Soft-modeling calibration

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 ... 

In the A-matrix approach, all absorbing constituents of a sample must be explicitly known to be included into the calibration procedure. As we will see in the following, with more soft modeling techniques, it will also be possible to account for unknown constituents without their explicit calibration. 

A disadvantage of this calibration method is the fact that the calibration coefficients (elements of the P matrix) have no physical meaning, since they do not reflect the spectra of the individual components. The usual assumptions about errorless independent variables (here, the absorbances) and error-prone dependent variables (here, concentrations) are not valid. Therefore, if this method of inverse calibration is used in coimection with OLS for estimating the P coefficients, there is only a slight advantage over the classical /C-matrix approach, due to the fact that a second matrix inversion is avoided. However, in coimection with more soft modeling methods, such as PCR or PLS, the inverse calibration approach is one of the most frequently used calibration tools. 

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

Visual inspection should be possible from plots of predicted versus measured concentrations, from principal component plots of loadings and scores in the case of soft modeling techniques, and by plotting the standard error of calibration (SEC) or the standard error of prediction (SEP(-y, Eq. (6.68)) from cross-validation in dependence on the number of eigenvalues or of principal components. 

In 1989, Harald Martens and Tormod Nt6s published their now seminal book. Multivariate Calibration (John Wiley Sons Ltd, 1989). Although the book s primary focus is on what we now refer to as chemometric modeling — soft modeling to be more precise — they comment on an aspect of modeling that is at the present time very relevant to the focus of this chapter. To quote ... 

The Leggett Model simulates lead biokinetics in liver with two compartments the first simulates rapid uptake of lead from plasma and a relatively short removal half-life (days) for transfers to plasma and to the small intestine by biliary secretion a second compartment simulates a more gradual transfer to plasma of approximately 10% of lead uptake in liver. Different transfer rates associated with each compartment are calibrated to reproduce patterns of uptake and retention of lead observed in humans, baboons, and beagles following intravenous injection, as well as blood-to-liver concentration ratios from data on chronically exposed humans. Similarly, the Leggett Model simulates lead biokinetics in three compartments of soft tissues, representing rapid, intermediate, and slow turnover rates (without specific physiologic correlates). 

Force-Curves and Force-Modulation Calibration. In figure l(a b), typical force-indentation curves obtained respectively on a rigid ( = 610 MPa) and a soft (E = 27 MPa) polymer are presented. The elastic modulus derived from the analysis of the force-indentation curves is compared to the bulk elastic modulus measured by DMA in figure 1(c). For this analysis, the used tip geometry was adapted to the maximum indentation depth reached during the experiment, Smax- For Smax Rj the spherical geometry was considered while, for Smax the conical one was used. For intermediate values, the paraboloid model was used. 

The finite-difference (FD) definition based on vertical ionization energy (IP) and electron affinity (EA) scales was added for experimental assessment (Lackner Zweig, 1983). For comparison, the softness based chemical hardness values based on sphere-charged model of Pearson was also employed (Pearson, 1997). In all cases the atomic values were computed upon hydrogen calibration to its experimental 6.45 eV value. All values are in electron-volts (Putz, 2008c). 

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