Calibration inverse

The classical direct or indirect calibration is carried out by OLS minimization according to Gauss. Error-free analytical values x are assumed or at 

The inverse calibration regresses the analytical values (concentrations), x, on the measured values, y. Although with it a prerequisite of the GAussian least squares minimization is violated because the y-values are not error-free, it has been proved that predictions with inverse calibration are more precise than those with the classical calibration (Centner et al. [1998]). This holds true particularly for multivariate inverse calibration. 

In chemometrics, the inverse calibration model is also denoted as the P-matrix model (the dimension of P is m x n)  

The calibration coefficients are elements of the matrix P which can be estimated by 

The analysis of an unknown sample is carried out by multiplication of the measured spectrum y by the P-matrix 

Difference between errors in (a) classical and (b) inverse calibration 

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The model of eq. (36.3) has the considerable advantage that X, the quantity of interest, now is treated as depending on Y. Given the model, it can be estimated directly from Y, which is precisely what is required in future application. For this reason one has also employed model (36.3) to the controlled calibration situation. This case of inverse calibration via Inverse Least Squares (ILS) estimation will be treated in Section 36.2.3 and has been treated in Section 8.2.6 for the case of simple straight line regression. 

In inverse calibration one models the properties of interest as a function of the predictors, e.g. analyte concentrations as a function of the spectrum. This reverses the causal relationship between spectrum and chemical composition and it is geared towards the future goal of estimating the concentrations from newly measured spectra. Thus, we write... 

The advantage of the inverse calibration approach is that we do not have to know all the information on possible constituents, analytes of interest and inter-ferents alike. Nor do we need pure spectra, or enough calibration standards to determine those. The columns of C (and P) only refer to the analytes of interest. Thus, the method can work in principle when unknown chemical interferents are present. It is of utmost importance then that such interferents are present in the Ccdibration samples. A good prediction model can only be derived from calibration data that are representative for the samples to be measured in the future. 

The suffix in T (nxA) and P (< xA) indicates that only the first A columns of T and P are used, A being much smaller than n and q. In principal component regression we use the PC scores as regressors for the concentrations. Thus, we apply inverse calibration of the property of interest on the selected set of factor scores ... 

In the case that the original variables, the measured values y, are used for inverse calibration, there are no significant advantages of the procedure apart from the fact that no second matrix inversion has to be carried out in the analysis step see Eq. (6.87). On the contrary, it is disadvantageous that the calibration coefficients (elements of the P-matrix) do not have any physical meaning because they do not reflect the spectra of the single species. In addition, multicollinearities may appear which can make inversion of the T-matrix difficult see Eq. (6.86). 

On the other hand, when latent variables instead of the original variables are used in inverse calibration then powerful methods of multivariate calibration arise which are frequently used in multispecies analysis and single species analysis in multispecies systems. These so-called soft modeling methods are based, like the P-matrix, on the inverse calibration model by which the analytical values are regressed on the spectral data ... 

Centner V, Massart DL, de Jong S (1998) Inverse calibration predicts better than classical calibration. Fresenius J Anal Chem 361 2... 

Like MLR, PCR [63] is an inverse calibration method. However, in PCR, the compressed variables (or PCs) from PCA are used as variables in the multiple linear regression model, rather than selected original X variables. In PCR, PCA is first done on the calibration x data, thus generating PCA scores (T) and loadings (P) (see Section 12.2.5), then a multiple linear regression is carried out according to the following model ... 

The inverse calibration method of Projection to Latent Structures (PLS, also known as partial least squares ), is very similar to PCR, and has been a highly utilized tool in PAT [1]. Like the PCR method, PLS uses the... 

Several PAT calibration strategies, especially those that are intended to support inverse calibration methods, rely heavily on data that is routinely collected from the deployed analyzer, as opposed to data collected from carefully designed experiments. Such data, often called happenstance data , can be very inexpensive. 

For inverse calibration methods, the fact that reference data (y) is never noise-free in practice allows irrelevant variation in the x variables to find its way into the calibration model. 

Recall thatm the example above the interest is in developing a predictive model for ecK onent A using spectroscopy. A response surface design is appropriate for the controllable variables because the model is to be used for prediction ani the relationship of some of the variables is considered to be complex. Ta it 2.4 also shows that the pressure and oxygen concentration cannot be comcoUed, but the variation is significant. In this case, a natural design for these 3WO variables also needs to be incorporated into the experimental scheme. M inverse calibration technique can then be used to develop a predictive mofM. 

If the system is not simple, an inverse calibration method can be employed where it is iKst necessary to obtain the spectra of the pure analytes. The three inverse methods discussed later in this chapter include multiple linear regression (MLR), jirincipal components regression (PCR), and partial least squares (PLS). Wlien using. MLR on data sees found in chemlstiy, variable. sciectson is... 

Partial least squares (PLS) and principal component regression (PCR) are the most widely used multivariate calibration methods in chemometrics. Both of these methods make use of the inverse calibration approach, where it i.s... 

FIGURE 5.63. Example of calibration and validation using the inverse calibration approach, (a) Initial inverse model form (b) estimating the regression vector (c) preaicting the concentrations of components 1 and 2. 

In the field of chemometrics, PCR and PLS are the most widely used of the inverse calibration methods. Tliese methods solve the matrix inversion problem inherent to the inverse methods by using a linear combination of variables in... 

The multivariate quantitative spectroscopic analysis of samples with complex matrices can be performed using inverse calibration methods, such as ILS, PCR and PLS. The term "inverse" means that the concentration of the analyte of interest is modelled as a function of the instrumental measurements, using an empirical relationship with no theoretical foundation (as the Lambert Bouguer-Beer s law was for the methods explained in the paragraphs above). Therefore, we can formulate our calibration like eqn (3.3) and, in contrast to the CLS model, it can be calculated without knowing the concentrations of all the constituents in the calibration set. The calibration step requires only the instrumental response and the reference value of the property of interest e.g. concentration) in the calibration samples. An important advantage of this approach is that unknown interferents may be present in the calibration samples. For this reason, inverse models are more suited than CLS for complex samples. 

ILS is a least-squares method that assumes the inverse calibration model given in eqn (3.4). For this reason it is often also termed multiple linear regression (MLR). In this model, the concentration of the analyte of interest, k, in sample i is regressed as a linear combination of the instrumental measurements at J selected sensors [5,16-19] ... 

In this book, we plot analytical response on the y-axis versus concentration on the x-axis. The inverse calibration (y = concentration, x = response) is said to provide a less biased estimate of concentration from a measured response. 

See V. Centner, D. L. Massart, and S. de Jong, Inverse Calibration Predicts Better Than Qassical Calibration, Fresenius J. Anal. Chem. 1998,361, 2 ... 

For most spectroscopic applications, the goal of multivariate calibration is to predict the concentration of a given analyte(s) in a future (prospective) sample using only its measured spectrum and a previously determined model. To do this, the inverse calibration method is used in which equation (12.2) is rewritten as... 

Inverse calibration. Although classical calibration is widely used, it is not always the most appropriate approach in analytical chemistry, for two main reasons. First, the ultimate aim is usually to predict the concentration (or factor) from the spectrum or chromatogram (response) rather than vice versa. There is a great deal of technical discussion of the philosophy behind different calibration methods, but in other areas of chemistry the reverse may be true, for example, can a response... 

The approach described above is related to classical calibration, but it is also possible to envisage an inverse calibration model since... 

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