Hard models

Figure 5-31 clearly features three minima at the correct positions, Aiesi=0, and the two rate constants used to generate the data ltest=0.03 and Xtest=0.1. A very interesting feature of the whole method is that the rate constants are completely independent. Each minimum, or rate constant, is defined on its own, completely independent of all the others. This is in clear contrast to normal, hard-modelling data fitting where the residuals are a function of all parameters together. 

One could argue whether PCR and PLS should be part of the chapter Model-Based Analyses or Model-Free Analyses. Both, PCR and PLS, are clearly not hard-model fitting methods in the way presented in Chapter 4, nor are they pure model-free analyses. They are somewhere in between, maybe closer to model-free analyses and that is the reason for discussing them here. 

Model-based nonlinear least-squares fitting is not the only method for the analysis of multiwavelength kinetics. Such data sets can be analyzed by so-called model-free or soft-modeling methods. These methods do not rely on a chemical model, but only on simple physical restrictions such as positiveness for concentrations and molar absorptivities. Soft-modeling methods are discussed in detail in Chapter 11 of this book. They can be a powerful alternative to hard-modeling methods described in this chapter. In particular, this is the case where there is no functional relationship that can describe the data quantitatively. These methods can also be invaluable aids in the development of the correct kinetic model that should be used to analyze the data by hard-modeling techniques. 

FIGURE 11.8 Effect of the hard-modeling constraint on a set of concentration profiles representing a protonation process in the presence of an interference (left profiles, unconstrained right profiles, constrained). Only the compounds involved in the protonation are constrained according to the physicochemical law. 

Biochemical processes are among the most challenging and interesting reaction systems. Due to the nature of the constituents involved, macromolecules such as nucleic acids or proteins, the processes to be analyzed do not follow a simple physicochemical model, and their mechanism cannot be easily predicted. For example, well-known reactions for simple molecules, e.g., protonation equilibria, increase in complexity for macromolecules due to the presence of polyelectrolytic effects or conformational transitions. Because the data analysis cannot be supported in a model-fitting procedure (hard-modeling methods), the analysis of these processes requires soft-modeling methods that can unravel the contributions of the process without the assumption of an a priori model. 

Most traditional approaches to classification in science are called discriminant analysis and are often also called forms of hard modelling . The majority of statistically based software packages such as SAS, BMDP and SPSS contain substantial numbers of procedures, referred to by various names such as linear (or Fisher) discriminant analysis and canonical variates analysis. There is a substantial statistical literature in this area. 

Quantitative models is a heterogeneous group of models expressed in mathematical language. This includes what can be called hard models of general applicability, e.g. thermodynamic models, quantum mechanical models, absolute rate theory, as well as soft models or local models, usually expressed in terms of analogy and similarity, e.g. linear free energy relationships (LFERs), correlations for spectroscopic structural determination, empirical determined kinetic models, and as we shall see, models obtained by statistical treatment of experimental data from properly designed experiments. 

In principle, it would be possible to determine the outcome of any chemical reaction if (a) The reaction mechanisms were known in detail, i.e. if all equilibrium constants and all rate constants of intermediary steps were known and (b) the initial concentrations of the reactants and the activity coefficients of all species involved were perfectly known. However, this is never the case in practice. It would be impossible to derive such a model by deduction from physical chemical theory without introducing drastic assumptions and simplifications. A consequence of this is, that the precision of any detailed prediction from such hard models will be low. In addition to this, physical chemical models rarely take interaction effects between experimental variables into account, which means that, in practice, such models will not be very useful for analysing the influence of experimental variables on synthetic operations. 

As was seen in the preceeding section on hard models it is evident that it will be very difficult to derive an analytical expression for the function/by theoretical means. For practical purposes, it is, however, possible to use experiments to establish sufficiently good local approximations of the function/. 

Environmental studies are very complex and often no hard model is known. Studying these systems is very expensive and extends over a long period of time. Two-way and three-way arrays are built as the study progresses and one hopes to find some kind of latent variables that are useful for interpretation. 

In Chapter 10, examples of the need for scatter plots of both types are given. Sometimes the underlying hard model is reflected very strongly in the PARAFAC or Tucker3 loadings... 

If we take an uncritical view and build on the scratch hardness model assuming that the volume displaced. A, in some way defines the wear rate of the whole assembly of asperities we produce a relationship for the abrasive wear per unit distance (8,23) ... 

Maeso, M. J., and Solana, 1. R., Corresponding-states principle for two-dimensional hard models of molecular fluids, Phys. Rev. E, 56, 466 71 (1997). 

The Grain Size Influence on the Load Effect of the Hardness Modeling Experimental Results... 

The described approaches are often named hard modeling as they rely on physicochemical information that is sample specific. A modification of the hard modehng became available indirect hard modehng (IHM) [23, 24] considers each... 

The soft models defined by the weights of the neural networks are capable of accommodating all types of relationships, being especially useful when the dependence of the retention behavior with the mobile phase variables is unknown. However, neural networks learn the relationships from the data themselves, and hence, more experimental points are needed with respect to hard-modeling methods. The use of neural networks is, therefore, only recommended for those cases where adequate theoretical or empirical models do not exist, such as retention modeling in MLC with four variables (e.g., pH, surfactant, modifier, and temperature). 

Coomans D and Massart DL (1992) Hard modelling in supervised pattern recognition. In Brereton RG (ed.) Multivariate Pattern Recognition in Chemometrics, pp. 249-288. Amsterdam Elsevier. 

The Higher Education Institutions in which teacher education takes place are driven by the standards agenda and managerialism and are themselves hardly models... 

Apart from discrete modelling of relaxation processes, ID and 2D Inverse Laplace Transformation (ILT) is gaining more and more interest. Moreover, soft and hard modelling data processing tools like PLS or multivariate curve resolution (MCR) are applied to low field NMR data. Special algorithms were developed for the needs in relaxation modelling, for example DOUBLESLICING l... 

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