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Limited number of principal components in PCA

4 messages · David L Carlson, William Armstrong, Michael Dewey +1 more

Original
David L Carlson
# 
Providing the data will help, but the first thing I noted is that you have more columns (variables) than rows (cases). PCA will return a maximum of (the number of columns) or (the number of rows-1) whichever is less. With 84 columns and 66 rows means you can get no more than 65 components. If the variables are highly correlated, you will get fewer components and that probably explains the reduction to 54. I would guess the variables are highly correlated and the first eigenvalue is very large.

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David L Carlson
Associate Professor of Anthropology
Texas A&M University
College Station, TX 77843-4352



-----Original Message-----
From: r-help-bounces at r-project.org [mailto:r-help-bounces at r-project.org] On Behalf Of Joshua Wiley
Sent: Friday, July 29, 2011 10:20 PM
To: William Armstrong
Cc: r-help at r-project.org
Subject: Re: [R] Limited number of principal components in PCA

Hi Billy,

Can you provide your data?  You could attach it as a text file or
provide it by pasting the output of:

dput(Q)

into an email.  It would help if we could reproduce what you are
doing.  You might also consider a list or forum that is more
statistics oriented than Rhelp, as your questions are more related to
the statistics than the software itself (but still, if you give us
data, you will probably get farther).

Cheers,

Josh

On Fri, Jul 29, 2011 at 11:33 AM, William Armstrong
<William.Armstrong at noaa.gov> wrote:

  
    

      
      
    

  


  


    

    
3 days later

    

      
      

        


  

        

      William Armstrong
    

    

      
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David and Josh,

Thank you for the suggestions.  I have attached a file ('q_values.txt') that
contains the values of the 'Q' variable.

David -- I am attempting an 'S' mode PCA, where the columns are actually the
cases (different stream gaging stations) and the rows are the variables (the
maximum flow at each station for a given year).  I think the format you are
referring to is 'R' mode, but I was under the impression that R (the
program, not the PCA mode) could handle the analyses in either format.  Am I
mistaken?

My first eigenvalue is:

            

      
      
      
    

        
[1] 17.77812

Does that value seem large enough to explain the reduction in principal
components from 65 to 54?

Also, the loadings on the first PC are not particularly high:

 > max(abs(unrotated_pca_q$rotation[1:84]))
[1] 0.1794776

Does that suggest that maybe the data are not very highly correlated?

Thank you both very much for your help.

Billy

http://r.789695.n4.nabble.com/file/n3719440/q_values.txt q_values.txt 

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      Michael Dewey
    

    

      
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At 19:07 04/08/2011, William Armstrong wrote:

            

      
      
      
    

        
try doing
table(complete.cases(q_values))
or whatever you are calling q_values.txt

Does that help?

Moral: when R does not do what you thought you told it to do it may 
still have done what you told it to do.

            

      
      
      
    

        
Michael Dewey
info at aghmed.fsnet.co.uk
http://www.aghmed.fsnet.co.uk/home.html

  
  


  


      

    

    

      
      

        


  

        

      Joshua Wiley
    

    

      
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Hi Billy,

Thanks for posting your data.  Okay, first off as Michael pointed out:

            

      
      
      
    

        
FALSE  TRUE
   12    54

shows that of the 66 rows in your data set, only 54 of them are
complete.  That means when you use na = na.omit, you are actually only
passing a data frame with 54 rows.  Second, prcomp() will not return
more components than observations.  Think of it this way, it is like
trying to connect 54 data points with 84 lines---53 lines will fit
perfectly (straight line connects two points), you are trying to go
way past that.

I have not heard (this is just an indicator of my ignorance, not their
lack of existence) of 'S' vs. 'R' mode PCA, but if you want the
columns to be the cases, just t()ranspose the data frame.

            

      
      
      
    

        
FALSE  TRUE
    9    75

So there are still only 75 possible observations to work with due to
missingness, but that is enough for only 66 variables

test_pca_Q2 <- prcomp(~ ., data = data.frame(t(Q)), scale = TRUE, retx = FALSE,
  na.action = na.omit)

            

      
      
      
    

        
[1] 66

so there are 66 SDs for the 66 principal components.

Regarding what the 'sdev' values are, they are the square root of the
eigen values of the correlation (in your case since you scaled)
matrix.  You can see this below:

## first ten (1:10) square roots of the eigen values of the correlation matrix
## of the complete cases of the transposed data set 'Q'

            

      
      
      
    

        
[1] 7.3267465 2.0349335 1.2913823 1.0750288 0.9035650 0.8301671 0.7370896
 [8] 0.7132530 0.6196836 0.5396176
## sdev from prcomp()

            

      
      
      
    

        
[1] 7.3267465 2.0349335 1.2913823 1.0750288 0.9035650 0.8301671 0.7370896
 [8] 0.7132530 0.6196836 0.5396176

You can also try the principal() function in package "psych".  It has
a lot of nice options, and I tend to use it for all this sort of
stuff.

Cheers,

Josh

        
On Thu, Aug 4, 2011 at 11:07 AM, William Armstrong <armstrwa at bc.edu> wrote: