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When we perform a principal components analysis (PCA) on a multivariate data set we are interested in finding orthogonal components that explain maximal variance in the data set. We can form a biplot of the data using the scores and the loadings, and the locations of the sample points in the biplot are an approximation of the Euclidean distance between the samples.

In PLS, we extract orthogonal components from a predictor data set that have maximal covariance with the response (vector or matrix). We also get scores and loadings as part of the analysis and can draw a biplot of these scores.

What, if any, dissimilarity is represented by the Euclidean distances on the biplot between sample points?

One of the reasons I ask is that with PCA, we can apply a transformation to the data prior to applying PCA such that the Euclidean distance between samples on the biplot approximates the Euclidean distance between samples in the transformed data, but in the untransformed data the distance represented is some other distance. For example, by applying the Hellinger transformation (rows are standardised by their row sum and then a square root transformation is applied) to the raw data, a PCA applied to the transformed data will reflect the Hellinger distances between the observations.

I wonder if a similar principal might hold for PLS?

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Tell me if this doesn't make sense (it probably doesn't). If a PCA biplot depicts ~ Euclidean distances between the samples with respect to the goal of PCA (max(var(X)), then wouldn't the same be true of PLS, but with respect to max(cov(y, X))? The problem I see with respect to applying a transformation (like Hellinger) is if you only apply it to one of the two matrices. – Patrick Jun 4 '14 at 3:30

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