LINEAR DISCRIMINANT

Parametric discriminant development methods employ the mean vectors and covariance matrices of each class of data to develop the separating discriminant. Linear discriminant analysis (LDA), based on Bayesian statistics, generates a linear discriminant function with the following form ... 

In previous sections, we introduced the linear SVM classification algorithm, which uses the training patterns to generate an optimum separation hyperplane. Such classifiers are not adequate for cases when complex relationships exist between input parameters and the class of a pattern. To discriminate linearly nonseparable classes of patterns, the SVM model can be fitted with nonlinear functions to provide efficient classifiers for hard-to-separate classes of patterns. 

The geometrical measurements previously extracted help the making decision system to decide for example whether the defect is linear or not. This defect discrimination into two categories is considered as a first attempt for defect classification. To this end, we define a linearity ratio (Ri) Rl =Length / width. If Rl is equal or near to "1", the defect is volumic, otherwise it is a linear defect. 

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. 

Woodruff and co-workers introduced the expert system PAIRS [67], a program that is able to analyze IR spectra in the same manner as a spectroscopist would. Chalmers and co-workers [68] used an approach for automated interpretation of Fourier Transform Raman spectra of complex polymers. Andreev and Argirov developed the expert system EXPIRS [69] for the interpretation of IR spectra. EXPIRS provides a hierarchical organization of the characteristic groups that are recognized by peak detection in discrete ames. Penchev et al. [70] recently introduced a computer system that performs searches in spectral libraries and systematic analysis of mixture spectra. It is able to classify IR spectra with the aid of linear discriminant analysis, artificial neural networks, and the method of fe-nearest neighbors. 

Alternatives to Multiple Linear Regression Discriminant Analysis, Neural Networks and Classification Methods... 

Discriminant emalysis is a supervised learning technique which uses classified dependent data. Here, the dependent data (y values) are not on a continuous scale but are divided into distinct classes. There are often just two classes (e.g. active/inactive soluble/not soluble yes/no), but more than two is also possible (e.g. high/medium/low 1/2/3/4). The simplest situation involves two variables and two classes, and the aim is to find a straight line that best separates the data into its classes (Figure 12.37). With more than two variables, the line becomes a hyperplane in the multidimensional variable space. Discriminant analysis is characterised by a discriminant function, which in the particular case of hnear discriminant analysis (the most popular variant) is written as a linear combination of the independent variables ... 

We will explore the two major families of chemometric quantitative calibration techniques that are most commonly employed the Multiple Linear Regression (MLR) techniques, and the Factor-Based Techniques. Within each family, we will review the various methods commonly employed, learn how to develop and test calibrations, and how to use the calibrations to estimate, or predict, the properties of unknown samples. We will consider the advantages and limitations of each method as well as some of the tricks and pitfalls associated with their use. While our emphasis will be on quantitative analysis, we will also touch on how these techniques are used for qualitative analysis, classification, and discriminative analysis. 

In [162] experiments on methane provided a linear pressure dependence of the contour width. This made it possible to find the dephasing cross-section and to discriminate between contributions of rotational and vibrational relaxation to the contour width. This was done under the above-mentioned simplifying assumption that they are additive. (Let us note that processing of experimental data on linear molecules was always performed under this assumption.) The points found by this method are shown in Fig. 3.15, curves (4) and (6). 

As can be seen from the above, the shape of the resolved rotational structure is well described when the parameters of the fitting law were chosen from the best fit to experiment. The values of estimated from the rotational width of the collapsed Q-branch qZE. Therefore the models giving the same high-density limits. One may hope to discriminate between them only in the intermediate range of densities where the spectrum is unresolved but has not yet collapsed. The spectral shape in this range may be calculated only numerically from Eq. (4.86) with impact operator Tj, linear in n. Of course, it implies that binary theory is still valid and that vibrational dephasing is not yet... 

This approach did not seem to be as satisfactory for those sulfamates having heteroatom substituents (hetero-sulfamates). Spillane suggested that the various electronic effects of the hetero-atoms probably introduce an additional variable that is apparently absent, or constant, for the carbosulfamates. Because molecular connectivity correlates structure with molecular volume and electronic effects, Spillane included molecular connectivity, (computed for the entire molecule, RNHSOO to the four variables, x, y, z, and V, and applied the statistical technique of linear-discrimination analysis to 33 heterosulfamates (10 sweet, 23 not sweet). A correlation of >80% was obtained for the x, z, x subset 5 of the 33... 

Linear, polynomial, or statistical discriminant functions (Fukunaga, 1990 Kramer, 1991 MacGregor et al., 1991), or adaptive connectionist networks (Rumelhart et al, 1986 Funahashi, 1989 Vaidyanathan and Venkatasub-ramanian, 1990 Bakshi and Stephanopoulos, 1993 third chapter of this volume, Koulouris et al), combine tasks 1 and 2 into one and solve the corresponding problems simultaneously. These methodologies utilize a priori defined general functional relationships between the operating data and process conditions, and as such they are not inductive. Nearest-neigh-... 

A systematic study was carried out using in parallel 50 standard solutions for each concentration of three natural colorants (curcumin, carminic acid, and caramel as yellow, red, and brown, respectively). No false positive results for synthetics were obtained up to concentrations of 15 and 20 ng/ml for natural red and yellow colorants, respectively, or 110 ng/ml for natural brown colorant. The concentrations have to be high enough to prove that the screening method is able to accurately discriminate natural and synthetic colorants. To make a clear interpretation of the quantitative UV-Vis spectrum, linear regression analysis was used. Quantitative UV-Vis analysis of a dye ° can be calculated according to the following formula ... 

More recently (2006) we performed and reported quantitative structure-activity relationship (QSAR) modeling of the same compounds based on their atomic linear indices, for finding fimctions that discriminate between the tyrosinase inhibitor compounds and inactive ones [50]. Discriminant models have been applied and globally good classifications of 93.51 and 92.46% were observed for nonstochastic and stochastic hnear indices best models, respectively, in the training set. The external prediction sets had accuracies of 91.67 and 89.44% [50]. In addition to this, these fitted models have also been employed in the screening of new cycloartane compounds isolated from herbal plants. Good behavior was observed between the theoretical and experimental results. These results provide a tool that can be used in the identification of new tyrosinase inhibitor compounds [50]. 

As with in vivo voltammetry, a variety of electrochemical techniques have been used for the stripping step. Because of its simplicity, linear sweep voltammetry has enjoyed widespread use however, the detection limit of this technique is limited by charging current. Differential pulse has become popular because it discriminates against the charging current to provide considerably lower detection limits. 

E. Marengo and R. Todeschini, Linear discriminant hierarchical clustering a modeling and cross-validable divisive clustering method. Chemom. Intell. Lab. Syst., 19 (1993) 43-51. 

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