Nonlinear calibration models

In calibration, the number of degrees of freedom depends on the number of parameters estimated by the given model. In case of the two-parametric model (Eq. 6.6) v = n — 2, in case of linear calibration through the coordinate origin (a = 0) v = n — 1, and in case of a three-parametric nonlinear calibration model... 

Validation of a chromatographic method demands that linearity be established to verify that the analyte response is linearly proportional to the concentration range of interest. Chromatographic methods allow the use of both linear and nonlinear models for the calibration data. Given the limitations of the nonlinear method, the analyst will most likely choose the linear model. The nonlinear calibration model may be necessary to achieve very low detection limits or to address specific method demands. 

Because nonlinear calibration needs higher expense both in experimental and computational respects, linear models are mostly preferred. 

ANNs need supervised learning schemes and can so be applied for both classification and calibration. Because ANNs are nonlinear and model-free approaches, they are of special interest in calibration. 

Other, related, questions are also important Having determined this in isolation, how does the data analyst determine this in real data, where unknown amounts of several effects may be present There is a similarity here to Richard s earlier point regarding the relationship between the amount of noise and the amount of nonlinearity. Here are more fertile areas for research into the behavior of calibration models. 

Draper and Smith [1] discuss the application of DW to the analysis of residuals from a calibration their discussion is based on the fundamental work of Durbin, et al in the references listed at the beginning of this chapter. While we cannot reproduce their entire discussion here, at the heart of it is the fact that there are many kinds of serial correlation, including linear, quadratic and higher order. As Draper and Smith show (on p. 64), the linear correlation between the residuals from the calibration data and the predicted values from that calibration model is zero. Therefore if the sample data is ordered according to the analyte values predicted from the calibration model, a statistically significant value of the Durbin-Watson statistic for the residuals in indicative of high-order serial correlation, that is nonlinearity. 

In a well-behaved calibration model, residuals will have a Normal (i.e., Gaussian) distribution. In fact, as we have previously discussed, least-squares regression analysis is also a Maximum Likelihood method, but only when the errors are Normally distributed. If the data does not follow the straight line model, then there will be an excessive number of residuals with too-large values, and the residuals will then not follow the Normal distribution. It follows, then, that a test for Normality of residuals will also detect nonlinearity. 

There are other mysteries in NIR (and other applications of chemometrics) that nonlinearity can also explain. For example, as indicated above, one is the difficulty of transferring calibration models between instruments, even of the same type. Where would our technological world be if a manufacturer of, say, rulers could not reliably transfer the calibration of the unit of length from one ruler to the next ... 

But given our discussion above, he should not be. So in this case it is only surprising that he is able to extrapolate the predictions - we think that it is inevitable, since he has found a way to utilize only those wavelengths where nonlinearity is absent. Now what we need are ways to extend this approach to samples more nearly like real ones. And if we can come up with a way to determine the spectral regions where all components are linearly related to their absorbances, the issue of not being able to extrapolate a calibration should go away. Surely it is of scientific as well as practical and commercial interest to understand the reasons we cannot extrapolate calibration models. And then devise ways to circumvent those limitations. 

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. 

The main limitation of this model [6,14] is that it assumes that the measured response at a given sensor is due entirely to the constituents considered in the calibration step, whose spectra are included in the matrix of sensitivities, S. Hence, in the prediction step, the response of the unknown sample is decomposed only in the contributions that are found in S. If the response of the unknown contains some contributions from constituents that have not been included in S (in addition to background problems and baseline effects), biased predicted concentrations may be obtained, since the system will try to assign this signal to the components in S. For this reason, this model can only be used for systems of known qualitative composition (e.g. gas-phase spectroscopy, some process monitoring or pharmaceutical samples), in which the signal of all the pure constituents giving rise to a response can be known. For the same reason, CLS is not useful for mixtures where interaction between constituents or deviations from the Lambert-Beer law (nonlinear calibration curves) occur. 

Kowalska and Urbanski [82] applied an ANN to compute calibration models for two XRF gauges. The performance of the ANN calibration was compared with that of a PLS model. It was found that the ANN calibration model exhibited better prediction ability in cases when the relationship between input and output variables was nonlinear. 

Diagnostic information can be obtained to determine whether the calibration model provides an adequate fit to the standards, e.g., nonlinearity or other kinds of model errors can be detected, or whether an unknown sample is adequately fitted by the calibration model. A large lack of fit is usually due to background signals different from those present in the calibration standards. This is what some people have called the false sample problem. For example, suppose a calibration model was developed for the spectroscopic determination of iron in dissolved carbon steel samples. This model might be expected to provide a poor performance in the determination of iron in stainless steel samples. In this case, a figure-of-merit calculated from the biased model would detect the false sample. ... 

Using artificial neural networks to develop calibration models is also possible. The reader is referred to the literature [68-70] for further information. Neural networks are commonly utilized when the data set maintains a large degree of nonlinearity. Additional multivariate approaches for nonlinear data are described in the literature [71, 72],... 

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