Nonlinear calibration

Section 6.2 gave a brief description of the baseline method for determining A, based on the use of chart paper with the ordinate calibrated nonlinearly in absorbance units. This technique can also be... 

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. ... 

According to Beer s law, a calibration curve of absorbance versus the concentration of analyte in a series of standard solutions should be a straight line with an intercept of 0 and a slope of ab or eb. In many cases, however, calibration curves are found to be nonlinear (Figure 10.22). Deviations from linearity are divided into three categories fundamental, chemical, and instrumental. 

Accuracy When spectral and chemical interferences are minimized, accuracies of 0.5-5% are routinely possible. With nonlinear calibration curves, higher accuracy is obtained by using a pair of standards whose absorbances closely bracket the sample s absorbance and assuming that the change in absorbance is linear over the limited concentration range. Determinate errors for electrothermal atomization are frequently greater than that obtained with flame atomization due to more serious matrix interferences. 

Standardizing the Method Equations 10.32 and 10.33 show that the intensity of fluorescent or phosphorescent emission is proportional to the concentration of the photoluminescent species, provided that the absorbance of radiation from the excitation source (A = ebC) is less than approximately 0.01. Quantitative methods are usually standardized using a set of external standards. Calibration curves are linear over as much as four to six orders of magnitude for fluorescence and two to four orders of magnitude for phosphorescence. Calibration curves become nonlinear for high concentrations of the photoluminescent species at which the intensity of emission is given by equation 10.31. Nonlinearity also may be observed at low concentrations due to the presence of fluorescent or phosphorescent contaminants. As discussed earlier, the quantum efficiency for emission is sensitive to temperature and sample matrix, both of which must be controlled if external standards are to be used. In addition, emission intensity depends on the molar absorptivity of the photoluminescent species, which is sensitive to the sample matrix. 

Conductometric Analysis Solutions of elec trolytes in ionizing solvents (e.g., water) conduct current when an electrical potential is applied across electrodes immersed in the solution. Conductance is a function of ion concentration, ionic charge, and ion mobility. Conductance measurements are ideally suited tor measurement of the concentration of a single strong elec trolyte in dilute solutions. At higher concentrations, conduc tance becomes a complex, nonlinear func tion of concentration requiring suitable calibration for quantitative measurements. 

Linear type columns are especially designed to have wider linear molecular weight ranges. These linear-type columns are highly recommended for correcting nonlinear sections of molecular weight calibration curves (Table 6.2). 

It is important to understand that this material will not be presented in a theoretical vacuum. Instead, it will be presented in a particular context, consistent with the majority of the author s experience, namely the development of calibrations in an industrial setting. We will focus on working with the types of data, noise, nonlinearities, and other sources of error, as well as the requirements for accuracy, reliability, and robustness typically encountered in industrial analytical laboratories and process analyzers. Since some of the advantages, tradeoffs, and limitations of these methods can be data and/or application dependent, the guidance In this book may sometimes differ from the guidance offered in the general literature. 

Whether this tendency of PLS to reject nonlinearities by pushing them onto the later factors which are usually discarded as noise factors will improve or degrade the prediction accuracy and robustness of a PLS calibration as compared to the same calibration generated by PCR depends very much upon the specifics of the data and the application. If the nonlinearities are poorly correlated to the properties which we are trying to predict, rejecting them can improve the accuracy. On the other hand, if the rejected nonlinearities contain information that has predictive value, then the PLS calibration may not perform as well as the corresponding PCR calibration that retains more of the nonlinearities and therefore is able to exploit the information they contain. In short, the only sure way to determine if PLS or PCR is better for a given calibration is to try both of them and compare the results. 

Use the adsorption theory (of Section 2-1-3) to explain why adsorptive stripping voltammetry results in nonlinear calibration plots. 

The main consequences are twice. First, it results in contrast degradations as a function of the differential dispersion. This feature can be calibrated in order to correct this bias. The only limit concerns the degradation of the signal to noise ratio associated with the fringe modulation decay. The second drawback is an error on the phase closure acquisition. It results from the superposition of the phasor corresponding to the spectral channels. The wrapping and the nonlinearity of this process lead to a phase shift that is not compensated in the phase closure process. This effect depends on the three differential dispersions and on the spectral distribution. These effects have been demonstrated for the first time in the ISTROG experiment (Huss et al., 2001) at IRCOM as shown in Fig. 14. 

Selectivity and Interference Selectivity means that only that species is measured which the analyst is looking for. A corollary is the absence of chemical interferents. A lack of selectivity is often the cause of nonlinearity of the calibration curve. Near infra-red spectroscopy is a technology that exemplifies how seemingly trivial details of the experimental set-up can frustrate an investigator s best intentions Ref. 124 discusses some factors that influence the result. 

Nonlinear calibration curves are not forbidden, but they do complicate things quite a bit more calibration points are necessary, and interpolation from signal to concentration is often tedious. It would be improper to apply... 

Figure 4.36. Cross validation between two HPLCs A stock solution containing two compounds in a fixed ratio was diluted to three different concentrations (1 10 20) and injected using both the 10 and the 20 /xl loop on both instruments. The steps observed at Amount = 100 (gray ellipses) can be explained with effective loop volumes of 9.3 and 20 pi (model 1) and 14.3 and 20 pi (model 2) instead of nominally 10 and 20 pi. This is irrelevant as both a sample and the calibration solution will be run using the same equipment configuration. The curved portion of the model 2 calibration function was fitted using Y = A /x this demonstrates the nonlinearity of the response at these high concentrations. The angle between the full and the dotted line indicates the bias that would obtain if a one-point calibration scheme were used.

Option (Valid) presents a graph of relative standard deviation (c.o.v.) versus concentration, with the relative residuals superimposed. This gives a clear overview of the performance to be expected from a linear calibration Signal = A + B Concentration, both in terms of (relative) precision and of accuracy, because only a well-behaved analytical method will show most of the residuals to be inside a narrow trumpet -like curve this trumpet is wide at low concentrations and should narrow down to c.o.v. = 5% and rel. CL = 10%, or thereabouts, at medium to high concentrations. Residuals that are not randomly distributed about the horizontal axis point either to the presence of outliers, nonlinearity, or errors in the preparation of standards. 

Dose, E. V, and Guiochon, G., Bias and Nonlinearity of Ultraviolet Calibration Curves Measured Using Diode-Array Detectors, Anal. Chem. 61, 1989,... 

Schwartz, L. M., Lowest Limit of Reliable Assay Measurement with Nonlinear Calibration, Ana/. Chem. 55, 1983, 1424-1426. 

The nonlinear universal molecular weight calibration curve may be expressed as shown in equation ( ). 

S.D. Oman, T. Naes and A. Zube, Detecting and adjusting for nonlinearities in calibration of near-infrared data using principal components. J. Chemom., 7 (1993) 195-212. 

S. Sekulics, B.R. Kowalski, Z.Y. Wang, et al., Nonlinear multivariate calibration methods in analytical chemistry. Anal. Chem., 65 (1993) A835-A845. 

Figure 2 Example of an apparently linear calibration curve drawn from nonlinear calibration data, calculated E > 0.999...

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