THE H-PRINCIPLE IN MODELING WITH APPLICATIONS TO CHEMOMETRICS

Heisenberg formulated certain rules and principles for describing and predicting physical systems. The H-principle is a mathematical formulation of these principles, when modelling data. One can show that it has two major benefits compared to other principles of modelling data: (a) it determines a proper balance between fit (how well the model fits the data) and the variance of a predictor derived from the model; (b) it optimizes the mean square error of prediction with respect to bias and prediction variance associated with the model. Application of the H-principle to modelling generally gives more stable predictions than other models, because it eliminates variables/components with low predictive abilities. In the special case of partial least squares (PLS) regression it gives the criteria of PLS. The H-principle is here applied in the principal components analysis context to the selection of variables. In the context of regression it is applied to stepwise regression and nonlinear PLS. It is also used to determine the number of variables/components to be used in the model.