Classical Calibration

One of the simplest problems is to determine die concentration of a single compound using the response at a single detector, for example a single spectroscopic wavelength or a chromatographic peak area. 

Mathematically a series of experiments can be performed to relate die concentration to spectroscopic measurements as follows  

A simple mediod for solving diis equation is to use the pseudo-inverse (see Chapter 2, Section 2.2.2.3, for an introduction)  

Absorbance at 335 nm for the PAH case study plotted against concentration of pyrene 

The quality of prediction can be determined by the residuals (or errors), i.e. die difference between the observed and predicted, i.e. x — x, the smaller, the better. Generally, the root mean error is calculated  

---

Classical calibration procedure can only be applied when all the species that contribute to the form of the spectra are known and can be included into the calibration. Additionally, there is the constraint that no interactions between the analytes and other species (e.g. solvent) or effects (e.g. of temperature) should occur. 

The inverse calibration regresses the analytical values (concentrations), x, on the measured values, y. Although with it a prerequisite of the GAussian least squares minimization is violated because the y-values are not error-free, it has been proved that predictions with inverse calibration are more precise than those with the classical calibration (Centner et al. [1998]). This holds true particularly for multivariate inverse calibration. 

Centner V, Massart DL, de Jong S (1998) Inverse calibration predicts better than classical calibration. Fresenius J Anal Chem 361 2... 

FIGURE 5.62. Example of calibration and validation using the classical calibration approach, (a) Initial classical model form (b) estimating concentrations (c) reconstructing the response vector (d) calculating the spectral residual (e) calculating the concentrational residual. 

In Section 5.2, the two classical calibration methods, direct and indirect CLS, are discussed. These methods work well with simple systems that adhere to a linear model (e.g., Beer s Law). Calibrating involves determining the spectra of the pure components and quantitation is achieved using regression. The distinction between these methods is in how the pure-component spectra are obtained. With DCLS they are measured directly with ICLS they are estimated from spectra of miiaures of the components. 

D. Grientschnig, Relation Between Prediction Errors of Inverse and Classical Calibration, Fresenius J. Anal. Chem. 2000,367, 497 J. Tellinghuisen, Inverse vs Classical Calibration for Small Data Sets, Fresenius J. Anal. Chem. 2000, 368, 585. 

See V. Centner, D. L. Massart, and S. de Jong, Inverse Calibration Predicts Better Than Classical Calibration, Fresenius J. Anal. Chem. 1998,361, 2 ... 

External standard quantitation involves the preparation of a classical calibration curve, as shown in Figure 4.6a. Standard samples are prepared at various concentrations over the desired range and analyzed. A calibration... 

This error can be represented as a percentage of the mean E% = 100 (E/x) = 24.1% in this case. It is always useful to check the original graph (Fig. 1) just to be sure, which appears a reasonable answer. Note that classical calibration is slightly illogical in analytical chemistry. The aim of calibration is to determine concentrations from spectral intensities, and not vice versa yet the calibration equation in this section involves fitting a model to determine a peak height from a known concentration. 

Inverse calibration. Although classical calibration is widely used, it is not always the most appropriate approach in analytical chemistry, for two main reasons. First, the ultimate aim is usually to predict the concentration (or factor) from the spectrum or chromatogram (response) rather than vice versa. There is a great deal of technical discussion of the philosophy behind different calibration methods, but in other areas of chemistry the reverse may be true, for example, can a response... 

It should be pointed out that the predictions for both methods described in this section differ from those obtained for the uncentred data. It is also useful to realise that it is also possible to use an intercept in models obtained using classical calibration the details have been omitted in this section for brevity. 

The approach described above is related to classical calibration, but it is also possible to envisage an inverse calibration model since... 

Table 5.3 Concentration of pyrene, absorbance at 335 nm and predictions of absorbance, using single parameter classical calibration.

This approach to calibration, although widely used throughout most branches of science, is nevertheless not always appropriate in all applications. We may want to answer the question can the absorbance in a spectrum be employed to determine the concentration of a compound . It is not the best approach to use an equation that predicts the absorbance from the concentration when our experimental aim is the reverse. In other areas of science the functional aim might be, for example, to predict an enzymic activity from its concentration. In the latter case univariate calibration as outlined in this section results in the correct functional model. Nevertheless, most chemists employ classical calibration and provided that the experimental errors are roughly normal and there are no significant outliers, all the different univariate methods should result in approximately similar conclusions. 

Although classical calibration is widely used, it is not always the most appropriate approach in chemistry, for two main reasons. First, the ultimate aim is usually to predict the concentration (or independent variable) from the spectrum or chromatogram (response) rather than vice versa. The second relates to error distributions. The errors in the response are often due to instrumental performance. Over the years, instruments have become more reproducible. The independent variable (often concentration) is usually determined by weighings, dilutions and so on, and is often by far the largest source of errors. The quality of volumetric flasks, syringes and so on has not improved dramatically over the years, whereas the sensitivity and reproducibility of instruments has increased manyfold. Classical calibration fits a model so diat all errors are in the response [Figure 5.4(a)], whereas a more appropriate assumption is that errors are primarily in the measurement of concentration [Figure 5.4(b)]. 

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... 

The classical calibration of NAA methods involves comparison of sample activities with those of co-irradiated standards of the same element. Especially at multi-element analysis, the need for a large number of standards has limited the sample throughput capacity. To circumvent this, alternative calibration techniques have been elaborated. The single comparator method makes multi-element determinations possible, by use of a single element standard (neutron flux monitor). The mass of the analyte is calculated by use of an experimentally determined element-specific factor (k-value), valid for the analytical equipment in question (Girardi et al., 1965 Linekin, 1973 Simonits et al., 1975). Later, a more generalized standardization method, based on accurately determined constants for the active compound nuclei (ko-factors), and applicable to various analytical equipments, has been proposed (Moens et al., 1984 De Code et al., 1987). 

The methodology presented so far corresponds to the classical calibration. A different approach is inverse calibration, where the functional relationship is directly modeled in the form required for prediction of concentrations in test samples x-=f y). This approach has proven useful in the context of mutlivariate calibration (see Chap 3.11). 

External standardization corresponds to the classical calibration procedure (Figure 3.163). The substance to be determined is used to prepare standard solutions with a known concentration. Measurements are made on standard solutions of different concentrations (calibration steps, calibration levels). For calibration, the peak areas determined are plotted against the concentrations of the different calibration levels. 

---