Univariate calibration models

It is well known that the accuracy of CE determinations using univariate calibration models, such as linear regression, relies on the selectivity of the electrophoretic data. Peaks of analytes must be baseline resolved and the occurrence of comigrations and minor impurities should be avoided. Note that peak contaminations lead to wrong integrations, and, consequently, the concentrations estimated from these data may be unreliable. 

In this way the child spectrum is transformed into a spectrum as if measured on the parent instrument. In a more refined implementation one establishes the highest correlating wavelength channel through quadratic interpolation and, subsequently, the corresponding intensity at this non-observed channel through linear interpolation. In this way a complete spectrum measured on the child instrument can be transformed into an estimate of the spectrum as if it were measured on the parent instrument. The calibration model developed for the parent instrument may be applied without further ado to this spectram. The drawback of this approach is that it is essentially univariate. It cannot deal with complex differences between dissimilar instruments. 

While in classical statistics (univariate methods) modelling regards only quantitative problems (calibration), in multivariate analysis also qualitative models can be created in this case classification is performed. 

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... 

An example of CQV of the batch cultivation of a vaccine has been demonstrated, where univariate (temperature, dissolved oxygen, pH) as well as spectroscopic tools were used to develop process models. The measurements were used for a consistency analysis of the batch process, providing better process understanding which includes the understanding of the variations in the data. MSPC analysis of four batches of data was performed to monitor the batch trajectories, and indicated that one batch had a deviation in the pH. From the MSPC information, combined with calibration models for the composition of the process based on NIR spectral data, improved monitoring and control systems can be developed for the process, consistent with concept of CQV. The data from the univariate sensors and NIR were also fused for a global analysis of the process with a model comprised of all the measurements. 

Multivariate calibration tools are used to construct models for predicting some characteristic of future samples. Chapter 5 begins with a discussion of the reasons for choosing multivariate over univariate calibration methods. The most widely used multivariate calibration tools are then presented in two categories classical and inverse methods. 

In traditional method validation, assessment of the calibration has been discussed in terms of linear calibration models for univariate systems, with an emphasis on the range of concentrations that conform to a linear model (linearity and the linear range). With modern methods of analysis that may use nonlinear models or may be multivariate, it is better to look at the wider picture of calibration and decide what needs to be validated. Of course, if the analysis uses a method that does conform to a linear calibration model and is univariate, then describing the linearity and linear range is entirely appropriate. Below I describe the linear case, as this is still the most prevalent mode of calibration, but where different approaches are required this is indicated. 

Figure 3.1 Univariate calibration. Direct model (left) and inverse model (right). The arrow indicates the direction in which the prediction is made.

The most common, straightforward multivariate calibration model is the natural extent of the univariate calibration, the linear equation for which is... 

For all the mentioned reasons, there is an ongoing tendency in spectroscopic studies to manipulate samples less and perform fewer experiments but to obtain more data in each of them and use more sophisticated mathematical techniques than simple univariate calibration. Hence multivariate calibration methods are being increasingly used in laboratories where instruments providing multivariate responses are of general use. Sometimes, these models may give less precise or less accurate results than those given by the traditional method of (univariate) analysis, but they are much quicker and cheaper than classical approaches. 

The main advantage of multivariate calibration based on CLS with respect to univariate calibration is that CLS does not require selective measurements. Selectivity is obtained mathematically by solving a system of equations, without the requirement for chemical or instrumental separations that are so often needed in univariate calibration. In addition, the model can use a large number of sensors to obtain a signal-averaging effect [4], which is beneficial for the precision of the predicted concentration, making it less susceptible to the noise in the data. Finally, for the case of spectroscopic data, the Lambert Bouguer Beer s law provides a sound foundation for the predictive model. 

The quality of a model depends on the quality of the samples used to calculate it (or, to say it using the univariate approach, the quality of any traditional univariate calibration cannot be better than the quality of the standards employed to measure the analyte). Although this statement is trivial, the discussion on how many samples and which samples are required to develop a good predictive model is still open, so only general comments will be given. Below, we consider that the quality of the measurement device fits the purpose of the analytical problem. 

Nevertheless, obtaining a calibration model with a good fit for the calibration samples is not an indication that the model will be useful for predicting future samples. The more factors are considered in the PLS model, the better is the fit, although, probably, the prediction of new samples is worse. Hence another set of known samples, the validation set, is required to test the models. This can be a small group of samples prepared within the overall domain of the calibration samples (remember that extrapolation is not allowed in regression models, not even in the univariate models), with different concentrations of the analyte (we want to test that the model quantifies it correctly) and different concentrations of the concomitants. Then, if the predictions are good, we can be reasonably sure that the model can avoid the undesirable influence of the concomitants. 

Calibration is the process by which a mathematical model relating the response of the analytical instrument (a spectrophotometer in this case) to specific quantities of the samples is constructed. This can be done by using algorithms (usually based on least squares regression) capable of establishing an appropriate mathematical relation such as single absorbance vs. concentration (univariate calibration) or spectra vs. concentration (multivariate calibration). 

Univariate calibration is specific to situations where the instrument response depends only on the target analyte concentration. With multivariate calibration, model parameters can be estimated where responses depend on the target analyte in addition to other chemical or physical variables and, hence, multivariate calibration corrects for these interfering effects. For the ith calibration sample, the model with a nonzero intercept can be written as... 

Physical and chemical effects can be combined for identification as sample matrix effects. Matrix effects alter the slope of calibration curves, while spectral interferences cause parallel shifts in the calibration curve. The water-methanol data set contains matrix effects stemming from chemical interferences. As already noted in Section 5.2, using the univariate calibration defined in Equation 5.4 requires an interference-free wavelength. Going to multivariate models can correct for spectral interferences and some matrix effects. The standard addition method described in Section 5.7 can be used in some cases to correct for matrix effects. Severe matrix effects can cause nonlinear responses requiring a nonlinear modeling method. 

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. 

From the scores of the calibration data, regression models can be made for predicting the concentrations of the four analytes in new samples. A regression model for tryptophan was made using the score vector of component four of the PARAFAC model as independent variable and tryptophan as dependent variable in a univariate regression model without offset. Component four was chosen because the estimated emission and excitation spectra (e.g. Figure 10.14) of that component resembled the pure spectra of tryptophan. The scores of this component were also the ones correlating the most with the known concentrations of tryptophan. 

Using the score of component four, a univariate regression model is built predicting concentration from scores of the calibration samples (without an offset). As can be seen in Figure 10.19 the model of calibration data is excellent. The unknown scores in the three test samples can subsequently be found simply by multiplying their score of component four with the regression coefficient. As expected, sample number five is not well predicted, whereas the two remaining ones are well predicted. 

Analytical chemists always face a problem in comparison of the performance between analytical instruments. There is no simple rule to justify which one is better because of the variations between the instrumental responses. In order to correct this, a standardization approach is generally adopted. However, a calibration model as developed on an instrument cannot be employed for the other instrument in the real situation. Walczak et al. [28] suggested a new standardization method for comparing the performance between two near-infrared (NIR) spectrometers in the wavelet domain. In their proposed method, the NIR spectra from two different spectrometers were transformed to the wavelet domain at resolution level (J — 1). Suppose and correspond to the NIR spectra from Instruments 1 and 2, respectively, in the wavelet domain. A univariate linear model is applied to determine the transfer parameters t between and... 

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