Non-linear calibration

BS 7118 Part 2 1989. Measurement of flaiid Flow Assessment of Uncertainty in the Calibration and Use of Flow Measurement Devices Non-linear Calibration Relationships. British Standards Institution, 1989. 

Non-linear curves may be treated using Equation 9 directly, using the techniques of non-linear least squares, when appropriate. (Note that a non-linear calibration curve does not necessarily imply non-linear least squares. The latter is necessary only if the problem is non-linear in the estimated parameters (16). For example, y = a+bx+cx and y = a+bx are both non-linear functions, but only the latter is non-linear in the parameters.)... 

Non-linear Hydrodynamic Volume Calibration Curve. The hydrodynamic calibration curve, log. V shown in Figure lb, is generated using the commercially available narrow MWD polystyrene standards listed in Table 1 and published values (28, 29) of the Mark-Houjjink parameters K and a for polystyrene in THF a 25°C, (K=1.6 x 10, o = 0.706 for > 10,000 and K = 9.0 x 10, a = 0.5 for Mw <10,000). The experimental data points composing the non-linear calibration curve were fitted with the phenomenologically based Yau-Malone equation.(30) This equation is derived from diffusion theory and is expressed as follows ... 

Another important characteristic of a detector is the linear dynamic range a range of analyte concentrations that produce a calibration line best described with a linear equation. Non-linear calibrations with subtle curvatures may be also used in the linear dynamic range of the detector. 

The non-linear calibration models are also acceptable. Figure 4.6 shows an example of a non-linear calibration approximated with a quadratic fit. Non-linear calibration curves are not acceptable if used to compensate for the detector saturation at a high concentration level. 

A linear regression or non-linear calibration ICV acceptance criterion is percent drift calculated as shown in Equation 5, Appendix 22. 

Linear calibration verification is always performed with a single ICV standard. Non-linear calibrations may require two ICV standards analyzed at different inflections of the curve. 

Linear regression and non-linear calibration analytical sequence... 

Non-linearities occur in some forms of spectroscopy, especially when the absorbance is high, and greater effort has been made to enhance the basic PLS method to include squared and other terms. However, the analytical chemist will probably prefer to improve the experimental method of acquiring data. Non-linear calibration is most valuable in other areas of chemistry, such as QSAR, where a purely additive linear model is not necessarily expected. 

In analytical applications of liquid chromatography the most common causes of peak asymmetry are mixed mechanisms of retention, incompatibility of the sample with the chromatographic mobile phase, or development of excessive void volume at the head of the column. In preparative applications of liquid chromatography and related techniques, column overload can also contribute to peak asymmetry. The causes of severe peak asymmetry in analytical applications should be identified and corrected because they are frequently accompanied by concentration-dependent retention, non-linear calibration curves and poor precision. In addition, peak asymmetry can significantly compromise column efficiency leading, in turn, to reduced resolution and lower peak capacity (see sections 2.5 and 2.6). 

Prerequisites for the Calibration Types. It depends on the design of the analytical procedure as to which regression parameters are meaningful and which results are acceptable. In other words, the model to be used for quantitation must be justified. For a singlepoint calibration (external standardization), a linear function, zero intercept, and the homogeneity of variances are required. The prerequisites for a linear multiple-point calibration are a linear function and in case of an unweighted calibration also the homogeneity of variances. A non-linear calibration requires only a continuous function. With respect to the 100%... 

Rantanen, J. Rasanen, E. Antikainen, O. Mannermaa, J.-P. Yliruusi, J. In-hne moisture measurements during granulation with a four wavelength near-infrared sensor an evaluation of proces-related variables and a development non-linear calibration model. Chemometrics Intell. Lab. Syst. 2001, 56, 51-58. 

As in thin-layer chromatography the calibration curves are usually only linear in a limited range, various linearisation functions are provided in the software program of this system, but also non-linear calibration curves for working in a wider concentration range 15). 

If an alcohol such as methanol can be separated by ion-exclusion chromatography using water as the eluent, why not do the reverse and separate water using a methanol eluent Stevens et al. [12] did just this. They added a small amount of sulfuric acid to the eluent and detected the chromatographic water peak by a decrease in conductivity. The main drawback with this method was a non-linear calibration curve with very poor detection sensitivity in some concentration regions. 

