Nonlinear calibration approaches

Nonlinear Calibration Approaches Spectral data can respond nonlinearly to process perturbations due to deviations of the Lambert-Beer law, to the nonlinear characteristics of light detectors or to interactions among analytes. Sources of nonlinear behavior and techniques for the detection of important nonlinear effects in spectral responses have been discussed in the literature [25, 76]. In order to cope with the nonlinear features of spectral data sets, different approaches have been applied to build calibration models. These calibration approaches have almost always been based on NN models and locally weighted regression (LWR) models. 

The approach described by Garland and collaborators (Garland and Powell, 1981 Min et al., 1978) is generally applicable and is different from the previous ones in that they use nonlinear calibration curves. By dividing both numerator and denominator by p, Eq. (1) was rewritten as... 

This approach is based on assuming (a) a Gaussian (but nonuniform) axial dispersion with a variance (T (y) (b) a Gaussian (but nonuniform) instantaneous MWD of variance a-liV) and average retention volume To(T) and (c) an (in general nonlinear) calibration M = Di(y) exp[-D2(T)y], The corrected chromatogram is obtained from... 

This approach can be used for multicomponent mixtures by applying matrix algebra. This is generally done with a software program and even nonlinear calibrations can be handled... 

Instead of a theoretical model that requires careful measurements of distances, angles, and so on, an experimental calibration approach is preferred. The experimental calibration estimates the model parameters based on the images of a calibration target as recorded by each camera. A linear imaging model that works well for most cases, the pinhole camera model, is based on geometrical optics. This leads to the following direct linear transform equations, where x,y are image coordinates, and X,Y,Z are object coordinates. This physics-based model cannot describe nonlinear phenomena such as lens distortions. 

Inner filter effects are often responsible for distorted emission spectra and nonlinear calibration curves between fluorescence intensity and fluorophore concentration. In a case of concentrated samples, in order to minimize or correct for IFE either instrumental or mathematical corrections can be done (Kao et al., 1998). Instrumental corrections are based on the observation that fluorescence intensity and fluorophore concentration are linear at low absorbance. In principle, any method that can lower the absorbance will reduce IFEs. Sample dilution (Guilbault, 1990 Lakowicz, 1983) is the most popular approach, but it may cause changes in conformation, bonding, solvation, and the degree of association. Also other chemical events may alter absorption-fluorescence processes and, there by, introduce large unknown errors (Holland, 1977). 

The camera model has a high number of parameters with a high correlation between several parameters. Therefore, the calibration problem is a difficult nonlinear optimization problem with the well known problems of instable behaviour and local minima. In out work, an approach to separate the calibration of the distortion parameters and the calibration of the projection parameters is used to solve this problem. 

We now consider how one extracts quantitative infonnation about die surface or interface adsorbate coverage from such SHG data. In many circumstances, it is possible to adopt a purely phenomenological approach one calibrates the nonlinear response as a fiinction of surface coverage in a preliminary set of experiments and then makes use of this calibration in subsequent investigations. Such an approach may, for example, be appropriate for studies of adsorption kinetics where the interest lies in die temporal evolution of the surface adsorbate density N. ... 

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. 

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. 

M. Blanco, J. Coello, H. Iturriaga, S. Maspoch and J. Pages, NIR calibration in nonlinear systems by different PLS approaches and artificial neural networks, Chemom. Intell. Lab. Syst., 50(1), 2000, 75-82. 

As a supplemental approach to determine the linearity of calibration curves that span a wide concentration range, many analytical chemists replot the data on a log-log scale. Although that approach makes the data points at the lower end of the concentration scale easier to see, the data are displayed in a nonlinear system which distorts the patterns that existed in a linear scale.24 The tendency, however, is to interpret the curve using a linear scale, so log-log plots are easily misinterpreted. 

Apart from the problem of nonlinearity, the calibration curve approach has another pitfall measured ion abundance ratios can change with time, leading to the possibility of significant errors since the calibration and sample measurements cannot be simultaneous (Schoeller, 1980). In order to minimize the effect of instrumental drift and to optimize precision, the National Bureau of Standards (NBS) proposed a bracketing protocol for the development of definitive (i.e., essentially bias-free and precise) IDMS methods (Cohen et al., 1980 White et al., 1982 Yap et al., 1983). It involves the measurement of each sample between measurements of calibration standards whose ion abundances most closely surround the ion abundance ratio of the sample. Measurements are made according to a strict protocol, used with samples prepared under restrictive conditions ... 

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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