[R] Error in principal component loadings calculation

[David L Carlson]

The sum of the squared loadings will always sum to 1 because they are standardized by dividing them by the standard deviation of each component. The terminology for principal components is not as consistent as we could hope. What princomp() calls loadings is really the structure matrix (the correlation between each variable and the component). The pattern matrix (often called the loadings) are the regression coefficients for computing the principal component scores. You are probably looking for the pattern matrix which is easy to obtain by multiplying by the standard deviations:

> set.seed(42)
> data <- matrix(runif(100), 20, 5)
> pc <- princomp(data, cor=TRUE)
> loadings(pc)

Loadings:
     Comp.1 Comp.2 Comp.3 Comp.4 Comp.5
[1,]  0.638  0.249        -0.260 -0.679
[2,]        -0.714  0.449  0.298 -0.444
[3,]  0.585 -0.152  0.522 -0.231  0.555
[4,]        -0.617 -0.543 -0.564       
[5,] -0.496  0.154  0.479 -0.687 -0.172

               Comp.1 Comp.2 Comp.3 Comp.4 Comp.5
SS loadings       1.0    1.0    1.0    1.0    1.0
Proportion Var    0.2    0.2    0.2    0.2    0.2
Cumulative Var    0.2    0.4    0.6    0.8    1.0
> rowSums(pc$loadings^2)
[1] 1 1 1 1 1
> # Notice that the column sums of the squared loadings all equal 0
> # Now multiply each loading by its standard deviation
> sweep(pc$loadings, 2, pc$sdev, "*")

Loadings:
     Comp.1 Comp.2 Comp.3 Comp.4 Comp.5
[1,]  0.765  0.275        -0.237 -0.531
[2,]        -0.787  0.427  0.271 -0.347
[3,]  0.701 -0.167  0.497 -0.211  0.434
[4,]        -0.680 -0.518 -0.515       
[5,] -0.594  0.169  0.456 -0.627 -0.134

               Comp.1 Comp.2 Comp.3 Comp.4 Comp.5
SS loadings     1.436  1.215  0.907  0.832  0.611
Proportion Var  0.287  0.243  0.181  0.166  0.122
Cumulative Var  0.287  0.530  0.712  0.878  1.000
> pc$sdev^2
   Comp.1    Comp.2    Comp.3    Comp.4    Comp.5 
1.4362072 1.2145055 0.9068555 0.8315685 0.6108632 
> # Now the sum of the squared loadings equals the 
> # squared standard deviation (aka the eigenvalues)

-------------------------------------
David L Carlson
Department of Anthropology
Texas A&M University
College Station, TX 77840-4352

-----Original Message-----
From: R-help [mailto:r-help-bounces at r-project.org] On Behalf Of Marcelo Kittlein
Sent: Monday, September 14, 2015 8:46 AM
To: r-help at r-project.org
Subject: [R] Error in principal component loadings calculation

Hi all

I have been using "princomp" to obtain the principal components of some 
data and find that the loadings returned by the function appear to have 
some error.

in a simple example if a calculate de pc for a random matrix I get that 
all loadings for the different components have the same proportion of 
variance

data <- matrix(runif(100), 20, 5)
pc <- princomp(data, cor=TRUE)
loadings(pc)

Loadings:
      Comp.1 Comp.2 Comp.3 Comp.4 Comp.5
[1,] -0.280  0.510  0.674 -0.217 -0.400
[2,]  0.529 -0.353        -0.694 -0.330
[3,] -0.111  0.563 -0.713 -0.336 -0.222
[4,] -0.530 -0.502 -0.178  0.140 -0.645
[5,] -0.590 -0.215        -0.582  0.516

                Comp.1 Comp.2 Comp.3 Comp.4 Comp.5
SS loadings       1.0    1.0    1.0    1.0    1.0
Proportion Var    0.2    0.2    0.2    0.2    0.2
Cumulative Var    0.2    0.4    0.6    0.8    1.0

This keep returning the same proportion of variance for each component 
regardless of the data used.

my R version is

 > R.Version()
$platform
[1] "x86_64-unknown-linux-gnu"

$arch
[1] "x86_64"

$os
[1] "linux-gnu"

$system
[1] "x86_64, linux-gnu"

$status
[1] ""

$major
[1] "3"

$minor
[1] "2.1"

$year
[1] "2015"

$month
[1] "06"

$day
[1] "18"

$`svn rev`
[1] "68531"

$language
[1] "R"

$version.string
[1] "R version 3.2.1 (2015-06-18)"

$nickname
[1] "World-Famous Astronaut"

some hint would be much appreciated.

Best regards

Marcelo Kittlein

	[[alternative HTML version deleted]]

______________________________________________
R-help at r-project.org mailing list -- To UNSUBSCRIBE and more, see
https://stat.ethz.ch/mailman/listinfo/r-help
PLEASE do read the posting guide http://www.R-project.org/posting-guide.html
and provide commented, minimal, self-contained, reproducible code.