Evaluation of empirical chemometric models for calibration and classification

Abstract

constant, g, is known and y can be easily calculated by (2x/g) 0.5 - if effects like air friction are ignored.  The relationship is well described by a relatively simple mathematical equation usually based on physical/chemical knowledge - but the parameters are not known. An example is Lambert-Beer's law. The concentration, c, of a light-absorbing substance is given by A/(a·L) with L being the path length, a the absorption coefficient, and A the absorbance defined by log(I0 /I) with I0 the incident light intensity, and I the light intensity after passing the sample. I0, I, and L can be measured easily but the absorption coefficient is in general not known. It has to be determined from a set of reference samples with known concentrations and by application of a regression method - a so called calibration procedure. This method becomes very powerful if many wavelengths in the IR or NIR range are used [2], and it is one of the main applications of chemometrics [3-5].  In many cases of practical interest no theoretically based mathematical equations exist for the relationship between x and y. We sometimes know but often only assume that a relationship exists. In this case we call the model 'EMPIRICAL' or 'DATA DRIVEN'. Examples are for instance modeling of the boiling point or the toxicity of chemical compounds by variables derived from the chemical structures (MOLECULAR DESCRIPTORS). Development of quantitative structure-property or structure-activity relationships (QSPR, QSAR) by this approach requires multivariate calibration methods. For such purely empirical models - often with many x-variables - the COMPLEXITY of the model and the PREDICTION PERFORMANCE have to be estimated very carefully and cautiously. Also the variability of used measures has to be estimated. Typical for problems in chemistry are a small number of cases (objects) and a large number of x-variables.