INLR, implicit non-linear latent variable regression

Abstract

A simple way to develop non‐linear PLS models is presented, INLR (implicit non‐linear latent variable regression). The paper shows that by simply added squared x‐variables x2a, both the square and cross terms of the latent variables are implicitly included in the resulting PLS model. This approach works when X itself is well modelled by a projection model T*PT. Hence, if a latent structure is present in X, it is not necessary to include the cross terms of the X‐variables in the polynomial expansion. Analogously, with cubic non‐linearities, expanding X with cubic terms x3a is sufficient. INLR is attractive in that all essential features of PLS are preserved i.e. (a) it can handle many noisy and collinear variables, (b) it is stable and gives reliable results and (c) all PLS plots and diagnostics still apply. The principles of INLR are outlined and illustrated with three chemical examples where INLR improved the modelling and predictions compared with ordinary linear PLS. © 1997 John Wiley & Sons, Ltd.

Use of Chemometrics and Laser-Induced Breakdown Spectroscopy for Quantitative Analysis of Major and Minor Elements in Aluminum Alloys

“…40,41 The problem with these approaches is that the complexity of the algorithms can increase significantly. The approach that gives the best performance on the current data set is the implicit nonlinear correction developed by Berglund, 42 while keeping the algorithm as simple as possible. This method consists of adding to the original spectral data the squares and the cubes of these data, in order to compensate for the quadratic and cubic signal dependence on concentration.…”

Incorporating interactions in multi‐block sequential and orthogonalised partial least squares regression

“…The present paper has focused on interactions, but the same idea could easily be extended also to incorporate non‐linear terms, either by using the data directly as proposed in the study by Berglund and Wold for regular PLS regression or using the principal components as indicated earlier. The simple idea is simply to add blocks that are based on products of variables.…”

Multivariate Analysis of the Sequence Dependence of Asparagine Deamidation Rates in Peptides

“…Since there was clearly some non-linear behavior in the deamidation halflives, two different schemes (the natural logarithm and the square root) were evaluated for linearizing the deamidation data. This is a common approach in chemometric modeling for data that extends over orders of magnitude (37,42,(44)(45)(46)(47). The natural logarithm of the deamidation half-lives provided a more effective linearization than the square root of the half-lives (Fig.…”