Selection of Variables for Regression

In the discussions above and in the examples previously described, it has been assumed that the variables to be included in the multivariate regression equation were known in advance. Either some theoretical considerations determine the variables or, as in many spectroscopic examples, visual inspection of the data provides an intuitive feel for the greater relevance of some variables compared with others. In such cases, serious problems associated with the selection of appropriate variables may not arise. The situation is not so simple where no sound theory exists and variable selection is not obvious. Then some formal procedure for choosing which variables to include in a regression analysis is important and the task may be far from trivial. 

As an easily managed example of multivariate data analysis we shall consider the spectral data presented in Table 11. These data represent the recorded absorbance of 14 standard solutions containing known amounts of tryptophan, measured at seven wavelengths, in the UV region under noisy conditions and in the presence of other absorbing species. Two test spectra, X and XZ, are also included. 

In order to improve the performance of the calibration model other information from the spectral data could be included. The absorbance at 2i, for example, is negatively correlated with tryptophan concentration and may serve to compensate for the interfering species present. Including A21 gives the bivariate model defined by 

By ordinary least squares regression. Equation (45) can be solved to provide 

Although the bivariate model performs considerably better than the univariate model, as evidenced by the smaller residuals, the calibration might be improved further by including more spectral data. The question arises as to which data to include. In the limit of course, all data will be used and the model takes the form 

The problems and procedures for selecting variables for regression analysis can be illustrated by considering the use of near-IR spectrometry for quantitative analysis. Despite its widespread use in manufacturing and process industries, the underlying theory regarding specific spectral transitions 

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