Principal-component discriminant

Figure 10.13 Principal components analysis scores plots (a) using all three example variables (first two principal components discriminations of the varieties, particularly sample Le2, would be very difficult) (b) using the best two variables, unweighted w selected [equation (10.31) in text] (discrimination of the varieties is now possible using only the first principal component) (c) discarding the best variable, unweighted w selected (linear discrimination of the varieties would not appear to be possible from this chart). Details of the variables are given in Table 10.3.

The previously mentioned data set with a total of 115 compounds has already been studied by other statistical methods such as Principal Component Analysis (PCA), Linear Discriminant Analysis, and the Partial Least Squares (PLS) method [39]. Thus, the choice and selection of descriptors has already been accomplished. 

Since that time thousands of QSARs, covering a wide and diverse range of end points, have been published [9] most of these have used MLR, but numerous other statistical techniques have also been used, such as partial least squares, principal component analysis, artificial neural networks, decision trees, and discriminant analysis [f4]. 

In the method of linear discriminant analysis, one therefore seeks a linear function of the variables, D, which maximizes the ratio between both variances. Geometrically, this means that we look for a line through the cloud of points, such that the projections of the points of the two groups are separated as much as possible. The approach is comparable to principal components, where one seeks a line that explains best the variation in the data (see Chapter 17). The principal component line and the discriminant function often more or less coincide (as is the case in Fig. 33.8a) but this is not necessarily so, as shown in Fig. 33.8b. 

Fig. 33.8. Situation where principal component (PC) and linear discriminant function (DF) are essentially the same (a) and very different (b).

SIMCA has inspired several related methods, such as DASCO [33] and CLASSY [34,35]. The latter has elements of the potential methods and SIMCA, while the former starts with the extraction of principal components, as in SIMCA, but then follows a quadratic discriminant rule. 

The combination of PCA and LDA is often applied, in particular for ill-posed data (data where the number of variables exceeds the number of objects), e.g. Ref. [46], One first extracts a certain number of principal components, deleting the higher-order ones and thereby reducing to some degree the noise and then carries out the LDA. One should however be careful not to eliminate too many PCs, since in this way information important for the discrimination might be lost. A method in which both are merged in one step and which sometimes yields better results than the two-step procedure is reflected discriminant analysis. The Fourier transform is also sometimes used [14], and this is also the case for the wavelet transform (see Chapter 40) [13,16]. In that case, the information is included in the first few Fourier coefficients or in a restricted number of wavelet coefficients. 

Multivariate chemometric techniques have subsequently broadened the arsenal of tools that can be applied in QSAR. These include, among others. Multivariate ANOVA [9], Simplex optimization (Section 26.2.2), cluster analysis (Chapter 30) and various factor analytic methods such as principal components analysis (Chapter 31), discriminant analysis (Section 33.2.2) and canonical correlation analysis (Section 35.3). An advantage of multivariate methods is that they can be applied in... 

While principal components models are used mostly in an unsupervised or exploratory mode, models based on canonical variates are often applied in a supervisory way for the prediction of biological activities from chemical, physicochemical or other biological parameters. In this section we discuss briefly the methods of linear discriminant analysis (LDA) and canonical correlation analysis (CCA). Although there has been an early awareness of these methods in QSAR [7,50], they have not been widely accepted. More recently they have been superseded by the successful introduction of partial least squares analysis (PLS) in QSAR. Nevertheless, the early pattern recognition techniques have prepared the minds for the introduction of modem chemometric approaches. 

The spectral pattern associated with principal component 1 is presented in Fig. 8. It provided the characteristic wavelengths which were the most discriminant to separate the... 

Subsequently 36 strains of aerobic endospore-forming bacteria, consisting of six Bacillus species and one Brevibacillus species could be discriminated using cluster analysis of ESMS spectra acquired in the positive ion mode (m/z 200-2000).57 The analysis was carried out on harvested, washed bacterial cells suspended in aqueous acidic acetonitrile. The cell suspensions were infused directly into the ionization chamber of the mass spectrometer (LCT, Micromass) using a syringe pump. Replicates of the experiment were performed over a period of six months to randomize variations in the measurements due to possible confounding factors such as instrumental drift. Principal components analysis (PCA) was used to reduce the dimensionality of the data, fol-... 

Figure 8.16 shows a principal component plot of that data the classification of which by MVDA was given in Fig. 8.11. It can be seen that a certain structure can be imagined which becomes clearer by the discrimination algorithm.

Because protein ROA spectra contain bands characteristic of loops and turns in addition to bands characteristic of secondary structure, they should provide information on the overall three-dimensional solution structure. We are developing a pattern recognition program, based on principal component analysis (PCA), to identify protein folds from ROA spectral band patterns (Blanch etal., 2002b). The method is similar to one developed for the determination of the structure of proteins from VCD (Pancoska etal., 1991) and UVCD (Venyaminov and Yang, 1996) spectra, but is expected to provide enhanced discrimination between different structural types since protein ROA spectra contain many more structure-sensitive bands than do either VCD or UVCD. From the ROA spectral data, the PCA program calculates a set of subspectra that serve as basis functions, the algebraic combination of which with appropriate expansion coefficients can be used to reconstruct any member of the... 

Different categories of Zonyl polymers are studied by ToF-SIMS both in the positive and negative ion mode. Studies have shown that, for each polymer, a specific fingerprint is obtained and the peaks corresponding to the specific chemical moieties of each polymer are detected (Figure 15.4). To represent this good selectivity, Principal Component Analysis is performed on the obtained spectra. The result clearly discriminates the different polymers. ToF-SIMS is then suited to the characterization of these materials. 

Fig. 8.4 Discriminant analyses of the principal chemical components in L. catta scent secretions by (a) gland, (b) season, and (c) individual, (a) Accurate classification of 97.5% of labial, scrotal, and brachial samples in = 77) by gland of origin (Wilks lambda = 0.003 P < 0.001). (b) Reliable differentiation of 100% of labial samples (n = 26) into prebreeding, breeding, and nonbreeding seasons (Wilks lambda = 0.018, P < 0.01). (c) Individual scent signatures in the scrotal secretions from seven males. LDA performed on 17 principal components correctly classified 100% of these samples to the individuals from which they were collected (Wilks lambda = 0.000, P < 0.002)...

The main classification methods for drug development are discriminant analysis (DA), possibly based on principal components (PLS-DA) and soft independent models for class analogy (SIMCA). SIMCA is based only on PCA analysis one PCA model is created for each class, and distances between objects and the projection space of PCA models are evaluated. PLS-DA is for example applied for the prediction of adverse effects by nonsteroidal anti-... 

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