Principal Component Score Calculations - Need Help

Question

Hi Team,

One of the clarifications request from professor on calculations of PCA score.

JMP documentation guide says that the calculation for the principal components depends on which matrix you select for extraction of principal components: –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. –For the on Covariances options, the ith principal component is a linear combination of the centered observations using the entries of the ith eigenvector as coefficients.

–For the Unscaled and Uncentered option, the ith principal component is a linear combination of the raw observations using the entries of the ith eigenvector as coefficients.

The professor is using the solubility. Jmp dataset and created saved One Principal Component,

I have attached the screen shots. He is trying to show in the class the calculation of Principal Component score manually using the Eigen Vector values. When we save the Principal component as a formula, it shows a formula along with a CONSTANT. And the coefficients in the formula are not matching with the Eigen vectors.  (He was using R earlier and now moving to JMP)

Can someone help me in how to derive the PCA scores from the Eigen Vectors and match with JMP output. JMP gives a formula along with a constant as well.

Appreciate your help in this regard.

thanks

Murali

jiancao · Accepted Answer

When PCA is performed on correlations JMP calculates scores using the standardized variables and then rearranges the expression. So,

PC1=0.37441*[ 1-Octanol -Mean(1-Octanol)]/Std (1-Octanol)+0.34834* [Ether -Mean(Ether)]/Std (Ether )+…..

=0.37441*[ 1-Octanol - 1.2665)/ 0.8219] +……

=0.45544*1-Octanol – 0.57695+……

The constant, -1.13263, is the sum of – 0.57695 and other Mean/Std.

jules · Accepted Answer

Hi Murali,

For demonstration purposes when teaching, using the 'Unscaled and Uncentered' is probably what he is going for. The eigenvectors will be the coefficients in the saved formula because there isn't any transformation happening to account for scale and center differences among the variables. That's a helpful start for students before introducing the value of that transformation and showing them the calculation @jiancao provided.

@jules