Limited number of principal components in PCA

Hi Billy,

Thanks for posting your data.  Okay, first off as Michael pointed out:
table(complete.cases(Q))
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.
table(complete.cases(t(Q)))
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)
length(test_pca_Q2$sdev)
[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'
sqrt(eigen(cor(na.omit(t(Q))))$values)[1:10]
[1] 7.3267465 2.0349335 1.2913823 1.0750288 0.9035650 0.8301671 0.7370896
 [8] 0.7132530 0.6196836 0.5396176
## sdev from prcomp()
test_pca_Q2$sdev[1:10]
[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
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:

unrotated_pca_q$sdev[1]^2
[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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Joshua Wiley
Ph.D. Student, Health Psychology
Programmer Analyst II, ATS Statistical Consulting Group
University of California, Los Angeles
https://joshuawiley.com/

Limited number of principal components in PCA

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