[Rd] summary( prcomp(*, tol = .) ) -- and 'rank.'
Martin Maechler
maechler at stat.math.ethz.ch
Thu Mar 24 18:09:27 CET 2016
Following from the R-help thread of March 22 on "Memory usage in prcomp",
I've started looking into adding an optional 'rank.' argument
to prcomp allowing to more efficiently get only a few PCs
instead of the full p PCs, say when p = 1000 and you know you
only want 5 PCs.
(https://stat.ethz.ch/pipermail/r-help/2016-March/437228.html
As it was mentioned, we already have an optional 'tol' argument
which allows *not* to choose all PCs.
When I do that,
say
C <- chol(S <- toeplitz(.9 ^ (0:31))) # Cov.matrix and its root
all.equal(S, crossprod(C))
set.seed(17)
X <- matrix(rnorm(32000), 1000, 32)
Z <- X %*% C ## ==> cov(Z) ~= C'C = S
all.equal(cov(Z), S, tol = 0.08)
pZ <- prcomp(Z, tol = 0.1)
summary(pZ) # only ~14 PCs (out of 32)
I get for the last line, the summary.prcomp(.) call :
> summary(pZ) # only ~14 PCs (out of 32)
Importance of components:
PC1 PC2 PC3 PC4 PC5 PC6 PC7 PC8
Standard deviation 3.6415 2.7178 1.8447 1.3943 1.10207 0.90922 0.76951 0.67490
Proportion of Variance 0.4352 0.2424 0.1117 0.0638 0.03986 0.02713 0.01943 0.01495
Cumulative Proportion 0.4352 0.6775 0.7892 0.8530 0.89288 0.92001 0.93944 0.95439
PC9 PC10 PC11 PC12 PC13 PC14
Standard deviation 0.60833 0.51638 0.49048 0.44452 0.40326 0.3904
Proportion of Variance 0.01214 0.00875 0.00789 0.00648 0.00534 0.0050
Cumulative Proportion 0.96653 0.97528 0.98318 0.98966 0.99500 1.0000
>
which computes the *proportions* as if there were only 14 PCs in
total (but there were 32 originally).
I would think that the summary should or could in addition show
the usual "proportion of variance explained" like result which
does involve all 32 variances or std.dev.s ... which are
returned from the svd() anyway, even in the case when I use my
new 'rank.' argument which only returns a "few" PCs instead of
all.
Would you think the current summary() output is good enough or
rather misleading?
I think I would want to see (possibly in addition) proportions
with respect to the full variance and not just to the variance
of those few components selected.
Opinions?
Martin Maechler
ETH Zurich
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