PLS can be applied to multidimensional dependent variables (Y), but when it is apply to a mono-dimensional Y, how is it different from a Principal Component Regression?
p.s. What is the meaning of the coefficients to the Y when it is mono-dimensional?
The algorithms used to calculate PCR and PLS are different, please check each algorithm to see the differences, for example, PCR uses PCA and uses singular value decomponsition (SVD) why PLS inflates or deflates the matrix. The meaning of the coefficients to the Y when it is mono-dimensional is the same when multi-dimensional, except that in multi-dimentional the Y scores can be related. These coefficients are then scaled to the reference values to have a determination capacity and they let you know where on the plane the scores of your reference values lie. For example, it helps to know if your validation data is far away from the calibration data in terms of Y, like X scores would help to know if your validation data is far away form calibration data in terms of X.
Hi Nicla,
In simpliest terms: PCR is an observed-variable approach after conducting factor analysis based on saturation, whereas PLS is 'technically' a latent-variable approach (I recommend PCR if your fairly confident in your 'monodimensionality' assumption.).
Hope that this may of some useful help,
Matthew
Hello Nicola,
Actually there is a major difference between PCR and PLS (even for one vector y); in PCR, the scores of the PCA used in regression are those that best explain the data X (the projection space only depends on X), in PLS the projection space of X is explicative of both X and y. Consequently, PLS usually achieves the same result as PCR with fewer number of latent variables.
In my opinion, the choice between them depends on the dimensions and the amount of noise (error) in your X and y data.
in addition to Jad Rizkallah 's answer: it could very well be that since PCR is concerned with capturing the most of the variance of X, it misses out on the relevant variance related to the prediction of the response variable (or this ends up in higher principal components). PLS creates latent variables that maximize the covariance between X and y, making sure that each latent variable is related to y.
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