B PLS algorithms

Multi-way Analysis With Applications in the Chemical Sciences 

Step 3 in Equation (3.45) (the deflation of X) is a regression step where X is regressed on ti. Hence, the residuals of that regression (E(1)) are orthogonal to ti. Because t2 is a linear combination of the columns of E(1), it follows that t2 is orthogonal to ti. 

Note that the algorithms in Equations (3.45) and (3.46) are noniterative, because the y is univariate. For multivariate Y, the PLS algorithms become iterative. There are alternative algorithms for partial least squares regression, depending on the size of X [De Jong Ter Braak 1994, Lindgren et al. 1993], Moreover, it is a matter of choice whether to deflate X and/or y [Burnham et al. 1996], 

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