robustbase (version 0.95-1)

fullRank: Remove Columns (or Rows) From a Matrix to Make It Full Rank

Description

From the QR decomposition with pivoting, (qr(x, tol) if \(n \ge p\)), if the matrix is not of full rank, the corresponding columns (\(n \ge p\)) or rows (\(n < p\)) are omitted to form a full rank matrix.

Usage

fullRank(x, tol = 1e-7, qrx = qr(x, tol=tol))

Value

a version of the matrix x, with less columns or rows if

x's rank was smaller than min(n,p).

If x is of full rank, it is returned unchanged.

Arguments

x

a numeric matrix of dimension \(n \times p\), or a similar object for which qr() works.

tol

tolerance for determining rank (deficiency). Currently is simply passed to qr.

qrx

optionally may be used to pass a qr(x, ..); only used when p <= n.

Author

Martin Maechler

See Also

qr; for more sophisticated rank determination, rankMatrix from package Matrix.

Examples

stopifnot(identical(fullRank(wood), wood))

## More sophisticated and delicate
dim(T <- tcrossprod(data.matrix(toxicity))) # 38 x 38
dim(T. <- fullRank(T)) # 38 x 10
if(requireNamespace("Matrix")) {
  rMmeths <- eval(formals(Matrix::rankMatrix)$method)
  rT. <- sapply(rMmeths, function(.m.) Matrix::rankMatrix(T., method = .m.))
  print(rT.) # "qr" (= "qrLinpack"): 13,  others rather 10
}
dim(T.2 <- fullRank(T, tol = 1e-15))# 38 x 18
dim(T.3 <- fullRank(T, tol = 1e-12))# 38 x 13
dim(T.3 <- fullRank(T, tol = 1e-10))# 38 x 13
dim(T.3 <- fullRank(T, tol = 1e-8 ))# 38 x 12
dim(T.) # default from above          38 x 10
dim(T.3 <- fullRank(T, tol = 1e-5 ))# 38 x 10 -- still

plot(svd(T, 0,0)$d, log="y", main = "singular values of T", yaxt="n")
axis(2, at=10^(-14:5), las=1)
## pretty clearly indicates that  rank 10  is "correct" here.