Linear discriminant analysis classification

Most of the supervised pattern recognition procedures permit the carrying out of stepwise selection, i.e. the selection first of the most important feature, then, of the second most important, etc. One way to do this is by prediction using e.g. cross-validation (see next section), i.e. we first select the variable that best classifies objects of known classification but that are not part of the training set, then the variable that most improves the classification already obtained with the first selected variable, etc. The results for the linear discriminant analysis of the EU/HYPER classification of Section 33.2.1 is that with all 5 or 4 variables a selectivity of 91.4% is obtained and for 3 or 2 variables 88.6% [2] as a measure of classification success. Selectivity is used here. It is applied in the sense of Chapter... 

There are many classification methods apart from linear discriminant analysis (Derde et al. [1987] Frank and Friedman [1989] Huberty [1994]). Particularly worth mentioning are the SIMCA method (Soft independent modelling of class analogies) (Wold [1976] Frank [1989]), ALLOC (Coomans et al. [1981]), UNEQ (Derde and Massart [1986]), PRIMA (Juricskay and Veress [1985] Derde and Massart [1988]), DASCO (Frank [1988]), etc. 

More recently, another linear discriminant analysis (LDA) model was constructed for a set of 157 compounds for which Pcaco-2 was measured [43]. This model, which applied DRAGON descriptors, achieved an accuracy of classification at 91 % for the training set and 84% for the test set. When this model was applied to predict a set of 241 drugs for which HIA data were available, good correlation (>81%) was achieved between the two ADME-Tox properties. 

Nevertheless, in most of the electronic tongue applications found in the literature, classification techniques like linear discriminant analysis (LDA) and partial least squares discriminant analysis (PLS-DA) have been used in place of more appropriate class-modeling methods. Moreover, in the few cases in which a class-modeling technique such as soft independent modeling of class analogy (SIMCA) is applied, attention is frequently focused only on its classification performance (e.g., correct classification rate). Use of such a restricted focus considerably underutilizes the significant characteristics of the class-modeling approach. 

In recent years, new methods have been introduced into chemistry for classification problems, and they have often been applied to food analytical data. The statistical linear discriminant analysis is still the most widely used method, as was noted in the previous section. 

In the same way as linear discriminant analysis is the most-used classification method, stepwise selection by LDA (SLDA) is the selection method that shows the greatest number of applications in food chemistry. 

Dudoit et al. also studied some more complex methods such a classification trees and aggregated classification trees. These methods did not appear to perform any better than diagonal linear discriminant analysis or nearest neighbor classification. Ben-Dor et al. (7J also compared several methods on several public datasets and found that nearest neighbor classification generally performed as well or better than more complex methods. 

Discriminant analysis (DA) performs samples classification with an a priori hypothesis. This hypothesis is based on a previously determined TCA or other CA protocols. DA is also called "discriminant function analysis" and its natural extension is called MDA (multiple discriminant analysis), which sometimes is named "discriminant factor analysis" or CD A (canonical discriminant analysis). Among these type of analyses, linear discriminant analysis (LDA) has been largely used to enforce differences among samples classes. Another classification method is known as QDA (quadratic discriminant analysis) (Frank and Friedman, 1989) an extension of LDA and RDA (regularized discriminant analysis), which works better with various class distribution and in the case of high-dimensional data, being a compromise between LDA and QDA (Friedman, 1989). 

Rezzi, S., Axelson, D. E., Heberger, K., Reniero, F., Mariani, C., and Guillou, C. (2005). Classification of olive oils using high throughput flow 1H NMR fingerprinting with principal component analysis, linear discriminant analysis and probabilistic neural networks. Anal. Chim. Acta 552,13-24. 

Linear discriminant analysis (LDA) is also a probabilistic classifier in the mold of Bayes algorithms but can be related closely to both regression and PCA techniques. A discriminant function is simply a function of the observed vector of variables (K) that leads to a classification rule. The likelihood ratio (above), for example, is an optimal discriminant for the two-class case. Hence, the classification rule can be stated as... 

Partial least square (PLS) regression model describes the dependences between two variables blocks, e.g. sensor responses and time variables. Let the X matrix represent the sensor responses and the Y matrix represent time, the X and Y matrices could be approximated to few orthogonal score vectors, respectively. These components are then rotated in order to get as good a prediction of y variables as possible [25], Linear discriminant analysis (LDA) is among the most used classification techniques. The method maximises the variance between... 

Figure 7.4 Authentication of monovarietal virgin olive oils results of applying stepwise linear discriminant analysis to volatile compounds. Classification was carried out by four volatiles (F)-2-hexenal, butyl acetate, (F)-3-hexenal, 2-methyl-3-buten-2-ol. F-to-Enter was 8.0 tolerance was upper 0.52 for all selected volatiles. Note A, cv. Arbequina C, cv. Coratina K, cv. Koroneiki P, cv. Picual (source SEXIA Group-Instituto de la Grasa, Seville, Spain).

This supervised classification method, which is the most used, accepts a normal multivariate distribution for the variables in each population ((Ai,..., A ) Xi) ), and calculates the classification functions minimising the possibility of incorrect classification of the observations of the training group (Bayesian type rule). If multivariate normality is accepted and equality of the k covariance matrices ((Ai,..., Xp) NCfti, X)), Linear Discriminant Analysis (LDA) calculates... 

Jurado, J.M., Alcazar, A., Pablos, F., Martin, M.J., Gonzalez, A.G. Classification of aniseed drinks by means of cluster, linear discriminant analysis and soft independent modelling of class analogy based on their Zn, B, Fe, Mg, Ca, Na and Si content. Talanta 66, 1350-1354 (2005)... 

Various classification approaches have been reported to be used successfully in conjunction with fragment descriptors for building classification SAR models the Linear Discriminant Analysis (LDA), the Partial Least Square Discriminant Analysis (PLS-DA), Soft Independent Modeling by Class Analogy (SIMCA), Artificial Neural Networks (ANN), ° Support Vector... 

The most popular classification methods are Linear Discriminant Analysis (LDA), Quadratic Discriminant Analysis (QDA), Regularized Discriminant Analysis (RDA), K th Nearest Neighbours (KNN), classification tree methods (such as CART), Soft-Independent Modeling of Class Analogy (SIMCA), potential function classifiers (PFC), Nearest Mean Classifier (NMC) and Weighted Nearest Mean Classifier (WNMC). Moreover, several classification methods can be found among the artificial neural networks. 

Fig. 1. Statistical classification strategy (SCS) a schematic road map of how the SCS method is developed for individual databases. GA ORS, genetic algorithm based optimal region selection LDA, linear discriminant analysis LOO, leave-one-out (method of cross-validation) coeff, coefficients.

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