The above relationship can also be used with binary systems. Its application permits a linear calibration graph to be obtained, which is preferable in most instances. We have re-calculated the non-linear calibration graphs presented earlier [236—238] using eqn. 3.10 and in all instances linear relationships were obtained. 

He continues "Often the calibration graphs are linear over a particular region but not over the region of Interest. Therefore, one should check the linearities and use a non-linear calibration graph as well." If fitting to a linear function Is desired (for simplicity s sake), the transformation function used for the amount variables need not be the same as that used on the response variables for equal variance. In this manner data can often be brought into conformity with the simpler linear function (O. 

The NMR experiment makes the direct observation of atoms possible. The integral of an NMR signal is strictly linearly proportional to the amount of atoms in the probe volume. The signals are a measure of molar ratios of molecules, independently of the molecular weight. There are no response factors such as those in UV-detection caused by varying extinctions dependent on molecular structures non-linear calibration curves such as those found with light-scattering detectors are unknown to NMR spectroscopy. 

With a linear calibration (on a log molecular mass scale) it should be possible to indicate an expected level of reproducibility, but with a non-linear calibration, the level of reproducibility will vary according to which part of the calibration (steep or shallow) is applicable. Also, for a non-linear calibration, the shape of the sample MMD will influence the level of reproducibility. 

Ma and Kim (1995) developed an on-line LC/TS/MS method for the analysis of PL molecular species in rat brain. After total lipid extraction, the extract was subjected to analysis with on-line reversed-phase HPLC and filament-on TS/MS. By using non-conventional HPLC conditions, partial separation of individual PL classes (PS, PI, PE and PC) and partial separation of molecular species within each class were achieved. By monitoring the retention time and the characteristic fragment ions (DG ions) formed in the filament-on TS process, individual molecular species in each PL class could be identified. Although non-linear calibration curves were observed for all DG ions monitored, even in the presence of an internal standard, semiquant-itative and quantitative results could still be obtained for a mixture of PLs. 

At each retention volume of w(V), the instantaneous MMDs in the detector cells were calculated by considering the contributions (toward that V) of aU the hypothetical molecular species in the distribution, as determined by the discrete w (V). From such instantaneous distributions, the ad hoc non-linear calibrations log M (V) and log M V) were calculated.From w(V), log M (V), and log M (V), the MMD estimates w(log M ) and w(log M ) were obtained their averages are presented in Table 2. As expected, wGog M ) accurately estimates M but underestimates M,v, while vv(log M ) accurately estimates but overestimates M . Thus, the global polydispersity is underestimated in both cases. 

In analytical chemistry we always try to arrange that Equation [8.19a] provides an adeqnate model for the relationship between the instrumental response (Y) and the concentration (or amount) of analyte (x) injected directly into the instrument (instrumental cahbration) or used to spike a blank matrix (method calibration, see Section 8.5). When analytical chemists speak of a linear calibration equation they refer to Equation [8.19a], a simple linear regression model that is linear in both the fitting parameters and also in the independent variable Equation [8.19b] might be referred to as a non-linear calibration equation by a chemist, although to a statistician it is an example of a simple linear regression model, i.e., it is hnear in aU of the fitting parameters. 

An ideal transducer would have an exactly linear relationship between pressure and output voltage. In reality, there will always be some deviation from the ideal linear relationship this is referred to as non-linearity. A best straight line (BSL) is fitted to the non-linear curve. The deviation from BSL is quoted in the specifications and expressed as a percent of full scale (FS). The non-linear calibration curve is determined in the ascending direction, i. e., with the pressure going from zero to full rating. 

For optimum sensor performance, the electron transfer process should be rapid and chemically reversible. With conventional metal electrodes it is unlikely that reversible electron transfer processes will change with analyte concentration. However, with CMEs it is likely this will occur. If so, then an extra degree of complexity is added to the interpretation of the CME sensor response in that non-linear calibration curves as well as non-linear standard addition curves will be obtained. Furthermore, the electron transfer process is also dependent on the chemical environment. For example, the nature and concentration of the counter-ion in conductive polymers have a significant influence. In such cases, matrix matching between standards and samples must be carried out. 

Figure 24.2 Non-linear calibration curve for biotin by post-column avidin reaction.

E. Alberttis, 1. Cortes, M. Alvarez, V. Cerd Non-linear calibration in single point flow titration of pro-tolytes. A comparison of model options by using delta Test, Anal. Chim. Acta 414 (2000) 221—237. 

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