Faq difference between a loading and a weighting

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Issue:

What is the difference between a loading and a weighting?

Possible Solutions:

When performing Principal Components Analysis (PCA), you get loadings, `P`, which are an orthonormal basis which can be used to calculate scores: `T = X*P` or to estimate data `X = T*P'`

These operations are invertible (repeating them gives the same result) because the loadings are the eigenvectors of `X'X`.

When using Partial Least Squares (PLS), you get loadings, `P`, but also weights, `W`, because the decomposition is based on `X'Y`. The weights and loadings must be used together to calculate scores: `T = X*W*pinv(P'*W)` From a phenomenological point of view, the weights represent features in `X` which are related to the original `Y` values. The loadings represent the features in `X` which are related to the scores, `T`, which are the given factor's estimate of `Y`.

Note, by the way, that the weights are the ones used to calculate the regression vector (that which is used to make a prediction). Loadings are only used when calculating scores and, of course, Hotelling's T2.

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