NNLS {DECIPHER} | R Documentation

## Sequential Coordinate-wise Algorithm for the Non-negative Least Squares Problem

### Description

Consider the linear system \bold{A} x = b where \bold{A} \in R\textsuperscript{m x n}, x \in R\textsuperscript{n}, and b \in R\textsuperscript{m}. The technique of least squares proposes to compute x so that the sum of squared residuals is minimized. `NNLS` solves the least squares problem \min{||\bold{A} x = b||\textsuperscript{2}} subject to the constraint x ≥ 0. This implementation of the Sequential Coordinate-wise Algorithm uses a sparse input matrix \bold{A}, which makes it efficient for large sparse problems.

### Usage

    NNLS(A,
         b,
         precision = sqrt(.Machine$double.eps),
         processors = 1,
         verbose = TRUE)

### Arguments

`A` | List representing the sparse matrix with integer components i and j, numeric component x. The fourth component, dimnames, is a list of two components that contains the names for every row (component 1) and column (component 2).
`b` | Numeric matrix for the set of observed values. (See details section below.)
`precision` | The desired accuracy.
`processors` | The number of processors to use, or `NULL` to automatically detect and use all available processors.
`verbose` | Logical indicating whether to display progress.

### Details

The input b can be either a matrix or a vector of numerics. If it is a matrix then it is assumed that each column contains a set of observations, and the output x will have the same number of columns. This allows multiple NNLS problems using the same \bold{A} matrix to be solved simultaneously, and greatly accelerates computation relative to solving each sequentially.

### Value

A list of two components:

`x` | The matrix of non-negative values that best explains the observed values given by b.
`res` | A matrix of residuals given by \bold{A} x - b.

### References

Franc, V., et al. (2005). Sequential coordinate-wise algorithm for the non-negative least squares problem. Computer Analysis of Images and Patterns, 407-414.

### See Also

`Array2Matrix`, `DesignArray`

### Examples

    # unconstrained least squares:
    A <- matrix(c(1, -3, 2, -3, 10, -5, 2, -5, 6), ncol=3)
    b <- matrix(c(27, -78, 64), ncol=1)
    x <- solve(crossprod(A), crossprod(A, b))

    # Non-negative least squares:
    w <- which(A > 0, arr.ind=TRUE)
    A <- list(i=w[,"row"], j=w[,"col"], x=A[w],
              dimnames=list(1:dim(A)[1], 1:dim(A)[2]))
    x_nonneg <- NNLS(A, b)

    # compare the unconstrained and constrained solutions:
    cbind(x, x_nonneg$x)

    # the input value "b" can also be a matrix:
    b2 <- matrix(b, nrow=length(b), ncol=2) # repeat b in two columns
    x_nonneg <- NNLS(A, b2) # solution is repeated in two output columns

[Package DECIPHER version 2.16.1 Index]