David L Carlson dcarlson at tamu.edu
Mon Sep 14 23:07:31 CEST 2015

```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)

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
Proportion Var    0.2    0.2    0.2    0.2    0.2
Cumulative Var    0.2    0.4    0.6    0.8    1.0
[1] 1 1 1 1 1
> # Notice that the column sums of the squared loadings all equal 0

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
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
> # 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

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
variance

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

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
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

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