principal component formulae

[#1]

What is the relationship, if any, between the loadings of variables, with respect to a principal component, and the coefficients of the variables in the formulae for each principal component?  I would have expected the loadings to correspond to the coefficients, but this does not seem to be the case.

[#2 · Accepted solution]

Hi,

The coefficients in the saved formulae for the principal components use the entries for the eigenvectors. Eigenvectors are related to loadings but are not the same.

This is all explained in more detail in the JMP help documentation here. It also depends whether you are performing PCA on unscaled, covariates, or correlations.

I hope that helps.

Phil

[#3]

Hello Phil,

Let's say I write down the eigenvalue for a principal component based on correlation, the eigenvector for a descriptor and the loading of the descriptor.  The loading is equal to the eigenvector multiplied by the square root of the eigenvalue.  However, this product does not seem to equal the coefficient for the descriptor in the formula for the P.C., even though I understood this coefficient to equal the loading.

[#4]

No, the coefficient is not equal to the loading. As it says in the help documentation "for the on Correlations option, the ith principal component is a linear combination of the centered and scaled observations using the entries of the ith eigenvector as coefficients."

If you use the Unscaled option (or if you use the Correlation option on already scaled and centred observations) the coefficients are just the eigenvectors. Otherwise the coefficients are the eigenvectors adjusted for centering and scaling. 

I think the attached example might help.