PCA (correlation) Biplot - correlation and angles

#1
Hi,

I have a question regarding a PCA correlation Biplot. As far as I understand, angles between lines are (approx.) correlations between corresponding variables. Especially, if the angle is about 90° between two lines, these two variables are uncorrelated. But how does the plot look like if I have three variables, which are all uncorrelated to each other?

Thanks
 
#2
In this case, you have to watch the 3-D plot, the 3rd variable will be perpendicular to the other 2 in the 3rd dimension, or you should watch the plots with "1st and 3rd" and "2nd and 3rd" PC and verifiy that your third variable is perpendicular to the other 2 in each plot.
However, to verify the correlation between variables, it is better to examine the correlation matrix, which explain it numerically
 
#3
Thank you already, axeler. That means, as soon as I have more than two variables within a "standard 2D PCA correlation Biplot", I can't anylonger interpret angles in terms of pairwise correlation? Isn't this odd, since >2 variables are rather standard in a PCA analysis and this angle-correlation-relationship is often mentioned?
 
#4
Well, you can use angles in loading plot (or biplot) to estimate pairwise correlation, but I think you should be careful in doing that: in addition to angles, you have also to take in account the relevance of variables in the PCs (i.e.: distances from axes). So two co-linear variables, one of which is near the origin and the other very far, could be very little correlated, most of all because the first isn't relevant for those PCs and the second has a great relevance. Remember that PCA is a qualitative analysis!
As an example, my usual software doesn't use lines to indicate the variables, but only dots, so my only way to estimate correlations is watching the distances between variables, possibly on more than two PCs.
I repeat that to assess correlation between variables, it is better to calculate a correlation matrix, which gives numerical values for correlations and doesn't have "spatial" problems