R: An implementation of least squares with non-negative and...
nnnpls
R Documentation
An implementation of least squares with non-negative and non-positive
constraints
Description
An implementation of an algorithm for linear least squares problems
with non-negative and non-positive
constraints based on the Lawson-Hanson
NNLS algorithm. Solves min{parallel A x - b parallel_2}
with the constraint x_i ≥ 0
if con_i ≥ 0 and x_i ≤ 0 otherwise, where
x, con in R^n, b in R^m, and A is an m \times n matrix.
Usage
nnnpls(A, b, con)
Arguments
A
numeric matrix with m rows and n columns
b
numeric vector of length m
con
numeric vector of length m where element i
is negative if and only if element i of the solution vector
x should be constrained to non-positive, as opposed to
non-negative, values.
Value
nnnpls returns
an object of class "nnnpls".
The generic accessor functions coefficients,
fitted.values, deviance and residuals extract
various useful features of the value returned by nnnpls.
An object of class "nnnpls" 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.
Source
This is an R interface to Fortran77 code distributed
with the book referenced below by Lawson CL, Hanson RJ (1995),
obtained from Netlib (file ‘lawson-hanson/all’), with some
trivial modifications to allow for the combination of constraints to
non-negative and non-positive values, and to return the variable
NSETP.
Lawson CL, Hanson RJ (1995). Solving Least Squares Problems. Classics
in Applied Mathematics. SIAM, Philadelphia.
See Also
nnls, the method "L-BFGS-B" for optim,
solve.QP, bvls
Examples
## 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
## 2 of the Gaussian shapes are non-negative and 1 is non-positive
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])
X[,2] <- -X[,2]
## 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 NNNPLS criteria
nnnpls_sol <- function(matdat, A) {
X <- matrix(0, nrow = 51, ncol = 3)
for(i in 1:ncol(matdat))
X[i,] <- coef(nnnpls(A,matdat[,i],con=c(1,-1,1)))
X
}
X_nnnpls <- nnnpls_sol(matdat,A)
## Not run:
## can solve the same problem with L-BFGS-B algorithm
## but need starting values for x and
## impose a very low/high bound where none is desired
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, -1000, 0), upper=c(1000,0,1000),
method="L-BFGS-B")$par
X
}
X_bfgs <- bfgs_sol(matdat,A)
## the RMS deviation under NNNPLS is less than under L-BFGS-B
sqrt(sum((X - X_nnnpls)^2)) < sqrt(sum((X - X_bfgs)^2))
## and L-BFGS-B is much slower
system.time(nnnpls_sol(matdat,A))
system.time(bfgs_sol(matdat,A))
## can also solve the same problem by reformulating it as a
## quadratic program (this requires the quadprog package; if you
## have quadprog installed, uncomment lines below starting with
## only 1 "#" )
# library(quadprog)
# quadprog_sol <- function(matdat, A) {
# X <- matrix(0, nrow = 51, ncol = 3)
# bvec <- rep(0, ncol(A))
# Dmat <- crossprod(A,A)
# Amat <- diag(c(1,-1,1))
# for(i in 1:ncol(matdat)) {
# dvec <- crossprod(A,matdat[,i])
# X[i,] <- solve.QP(dvec = dvec, bvec = bvec, Dmat=Dmat,
# Amat=Amat)$solution
# }
# X
# }
# X_quadprog <- quadprog_sol(matdat,A)
## the RMS deviation under NNNPLS is about the same as under quadprog
# sqrt(sum((X - X_nnnpls)^2))
# sqrt(sum((X - X_quadprog)^2))
## and quadprog requires about the same amount of time
# system.time(nnnpls_sol(matdat,A))
# system.time(quadprog_sol(matdat,A))
## End(Not run)