Multivariate problem, description

There are apparently many multivariate statistical methods partly overlapping in scope [11]. For most problems occurring in practice, we have found the use of two methods sufficient, as discussed below. The first method is called principal component analysis (PCA) and the second is the partial least-squares projection to latent structures (PLS). A detailed description of the methods is given in Appendix A. In the following, a brief description is presented. 

Model-based fitting of measured data can be a rather complex process, particularly if there are many parameters to be fitted to many data points. Multivariate measurements can produce very large data matrices, especially if spectra are acquired at many wavelengths. Such data sets may require many parameters for a quantitative description. It is crucial to deal with such large numbers of parameters in efficient ways, and we will describe how this can be done. Large quantities of data are no longer a problem on modem computers, since inexpensive computer memory is easily accessible. 

In the resolution of any multicomponent system, the main goal is to transform the raw experimental measurements into useful information. By doing so, we aim to obtain a clear description of the contribution of each of the components present in the mixture or the process from the overall measured variation in our chemical data. Despite the diverse nature of multicomponent systems, the variation in then-related experimental measurements can, in many cases, be expressed as a simple composition-weighted linear additive model of pure responses, with a single term per component contribution. Although such a model is often known to be followed because of the nature of the instrumental responses measured (e.g., in the case of spectroscopic measurements), the information related to the individual contributions involved cannot be derived in a straightforward way from the raw measurements. The common purpose of all multivariate resolution methods is to fill in this gap and provide a linear model of individual component contributions using solely the raw experimental measurements. Resolution methods are powerful approaches that do not require a lot of prior information because neither the number nor the nature of the pure components in a system need to be known beforehand. Any information available about the system may be used, but it is not required. Actually, the only mandatory prerequisite is the inner linear structure of the data set. The mild requirements needed have promoted the use of resolution methods to tackle many chemical problems that could not be solved otherwise. 

Another problem that has been tackled by multivariate statistical methods is the characterization of the solvation capability of organic solvents based on empirical parameters of solvent polarity (see Chapter 7). Since such empirical parameters of solvent polarity are derived from carefully selected, strongly solvent-dependent reference processes, they are molecular-microscopic parameters. The polarity of solvents thus defined cannot be described by macroscopic, bulk solvent characteristics such as relative permittivities, refractive indices, etc., or functions thereof. For the quantitative correlation of solvent-dependent processes with solvent polarities, a large variety of empirical parameters of solvent polarity have been introduced (see Chapter 7). While some solvent polarity parameters are defined to describe an individual, more specific solute/solvent interaetion, others do not separate specific solute/solvent interactions and are referred to as general solvent polarity scales. Consequently, single- and multi-parameter correlation equations have been developed for the description of all kinds of solvent effects, and the question arises as to how many empirical parameters are really necessary for the correlation analysis of solvent-dependent processes such as chemical equilibria, reaction rates, or absorption spectra. 

PCA and its application to chemometric problems have been reviewed by different authors [18,19], and a detailed description is beyond the scope of this article. However, we can introduce some of the basic concepts that will be used along this work. Basically, the PCA provides an approximations of the original multivariate description (the X matrix) in terms of two small matrices called scores (T) and loadings (P). Assuming that X is mean centered ... 

The states of a dynamic system are simply the variables that appear in the time differential. The time-domain differential equation description of multivariable systems can be used instead of Laplace-domain transfer functions. Naturally, the two are related, and we derive these relationships below. State variables are very popular in electrical and mechanical engineering control problems, which tend to be of lower order (fewer differential equations) than chemical engineering control problems. Transfer function representation is more useful in practical process control problems because the matrices are of lower order than would be required by a state variable representation. For example, a distillation column can be represented by a 2X2 transfer function matrix. The number of state variables of the column might be 200. 

A couple of problems exist with most of the shape descriptors mentioned above. The first is their nonspecific and nondirectional nature. In a single value, one cannot simultaneously enc e information about both the direction and the form of the space the molecule occupies. Thus, a parameterized or at least a multivariate, description of shape is necessary. This is a feature of some of the descriptors mentioned, but it is usually not easy to interpret. 

If one of the pair of variables is correlated with a dependent variable with a correlation coefficient of 0.7 this may well be very useful in the description of the property that we are interested in. If the variable that is retained in the data set from that pair is one that correlates with the dependent (Zi in Fig. 3.7) then all is well. If, however, X was discarded and X2 retained then this parameter may now be completely uncorrelated (0 = 90°) with the dependent variable. Although this is an idealized case and perhaps unlikely to happen so disastrously in a multivariate data set, it is still a situation to be aware of. One way to approach this problem is to keep a list of all the sets of correlated variables that were in the starting set. Figure 3.8 shows a diagram of the correlations between a set of parameters before and after treatment with the CORCHOP procedure. If no satisfactory correlations with activity are found in the de-correlated set, individual variables can be re-examined using a diagram such as Fig. 3.8. A list of such correlations may also assist when attempts are made to explain correlations in terms of mechanism or chemical features. 

This is a description of a system at steady state in which first order reaction takes place, the rate constant s being different in each environment. The transform of the multivariable joint p.d.f can then be readily obtained since the system of equations is linear. Moreover, obtaining the transform of a marginal p.d.f. simply requires assigning no reaction terms (s = 0) in all environments except in the region of interest and solving the problem far the properly dimensionalized tracer concentration at... 

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