# nnnpls: An implementation of least squares with non-negative and... In nnls: The Lawson-Hanson algorithm for non-negative least squares (NNLS)

## 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

 1 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.

## 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.

nnls, the method "L-BFGS-B" for optim, solve.QP, bvls

## Examples

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 ## 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)

### Example output

[1] TRUE
user  system elapsed
0.001   0.000   0.001
user  system elapsed
0.018   0.000   0.017

nnls documentation built on May 2, 2019, 3:13 p.m.