### How accurate is Beer's Law in the analysis of NIR Data?

#### Abstract

To construct a predictor of the proportional presence of some key component in a material, we assume that the near infrared (NIR) responses of the various component proportions are given by Beer's law. Using a set of NIR spectra of milk powder spiked with different amounts of casein, this assumption is tested by comparing different properties of this set of spectra with the equivalent properties of simulated spectra obtained by combining the spectrum of unspiked milk powder with the spectrum of casein, in the same proportions. The latter set of spectra corresponds to the situation where Beer's law holds exactly with no cross-interaction between the two linearly independent components, and, thereby, described by a rank-two row matrix. The former corresponds to what happens in practice with the possibility of cross-NIR-interaction occurring between the components. The degree to which Beer's law is likely to fail is examined. It is shown how spiking allows the identification of the casein wavelengths that are strongly independent of the NIR spectral responses of the other components in the milk powder. However, there are slight deviations from Beer's law and these are, in part, explained by diffraction effects.

**References**- B. Anderssen, F. de Hoog, and M. Hegland. A stable finite difference ansatz for higher order differentiation of non-exact data.
*Bull. Austral. Math. Soc.*58:223–232, 1998. doi:10.1017/S0004972700032196 - R. S. Anderssen and F. R. de Hoog. Finite-difference methods for the numerical differentiation of non-exact data.
*Computing*33:259–267, 1984. doi:10.1007/BF02242272 - R. S. Anderssen, F. R. de Hoog, I. J. Wesley, and A. B. Zwart. How much of a near infrared spectrum is useful? Sparse regularization–-let the data decide! In S. McCue, T. Moroney, D. Mallet, and J. Bunder (Eds),
*Proceedings of the 16th Biennial Computational Techniques and Applications Conference, CTAC-2012*,*ANZIAM J.*54:C788–C808, 2012. http://journal.austms.org.au/ojs/index.php/ANZIAMJ/article/view/6744 - R. S. Anderssen and M Hegland. For numerical differentiation, dimensionality can be a blessing!
*Math. Comput.*68(227):1121–1141, 1999. doi:10.1090/S0025-5718-99-01033-9 - R. S. Anderssen and M. Hegland. Derivative spectroscopy: An enhanced role for numerical differentiation.
*J. Integral Equat. Appl.*22:355–367, 2010. doi:10.1216/JIE-2010-22-3-355 - D. J. Dahm. Explaining some light scattering properties of milk using representative layer theory.
*J. Near Infrared Spec.*21:323–339, 2013. doi:10.1255/jnirs.1071 - T. Naes, T. Isaksson, T. Fearn, and T. Davies.
*A User-Friendly Guide to Multivariate Calibration and Classification*. NIR Publications, Chichester, UK, 2002. doi:10.1002/cem.815

DOI: http://dx.doi.org/10.21914/anziamj.v56i0.9291

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