Multivariate regression methods

In order to overcome, or at least minimise, such drawbacks we can resort to the use of chemometric techniques (which will be presented in the following chapters of this book), such as multivariate experimental design and optimisation and multivariate regression methods, that offer great possibilities for simplifying the sometimes complex calibrations, enhancing the precision and accuracy of isotope ratio measurements and/or reducing problems due to spectral overlaps. 

On the other hand, the huge efforts made by atomic spectroscopists to resolve interferences and optimise the instrumental measurement devices to increase accuracy and precision led to a point where many of the difficulties that have to be solved nowadays cannot be described by simple univariate, linear regression methods (Chapter 1 gives an extensive review of some typical problems shown by several atomic techniques). Sometimes such problems cannot even be addressed by multivariate regression methods based on linear relationships, as is the case for the regression methods described in the previous two chapters. 

As many of these difficulties are currently overcome in molecular spectroscopy using multivariate regression methods (or multivariate chemometrics), I proposed that these three ladies should apply them to their spectra. The first reaction on their faces seemed something like . . . this crazy chemometrician guy. . . , but after some discussions we agreed to work together and see what might be achieved. It was not easy to get the spectral raw data that we needed from our old Perkin-Elmer device and Monica and Maria spent much time on this until they found a way to do it. 

Another method that can be used to quickly extract useful chemical information from an infrared image dataset is MCR 50-52,54,56-58 In some cases, this method can be used to obtain the concentration and absorbance spectra for each constituent in the original dataset. However, if the goal is not necessarily to resolve the constituents spectra, but rather to empirically classify them, a regression method may be more appropriate.53,59,60 The most prominent multivariate regression methods include PLS and ANNs. 

Kalivas, J.H., Interrelationships of multivariate regression methods using the eigenvector basis sets, J. Chemom., 13, 111-132, 1999. 

TABLE 16.1. Multivariate regression methods used to predict cheese ripening time... 

Multivariate Regression Methods. The main goal of this study was to build a multivariate model for the reliable prediction of a property of interest y (cheese ripening time) from a number of predictor variables, xi, X2. .. (peak area of casein and peptide obtained by CE). This model should describe the measured x and y data of the calibration set (cheese samples at different ripening time). In particular, in this research, the PCR and PLS methods were evaluated. 

The problem of over-optimistic estimates of model quality is a general one for all multivariate regression methods, and a number of model quality diagnostics have been developed that do not rely on parametric assumptions, to both limit the model fitting process and to assess the ability of the model to generalise beyond the training set. 

Partial least squares regression A multivariate regression method that uses an algorithm that maximises the covariance between a number of independent variables (X-variables e.g., MIR absorbance values) and one or several dependent variables (Y-variables e.g., lake-water pH). This regression method summarises all X-variables containing the same information into latent variables, partial least square components (similar to principal components). 

The other technique, the support vector machines (SVM), is emerging as a powerful method to perform both classification and regression tasks. It can be employed as such or combined with other multivariate regression methods, such as PLS. SVM is not a natural computation method itself because it performs deterministic calculations so that randomness in the results is avoided. However, it derives from the automatic artificial learning field (some of the most relevant developers worked in ANNs as well) and there is a fairly close relationship with multilayer perceptrons (perceptrons will be introduced in the next section). Therefore, SVM has been included in this chapter. 

In the preceding sections of this chapter, several multivariate regression methods have been presented. All of these methods require standard mixmre spectra. The construction of the mixtures will now be addressed in general, but more detailed discussion can be found elsewhere [23]. 

---