classPC function

Compute Classical Principal Components via SVD or Eigen

Compute classical principal components (PC) via SVD (svd

or eigenvalue decomposition (eigen) with non-trivial rank determination.

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 <= p.
• scores: logical indicating

Author(s)

Valentin Todorov; efficiency tweaks by Martin Maechler

Returns

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 $p * k$ matrix.

• scores: if the scores argument was true, the $n * k$ 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.

See Also

In spirit very similar to '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

cpc1 <- classPC(cbind(1:3)) # 1-D matrix
stopifnot(cpc1$rank == 1,
          all.equal(cpc1$eigenvalues, 1),
          all.equal(cpc1$loadings, 1))