Imagine a PCA with only 2 variables (V1 and V2), the variables factor map show a 90° between this two variables but both of the are explained equally by the first and second axis.
I would like to know if a treatment of PCA data (such as a translation) to induce the rotation of graphics is feasible and even "rightful" ?
I'm interested into this translation to better explain the two axis.
The attached image would maybe help to understand my issue.
I would really appreciate if you could describe a procedure on R (if you have one) and even more with the package FactoMineR.
Thanking you in advance for your answer and time.
NB: of course my actual data contains more than two variables.
Rotating axes is almost always permissible to do - what I am not sure of is why you would want to do this and what you mean by legitimate.
First, I think that the two arrow picture is a little bit misleading about what PCA actually does, which perhaps leads to confusion.
If we imagined a multdimensional data set, the rotation of the axes is completely arbitrary, so you are free to rotate them however you want.
PCA is in fact just a rotation of the axes in space to explain all of the observed variance in the data set. The PCA's first component is the direction in which most of the most variability in the data lies. The second component is 90 degrees to the first component and gives the direction in which the most variability lies after the first principle component has been accounted for. This goes on until you run out of dimensions (determined by the number of variables).
Thus, if your data lies equally on the first two principal components, then it means that you have a cloud of data in which there is equal variability in 2 dimensions that are 90 degrees to one another. Rotating the axes further is completely legitimate to do, but it will not reveal any simpler structure.
A different approach to explaining the data is factor analysis, in which the common variance (rather than the total variance) is modeled, and for that rotation is more typically done. I recommend that you look at the psych package or the GPArotation pacakges if you want to see the kinds of rotations that can be done.
Thank you for your answer.
I'm aware that it will not reveal simplest structure, but it is more simple to explain the horizontal axis with the variable V1 and the vertical one with the V2. (I know that an example with only 2 variables mean nothing, but I tried to simplify more than I can the question).
Concerning legitimacy, I was quite confident that, by rotate, we don't change the information given by PCA, but I think by doing that we do change the inertia of each axis (probably by de creasing them, because I expect the script to give me the most elevated inertia at the end).
I'm just not sure on this last point and furthermore I considered that the inertia is a crucial information that can not be changed... (as a p-value of ANOVA does not change even if you mix the analyzed histograms, if I may compare PCA with an hypothesis test).
It is for this reason that I asked about the legacy of this rotation.
Anyway, thank you very much again for your answer.
I'll go forward with the packages you wrote-down at the end. Finally, thank you for the factor analysis idea, but I already compute one on my data (that is an helpful complementary tool by the way).
Factorial axes rotation has a long history in PCA and you should be aware that several kind of methods have been proposed depending on the criterion used, the most legitimate being the orthogonal rotations.
Those features are commonly provided by commercial package such as SAS or SPSS
As far as I know, there is no such an advanced function in FactoMiner.
However in R, at least one package (http://cran.r-project.org/web/packages/psych/index.html) provide such a function for Varimax rotation :
# Varimax Rotated Principal Components
# retaining 5 components
library(psych)
fit
Yes, it's fine to rotate your axes. If you perform an orthogonal rotation, the statistics are still easy to calculate. You just need to be clear how the rotations were performed when you publish your results. You can also reflect the axes to provide a more interpretable soln. Oblique rotations are also useful and the method typically depends on the type of data you're measuring and your overall objective.
Orthogonal rotations assumed that factors (with PCA, your components) are uncorrelated while oblique rotations assumed that factors are correlated.
Consequently, orthogonal rotations are simpler to interpret and understand than oblique rotations.
For a pretty discussion on this topic, please read the column written by James Dean Brown in JALT Testing & Evaluation SIG Newsletter. 13 (3) November 2009 (p. 20
-25), available at http://jalt.org/test/PDF/Brown31.pdf
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