Remove Columns (or Rows) From a Matrix to Make It Full Rank
From the QR decomposition with pivoting, (qr(x, tol) if
n ≥ p), if
the matrix is not of full rank, the corresponding columns (n >= p) or rows (n < p) are omitted to form a full rank matrix.
fullRank(x, tol = 1e-7, qrx = qr(x, tol=tol))
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.
This is useful for robustness algorithms that rely on X matrices
of full rank, e.g., adjOutlyingness.
This also works for numeric data frames and whenever qr() works correctly.
Martin Maechler
qr; for more sophisticated rank determination,
rankMatrix from package Matrix.
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.Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.