The objective function in PLS is a complex one where the method is capturing covariance and correlation between two spaces (and X and Y space). With this combined objective PLS to identifies spaces where co-variation in the X space is correlated with co-variation in the Y space. The resulting model will the possess enough information to reconstruct the X space since part of the model is dedicated to capturing the covariance structure of X. Which in the case where the X space is of higher effective rank than Y will lead to a null space i.e. a sub-space of the latent space where X variation occurs that is orthogonal to the variations in Y. This rank discrepancy makes it sometimes challenging to determine the directions of X that are strictly correlated with the Y space (the perpendicular space to the null-space). And specifically in spectroscopy, this interpretation is sometimes necessary for the application. The use of Orthogonal Signal Correction as a filter to PLS was proposed by (Trygg & Wold, 2002) as a method to better interpret the features in X that are strictly correlated with the Y. The effectiveness of the method relies on the success of the filter operation, if the orthogonal signal correction filter is not appropriately, the solution found by OPLS (i.e. the features of X strictly related to Y) will deteriorate. In a follow up publication (Yu & MacGregor, 2004) introduce the use of Canonical correlation analysis as an alternative way to identify the features of X that are strictly correlated with the Y space, in their paper they analytically demonstrate the equivalence between OPLS and PLS-CCA and furthermore show how the solution found with PLS-CCA is less prone to deterioration since this last one relies on extracting the desired information, rather than removing the undesired one (as OPLS does). This work presents a discussion of these concepts leading to the recommendation of using PLS-CCA as an improved route.
The identification of the co-variational spaces in X that are strictly related to the Y enables the development of MSPC approaches where the scores can be strictly related to a CQA answering a prevalent question in the application of MSPC to pharmaceutical processes (how to interpret the position in the scores space as it strictly relates to CQA’s). One can argue that an MSPC solution that uses the co-variant scores for monitoring can be postulated as a direct way to monitor the process space in directions that will impact CQA’s and therefore can be more easily incorporated into a control strategy to monitor the health of the process.
References
Trygg, J., & Wold, S. (2002). Orthogonal projections to latent structures (O‐PLS). Journal of Chemometrics: A Journal of the Chemometrics Society, 16(3), 119-128.
Yu, H., & MacGregor, J. F. (2004). Post processing methods (PLS–CCA): simple alternatives to preprocessing methods (OSC–PLS). Chemometrics and intelligent laboratory systems, 73(2), 199-205.
