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prcomp eigenvalues

4 messages · Rebecca Young, Sundar Dorai-Raj, Brian Ripley +1 more

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Rebecca Young
# 
Hello,

Can you get eigenvalues in addition to eigevectors using prcomp?  If so how?
I am unable to use princomp due to small sample sizes.
Thank you in advance for your help!
Rebecca Young

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Rebecca Young
Graduate Student
Ecology & Evolutionary Biology, Badyaev Lab
University of Arizona
1041 E Lowell
Tucson, AZ 85721-0088
Office: 425BSW
rlyoung at email.arizona.edu
(520) 621-4005
Sundar Dorai-Raj
#
Rebecca Young wrote:
Hi, Rebecca,

 From ?prcomp:

      The calculation is done by a singular value decomposition of the
      (centered and possibly scaled) data matrix, not by using 'eigen'
      on the covariance matrix.  This is generally the preferred method
      for numerical accuracy. ...

So you can get the singular values, but not the eigenvalues. You could 
use ?princomp if you really want the eigenvalues. In either case, you 
read the code to see how this is done.

x <- matrix(rnorm(1000), 100, 10)

# eigenvalues
v <- cov.wt(x)
ev <- eigen(v$cov * (1 - 1/v$n.obs), symmetric = TRUE)$values
ev[ev < 0] <- 0
princomp(x)$sdev
sqrt(ev)

# singular values
sv <- svd(scale(x, center = TRUE, scale = FALSE), nu = 0)
prcomp(x)$sdev
sv$d/sqrt(max(1, nrow(x) - 1))

HTH,

--sundar
Brian Ripley
# 
The eigenvalues are the squares of the singular values (although you need 
to watch the scalings used, in particular n vs n-1).  (This is standard 
theory.)

Since both are non-negative, given one you can get the other.
On Tue, 2 Aug 2005, Sundar Dorai-Raj wrote:

            

      
      
      
    

        

      
      
    

        

  
    

      
      
    

  


  


      

    

    

      
      

        


  

        

      Jari Oksanen
    

    

      
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On Tue, 2005-08-02 at 19:06 -0700, Rebecca Young wrote:

            

      
      
      
    

        
Rebecca, 

This answer is similar as some others, but this is simpler.

You have two separate problems: running PCA and getting eigenvalues. The
first is easy to solve: use prcomp instead of princomp (which only
exists for  historic reasons).  Function prcomp can handle cases with
more columns than rows. 

pc <- prcomp(x)

Above I assumed that your data are called x (or you can first make x,
say: x <- rcauchy(200); dim(x) <- c(20,10) -- which puts a funny twist
to comments on variances and standard deviations below).

This saves something that are called 'sdev' or standard deviations, and
you can get values that are (proportional to) eigenvalues simply by
taking their squares:

ev <- pc$sdev^2

These may be good enough for you (they would be good enough for me).
However, if you want to exactly replicate the numbers in some other
piece of software, you may need to multiply these by some constant. If
you don't need this, you may stop reading here.

The eigenvalues above are related to usual 'unbiased' variance so that
the following results are approximately equal:

sum(ev)
sum(apply(x, 2, var))

If you want to get eigenvalues related to biased estimate of variance,
you can do

eb <- (1-1/nrow(x))*ev

Function princomp uses these, as do some other software, but prcomp
works hard and carefully to get the eigenvalues it uses instead of
biased values (that would come naturally and directly in the algorithm
it uses). 

Some programs relate their eigenvalues to the sum of squares, and you
can get these by

es <- (nrow(x) - 1) * ev

Finally, some popular programs in ecology (your affiliation) use
proportional eigenvalues which you can get with:

ev/sum(ev)

cheers, jari oksanen