Matrices multivariate methods

To obtain this matrix by the multivariate method, we first generate two absorptivity vectors ap and a2j from a known concentration matrix in parts per million... 

Matrix (3-72) is essentially the same as mahix (3-70), but it is not exactly the same because it was obtained by the multivariate method from a different data set. 

Calibration Most process analyzers are designed to monitor concentration and/or composition. This requires a calibration of the analyzer with a set of prepared standards or from well-characterized reference materials. The simple approach must always be adopted first. For relatively simple systems the standard approach is to use a simple linear relationship between the instrument response and the analyte/ standard concentration [27]. In more complex chemical systems, it is necessary to adopt either a matrix approach to the calibration (still relying on the linearity of the Beer-Lambert law) using simple regression techniques, or to model the concentration and/or composition with one or more multivariate methods, an approach known as chemometrics [28-30]. 

Multivariate methods, on the other hand, resolve the major sources by analyzing the entire ambient data matrix. Factor analysis, for example, examines elemental and sample correlations in the ambient data matrix. This analysis yields the minimum number of factors required to reproduce the ambient data matrix, their relative chemical composition and their contribution to the mass variability. A major limitation in common and principal component factor analysis is the abstract nature of the factors and the difficulty these methods have in relating these factors to real world sources. Hopke, et al. (13.14) have improved the methods ability to associate these abstract factors with controllable sources by combining source data from the F matrix, with Malinowski s target transformation factor analysis program. (15) Hopke, et al. (13,14) as well as Klelnman, et al. (10) have used the results of factor analysis along with multiple regression to quantify the source contributions. Their approach is similar to the chemical mass balance approach except they use a least squares fit of the total mass on different filters Instead of a least squares fit of the chemicals on an individual filter. 

The term factor is a catch-all for the concept of an identifiable property of a system whose quantity value might have some effect on the response. Factor tends to be used synonymously with the terms variable and parameter, although each of these terms has a special meaning in some branches of science. In factor analysis, a multivariate method that decomposes a data matrix to identify independent variables that can reconstitute the observed data, the term latent variable or latent factor is used to identify factors of the model that are composites of input variables. A latent factor may not exist outside the mathematical model, and it might not therefore influence... 

Other multivariate methods have been applied to ICP spectra for quantitative measurements. As examples, they include multicomponent spectral fitting (which is incorporated in several commercial instrument software) [81] matrix projection, which avoids measurement of background species [94,95] generalised standard additions [96] and Bayesian analysis [97]. 

The third noteworthy feature of near-infrared spectra presented in Figure 13.3 is the uniqueness of the spectral patterns for each analyte. Although the spectral features are highly overlapping, the spectrum for glucose is notably unique relative to the others. The uniqueness of each spectrum provides the selectivity that is required for sound analytical measurements. However, the extensive overlap dictates that an analysis of the full spectrum is needed to extract the unique spectral signature for the targeted analyte relative to the sample matrix. Powerful multivariate methods are available for this purpose, as described in Chapter 12. 

Preliminary data analysis carried out for the spectral datasets were functional group mapping, and/or hierarchical cluster analysis (HCA). This latter method, which is well described in the literature,4,9 is an unsupervised approach that does not require any reference datasets. Like most of the multivariate methods, HCA is based on the correlation matrix Cut for all spectra in the dataset. This matrix, defined by Equation (9.1),... 

Contributions from molecular ions are typically more difficult to correct quantitatively because there may be many molecular ions important over a short mass range the molecular ion intensity varies, depending on the sample matrix and the intensities may vary over time more dramatically than elemental ion signals. Multivariant methods including multiple linear regression [150,151], principal component analysis [152], and multicomponent analysis [153] have been used. Improvements in detection limits by up to two orders of magnitude... 

Variable selection is particularly important in LC-MS and GC-MS. Raw data form what is sometimes called a sparse data matrix, in which the majority of data points are zero or represent noise. In fact, only a small percentage (perhaps 5% or less) of die measurements are of any interest. The trouble with this is that if multivariate methods are applied to the raw data, often the results are nonsense, dominated by noise. Consider the case of performing LC-MS on two closely eluting isomers, whose fragment ions are of principal interest. The most intense peak might be the molecular... 

