Parsimonious rational function regression in multivariate calibrations

Recently, Taavitsainen introduced both rational function ridge and PLS regression methods. These approaches seem to cope with moderate nonlinearities and to some extent also with multiplicative spectral interferences as demonstrated by the authors more recently. The aim is to demonstrate the flexibility and robustness of two new additions to the rational function methods, namely the use of least absolute shrinkage and selection operator (LASSO), and elastic net (ENET) instead of ridge regression in the linear regression step of the original method. Previous work has been based on methods that use L2 regularization, i.e. ridge regression, to cope with collinearity and noise removal. In this work, we introduce more parsimonious methods for rational function modeling that use rather L1 regularization or the combination of L1 and L2 regularization. These methods provide interesting opportunities for model development and robustification. The key point is to illustrate the flexibility of rational function regression using ENET because it bridges the gap between L1 and L2 methods. The former suffers from spectral noise, which is well known in the context of normal LASSO approaches. The latter has been occasionally troublesome because it involves the use of a broad spectral range that can potentially pick up new spectral interferences that were not present in the calibration set. While this is not of concern in many cases, there are examples where the broad spectral approach is less robust in the prediction of future samples than a method that carefully selects narrower spectral bands or regions. Both L1 and L2 regularizations are extremely useful and this is exactly what is being balanced in elastic net. This property in the context of rational function approach is extremely valuable. In addition, we demonstrate that rational function ENET can include and, in fact, includes also normal ENET regression solutions (where the denominator is 1). These points are exemplified using a multipoint NIR instrument with two 5-channel measurement probes. Models were developed using data from two channels from the first probe while other three channels and the second measurement probe weree used as an independent test set. This also addresses one or two of the potential applications, namely, instrument standardization and calibration transfer. We also briefly demonstrate how to interpret and visualize rational function models efficiently. In summary, this work addresses new and interesting directions in developing calibration models in the field of spectroscopy and multipoint measurements.