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I have a question about PCA. I know that if you have correlated variables (x1, x2, x3, x4), its good to go do the PCA so that you can have new uncorrelated variables (pc1, pc2) used instead of the original correlated variables. My question is that if you have pc1, pc2 (for example), the loadings do not have to be influenced by the most highly correlated variables (let’s say highly correlated variables are x1 and x2). You can have high influence from x1 and x3 for pc1. The highly correlated variables do not need to have high influence for pc1, pc2, pc3, or pc4.

can you please confirm?

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So if I understand the question correctly, you want to know if uncorrelated variables influence the result of PCA, the answer is yes, this is sort of the high level idea of PCA, the more uncorrelated original variables will end up contributing to different principal components. The short version is that principal components are the eigenvectors of the covariance matrix of the original data, and the variance associated with each eigenvector is the associated eigenvalue.

Another way of thinking about this: if you have 2 perfectly correlated variables, say x2 = 2*x1, you can get rid of x2 completely and not lose any ability to make predictions. The less the two variables are correlated, the more you want to retain both of them.

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