Spectroscopic calibrations and wavelength selection

Quantitative spectroscopic multicomponent analysis is based on the direct proportionality between absorbance at a particular wavelength (a) and concentration of the chemical constituent (c). This relationship is known as the Beer-Lambert law [3]. For multicomponent systems, the measured absorbance at the jth wavelength equals the sum of individual contributions from the n responding components and is expressed as 

In order to predict concentrations of unknown samples, equation (1) must be solved for K. The most common approach is to approximate K by least squares using 

Equations (2) and (3) outline the classical calibration and prediction approach and the combination is often referred to as K-matrix analysis. The K-matrix analysis approach requires quantitative calibration for all n components of the chemical system, even if they are of no interest for future predictions. Solution of equation 

An alternative method, often referred to as the P-matrix analysis, avoids disadvantages present with K-matrix analysis by viewing concentration as a function of spectroscopic responses. Mathematically, 

A general requirement for P-matrix analysis is n = rank(R). Unfortcmately, for most practical cases, the rank of R is greater than the number of components, i.e., rank(R) n, and rank(R) = min(m, p). Thus, P-matrix analysis is associated with the problem of substituting R with an R that produces rank(R ) = n. This is mostly done by orthogonal decomposition methods, such as principal components analysis, partial least squares (PLS), or continuum regression [4]. Dimension requirements of involved matrices for these methods are m n, and p n. If the method of least squares is used, additional constraints on matrix dimensions are needed [4]. The approach of P-matrix analysis does not require quantitative concentration information of all constituents. Specifically, calibration samples with known concentrations of analytes under investigation satisfy the calibration needs. The method of PLS will be used in this chapter for P-matrix analysis. 

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