robustbase (version 0.92-5)

classPC: Compute Classical Principal Components via SVD or Eigen

Description

Compute classical principal components (PC) via SVD (svd or eigenvalue decomposition (eigen) with non-trivial rank determination.

Usage

classPC(x, scale = FALSE, center = TRUE, signflip = TRUE,
        via.svd = n > p, scores = FALSE)

Arguments

x

a numeric matrix.

scale

logical indicating if the matrix should be scaled; it is mean centered in any case (via scale(*, scale=scale)c

center

logical or numeric vector for centering the matrix.

signflip

logical indicating if the sign(.) of the loadings should be determined should flipped such that the absolutely largest value is always positive.

via.svd

logical indicating if the computation is via SVD or Eigen decomposition; the latter makes sense typically only for n

scores

logical indicating

Value

• a list with components
• rank
  the (numerical) matrix rank of x; an integer number, say $k$, from 0:min(dim(x)). In the $n > p$ case, it is rankMM(x).
• eigenvalues
  the $k$ eigenvalues, in the $n > p$ case, proportional to the variances.
• loadings
  the loadings, a $p \times k$ matrix.
• scores
  if the scores argument was true, the $n \times k$ matrix of scores, where $k$ is the rank above.
• center
  a numeric $p$-vector of means, unless the center argument was false.
• scale
  if the scale argument was not false, the scale used, a $p$-vector.

concept

PCA

See Also

In spirit very similar to R's standard prcomp and princomp, one of the main differences being how the rank is determined via a non-trivial tolerance.

Examples

set.seed(17)
x <- matrix(rnorm(120), 10, 12) # n < p {the unusual case}
pcx  <- classPC(x)
(k <- pcx$rank) # = 9  [after centering!]
pc2  <- classPC(x, scores=TRUE)
pcS  <- classPC(x, via.svd=TRUE)
all.equal(pcx, pcS, tol = 1e-8)
## TRUE: eigen() & svd() based PC are close here
pc0 <- classPC(x, center=FALSE, scale=TRUE)
pc0$rank # = 10  here *no* centering (as E[.] = 0)

## Loadings are orthnormal:
zapsmall( crossprod( pcx$loadings ) )

## PC Scores are roughly orthogonal:
S.S <- crossprod(pc2$scores)
print.table(signif(zapsmall(S.S), 3), zero.print=".")
stopifnot(all.equal(pcx$eigenvalues, diag(S.S)/k))

## the usual n > p case :
pc.x <- classPC(t(x))
pc.x$rank # = 10, full rank in the n > p case
stopifnot(classPC(x, center=FALSE)$rank == min(dim(x)))
ii <- names(pcx); ii <- ii[ii != "scores"]
stopifnot(all.equal(pcx[ii], pc2[ii], tol=0),
	  all.equal(pcx, pcS, tol=1e-8),
	  length(pc.x$center) == 10, identical(pc0$center, FALSE),
          all.equal(crossprod(pcx $loadings), diag(9)),
          all.equal(crossprod(pc.x$loadings), diag(10)),
          all.equal(colSums(abs(pcx$loadings)),
                    c(2.69035673, 2.78449399, 3.00148438,
                      2.9016688,  2.49400759, 2.90477204,
                      3.01639807, 2.4217181, 2.64665957)),
	  length(pc0$scale) == 12)