[R] prcomp - arbitrary direction of the returned principal components

[Chris Evans]

I agree with this and I'm not very sure why you feel you need the signs
fixed one way: they are arbitrary and how they come out is generally a
function of things in the handling of rounding as values hit the limit
of the finite arithmetic in the particular program and OS and hardware
on which it's running.  Can't you just say that?!

If you must have it aligned one way then I think the only thing you can
do is to select the row/item/variable with the highest absolute loading
on the component and set that to be positive (say, whether you choose
positive or negative is up to you).  I think this does it:

### create some data to analyse
set.seed(12345)
n <- 500 # number of observations
k <- 8 # number of variables
fuzz <- .1 # used to add noise
varLatent <- rnorm(n) # create the values of the dominant PC
vecLoadings <- c(rep(1, k/2), rep(-1, k/2)) # binary loadings 

### make the raw data
matData <- matrix(rep(NA, n * k), ncol = k)
for (i in 1:k) {
  matData[, i] <- vecLoadings[i] * varLatent + rnorm(n, fuzz)
}
head(matData)

### get the PCA
matPrcomp <- prcomp(matData)$rotation
round(matPrcomp, 2)
# PC1   PC2   PC3   PC4   PC5   PC6   PC7   PC8
# [1,]  0.34  0.00 -0.74  0.32 -0.38 -0.25  0.15  0.00
# [2,]  0.37 -0.19  0.49  0.33 -0.57  0.38  0.07 -0.01
# [3,]  0.34  0.80  0.14  0.11  0.10 -0.05 -0.16 -0.43
# [4,]  0.33 -0.47 -0.16  0.10  0.34  0.17 -0.61 -0.35
# [5,] -0.35  0.15 -0.15 -0.34 -0.61  0.10 -0.57 -0.08
# [6,] -0.37 -0.25  0.11  0.13 -0.14 -0.30  0.27 -0.77
# [7,] -0.35  0.07  0.17  0.72  0.04 -0.32 -0.38  0.28
# [8,] -0.38  0.17 -0.32  0.33  0.14  0.75  0.16 -0.12

### find the sign of the maximum absolute loading for each component
vecMaxItemSigns <- apply(matPrcomp, 2,
function(x){sign(x[(which.max(abs(x)))])})
vecMaxItemSigns
# PC1 PC2 PC3 PC4 PC5 PC6 PC7 PC8 
# -1   1  -1   1  -1   1  -1  -1

### now use that to create a new PCA where the strongest loadings are
always positive
newMatPrcomp <- matrix(rep(NA, k * k), ncol = k)
colnames(newMatPrcomp) <- colnames(matPrcomp)
rownames(newMatPrcomp) <- rownames(matPrcomp)
for (i in 1:k) {
  newMatPrcomp[, i] <- matPrcomp[, i] * vecMaxItems[i]
}
round(newMatPrcomp, 2)
# PC1   PC2   PC3   PC4   PC5   PC6   PC7   PC8
# [1,] -0.34  0.00  0.74  0.32  0.38 -0.25 -0.15  0.00
# [2,] -0.37 -0.19 -0.49  0.33  0.57  0.38 -0.07  0.01
# [3,] -0.34  0.80 -0.14  0.11 -0.10 -0.05  0.16  0.43
# [4,] -0.33 -0.47  0.16  0.10 -0.34  0.17  0.61  0.35
# [5,]  0.35  0.15  0.15 -0.34  0.61  0.10  0.57  0.08
# [6,]  0.37 -0.25 -0.11  0.13  0.14 -0.30 -0.27  0.77
# [7,]  0.35  0.07 -0.17  0.72 -0.04 -0.32  0.38 -0.28
# [8,]  0.38  0.17  0.32  0.33 -0.14  0.75 -0.16  0.12


Apologies for the coding: I'm a better therapist than coder and
it's a while since I've done much in base R like this.  Quite fun
to get back to it!  R artistes can probably do that in four lines!

But I'm not convinced doing this to "fix" the signs is really 
worth it however many lines one uses to code it!!

Very best all,

Chris

 On Thu, 2022-10-13 at 09:58 +0530, Ashim Kapoor wrote:
> Dear Aaron,
> 
> Many thanks for your reply.
> 
> Please allow me to illustrate my query a bit.
> 
> I take some data, throw it to prcomp and extract the x data frame
> from prcomp.
> 
> From ?prcomp:
> 
>        x: if ‘retx’ is true the value of the rotated data (the
> centred
>           (and scaled if requested) data multiplied by the ‘rotation’
>           matrix) is returned.  Hence, ‘cov(x)’ is the diagonal
> matrix
>           ‘diag(sdev^2)’.  For the formula method, ‘napredict()’ is
>           applied to handle the treatment of values omitted by the
>           ‘na.action’.
> 
> I consider x[,1] as my index. This makes sense as x[,1] is the
> projection of the data on the FIRST principal component.
> Now this x[,1] can be a high +ve number or a low -ve number. I can't
> ignore the sign.
> 
> If I ignore the sign by taking the absolute value, the HIGH / LOW
> stress values will be indistinguishable.
> 
> Hence I do not think using absolute values of x[,1] is the solution.
> Yes it will make the results REPRODUCIBLE but that will be at the
> cost
> of losing information.
> 
> Any other idea ?
> 
> Many thanks,
> Ashim
> 
> On Wed, Oct 12, 2022 at 5:23 PM Ebert,Timothy Aaron <tebert using ufl.edu>
> wrote:
> > 
> > Use absolute value
> > 
> > Tim
> > 
> > -----Original Message-----
> > From: R-help <r-help-bounces using r-project.org> On Behalf Of Ashim
> > Kapoor
> > Sent: Wednesday, October 12, 2022 7:48 AM
> > To: R Help <r-help using r-project.org>
> > Subject: [R] prcomp - arbitrary direction of the returned principal
> > components
> > 
> > [External Email]
> > 
> > Dear R experts,
> > 
> > From ?prcomp,
> > 
> > ---- snip -----
> > Note:
> > 
> >      The signs of the columns of the rotation matrix are arbitrary,
> > and
> >      so may differ between different programs for PCA, and even
> > between
> >      different builds of R.
> > ---- snip ------
> > 
> > My problem is that I am building an index based on Principal
> > Components Analysis.
> > When the index is high it should indicate stress in the market. Due
> > to the arbitrary sign sometimes I get an index which is HIGH when
> > there is stress and sometimes I get  the OPPOSITE - an index which
> > is LOW when there is stress.
> > This program is shared with other people who may have a different
> > build of R.
> > 
> > I can forcefully use a NEGATIVE sign to FLIP the index when it is
> > LOW.
> > That works.
> > 
> > Now my query is : Just like we do set.seed(1234) and force the
> > pattern of generation of random number and make it REPRODUCIBLE,
> > can I do something like :
> > 
> > set.direction.for.vector.in.pca(1234)
> > 
> > Now each time I do prcomp it should choose the SAME ( high or low )
> > direction of the principle component on ANY computer having ANY
> > version of R installed.
> > 
> > That's what I want. I don't want the the returned principal
> > component to be HIGH(LOW) on my computer and LOW(HIGH) on someone
> > else's computer.
> > That would confuse the people the code is shared with.
> > 
> > Is this possible ? How do people deal with this ?
> > 
> > Many thanks,
> > Ashim
> > 
> > 
> 
> ______________________________________________
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-- 
Chris Evans (he/him)        <chris using psyctc.org>     
Visiting Professor, UDLA, Quito, Ecuador & Honorary
Professor, University of Roehampton, London, UK.
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