Example Multivariate Distances

FIGURE 3.8 NIR reflectance spectra of 83 aliquots of sulfamethoxazole powder. 

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The HCA method, which uses any of a variety of multivariate distance calculations to identify similar spectra, has found little use in Raman spectroscopy, although it could be of use in the growing analysis of complicated systems in which a large heterogeneous sample set is being analyzed. A study of spruce needles by Krizova et al. [54] and an investigation of cancerous skin lesions by Fendel and Schrader [55] are two examples showing the modest power of HCA. 

One common query to the CSD (Table 2) involves the classification of fragment conformations - a recognition of the different 3-D shapes that are exhibited by a given 2-D chemical substructure. Here the number of geometrical parameters required to define the 3-D conformation is normally >2. For example, 15 distances or six torsion angles provide suitable descriptors for cyclohexane, and we must resort to multivariate statistical methods to analyse the appropriate G-matrix. Two techniques may be selected within GSTAT principal component analysis and cluster analysis. 

A great variety of different methods for multivariate classification (pattern recognition) is available (Table 5.6). The conceptually most simply one is fc-NN classification (Section 5.3.3), which is solely based on the fundamental hypothesis of multivariate data analysis, that the distance between objects is related to the similarity of the objects. fc-NN does not assume any model of the object groups, is nonlinear, applicable to multicategory classification, and mathematically very simple furthermore, the method is very similar to spectral similarity search. On the other hand, an example for a rather sophisticated classification method is the SVM (Section 5.6). 

HCA is a common tool that is used to determine the natural grouping of objects, based on their multivariate responses [75]. In PAT, this method can be used to determine natural groupings of samples or variables in a data set. Like the classification methods discussed above, HCA requires the specification of a space and a distance measure. However, unlike those methods, HCA does not involve the development of a classification rule, but rather a linkage rule, as discussed below. For a given problem, the selection of the space (e.g., original x variable space, PC score space) and distance measure (e.g.. Euclidean, Mahalanobis) depends on the specific information that the user wants to extract. For example, for a spectral data set, one can choose PC score space with Mahalanobis distance measure to better reflect separation that originates from both strong and weak spectral effects. 

Comparison and ranking of sites according to chemical composition or toxicity is done by multivariate nonparametric or parametric statistical methods however, only descriptive methods, such as multidimensional scaling (MDS), principal component analysis (PCA), and factor analysis (FA), show similarities and distances between different sites. Toxicity can be evaluated by testing the environmental sample (as an undefined complex mixture) against a reference sample and analyzing by inference statistics, for example, t-test or analysis of variance (ANOVA). 

The following examples demonstrate the usefulness of multivariate methods in the evaluation of field ecological data and laboratory multispecies toxicity tests. In each of the examples, several multivariate techniques were used — generally Euclidean and cosine distances (Figure 11.29), principal components, and nonmetric clustering and association analysis. 

The Mahalanobis distance occurs frequently in chemometric analysis, not only in cluster analysis but also in multivariate calibration, in discriminant analysis, and in modelling. It is appropriate to consider it further here, and the following account is based on the tutorial article by De Maesschalck et al Table 4.4 details ten objects A. .. J) described by two, mean-centred, variables (jcj and JC2). These data are illustrated graphically in Figure 4.6, and it is immediately apparent that the two variables exhibit high, positive correlation. In Figure 4.6(b), contours (circles) of equal Euclidean distance from the centroid are displayed and, for example, objects C and F have similar distance values. This is not the case if Mahalanobis distances are calculated and examined. 

Most uses of flow techniques involve the quantitative determination of some target species. This chapter describes various ways of using flow techniques with quantitative purposes, such as calibration curves, based on peak height or peak area, and titrations, based on distance between equivalence points or single-point method. Stopped-flow technique can be used for both, quantitative approach, for example in kinetic methods, or for qualitative determinations inasmuch as it allows spectral and potential scans to be performed. Multiparameter analysis is presented in two forms to be carried out, by multivariate chemometric techniques or by applying sandwich technique. Finally, smart systems are presented as a step forward in automation, commonly used in multiparameter analysis. 

Dynamics-Multivariate Plane Multivariate interactions are quantified through use of the RGA. The value of fij is a fimction of the distance of the calculated RGA from the ideal case of the identity matrix. This result is not a function of dynamic difficulty, as measured by fio, since the ideal RGA is not a function of model structure. While fij can be computed as a dynamic quantity by considering frequency dependence, it is only the magnitude of the dynamic components that contributes to fi[. For example, RHP and LHP-zeros of similar magnitude will contribute equally to /x/ since their difference lies in their phase characteristics, as will be detected by /xp, not in magnitude. 

Provided that the system (regression) parameters for a chromatographic method are known and the molecular descriptors for a new analyte are available, retention and thus selectivity should be predictable with the help of the LFER approach. Correlations of the calculated and the measured selectivities for nine pairs of solutes under six eluent compositions on a Merck Lichrospher 100 RP-18e column are shown in Fig. 6a for water/acetonitrile eluents and in Fig. 6b for water/ methanol eluents (taken from [14]). This plot, however, does not illustrate a real prediction of solute retention, since the depicted measured retention data served at the same time as a basis for the calculation of the theoretical selectivities. Even under these less critical conditions, relative deviations (expressed as the distances from the best linear fit in the x-coordinate) reach values of up to 20%, due to the significant residues from the multivariate regression (see confidence ranges in Fig. 4). To discuss the impact of these uncertainties on the quality of a possible prediction, a real example is discussed here. Starting from a selectivity of... 

Another approach to using spectra to identify cell type is a method called principal component analysis (PCA). This method is commonly used in multivariate analysis to reduce the number of variables and allow visualization of complex data sets in two or three dimensions [16,17]. We use our Raman spectra as examples and give a geometric interpretation of PCA to explain the process. We first assume that each of the 1340 spectral channels is a dimension in a 1340 hyperspace. As a result, any spectmm can be represented by a single point in this space. The distance between points in this hyperspace is determined by calculating the Euclidian distance... 

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