I would like to introduce two supplementary variables into a PCA I'm conducting on a set of data measuring concentration in different material phases.
However I'm unclear as to how to interpret the correlation of the "supplementary" variables to the principal component (as opposed to how I would interpret active variable loadings, and eventually the relationship between the PC and the scores).
Clearly the factor loadings and sq. cosines of the variables are different depending on whether I treat them as active or supplementary. Specifically when variable "A" and "B" are included as active variables they have a strong correlation with PC2, but when treated as supplementary they do not link well to the first few PCs (and explain very little of the total variance). Based on my current understanding, the two different scenarios impact my interpretation of the scores.
I realize this is likely a simple question, but would appreciate any help to clarify my understanding so I can make sure I proceed correctly.
To follow up then, one is investigating the relationships of the passive variables (eg: w1,w2,w3) with each other in the context of the PC structure. So if the loadings of all active variable (eg: v1, v2, v3) is high on PC1, but the passive variables have minimal calculated loadings on PC1 - how would one interpret the relationship between the passive variable and the active variable? There is no relationship between active and passive variables?
more info here are poor quality charts - I will try and improve them. The first shows the red points as the active variables, and the second poor quality image shows the 3 variables treated as supplementary variables. I believe the scales are similar. The factor loadings and sq. cosines are different of course.