Parsimonious and robust multivariate calibration with rational function Least Absolute Shrinkage and Selection Operator and rational function Elastic Net

This paper presents new methods for multivariate calibration. A unique aspect is that this approach uses rational functions with either Least Absolute Shrinkage and Selection Operator (LASSO) or Elastic Net (ENET), and builds parsimonious models in an automated way via cross-validation. Rational function modeling provides robustness, as will be briefly demonstrated. Interestingly, rational function models are also flexible, in that occasionally they are reduced to ordinary linear models based on cross-validation. Thus, model complexity is not forced to take the form of rational functions.

Additional benefits arise from the use of LASSO and ENET. While LASSO uses only ℓ₁ norm on regression coefficients, ENET combines the best of both worlds by using ℓ₁ and ℓ₂ norms. The former (ℓ₁) provides variable selection while the latter (ℓ₂) handles collinearity via shrinkage of regression coefficients. Rational functions are highly collinear if full rank is used and, thus, not necessarily robust unless either ℓ₁ or ℓ₂ norm is used on the regression coefficients. The use of ℓ₁ norm allows for a more parsimonious model that can potentially be more robust. This is contrary to the use of a broadband spectrum that is likely to be contaminated at some point in the future by unknown spectral interferences. The real benefits seem to originate from the combination of rational functions and ENET. Note that LASSO solutions form a subset of ENET solutions and are thus included in ENET.