Linear calibration models

There is a general agreement that at least the following validation parameters should be evaluated for quantitative procedures selectivity, calibration model (linearity), stability, accuracy (bias, precision) and limit of quantification. Additional parameters which might have to be evaluated include limit of detection, recovery, reproducibility and ruggedness (robustness) [2,4-10,12],... 

Calibration model Linear Nonhnear (usually 4/5 PL model)... 

A solvent free, fast and environmentally friendly near infrared-based methodology was developed for the determination and quality control of 11 pesticides in commercially available formulations. This methodology was based on the direct measurement of the diffuse reflectance spectra of solid samples inside glass vials and a multivariate calibration model to determine the active principle concentration in agrochemicals. The proposed PLS model was made using 11 known commercial and 22 doped samples (11 under and 11 over dosed) for calibration and 22 different formulations as the validation set. For Buprofezin, Chlorsulfuron, Cyromazine, Daminozide, Diuron and Iprodione determination, the information in the spectral range between 1618 and 2630 nm of the reflectance spectra was employed. On the other hand, for Bensulfuron, Fenoxycarb, Metalaxyl, Procymidone and Tricyclazole determination, the first order derivative spectra in the range between 1618 and 2630 nm was used. In both cases, a linear remove correction was applied. Mean accuracy errors between 0.5 and 3.1% were obtained for the validation set. 

Purpose Generate a data set that superimposes normally distributed noise on a linear calibration model to study the effects of the adjustable parameters. A whole calibration—measurement—evaluation sequence can be optimized for quality of the results and total costs. 

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. 

As mentioned above, the random character of the input and output variables are of importance with regard to the calibration model and its estimation by calculus of regression. Because of the different character of the analytical quantity x in the calibration step (no random variables but fixed variables which are selected deliberately) and in the evaluation step (random variables like the measured values), the closed loop of Fig. 6.1 does not correctly describe the situation. Instead of this, a linear progress as shown in Fig. 6.2 takes place. 

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... 

A calibration procedure has to be validated with regard to general and specific requirements under which the calibration model has been developed. For this purpose, it is important to test whether the conditions represented in Fig. 6.6 are fulfilled. On the other hand, it is to assure by experimental studies that certain performance features (accuracy, precision, sensitivity, selectivity, specificity, linearity, working range, limits of detection and of quantification, robustness, and ruggedness, see Chap. 7) fulfil the expected requirements. 

Commercial software packages are usually able to represent graphically the residual errors (deviations) of a given calibration model which can be examined visually. Typical plots as shown in Fig. 6.8 may give information on the character of the residuals and therefore on the tests that have to be carried out, such as randomness, normality, linearity, homoscedasticity, etc. 

When a blank appears, it has to be estimated from a sufficiently large number of blank measurements and the measured values must be corrected in this respect. To ensure the adequateness of the SA calibration model, p >2 additions should be carried out. Only in the case when it is definitely known that the linear model holds true, then one single addition (ft times repeated) may be carried out. In general, linearity can be tested according to Eqs. (6.49)-(6.51). 

The classical multivariate calibration represents the transition of common single component analysis from one dependent variable y (measured value) to m dependent variables (e.g., wavelengths or sensors) which can be simultaneously included in the calibration model. The classical linear calibration (Danzer and Currie [1998] Danzer et al. [2004]) is therefore represented by the generalized matrix relation... 

Some time ago we wrote an article entitled Linearity in Calibration [1], in which we presented some unexpected results when comparing a calibration model using MLR with the model found using PCR. That column generated an active response, so we are discussing the subject in some detail, spread over several columns. The first part of these discussions have been published [2] this chapter is the continuation of that one. 

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. 

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. 

Computation Effects Selection of the calibration model will influence results. For example, fitting a linear calibration function to data that are non-linear will result in increased uncertainty in values predicted by using the calibration function. 

A quantitative procedure should be validated for selectivity, calibration model, stability, accuracy (bias, precision), linearity, and limit of quantification (LOQ). Additional... 

FIGURE 4.25 PLS2 works with X- and K-matrix in this scheme both have three dimensions. t and u are linear latent variables with maximum covariance of the scores (inner relation) the corresponding loading vectors are p und q. The second pair of x- and y-components is not shown. A PLS2 calibration model allows a joint prediction of all y-variables from the x-variables via x- and y-scores. 

An extension of linear regression, multiple linear regression (MLR) involves the use of more than one independent variable. Such a technique can be very effective if it is suspected that the information contained in a single dependent variable (x) is insufficient to explain the variation in the independent variable (y). In PAT, such a situation often occurs because of the inability to find a single analyzer response variable that is affected solely by the property of interest, without interference from other properties or effects. In such cases, it is necessary to use more than one response variable from the analyzer to build an effective calibration model, so that the effects of such interferences can be compensated. 

This method can be considered a calibration transfer method that involves a simple instrument-specific postprocessing of the calibration model outputs [108,113]. It requires the analysis of a subset of the calibration standards on the master and all of the slave instmments. A multivariate calibration model built using the data from the complete calibration set obtained from the master instrument is then applied to the data of the subset of samples obtained on the slave instruments. Optimal multiplicative and offset adjustments for each instrument are then calculated using linear regression of the predicted y values obtained from the slave instrument spectra versus the known y values. 

Standardizing the predicted values is a simple, useful choice that ensures smooth calibration transfer in situations (a) and (b) above. The procedure involves predicting samples for which spectra have been recorded on the slave using the calibration model constructed for the master. The predicted values, which may be subject to gross errors, are usually highly correlated with the reference values. The ensuing mathematical relation, which is almost always linear, is used to correct the values subsequently obtained with the slave. 

Gi cn th.iu ail tiircc assiiiiiptions hold (linearity, linear addithin, and all pure spectra known), CIS has an advantage over the inverse methods (see Section 5.3) in that the calibration models are often easier to determine. For a simple system with three components, calibration may be as simple as obtaining the spectra of the three pure components. 

---