R help - Nov 2022 - Axis scaling for PCA biplot

Hi there,

I'm puzzled about the axis scaling in the PCA biplot.

Here's an example.

library(pdfCluster) # package cepp seems to have the same data set.

data(oliveoil) # 572 observations 10 variables

olive <- oliveoil[,3:10] # numerical variables

prolive <- princomp(olive)
summary(prolive)
# Importance of components:
#???????????????????????????? Comp.1?????? Comp.2 Comp.3?????? Comp.4
# Standard deviation???? 479.7299024 150.82827868 45.394449751 27.522646558
# Proportion of Variance?? 0.8970072?? 0.08866821? 0.008031707 0.002952451
# Cumulative Proportion??? 0.8970072?? 0.98567544? 0.993707152 0.996659603
#???????????????????????????? Comp.5?????? Comp.6 Comp.7?????? Comp.8
# Standard deviation???? 24.78169442 1.196956e+01 7.1390744088 6.9756965249
# Proportion of Variance? 0.00239367 5.584168e-04 0.0001986489 0.0001896608
# Cumulative Proportion?? 0.99905327 9.996117e-01 0.9998103392 1.0000000000

plot(prolive$scores)

# Scaling of this plot reproduces the variances of the components given 
in the summary,

# as does cov(prolive$scores). This seems all fine, however...

biplot(prolive)

I have no idea what the numbers on the axes of the biplot are, at least 
not the larger ones. Chances are the smaller ones indicate the loadings. 
The larger ones are neither the same as in the first plot, nor are they 
standardised to one, but they seem to be standardised somehow, as the 
range on x- and y-axis looks the same, which it shouldn't be if 
variances represented the PCA eigenvalues.

Can anyone explain this to me?

Actually the help page of biplot.princomp says something on this, but I 
don't get my head around it:

"scale

The variables are scaled by |lambda ^ scale| and the observations are 
scaled by |lambda ^ (1-scale)| where |lambda| are the singular values as 
computed by |princomp 
<https://www.rdocumentation.org/link/princomp?package=stats&version=3.6.2>|.
Normally |0 <= scale <= 1|, and a warning will be issued if the 
specified |scale| is outside this range."

The default value of scale seems to be 1, but then (1-scale) is zero so 
I'd assume data to be unscaled, but that should have reproduced the 
"plot" scale, shouldn't it?

Thanks,

Christian

-- 
Christian Hennig
Dipartimento di Scienze Statistiche "Paolo Fortunati",
Universita di Bologna, phone +39 05120 98163
christian.hennig at unibo.it

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? Tue, 15 Nov 2022 12:53:30 +0100
Christian Hennig <christian.hennig at unibo.it> ?????:
> The default value of scale seems to be 1, but then (1-scale) is zero
> so I'd assume data to be unscaled, but that should have reproduced
> the "plot" scale, shouldn't it?
I think this might be a documentation error. What actually happens
inside biplot.princomp is a bit different [1]:

   lam <- x$sdev[choices]
   n <- x$n.obs %||% 1
   lam <- lam * sqrt(n)
   
   if(scale != 0) lam <- lam^scale else lam <- 1
   
   biplot.default(t(t(scores[, choices]) / lam),
		   t(t(x$loadings[, choices]) * lam), ...)

Every score is divided by standard deviations (square roots of the
eigenvalues of the covariance matrix) and the square root of the number
of samples to the power of `scale`, while the loadings are multiplied
by the same numbers.

I am not sure why the singular values have to be multiplied by sqrt(n)
(the same thing happens in biplot.prcomp), but other than that, I think
that the documentation should say "the observations are scaled by
|lambda ^ -scale|", not (1-scale).

Passing scale = 0, on the other hand, should reproduce the "plot"
scale.

-- 
Best regards,
Ivan

[1]:
https://github.com/r-devel/r-svn/blob/c5f16db96ad76504893a1070d3b46b4093bb49c4/src/library/stats/R/biplot.R#L80-L95