nnls | R Documentation |
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.
nnls(A, b)
A |
numeric matrix with |
b |
numeric vector of length |
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 |
bound |
vector of the indices of |
nsetp |
the number of elements of |
This is an R interface to the Fortran77 code distributed with the book referenced below by Lawson CL, Hanson RJ (1995), obtained from Netlib (file ‘lawson-hanson/all’), with a trivial modification to return the variable NSETP.
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.
nnnpls, the method "L-BFGS-B"
for optim,
solve.QP, bvls
## 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))
## Not run:
## 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))
## End(Not run)
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