Principal Component Terminology Confusion

Mar 2008

I am currently learning about Yield Curve forecasting using Principal Component Analysis.

However I see the term Loadings used/explained in different fashions depending on the source.

Using the example of a 3-dimensional dataset displaying spheroid characteristics:

The Principle Components (PCs) are the directions of the largest variation in the spheroid with PC1, PC2, PC3 having descending magnitudes of variation.

The direction Principle Component can determined by applying the spectral theorem to the covariance matrix of the dataset dimensions which results in a matrix of eigenvectors and a diagonal matrix of eigenvalues.

The raw data can be rotated by the transpose of the eigenvector matrix such that the original dataset (\(\displaystyle X\)) can rotated to a new frame of reference the axes with directions PC1, PC2, PC3:

\(\displaystyle Y = A^T(X-\mu)\)

One of my sources calls each of the vectors of the transposed eigenvector matrix (\(\displaystyle A^T\)), "the ith vector of loadings" but in another source I read "\(\displaystyle A\) is a square matrix who's columns are the PCs".

So between loadings, principle components and eigenvectors I am a little lost as all of these terms seem to mean the same thing.

My guess right now is, the eigenvectors point in the direction of the principle components. When the eigenvectors are put into a matrix, then the matrix is called the loadings.

Am I using the correct terminology?