Multivariate calibration methods

Calibration Most process analyzers are designed to monitor concentration and/or composition. This requires a calibration of the analyzer with a set of prepared standards or from well-characterized reference materials. The simple approach must always be adopted first. For relatively simple systems the standard approach is to use a simple linear relationship between the instrument response and the analyte/ standard concentration [27]. In more complex chemical systems, it is necessary to adopt either a matrix approach to the calibration (still relying on the linearity of the Beer-Lambert law) using simple regression techniques, or to model the concentration and/or composition with one or more multivariate methods, an approach known as chemometrics [28-30]. 

This method can be considered a calibration transfer method that involves a simple instrument-specific postprocessing of the calibration model outputs [108,113]. It requires the analysis of a subset of the calibration standards on the master and all of the slave instmments. A multivariate calibration model built using the data from the complete calibration set obtained from the master instrument is then applied to the data of the subset of samples obtained on the slave instruments. Optimal multiplicative and offset adjustments for each instrument are then calculated using linear regression of the predicted y values obtained from the slave instrument spectra versus the known y values. 

Standardizing the spectral response is mathematically more complex than standardizing the calibration models but provides better results as it allows slight spectral differences - the most common between very similar instruments - to be corrected via simple calculations. More marked differences can be accommodated with more complex and specific algorithms. This approach compares spectra recorded on different instruments, which are used to derive a mathematical equation, allowing their spectral response to be mutually correlated. The equation is then used to correct the new spectra recorded on the slave, which are thus made more similar to those obtained with the master. The simplest methods used in this context are of the univariate type, which correlate each wavelength in two spectra in a direct, simple manner. These methods, however, are only effective with very simple spectral differences. On the other hand, multivariate methods allow the construction of matrices correlating bodies of spectra recorded on different instruments for the above-described purpose. The most frequent choice in this context is piecewise direct standardization... 

Multivariate calibration (chemometrics) methods must be used to develop acceptable methods. 

The remaining chapters of the book introduce some of the advanced topics of chemometrics. The coverage is fairly comprehensive, in that these chapters cover some of the most important advanced topics. Chapter 6 presents the concept of robust multivariate methods. Robust methods are insensitive to the presence of outliers. Most of the methods described in Chapter 6 can tolerate data sets contaminated with up to 50% outliers without detrimental effects. Descriptions of algorithms and examples are provided for robust estimators of the multivariate normal distribution, robust PC A, and robust multivariate calibration, including robust PLS. As such, Chapter 6 provides an excellent follow-up to Chapters 3, 4, and 5. 

For inttoductory purposes multiple linear regression (MLR) is used to relate the experimental response to the conditions, as is common to most texts in this area, but it is important to realise that odter regression methods such as partial least squares (PLS) are applicable in many cases, as discussed in Chapter 5. Certain designs, such as dtose of Section 2.3.4, have direct relevance to multivariate calibration. In some cases multivariate methods such as PLS can be modified by inclusion of squared and interaction terms as described below for MLR. It is important to remember, however, diat in many areas of chemistry a lot of information is available about a dataset, and conceptually simple approaches based on MLR are often adequate. 

The least-squares procedure just described is an example of a univariate calibration procedure because only one response is used per sample. The process of relating multiple instrument responses to an analyte or a mixture of analytes is known as multivariate calibration. Multivariate calibration methods have become quite popular in recent years as new instruments become available that produce multidimensional responses (absorbance of several samples at multiple wavelengths, mass spectrum of chromatographically separated components, and so forth). Multivariate calibration methods are very powerful. They can be used to determine multiple components in mixtures simultaneously and can provide redundancy in measurements to improve precision. Recall that repeating a measurement N times provides a Vn improvement in the precision of the mean value. These methods can also be used to detect the presence of interferences that would not be identified in a univariate calibration. 

The prediction of Y-data of unknown samples is based on a regression method where the X-data are correlated to the Y-data. The multivariate methods, usually used for such a calibration, are principal component regression (PCR) and partial least squares regression (PLS). Both methods are based on the assumption of linearity and can deal with co-linear data. The problem of co-linearity is solved in the same way as the formation of a PCA plot. The X-variables are added together into latent variables, score vectors. These vectors are independent since they are orthogonal to each other and they can therefore be used to create a calibration model. 

The similarity between the diffuse reflectance FTIR spectra of aristeromycin and neplanocin A (Figure 2) and the complexity of the supernatant background FTIR spectra (see Figure 3) necessitates the use of multivariate methods for quantitative analysis of the components. Training data for the construction of multivariate calibration models and test data for model validation were provided by HPLC and FTIR analysis of the S. citricolor mutant fermentation supernatants with a total of 48 samples taken from duplicate fermentations after 3 and 6 days incubation. 

In the case of quantitative analysis, the amount of, or the exact relation between, the constituents of a compound or a mixture have to be estabhshed. The direct relation between the properties of a specimen and the concentration of its constituents could also be the aim of the quantitative investigation. In the latter case so-called calibration models have to be established, the corresponding model parameters have to be estimated, and they have to be confirmed by statistical methods. Calibration models estabhshed this way may then be used to determine, on a statistically verified basis, the concentration of constituents of an analyte within the calibrated range. AU multivariate methods for quantitative analysis mentioned in Fig. 22.2 are employed for the evaluation of spectra. 

