Inverse calibration model

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

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 ... 

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] ... 

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

The methods of soft modeling are based on the inverse calibration model where concentrations are regressed on spectral data ... 

PLS Approach Details of the PLS method were given earlier in this chapter. In multicomponent analysis, we obtain the following equations for the decomposition of the absorbance matrix. 4 (the former X matrix) and the concentration matrix C (formerly the Y matrix) according to the inverse calibration model in Eq. (6.87) ... 

In the case of an inverse calibration model, one can interpret the model in Eq. (6.88) as regressing the concentrations of a single... 

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. 

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 ... 

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. 

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. 

The DCLS method can be applied to simple systems where all of the pure-component spectra can be measured. To construct the DCLS model, the pure-component spectra are measured at unit concentration for each of the analytes in the mixture. Tliese are used to form a matrix of pure spectra (S) and the model is then constructed as the pseudo-inverse of this S matrix. This calibration model is used to predict the concentrations in unknown samples. 

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. 

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. 

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... 

Constrained Regularization (CR) To understand constrained regularization, multivariate calibration can be viewed as an inverse problem. Given the inverse mixture model for a single analyte... 

Including the intercept. In many situations it is appropriate to include extra terms in the calibration model. Most commonly an intercept (or baseline) term is included to give an inverse model of the form... 

Faber, N.M., Efficient computation of net analyte signal vector in inverse multivariate calibration models, Anal. Chem., 70, 5108-5110, 1998. 

As previously noted, in a typical process analytical application, the measured data set might consist of spectral data recorded at a number of wavelengths much higher than the number of samples. The rank, R, of the measured matrix of spectra will be equal to or smaller than the number of the samples N. This causes rank deficiency in X, and the direct calculation of a regression or calibration model by use of the matrix inverse using Equation 8.85 and Equation 8.86 is problematic. 

Multivariate techniques are inverse calibration methods. In normal least-squares methods, often called classical least-squares methods, the system response is modeled as a function of analyte concentration. In inverse methods, the concentrations are treated as functions of the responses. The latter has some advantages in that concentrations can be accurately predicted even in the presence of chemical and physical sources of interference. In classical methods, all components in the system need to be considered in the mathematical model produced (regression equation). 

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