Orthogonal PLS

Although Equation (7.20a) has the same form as Equation (7.18), their calculation procedures are different from each other. The PLS loading vectors do not have strict orthogonality. PLS becomes PCR if the PLS loadings are replaced by the PCA loadings. 

Thus, we see that CCA forms a canonical analysis, namely a decomposition of each data set into a set of mutually orthogonal components. A similar type of decomposition is at the heart of many types of multivariate analysis, e.g. PCA and PLS. Under the assumption of multivariate normality for both populations the canonical correlations can be tested for significance [6]. Retaining only the significant canonical correlations may allow for a considerable dimension reduction. 

The purpose of Partial Least Squares (PLS) regression is to find a small number A of relevant factors that (i) are predictive for Y and (u) utilize X efficiently. The method effectively achieves a canonical decomposition of X in a set of orthogonal factors which are used for fitting Y. In this respect PLS is comparable with CCA, RRR and PCR, the difference being that the factors are chosen according to yet another criterion. 

In principle, in the absence of noise, the PLS factor should completely reject the nonlinear data by rotating the first factor into orthogonality with the dimensions of the x-data space which are spawned by the nonlinearity. The PLS algorithm is supposed to find the (first) factor which maximizes the linear relationship between the x-block scores and the y-block scores. So clearly, in the absence of noise, a good implementation of PLS should completely reject all of the nonlinearity and return a factor which is exactly linearly related to the y-block variances. (Richard Kramer)... 

The described projection method with scores and loadings holds for all linear methods, such as PCA, LDA, and PLS. These methods are capable to compress many variables to a few ones and allow an insight into the data structure by two-dimensional scatter plots. Additional score plots (and corresponding loading plots) provide views from different, often orthogonal, directions. 

In literature, PLS is often introduced and explained as a numerical algorithm that maximizes an objective function under certain constraints. The objective function is the covariance between x- and y-scores, and the constraint is usually the orthogonality of the scores. Since different algorithms have been proposed so far, a natural question is whether they all maximize the same objective function and whether their results lead to comparable solutions. In this section, we try to answer such questions by making the mathematical concepts behind PLS and its main algorithms more transparent. The main properties of PLS have already been summarized in the previous section. 

Earlier, it was mentioned that due to the orthogonality constraints of scores and loadings, as well as the variance-based criteria for their determination, it is rare that PCs and LVs obtained from a PC A or PLS model correspond to pure chemical or physical phenomena. However, if one can impose specihc constraints on the properties of the scores and or loadings, they can be rotated to a more physically meaningful form. The multivariate curve resolution (MCR) method attempts to do this for spectral data. 

The number of active sources is estimated by cross-validation, l.e. it is the optimal number of PLS components. The latent variables of the PLS model would correspond to the eigenvectors of the TTFA model. The linear combination of the latent variables in the inner relationship gives the estimate for the source profiles. PLS calculates the orthogonal linear combinations and the rotation in one step. Also, it solves for all the sources in the same model. 

---