Explained variation in PLS vs PCA

Question

A lot of research articles outline that the number of extracted factors by PLS (partial least squares) is less than the number of extracted factors by PCA (principal component analysis). However, the explained variation is greater for PLS compared to PCA. This is due to the fact that PLS extracts the factors such that the covariance between X (predictors) and Y (responses) is maximum?

Answer 1

You have to distinguish two different explained variances here:

• The explained variance in $\mathbf X$ and
• The explained variance in $\mathbf Y$.

PCA by construction yields the maximum explained variance in $\mathbf X$. A PLS model (with same number of components/latent variables) may explain the same amount of $\mathbf X$ variance or less.

OTOH, the PCA doesn't optimize the explained variance in $\mathbf Y$, and PLS may explain more variance in $\mathbf Y$ than the PCA model (again with same number of components/latent variables).

You may even achieve the same or more explained variance in $\mathbf Y$ with a PLS model that has somewhat fewer latent variables than the corresponding PCA/PCR model - but there is no guarantee here.