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I am doing correlation-based principal component analysis (PCA) where the eigenvalues are extracted from the correlation matrix instead of the covariance matrix.

Now, the issue is to understand how the scores are being calculated in correlation-based PCA?

amoeba
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barak
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    Please look [this thread](http://stats.stackexchange.com/q/62677/3277). PCA based on correlations is the PCA of the standardized variables - that is, PCA of another dataset. It is different PCA. A component from the correlation-based PCA is not related in any knowingly clear fashion to the corresponding component of the covariance-based PCA. Getting "back" from corr-based components to the unstandardized variables just makes no sense. – ttnphns Oct 26 '15 at 10:17
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    thanks for the replay i will be more specific so you can understand what i want. once i did the correlation-based PCA and I'm getting my loading now i want to project a new observation not from the sample on the PC's and see where it's stand next to the sample observation(scores) how can i achieve that? – barak Oct 26 '15 at 14:05
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    You want to compute a componet score for new coming observations? It is very simple. A new observation should be standardized by the means and st.deviations of the old sample. Then you multiply it by the eigenvectors which you obtained in your PCA. – ttnphns Oct 26 '15 at 15:08
  • **Correlation-based PCA is equivalent to the covariance-based PCA on a standardized data matrix.** This is because correlation matrix equals the covariance matrix of standardized data. Indeed, corr between any $x$ and $y$ is equal to the cov between standardized $\tilde x$ and $\tilde y$: $$Corr(x,y)=\frac{Cov(x,y)}{\sqrt{Var(x)\cdot Var(y)}}=Cov\left(\frac{x}{\sqrt{Var(x)}},\frac{y}{\sqrt{Var(y)}}\right),$$ and that's $Cov(\tilde x,\tilde y)$. [cont.] So you need to standardize your data matrix (i.e. subtract column means and divide by std) and then to compute the scores of the cov-based PCA. – amoeba Oct 30 '15 at 10:08

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