Inverse calibration methods

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... 

With this background infonnation on the inverse methods, it is instructive to examine the calculations for the inverse model in more detail. In Equation 5-23, the key to the model-building step is the inversion of the matrix CR ). This is a squire matrix with number of rows and columns equal to the number of measurement variables (nvars). From theory, a number of independent samples in the calibration set greater than or equal to nvars is needed in order to invert this matrix. For most analytical measurement systems, nvars (e.g., number of wavelengths) is greater than the number of independent samples and therefore RTr cannot be directly inverted. However, with a transformation, calculating she pseudo-inverse of R (R is possible. How this transformation is accomplished distinguishes the different inverse methods. 

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. 

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... 

Multivariate calibration tools are used to construct models for predicting some characteristic of future samples. Chapter 5 begins with a discussion of the reasons for choosing multivariate over univariate calibration methods. The most widely used multivariate calibration tools are then presented in two categories classical and inverse methods. 

With respect to the spreading calibration, several methods have been suggested e.g. (6-1 ) Numerous techniques have been proposed for solving the inverse filtering problem represented by Equation 1, with different degrees of success e.g. (it,15-19) Only references (M, (l8) and (I9) make no assumptions on the shape of g(t,x). 

Most chemometricians prefer inverse methods, but most traditional analytical chemistry texts introduce the classical approach to calibration. It is important to recognise that there are substantial differences in terminology in the literature, the most common problem being the distinction between V and y variables. In many areas of analytical chemistry, concentration is denoted by V, the response (such as a spectroscopic peak height) by y However, most workers in the area of multivariate calibration have first been introduced to regression methods via spectroscopy or chromatography whereby the experimental data matrix is denoted as 6X , and the concentrations or predicted variables by y In this paper we indicate the experimentally observed responses by V such as spectroscopic absorbances of chromatographic peak areas, but do not use 6y in order to avoid confusion. 

Calibration equations can be obtained, as follows, using inverse methods. First, select the absorbances of the 25 spectra at these four wavelengths to give an A matrix with four columns and 25 rows. Second, obtain the corresponding C matrix consisting of the relevant concentrations (Table 6). The aim is to find coefficients B relating X and C by... 

Calibration equations can be obtained, as follows, using inverse methods. 

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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To take account of interactions between individual components (association, nonlinearities), calibration using multivariate data analysis is often also carried out with mixtures rather than pure substances. Despite this fact, limitations to this method of assessment are encountered quickly. Therefore, the so-called inverse method using the g-matrix is employed, and either principal component regression (PCR) or the partial least squares (PLS) method is used [6, [114], [116]. In both methods, calibration is carried out not with pure substances, but with various mixtures, which must cover the expected concentration range of all components. Within limits, this can allow for non-linearities ... 

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. 

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 ... 

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... 

In Section 12.3.2, the fundamental differences between direct and inverse modeling methods were discussed. As will be discussed here, this distinction is not just a convenient means for classifying quantitative regression methods, but has profound implications regarding calibration strategy and supporting infrastructure. 

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. 

Inversion pulses can be calibrated either by the crude method which consists of applying the pulse to an isolated multiplet and searching the... 

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... 

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. 

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