Random variables, chemometrics

Nonlinear mapping (NLM) as described by Sammon (1969) and others (Sharaf et al. 1986) has been popular in chemometrics. Aim of NLM is a two-(eventually a one- or three-) dimensional scatter plot with a point for each of the n objects preserving optimally the relative distances in the high-dimensional variable space. Starting point is a distance matrix for the m-dimensional space applying the Euclidean distance or any other monotonic distance measure this matrix contains the distances of all pairs of objects, due. A two-dimensional representation requires two map coordinates for each object in total 2n numbers have to be determined. The starting map coordinates can be chosen randomly or can be, for instance, PC A scores. The distances in the map are denoted by d t. A mapping error ( stress, loss function) NLm can be defined as... 

Variable selection is an optimization problem. An optimization method that combines randomness with a strategy that is borrowed from biology is a technique using genetic algorithms—a so-called natural computation method (Massart et al. 1997). Actually, the basic structure of GAs is ideal for the purpose of selection (Davis 1991 Hibbert 1993 Leardi 2003), and various applications of GAs for variable selection in chemometrics have been reported (Broadhurst et al. 1997 Jouan-Rimbaud et al. 1995 Leardi 1994, 2001, 2007). Only a brief introduction to GAs is given here, and only from the point of view of variable selection. 

D. Jouan-Rimbaud, D.L. Massart, O.E. de Noord, Random Correlation in Variable Selection for Multivariate Calibration with a Genetic Algorithm, Chemometrics and Intelligent Laboratory Systems, 35 (1996), 213-220. 

Chemometrics mainly focuses on the chemical model, rather than on random effects as is typical for statistics. A chemometric approach does not exclude theory or human experience from problem solving they tell us how to define the problem, what to measure and how to pre-process measurement data. The basic hypothesis suggests that complicated chemical systems can be described (characterized) by a set of (measured) variables and that models (latent variables) can help to find the essential information. Selection or creation of appropriate problem-relevant features is often more important than the method which is applied for data interpretation. 

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