# Correlation between an original variable and a principal component

I am looking at this link http://strata.uga.edu/8370/lecturenotes/principalComponents.html where it says

In interpreting the principal components, it is often useful to know the correlations of the original variables with the principal components. The correlation of variable $X_i$ and principal component $Y_j$ is

$$r_{ij} = \sqrt{a_{ij}^2\times \mathrm{var}(Y_j)/s_{ii}}$$

where $a_{ij}$ is the $i$-th's variable principal component weight on on principal component $j$ and $Y_j$ is the $j$-th principal component score and I am not sure what $s_{ii}$ is.

I do a simple PCA on the iris dataset below and would like to calculate the correlation of sepal width and legth to PC score 1 and PC score 2 but what is $s_{ii}$?

The $\mathrm{var}(Y_j)$ is the variance of the principal component score. I have that in the VarofSCores object below. In that link the author says $S_Y$ is the varcov matrix of the scores so if $s_{ii}$ is the diagonal of $S_Y$ then that is the same value I have in VarOfSCores.

data(iris)
names(iris)
dat = data.frame(iris$Sepal.Width, iris$Sepal.Length)
pca= prcomp(dat)
PC = pca$rotation VarOfScores = pca$sd^2
scores = pca$x #correlation of sepal width to score 1 sqrt(PC[1,1]^2* VarOfScores/?? ) #correlation of sepal width to score 2 sqrt(PC[1,2]^2* VarOfScores/?? ) #correlation of sepal LENGTH to score 1 sqrt(PC[2,1]^2* VarOfScores/?? ) #correlation of sepal LENGTH to score 2 sqrt(PC[2,2]^2* VarOfScores/?? )  Also - Why does the author say "loadings" instead of "principal components". The eigenvectors are "principal components" not "loadings" and the data times the eigenvectors are "scores". I think that is poor terminology. See here http://www.cs.princeton.edu/courses/archive/spr08/cos424/scribe_notes/0424.pdf where the author states on on page 4 "V" is the principal components. • s is diagonal matrix of eigenvalues. Therefore, s_{ii} gives the eigenvalues of the sample covariance matrix. – ARAT Oct 24, 2017 at 2:00 ## 2 Answers Correlation coefficient between variable$X_i$and principal component$Y_j$is given by $$r=v_{ij}\cdot \mathrm{std}(Y_j) / \mathrm{std}(X_i)=v_{ij}\cdot \sqrt{e_{j}} / \mathrm{std}(X_i),$$ where$v_{ij}$is an$i$-th element of the$j$-th unit-length eigenvector of the covariance matrix,$e_j=\mathrm{var}(Y_j)$is the corresponding eigenvalue which gives variance of this PC, and$\mathrm{std}(X_i)$is standard deviation of$X_i$. Please see my answer to How to find which variables are most correlated with the first principal component? for the derivation and additional explanations. Note that$v_{ij}\cdot \sqrt{e_{j}}$are called "loadings"$L_{ij}$. I think the formula in your quote says the same thing:$a_{ij}$are the elements of the eigenvectors,$\mathrm{var}(Y_i)$are the respective eigenvalues, and$s_{ii}$stands for the diagonal elements of the original covariance matrix, i.e. for the variances of$\mathrm{var}(X_i)\$.

Regarding terminology, please see What exactly is called "principal component" in PCA? and Loadings vs eigenvectors in PCA: when to use one or another?