NNLS: Sequential Coordinate-wise Algorithm for the Non-negative...

In DECIPHER: Tools for curating, analyzing, and manipulating biological sequences

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

Example output

Loading required package: Biostrings
Loading required package: BiocGenerics
Loading required package: parallel

Attaching package: 'BiocGenerics'

The following objects are masked from 'package:parallel':

    clusterApply, clusterApplyLB, clusterCall, clusterEvalQ,
    clusterExport, clusterMap, parApply, parCapply, parLapply,
    parLapplyLB, parRapply, parSapply, parSapplyLB

The following objects are masked from 'package:stats':

    IQR, mad, sd, var, xtabs

The following objects are masked from 'package:base':

    Filter, Find, Map, Position, Reduce, anyDuplicated, append,
    as.data.frame, basename, cbind, colMeans, colSums, colnames,
    dirname, do.call, duplicated, eval, evalq, get, grep, grepl,
    intersect, is.unsorted, lapply, lengths, mapply, match, mget,
    order, paste, pmax, pmax.int, pmin, pmin.int, rank, rbind,
    rowMeans, rowSums, rownames, sapply, setdiff, sort, table, tapply,
    union, unique, unsplit, which, which.max, which.min

Loading required package: S4Vectors
Loading required package: stats4

Attaching package: 'S4Vectors'

The following object is masked from 'package:base':

    expand.grid

Loading required package: IRanges
Loading required package: XVector

Attaching package: 'Biostrings'

The following object is masked from 'package:base':

    strsplit

Loading required package: RSQLite

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1    1   17
2   -4    0
3    7    5

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DECIPHER documentation built on Nov. 8, 2020, 8:30 p.m.