Mathematical Description of PLS

X and Y denote the scaled and mean centered matrices of the variations in the X and Y space, respectively. Scaling to unit variance is usually employed. PLS modelling involves the factorization of the matrices X and Y into matrices of scores and loadings. In matrix notation, the model 

T and U are matrices of score vectors, P and C are the transposed matrices of the loading vectors, E and F are matrices of residuals. The inner relation which describes the correlation between the scores is defined by D which is a diagonal matrix in which the elements, dH, are the correlation coefficients of the linear relation between the scores. H is the matrix of residuals from the correlation fit. 

however, not necessary to compute eigenvectors by diagonalization of matrices. The PLS algorithm is based upon the NIPALS algorithm which makes it possible to iteratively determine one PLS dimension at a time. For details of the PLS algorithm, see [75]. 

Recently, PLS modelling involving non-linear inner relation has been described [76]. 

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Mathematical Description of PLS and PCR In this section, the PCR algorithm is described as a two-step procedure of PCA followed by MLR. Although in practice the steps are combined, we feel tliis is tlie most intuitive approach to understanding the algorithm. This description of PCR is followed by a brief discussion of the differences between PLS and PCR. 

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