Calibration matrix classical least-squares

To produce a calibration using classical least-squares, we start with a training set consisting of a concentration matrix, C, and an absorbance matrix, A, for known calibration samples. We then solve for the matrix, K. Each column of K will each hold the spectrum of one of the pure components. Since the data in C and A contain noise, there will, in general, be no exact solution for equation [29]. So, we must find the best least-squares solution for equation [29]. In other words, we want to find K such that the sum of the squares of the errors is minimized. The errors are the difference between the measured spectra, A, and the spectra calculated by multiplying K and C ... 

Classical least-squares (CLS), sometimes known as K-matrix calibration, is so called because, originally, it involved the application of multiple linear regression (MLR) to the classical expression of the Beer-Lambert Law of spectroscopy ... 

There are several mathematical limitations inherent in the inverse least squares method. The number of frequencies employed cannot exceed the number of calibration standards in the training set. The selection of frequencies is further limited by the problem of collinearity that is, the solution of the matrix equation tends to become unstable as more frequencies that correspond to absorptions of a particular component x are included because the absorbances measured at these frequencies will change in a collinear manner with changes in the concentration of x. Thus, the possibilities for averaging out errors through the use of over-determination are greatly reduced by comparison with the classical least squares method, in which there are no limitations on the number of frequencies employed. 

The p and k matrix methods are two classical least squares approaches to multicomponent calibration. There are techniques based on factor analysis, however, that are increasingly popular these include the... 

However, multicomponent quantitative analysis is the area we are concerned with here. Regression on principle components, by PCR or PLS, normally gives better results than the classical least squares method in equation (10.8), where collinearity in the data can cause problems in the matrix arithmetic. Furthermore, PLS or PCR enable a significant part of the noise to be filtered out of the data, by relegating it to minor components which play no further role in the analysis. Additionally, interactions between components can be modelled if the composition of the calibration samples has been well thought out these interactions will be included in the significant components. 

This method of quantitative analysis is known as K matrix, or classic least squares (CLS). It has the advantage of being able to use large regions of the spectrum, or even the entire spectrum, for calibration to gain an averaging effect for the predictive accuracy of the final model. One interesting side effect is that if the entire spectrum is used for calibration, the rows of the K matrix are actually spectra of the absorptivities for each of the constituents. These will actually look very similar to the pure constituent spectra. 

In the MCR framework, there are few cases in which the quantitative analysis is based on the acquisition of a single spectrum per sample, as is the case for classical first-order multivariate calibration methods, such as partial least squares (PLS), seen in other chapters of this book. There are some instances in which quantitation of compounds in a sample by MCR can be based on a single spectrum, that is, a row of the D matrix and the related row of the C matrix. Sometimes, this is feasible when the compounds to be determined provide a very high signal compared with the rest of the substances in the food sample, for example colouring additives in drinks determined by ultraviolet—visible (UV-vis) spectroscopy [26,27]. Recently, these examples have increased due to the incorporation of a new cmistraint in MCR, the so-caUed correlation constraint [27,46,47], which introduces an internal calihratimi step in the calculation of the elements of the concentradmi profiles in the matrix C related to the analytes to be quantified. This calibration step helps to obtain real concentration values and to separate in a more efficient way the information of the analytes to be quantified from that of the interferences. 

---