in 1:ncol(matdat))  
    X[i,] <-  optim(startval, fn = fn1, b=matdat[,i], A=A,
                   lower = rep(0,3), method="L-BFGS-B")$par
    X
}
X_bfgs <- bfgs_sol(matdat,A) 

## the RMS deviation under NNLS is less than under L-BFGS-B 
sqrt(sum((X - X_nnls)^2)) < sqrt(sum((X - X_bfgs)^2))

## and L-BFGS-B is much slower 
system.time(nnls_sol(matdat,A))
system.time(bfgs_sol(matdat,A))

## can also solve the same problem by reformulating it as a
## quadratic program (this requires the quadprog package; if you
## have quadprog installed, uncomment lines below starting with
## only 1 "#" )

# library(quadprog)

# quadprog_sol <- function(matdat, A) {
#  X <- matrix(0, nrow = 51, ncol = 3)
#  bvec <- rep(0, ncol(A))
#  Dmat <- crossprod(A,A)
#  Amat <- diag(ncol(A))
#  for(i in 1:ncol(matdat)) { 
#    dvec <- crossprod(A,matdat[,i]) 
#    X[i,] <- solve.QP(dvec = dvec, bvec = bvec, Dmat=Dmat,
#                      Amat=Amat)$solution
#  }
#  X
# }
# X_quadprog <- quadprog_sol(matdat,A) 

## the RMS deviation under NNLS is about the same as under quadprog 
# sqrt(sum((X - X_nnls)^2))
# sqrt(sum((X - X_quadprog)^2))

## and quadprog requires about the same amount of time 
# system.time(nnls_sol(matdat,A))
# system.time(quadprog_sol(matdat,A))

}