Copolymers are comprised of chains containing two or more different types of monomers. The composition of copolymers may be quantitatively determined by using infrared spectroscopy [3, 8]. Distinctive representative modes for the polymers may be identified. For example, in the case of vinyl chloride-vinyl acetate copolymers, the ratio of the absorbance of the acetate mode at 1740 cm to that of the vinyl chloride methylene bending mode at 1430 cm can be used for quantitative analysis. Copolymers of known composition may be used for calibration. The multivariate methods described earlier in Chapter 3 may also be applied. Care must be exercised because the position and shapes of the infrared bands of the components of copolymers may be affected by the sequencing of the constituent monomers. 

When the laboratory value is plotted against the NIR predicted value for the calibration sample set it may well be noted that some points lie well away from the computed regression line. This will, of course, reduce the correlation between laboratory and NIR data and increase the SEC or SEP. These samples may be outliers. The statistic hi describes the leverage or effect of an individual sample upon a regression. If a particular value of hi is exceeded this may be used to determine an outlier sample. Evaluation criteria for selecting outliers, howevei are somewhat subjective so there is a requirement for expertise in multivariate methods to make outlier selection effective. 

Inverse least squares in an example of a multivariate method. In this type of model, the dependent variable (concentration) is solved by calculating a solution from multiple independent variables (in this case, the responses at the selected wavelengths). It is not possible to work backwards from the concentration value to the independent spectral response values because an infinite number of possible solutions exist. However, the main advantage of a multivariate method is the ability to calibrate for a constituent of interest without having to account for any interferences in the spectra. 

In a practical sense, outliers are those samples that have unique character so as to make them recognizably (statistically) different from a designated sample population. Evaluation criteria for selecting outliers are often subjective therefore there is a requirement that some expertise in multivariate methods by employed prior to discarding any samples from a calibration set. 

This chapter consists of two distinct parts. In the first part, a cursory overview of chemometric methods, as applicable to analysis of and quantitation with NIR data is presented. In Section 10.2 common methods for preprocessing NIR spectra are described. Section 10.3 discusses multivariate methods for developing predictive calibration models with NIR spectra. In Section 10.4, strategies for sample and model validation are presented that exploit the multivariate nature of NIR spectra. The performance of the multivariate methods discussed in Section 10.2 through Section 10.4 are applied to a set of 80 NIR reflectance data of corn flour for determination of four physical properties moisture, oil, protein, and starch. 

The chemometric method is an essential tool for analyzing complicated NIR spectra of biomedical samples that always show overlapping absorption bands. In the construction of a calibration model, PLS regression is the most popular multivariate method [38,39], In general, PLS is a powerful method, but when it is applied to NIR spectra of very complex samples consisting of a number of components, it does not always yield good results. This is because such NIR spectra usually contain interface signals, such as those due to water and other components. 

However, most of the available simple analyzing systems doesn t show a sufficiently good selectivity and interference by other components in the sample has to be considered. When the presence of such other components is established and their concentration as well as their impact are known, a mathematical correction can be made, but when the concentrations of these interfering compounds are unknown other measures have to be taken. The most obvious way out is too install a series of analyzers with different selectivities for the different components and to apply so-called multivariate calibration techniques. In the domain of chemometry, these techniques have shown a fast development during the last two decades. Apart from the techniques that are appropriate for linear calibration systems, methods for the calibration of systems... 

Multiwavelength methods. Least squares curve fitting techniques may be used in the determination of multicomponent mixtures with overlapping spectral features. Two classical quantitation methods, the Classical Least Squares (CLS) mode and the Inverse Least Squares (ILS) model, are applied when wavelength selection is not a problem. CLS is based on Beer s law and uses large regions of the spec-tram for calibration but cannot cope with mixtures of interacting constituents. ILS (multivariate method) can accurately build models for complex mixtures when only some of the constituent concentrations are known. 

Multivariate calibration has the largest number of applications of chemometric methods in routine analysis for instance it became a widely used technique in quantitative analy.sis of complex mixtures by IR or UV detectors, especially in food chemistry and environmental analytical chemistry. A number of textbooks, tutorials, and reviews " have been published in this field, and software is offered by the instrument manufacturers. Applications of multivariate calibration is very widespread, ranging for instance from the determination of moisture in mushrooms to research for non-invasive measurement techniques of glucose in human blood. The multivariate methods mostly applied are PLS, PCR. and recently also neural networks, Typical calibration models describe the relationship between a set of x-variables (UV or IR absorbances) and one y-variable (concentration of one substance) although it is possible to derive models that simultaneously predict a set of y-variables. 

Closely related to multivariate calibration are the applications of multivariate methods for investigations of structure-property or stmcture-activity relationships (see Quantitative Structure-Activity Relationships in Drug Design and Quantitative Structure-Property Relationships (QSPR)). 

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