[Rd] summary( prcomp(*, tol = .) ) -- and 'rank.'

[peter dalgaard]

As I see it, the display showing the first p << n PCs adding up to 100% of the variance is plainly wrong. 

I suspect it comes about via a mental short-circuit: If we try to control p using a tolerance, then that amounts to saying that the remaining PCs are effectively zero-variance, but that is (usually) not the intention at all. 

The common case is that the remainder terms have a roughly _constant_, small-ish variance and are interpreted as noise. Of course the magnitude of the noise is important information.  

-pd

> On 25 Mar 2016, at 00:02 , Steve Bronder <sbronder at stevebronder.com> wrote:
> 
> I agree with Kasper, this is a 'big' issue. Does your method of taking only
> n PCs reduce the load on memory?
> 
> The new addition to the summary looks like a good idea, but Proportion of
> Variance as you describe it may be confusing to new users. Am I correct in
> saying Proportion of variance describes the amount of variance with respect
> to the number of components the user chooses to show? So if I only choose
> one I will explain 100% of the variance? I think showing 'Total Proportion
> of Variance' is important if that is the case.
> 
> 
> Regards,
> 
> Steve Bronder
> Website: stevebronder.com
> Phone: 412-719-1282
> Email: sbronder at stevebronder.com
> 
> 
> On Thu, Mar 24, 2016 at 2:58 PM, Kasper Daniel Hansen <
> kasperdanielhansen at gmail.com> wrote:
> 
>> Martin, I fully agree.  This becomes an issue when you have big matrices.
>> 
>> (Note that there are awesome methods for actually only computing a small
>> number of PCs (unlike your code which uses svn which gets all of them);
>> these are available in various CRAN packages).
>> 
>> Best,
>> Kasper
>> 
>> On Thu, Mar 24, 2016 at 1:09 PM, Martin Maechler <
>> maechler at stat.math.ethz.ch
>>> wrote:
>> 
>>> 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
>>> 
>>> ______________________________________________
>>> R-devel at r-project.org mailing list
>>> https://stat.ethz.ch/mailman/listinfo/r-devel
>>> 
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-- 
Peter Dalgaard, Professor,
Center for Statistics, Copenhagen Business School
Solbjerg Plads 3, 2000 Frederiksberg, Denmark
Phone: (+45)38153501
Office: A 4.23
Email: pd.mes at cbs.dk  Priv: PDalgd at gmail.com