We begin the discussion of multivariate methods, however, by returning to the topic of symmetry in the chemical fragment, introduced in Chapter 2. The presence of symmetry in the 2 D topological representation of the fragment has implications for the structure of the data matrix G(Af,Ap) that are related to the symmetry of... 

The vectors vj are used to construct a data matrix which is analysed with multivariate methods. 

Fig. 3. Comparison of some multivariate methods used in QSAR relating a matrix of biologieal variables, Y, to a matrix of deseriptor variables, X PCRA (prineipal component regression analysis) PLS (partial least-squares method) PCA (principal component analysis according to the Weiner/Malinowski approach) MRA (multivariate regression analysis) and CCA (canonical correlation analysis).

A graphical representation is less easy for three variables and no longer possible for four or more it is here that computer analysis is particularly valuable in finding patterns and relationships. Matrix algebra is needed in order to describe the methods of multivariate analysis fully. No attempt will be made to do this here. The aim is to give an appreciation of the purpose and power of multivariate methods. Simple data sets will be used to illustrate the methods and some practical applications will be described. 

Although colorimetric methods were the earliest to be used for pesticide analysis [203], competitive spectroscopic methodologies for the determination of these pollutants were not developed until the last decade. The spectroscopic determination of several pesticides in mixtures has been the major hindrance, especially when their analytical characteristics are similar and their signals overlap as a result. Multivariate calibration has proved effective with a view to developing models for qualitative and quantitative prediction from spectroscopic data. Thus, partial least squares (PLS) and principal component regression (PCR) have been used as calibration models for the spectrofluorimetric determination of three pesticides (carbendazim, fuberidazole, and thiabendazole) [204]. A three-dimensional excitation-emission matrix fluorescence method has also been used for this purpose (Table 18.3) [205]. 

The variable of the data matrix yielded by supervised analysis are the real concentrations of the detected metabolites, and the appUcatimi of multivariate methods can directly lead to the understanding of the metabolic system under study. 

The current widespread interest in MFC techniques was initiated by pioneering research performed by two industrial groups in the 1970s. Shell Oil (Houston, TX) reported their Dynamic Matrix Control (DMC) approach in 1979, while a similar technique, marketed as IDCOM, was published by a small French company, ADERSA, in 1978. Since then, there have been over one thousand applications of these and related MFC techniques in oil refineries and petrochemical plants around the world. Thus, MFC has had a substantial impact and is currently the method of choice for difficult multivariable control problems in these industries. However, relatively few applications have been reported in other process industries, even though MFC is a veiy general approach that is not limited to a particular industiy. 

One important class of nonlinear programming techniques is called quadratic programming (QP), where the objective function is quadratic and the constraints are hnear. While the solution is iterative, it can be obtained qmckly as in linear programming. This is the basis for the newest type of constrained multivariable control algorithms called model predic tive control. The dominant method used in the refining industiy utilizes the solution of a QP and is called dynamic matrix con-... 

Errors due to nonspectral interferences can be reduced via matrix matching, the method of standard additions (and its multivariant extensions), and the use of internal standards. ... 

The aim of all the foregoing methods of factor analysis is to decompose a data-set into physically meaningful factors, for instance pure spectra from a HPLC-DAD data-set. After those factors have been obtained, quantitation should be possible by calculating the contribution of each factor in the rows of the data matrix. By ITTFA (see Section 34.2.6) for example, one estimates the elution profiles of each individual compound. However, for quantitation the peak areas have to be correlated to the concentration by a calibration step. This is particularly important when using a diode array detector because the response factors (absorptivity) may considerably vary with the compound considered. Some methods of factor analysis require the presence of a pure variable for each factor. In that case quantitation becomes straightforward and does not need a multivariate approach because full selectivity is available. 

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